Notes on Data Structures and Programming Techniques (CPSC 223

Notes on Data Structures and Programming Techniques (CPSC 223
Notes on Data Structures and Programming
Techniques (CPSC 223, Spring 2015)
James Aspnes
2017-06-28T17:43:54-0400
Contents
1 Course administration
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Resources . . . . . . . . . . . . . . . . . . . . .
1.1.2 Documentation . . . . . . . . . . . . . . . . . .
1.1.3 Questions and comments . . . . . . . . . . . .
1.2 Lecture schedule . . . . . . . . . . . . . . . . . . . . .
1.2.1 Topics by date . . . . . . . . . . . . . . . . . .
1.2.2 Topics not covered in 2015 . . . . . . . . . . .
1.3 Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 On-line course information . . . . . . . . . . .
1.3.2 Meeting times . . . . . . . . . . . . . . . . . . .
1.3.3 Synopsis of the course . . . . . . . . . . . . . .
1.3.4 Prerequisites . . . . . . . . . . . . . . . . . . .
1.3.5 Textbook . . . . . . . . . . . . . . . . . . . . .
1.3.6 Course requirements . . . . . . . . . . . . . . .
1.3.7 Staff . . . . . . . . . . . . . . . . . . . . . . . .
1.3.7.1 Instructor . . . . . . . . . . . . . . . .
1.3.7.2 Teaching Fellows . . . . . . . . . . . .
1.3.7.3 Peer tutors . . . . . . . . . . . . . . .
1.3.8 Use of outside help . . . . . . . . . . . . . . . .
1.3.9 Clarifications for homework assignments . . . .
1.3.10 Late assignments . . . . . . . . . . . . . . . . .
1.4 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Why should you learn to program in C? . . . .
1.4.2 Why should you learn about data structures and
ming techniques? . . . . . . . . . . . . . . . . .
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2 The Zoo and the Zoo Annex
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2.1 Getting an account . . . . . . . . . . . . . . . . . . . . . . . . . . 20
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3 The Linux programming environment
3.1 The shell . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Getting a shell prompt in the Zoo . . . . . . . . .
3.1.2 The Unix filesystem . . . . . . . . . . . . . . . . .
3.1.3 Unix command-line programs . . . . . . . . . . . .
3.1.4 Stopping and interrupting programs . . . . . . . .
3.1.5 Running your own programs . . . . . . . . . . . .
3.1.6 Redirecting input and output . . . . . . . . . . . .
3.2 Text editors . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Writing C programs with Emacs . . . . . . . . . .
3.2.1.1 My favorite Emacs commands . . . . . .
3.2.2 Using Vi instead of Emacs . . . . . . . . . . . . . .
3.2.2.1 My favorite Vim commands . . . . . . . .
3.2.2.1.1 Normal mode . . . . . . . . . . .
3.2.2.1.2 Insert mode . . . . . . . . . . .
3.2.2.2 Settings . . . . . . . . . . . . . . . . . . .
3.3 Compilation tools . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 The GNU C compiler gcc . . . . . . . . . . . . . .
3.3.2 Make . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2.1 Make gotchas . . . . . . . . . . . . . . . .
3.4 Debugging tools . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Debugging in general . . . . . . . . . . . . . . . . .
3.4.2 Assertions . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 The GNU debugger gdb . . . . . . . . . . . . . . .
3.4.3.1 My favorite gdb commands . . . . . . . .
3.4.3.2 Debugging strategies . . . . . . . . . . . .
3.4.3.3 Common applications of gdb . . . . . . .
3.4.3.3.1 Watching your program run . .
3.4.3.3.2 Dealing with failed assertions . .
3.4.3.3.3 Dealing with segmentation faults
3.4.3.3.4 Dealing with infinite loops . . .
3.4.3.3.5 Mysterious variable changes . .
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2.4
Getting into the room . . . . . . . . . . . . .
Remote use . . . . . . . . . . . . . . . . . . .
2.3.1 Access using FastX . . . . . . . . . . .
2.3.1.1 Getting a license key . . . .
2.3.1.2 FastX in the Zoo Annex . . .
2.3.1.3 Using FastX from Windows .
2.3.1.4 Using FastX from OSX . . .
2.3.2 Terminal access . . . . . . . . . . . . .
2.3.3 GUI access . . . . . . . . . . . . . . .
How to compile and run programs . . . . . .
2.4.1 Creating the program . . . . . . . . .
2.4.2 Compiling and running a program . .
2.4.3 Some notes on what the program does
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4 The C programming language
4.1 Structure of a C program . . . . . . . . . . . . . . . . . . . .
4.2 Numeric data types . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Integer types in C . . . . . . . . . . . . . . . . . . . .
4.2.1.1 Basic integer types . . . . . . . . . . . . . . .
4.2.1.2 C99 fixed-width types . . . . . . . . . . . . .
4.2.2 size_t and ptrdiff_t . . . . . . . . . . . . . . . . .
4.2.2.1 Integer constants . . . . . . . . . . . . . . . .
4.2.2.1.1 Naming constants . . . . . . . . . .
4.2.2.2 Integer operators . . . . . . . . . . . . . . . .
4.2.2.2.1 Arithmetic operators . . . . . . . .
4.2.2.2.2 Bitwise operators . . . . . . . . . .
4.2.2.2.3 Logical operators . . . . . . . . . . .
4.2.2.2.4 Relational operators . . . . . . . . .
4.2.2.3 Converting to and from strings . . . . . . . .
4.2.3 Floating-point types . . . . . . . . . . . . . . . . . . .
4.2.3.1 Floating point basics . . . . . . . . . . . . .
4.2.3.2 Floating-point constants . . . . . . . . . . .
4.2.3.3 Operators . . . . . . . . . . . . . . . . . . . .
4.2.3.4 Conversion to and from integer types . . . .
4.2.3.5 The IEEE-754 floating-point standard . . . .
4.2.3.6 Error . . . . . . . . . . . . . . . . . . . . . .
4.2.3.7 Reading and writing floating-point numbers .
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3.7
Valgrind . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4.1 Compilation flags . . . . . . . . . . . . . .
3.4.4.2 Automated testing . . . . . . . . . . . . . .
3.4.4.3 Examples of some common valgrind errors
3.4.4.3.1 Uninitialized values . . . . . . . .
3.4.4.3.2 Bytes definitely lost . . . . . . . .
3.4.4.3.3 Invalid write or read operations .
3.4.5 Not recommended: debugging output . . . . . . . .
Performance tuning . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Timing under Linux . . . . . . . . . . . . . . . . . .
3.5.2 Profiling with gprof . . . . . . . . . . . . . . . . . .
3.5.2.1 Effect of optimization during compilation .
Version control . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Setting up Git . . . . . . . . . . . . . . . . . . . . .
3.6.2 Editing files . . . . . . . . . . . . . . . . . . . . . . .
3.6.3 Renaming files . . . . . . . . . . . . . . . . . . . . .
3.6.4 Adding and removing files . . . . . . . . . . . . . . .
3.6.5 Recovering files from the repository . . . . . . . . .
3.6.6 Undoing bad commits . . . . . . . . . . . . . . . . .
3.6.7 Looking at old versions . . . . . . . . . . . . . . . .
3.6.8 More information about Git . . . . . . . . . . . . . .
Submitting assignments . . . . . . . . . . . . . . . . . . . .
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4.9
4.2.3.8 Non-finite numbers in C . . . . . . . . . . . . . .
4.2.3.9 The math library . . . . . . . . . . . . . . . . .
Operator precedence . . . . . . . . . . . . . . . . . . . . . . . . .
Programming style . . . . . . . . . . . . . . . . . . . . . . . . . .
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Variables as names . . . . . . . . . . . . . . . . . . . . . .
4.5.2.1 Variable declarations . . . . . . . . . . . . . . .
4.5.2.2 Variable names . . . . . . . . . . . . . . . . . . .
4.5.3 Using variables . . . . . . . . . . . . . . . . . . . . . . . .
4.5.4 Initialization . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.5 Storage class qualifiers . . . . . . . . . . . . . . . . . . . .
4.5.5.1 Scope and extent . . . . . . . . . . . . . . . . . .
4.5.5.1.1 Additional qualifiers for global variables
4.5.6 Marking variables as constant . . . . . . . . . . . . . . . .
4.5.6.1 Pointers to const . . . . . . . . . . . . . . . . .
Input and output . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Character streams . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Reading and writing single characters . . . . . . . . . . .
4.6.3 Formatted I/O . . . . . . . . . . . . . . . . . . . . . . . .
4.6.4 Rolling your own I/O routines . . . . . . . . . . . . . . .
4.6.5 File I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Statements and control structures . . . . . . . . . . . . . . . . . .
4.7.1 Simple statements . . . . . . . . . . . . . . . . . . . . . .
4.7.2 Compound statements . . . . . . . . . . . . . . . . . . . .
4.7.2.1 Conditionals . . . . . . . . . . . . . . . . . . . .
4.7.2.2 Loops . . . . . . . . . . . . . . . . . . . . . . . .
4.7.2.2.1 The while loop . . . . . . . . . . . . . .
4.7.2.2.2 The do..while loop . . . . . . . . . . . .
4.7.2.2.3 The for loop . . . . . . . . . . . . . . .
4.7.2.2.4 Loops with break, continue, and goto .
4.7.2.3 Choosing where to put a loop exit . . . . . . . .
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.1 Function definitions . . . . . . . . . . . . . . . . . . . . .
4.8.2 When to write a function . . . . . . . . . . . . . . . . . .
4.8.3 Calling a function . . . . . . . . . . . . . . . . . . . . . .
4.8.4 The return statement . . . . . . . . . . . . . . . . . . . .
4.8.5 Function declarations and modules . . . . . . . . . . . . .
4.8.6 Static functions . . . . . . . . . . . . . . . . . . . . . . . .
4.8.7 Local variables . . . . . . . . . . . . . . . . . . . . . . . .
4.8.8 Mechanics of function calls . . . . . . . . . . . . . . . . .
Pointers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9.1 Memory and addresses . . . . . . . . . . . . . . . . . . . .
4.9.2 Pointer variables . . . . . . . . . . . . . . . . . . . . . . .
4.9.2.1 Declaring a pointer variable . . . . . . . . . . . .
4.9.2.2 Assigning to pointer variables . . . . . . . . . .
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4.9.2.3 Using a pointer . . . . . . . . . . . .
4.9.2.4 Printing pointers . . . . . . . . . . .
4.9.3 The null pointer . . . . . . . . . . . . . . . .
4.9.4 Pointers and functions . . . . . . . . . . . . .
4.9.5 Pointer arithmetic and arrays . . . . . . . . .
4.9.5.1 Arrays . . . . . . . . . . . . . . . .
4.9.5.2 Arrays and functions . . . . . . . .
4.9.5.3 Multidimensional arrays . . . . . . .
4.9.5.4 Variable-length arrays . . . . . . . .
4.9.6 Void pointers . . . . . . . . . . . . . . . . . .
4.9.6.1 Alignment . . . . . . . . . . . . . .
4.9.7 Run-time storage allocation using malloc . .
4.9.8 Function pointers . . . . . . . . . . . . . . . .
4.9.8.1 Function pointer declarations . . . .
4.9.8.2 Callbacks . . . . . . . . . . . . . . .
4.9.8.3 Dispatch tables . . . . . . . . . . . .
4.9.9 The restrict keyword . . . . . . . . . . . . . .
Strings . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10.1 C strings . . . . . . . . . . . . . . . . . . . .
4.10.2 String constants . . . . . . . . . . . . . . . .
4.10.3 String buffers . . . . . . . . . . . . . . . . . .
4.10.3.1 String buffers and the perils of gets
4.10.4 Operations on strings . . . . . . . . . . . . .
4.10.5 Finding the length of a string . . . . . . . . .
4.10.5.1 The strlen tarpit . . . . . . . . . . .
4.10.6 Comparing strings . . . . . . . . . . . . . . .
4.10.7 Formatted output to strings . . . . . . . . . .
4.10.8 Dynamic allocation of strings . . . . . . . . .
4.10.9 Command-line arguments . . . . . . . . . . .
Structured data types . . . . . . . . . . . . . . . . .
4.11.1 Structs . . . . . . . . . . . . . . . . . . . . .
4.11.1.1 Operations on structs . . . . . . . .
4.11.1.2 Layout in memory . . . . . . . . . .
4.11.1.3 Bit fields . . . . . . . . . . . . . . .
4.11.2 Unions . . . . . . . . . . . . . . . . . . . . . .
4.11.3 Enums . . . . . . . . . . . . . . . . . . . . . .
4.11.3.1 Specifying particular values . . . . .
4.11.3.2 What most people do . . . . . . . .
4.11.3.3 Using enum with union . . . . . . .
Type aliases using typedef . . . . . . . . . . . . . .
4.12.1 Opaque structs . . . . . . . . . . . . . . . . .
Macros . . . . . . . . . . . . . . . . . . . . . . . . . .
4.13.1 Macros with arguments . . . . . . . . . . . .
4.13.1.1 Multiple arguments . . . . . . . . .
4.13.1.2 Perils of repeating arguments . . . .
4.13.1.3 Variable-length argument lists . . .
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4.13.1.4 Macros vs. inline functions . . . .
4.13.2 Macros that include other macros . . . . . .
4.13.3 More specialized macros . . . . . . . . . . .
4.13.3.1 Multiple expressions in a macro .
4.13.3.2 Non-syntactic macros . . . . . . .
4.13.3.3 Multiple statements in one macro
4.13.3.4 String expansion . . . . . . . . . .
4.13.3.5 Big macros . . . . . . . . . . . . .
4.13.4 Conditional compilation . . . . . . . . . . .
4.13.5 Defining macros on the command line . . .
4.13.6 The #if directive . . . . . . . . . . . . . . .
4.13.7 Debugging macro expansions . . . . . . . .
4.13.8 Can a macro call a preprocessor command?
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5 Data structures and programming techniques
5.1 Asymptotic notation . . . . . . . . . . . . . . . . . . .
5.1.1 Two sorting algorithms . . . . . . . . . . . . .
5.1.2 Big-O to the rescue . . . . . . . . . . . . . . . .
5.1.3 Asymptotic cost of programs . . . . . . . . . .
5.1.4 Other variants of asymptotic notation . . . . .
5.2 Linked lists . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Stacks . . . . . . . . . . . . . . . . . . . . . . .
5.2.1.1 Building a stack out of an array . . .
5.2.2 Queues . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Looping over a linked list . . . . . . . . . . . .
5.2.4 Looping over a linked list backwards . . . . . .
5.2.5 Deques and doubly-linked lists . . . . . . . . .
5.2.5.1 Alternate implementation using a ring
5.2.6 Circular linked lists . . . . . . . . . . . . . . .
5.2.7 What linked lists are and are not good for . . .
5.2.8 Further reading . . . . . . . . . . . . . . . . . .
5.3 Abstract data types . . . . . . . . . . . . . . . . . . .
5.3.1 A sequence type . . . . . . . . . . . . . . . . .
5.3.1.1 Interface . . . . . . . . . . . . . . . .
5.3.1.2 Implementation . . . . . . . . . . . .
5.3.1.3 Compiling and linking . . . . . . . . .
5.3.2 Designing abstract data types . . . . . . . . . .
5.3.2.1 Parnas’s Principle . . . . . . . . . . .
5.3.2.2 When to build an abstract data type
5.4 Hash tables . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Dictionary data types . . . . . . . . . . . . . .
5.4.2 Basics of hashing . . . . . . . . . . . . . . . . .
5.4.3 Resolving collisions . . . . . . . . . . . . . . . .
5.4.3.1 Chaining . . . . . . . . . . . . . . . .
5.4.3.2 Open addressing . . . . . . . . . . . .
5.4.4 Choosing a hash function . . . . . . . . . . . .
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5.4.4.1 Division method . . . . . . . . . . . . . . . . . .
5.4.4.2 Multiplication method . . . . . . . . . . . . . . .
5.4.4.3 Universal hashing . . . . . . . . . . . . . . . . .
5.4.5 Maintaining a constant load factor . . . . . . . . . . . . .
5.4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.6.1 A low-overhead hash table using open addressing
5.4.6.2 A string to string dictionary using chaining . . .
5.5 Generic containers . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Generic dictionary: interface . . . . . . . . . . . . . . . .
5.5.2 Generic dictionary: implementation . . . . . . . . . . . .
5.6 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Example of recursion in C . . . . . . . . . . . . . . . . . .
5.6.2 Common problems with recursion . . . . . . . . . . . . .
5.6.2.1 Omitting the base case . . . . . . . . . . . . . .
5.6.2.2 Blowing out the stack . . . . . . . . . . . . . . .
5.6.2.3 Failure to make progress . . . . . . . . . . . . .
5.6.3 Tail-recursion and iteration . . . . . . . . . . . . . . . . .
5.6.3.1 Binary search: recursive and iterative versions .
5.6.4 Mergesort: a recursive sorting algorithm . . . . . . . . . .
5.6.5 Asymptotic complexity of recursive functions . . . . . . .
5.7 Binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.1 Tree basics . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.2 Binary tree implementations . . . . . . . . . . . . . . . .
5.7.3 The canonical binary tree algorithm . . . . . . . . . . . .
5.7.4 Nodes vs leaves . . . . . . . . . . . . . . . . . . . . . . . .
5.7.5 Special classes of binary trees . . . . . . . . . . . . . . . .
5.8 Heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.1 Priority queues . . . . . . . . . . . . . . . . . . . . . . . .
5.8.2 Expensive implementations of priority queues . . . . . . .
5.8.3 Structure of a heap . . . . . . . . . . . . . . . . . . . . . .
5.8.4 Packed heaps . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.5 Bottom-up heapification . . . . . . . . . . . . . . . . . . .
5.8.6 Heapsort . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.7 More information . . . . . . . . . . . . . . . . . . . . . . .
5.9 Binary search trees . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9.1 Searching for a node . . . . . . . . . . . . . . . . . . . . .
5.9.2 Inserting a new node . . . . . . . . . . . . . . . . . . . . .
5.9.3 Deleting a node . . . . . . . . . . . . . . . . . . . . . . . .
5.9.4 Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Augmented trees . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Balanced trees . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11.1 Tree rotations . . . . . . . . . . . . . . . . . . . . . . . . .
5.11.2 AVL trees . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11.2.1 Sample implementation . . . . . . . . . . . . . .
5.11.3 2–3 trees . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.11.4 Red-black trees . . . . . . . . . . . . . . . . . . . . . . . .
5.11.5 B-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11.6 Splay trees . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11.6.1 How splaying works . . . . . . . . . . . . . . . .
5.11.6.2 Analysis . . . . . . . . . . . . . . . . . . . . . .
5.11.6.3 Other operations . . . . . . . . . . . . . . . . . .
5.11.6.4 Top-down splaying . . . . . . . . . . . . . . . . .
5.11.6.5 An implementation . . . . . . . . . . . . . . . .
5.11.6.6 More information . . . . . . . . . . . . . . . . .
5.11.7 Scapegoat trees . . . . . . . . . . . . . . . . . . . . . . . .
5.11.8 Skip lists . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11.9 Implementations . . . . . . . . . . . . . . . . . . . . . . .
5.12 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . .
5.12.2 Why graphs are useful . . . . . . . . . . . . . . . . . . . .
5.12.3 Operations on graphs . . . . . . . . . . . . . . . . . . . .
5.12.4 Representations of graphs . . . . . . . . . . . . . . . . . .
5.12.4.1 Adjacency matrices . . . . . . . . . . . . . . . .
5.12.4.2 Adjacency lists . . . . . . . . . . . . . . . . . . .
5.12.4.2.1 An implementation . . . . . . . . . . .
5.12.4.3 Implicit representations . . . . . . . . . . . . . .
5.12.5 Searching for paths in a graph . . . . . . . . . . . . . . .
5.12.5.1 Implementation of depth-first and breadth-first
search . . . . . . . . . . . . . . . . . . . . . . . .
5.12.5.2 Combined implementation of depth-first and
breadth-first search . . . . . . . . . . . . . . . .
5.12.5.3 Other variations on the basic algorithm . . . . .
5.13 Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . .
5.13.1 Memoization . . . . . . . . . . . . . . . . . . . . . . . . .
5.13.2 Dynamic programming . . . . . . . . . . . . . . . . . . . .
5.13.2.1 More examples . . . . . . . . . . . . . . . . . . .
5.13.2.1.1 Longest increasing subsequence . . . . .
5.13.2.1.2 All-pairs shortest paths . . . . . . . . .
5.13.2.1.3 Longest common subsequence . . . . .
5.14 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.14.1 Generating random values in C . . . . . . . . . . . . . . .
5.14.1.1 The rand function from the standard library . .
5.14.1.1.1 Supplying a seed with srand . . . . . .
5.14.1.2 Better pseudorandom number generators . . . .
5.14.1.3 Random numbers without the pseudo . . . . . .
5.14.1.4 Range issues . . . . . . . . . . . . . . . . . . . .
5.14.2 Randomized algorithms . . . . . . . . . . . . . . . . . . .
5.14.2.1 Randomized search . . . . . . . . . . . . . . . .
5.14.2.2 Quickselect and quicksort . . . . . . . . . . . . .
5.14.3 Randomized data structures . . . . . . . . . . . . . . . . .
5.14.3.1 Skip lists . . . . . . . . . . . . . . . . . . . . . .
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5.14.3.2 Universal hash families . . . . . . . . . . .
5.15 String processing . . . . . . . . . . . . . . . . . . . . . . . .
5.15.1 Radix search . . . . . . . . . . . . . . . . . . . . . .
5.15.1.1 Tries . . . . . . . . . . . . . . . . . . . . .
5.15.1.1.1 Searching a trie . . . . . . . . . .
5.15.1.1.2 Inserting a new element into a trie
5.15.1.1.3 Implementation . . . . . . . . . .
5.15.1.2 Patricia trees . . . . . . . . . . . . . . . . .
5.15.1.3 Ternary search trees . . . . . . . . . . . . .
5.15.1.4 More information . . . . . . . . . . . . . .
5.15.2 Radix sort . . . . . . . . . . . . . . . . . . . . . . . .
5.15.2.1 Bucket sort . . . . . . . . . . . . . . . . . .
5.15.2.2 Classic LSB radix sort . . . . . . . . . . . .
5.15.2.3 MSB radix sort . . . . . . . . . . . . . . . .
5.15.2.3.1 Issues with recursion depth . . . .
5.15.2.3.2 Implementing the buckets . . . . .
5.15.2.3.3 Further optimization . . . . . . .
5.15.2.3.4 Sample implementation . . . . . .
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6 Other topics not covered in detail in 2015
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6.1 More applications of function pointers . . . . . . . . . . . . . . . 378
6.1.1 Iterators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
6.1.1.1 Option 1: Function that returns a sequence . . . 379
6.1.1.2 Option 2: Iterator with first/done/next operations380
6.1.1.3 Option 3: Iterator with function argument . . . 381
6.1.2 Closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
6.1.3 Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
6.2 Suffix arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
6.2.1 Why do we want to do this? . . . . . . . . . . . . . . . . . 388
6.2.2 String search algorithms . . . . . . . . . . . . . . . . . . . 388
6.2.3 Suffix trees and suffix arrays . . . . . . . . . . . . . . . . 388
6.2.3.1 Building a suffix array . . . . . . . . . . . . . . . 389
6.2.3.2 Searching a suffix array . . . . . . . . . . . . . . 390
6.3 Burrows-Wheeler transform . . . . . . . . . . . . . . . . . . . . . 390
6.3.1 Suffix arrays and the Burrows-Wheeler transform . . . . . 393
6.3.2 Sample implementation . . . . . . . . . . . . . . . . . . . 393
6.4 C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
6.4.1 Hello world . . . . . . . . . . . . . . . . . . . . . . . . . . 398
6.4.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
6.4.3 Function overloading . . . . . . . . . . . . . . . . . . . . . 400
6.4.4 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
6.4.5 Operator overloading . . . . . . . . . . . . . . . . . . . . . 404
6.4.6 Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
6.4.7 Exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . 407
6.4.8 Storage allocation . . . . . . . . . . . . . . . . . . . . . . 408
6.4.8.1 Storage allocation inside objects . . . . . . . . . 410
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6.6
6.7
6.8
6.4.9 Standard library . . . . . . . . . . . . . . . . . .
6.4.10 Things we haven’t talked about . . . . . . . . . .
Testing during development . . . . . . . . . . . . . . . .
6.5.1 Unit tests . . . . . . . . . . . . . . . . . . . . . .
6.5.1.1 What to put in the test code . . . . . .
6.5.1.2 Example . . . . . . . . . . . . . . . . .
6.5.2 Test harnesses . . . . . . . . . . . . . . . . . . .
6.5.2.1 Module interface . . . . . . . . . . . . .
6.5.2.1.1 stack.h . . . . . . . . . . . . .
6.5.2.2 Test code . . . . . . . . . . . . . . . . .
6.5.2.2.1 test-stack.c . . . . . . . . . . .
6.5.2.3 Makefile . . . . . . . . . . . . . . . . . .
6.5.2.3.1 Makefile . . . . . . . . . . . . .
6.5.3 Stub implementation . . . . . . . . . . . . . . . .
6.5.3.1 stack.c . . . . . . . . . . . . . . . . . .
6.5.4 Bounded-space implementation . . . . . . . . . .
6.5.4.1 stack.c . . . . . . . . . . . . . . . . . .
6.5.5 First fix . . . . . . . . . . . . . . . . . . . . . . .
6.5.6 Final version . . . . . . . . . . . . . . . . . . . .
6.5.6.1 stack.c . . . . . . . . . . . . . . . . . .
6.5.7 Moral . . . . . . . . . . . . . . . . . . . . . . . .
6.5.8 Appendix: Test macros . . . . . . . . . . . . . .
Algorithm design techniques . . . . . . . . . . . . . . . .
6.6.1 Basic principles of algorithm design . . . . . . .
6.6.2 Specific techniques . . . . . . . . . . . . . . . . .
6.6.3 Example: Finding the maximum . . . . . . . . .
6.6.4 Example: Sorting . . . . . . . . . . . . . . . . . .
Bit manipulation . . . . . . . . . . . . . . . . . . . . . .
Persistence . . . . . . . . . . . . . . . . . . . . . . . . .
6.8.1 A simple solution using text files . . . . . . . . .
6.8.2 Using a binary file . . . . . . . . . . . . . . . . .
6.8.3 A version that updates the file in place . . . . .
6.8.4 An even better version using mmap . . . . . . .
6.8.5 Concurrency and fault-tolerance issues: ACIDity
7 What next?
7.1 What’s wrong with C
7.2 What C++ fixes . . .
7.3 Other C-like languages
7.4 Scripting languages . .
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8 Assignments
8.1 Assignment 1, due Thursday 2015-01-29,
8.1.1 Bureaucratic part . . . . . . . . .
8.1.2 A rotten cipher . . . . . . . . . .
8.1.3 Your task . . . . . . . . . . . . .
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at 11:00pm
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8.2
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8.9
8.1.4 Hints . . . . . . . . . . . . . . . . . . . . . . .
8.1.5 Testing your assignment . . . . . . . . . . . . .
8.1.6 Submitting your assignment . . . . . . . . . . .
8.1.7 Sample solution . . . . . . . . . . . . . . . . . .
Assignment 2, due Wednesday 2015-02-04, at 11:00pm
8.2.1 Opening a safe . . . . . . . . . . . . . . . . . .
8.2.2 Submitting your assignment . . . . . . . . . . .
8.2.3 Valgrind . . . . . . . . . . . . . . . . . . . . . .
8.2.4 Sample solution . . . . . . . . . . . . . . . . . .
Assignment 3, due Wednesday 2015-02-11, at 11:00pm
8.3.1 Quadratic letter sequences . . . . . . . . . . . .
8.3.2 Your task . . . . . . . . . . . . . . . . . . . . .
8.3.3 Submitting your assignment . . . . . . . . . . .
8.3.4 Sample solution . . . . . . . . . . . . . . . . . .
Assignment 4, due Wednesday 2015-02-18, at 11:00pm
8.4.1 An ASCII art compositor . . . . . . . . . . . .
8.4.2 Submitting your assignment . . . . . . . . . . .
8.4.3 Notes . . . . . . . . . . . . . . . . . . . . . . .
8.4.3.1 Input . . . . . . . . . . . . . . . . . .
8.4.3.2 Output . . . . . . . . . . . . . . . . .
8.4.3.3 General . . . . . . . . . . . . . . . . .
8.4.4 Sample solution . . . . . . . . . . . . . . . . . .
Assignment 5, due Wednesday 2015-02-25, at 11:00pm
8.5.1 Build a Turing machine! . . . . . . . . . . . . .
8.5.2 Example . . . . . . . . . . . . . . . . . . . . . .
8.5.3 Your task . . . . . . . . . . . . . . . . . . . . .
8.5.4 Submitting your assignment . . . . . . . . . . .
8.5.5 Sample solution . . . . . . . . . . . . . . . . . .
Assignment 6, due Wednesday 2015-03-25, at 11:00pm
8.6.1 Sinking ships . . . . . . . . . . . . . . . . . . .
8.6.2 Things to watch out for . . . . . . . . . . . . .
8.6.3 The testShips program . . . . . . . . . . . . .
8.6.4 Submitting your assignment . . . . . . . . . . .
8.6.5 Provided source files . . . . . . . . . . . . . . .
8.6.6 Sample solution . . . . . . . . . . . . . . . . . .
Assignment 7, due Wednesday 2015-04-01, at 11:00pm
8.7.1 Solitaire with big cards . . . . . . . . . . . . .
8.7.2 Explanation of the testing program . . . . . . .
8.7.3 Submitting your assignment . . . . . . . . . . .
8.7.4 Sample solution . . . . . . . . . . . . . . . . . .
Assignment 8, due Wednesday 2015-04-08, at 11:00pm
8.8.1 An ordered set . . . . . . . . . . . . . . . . . .
8.8.2 The testOrderedSet wrapper . . . . . . . . .
8.8.3 Submitting your assignment . . . . . . . . . . .
8.8.4 Sample solution . . . . . . . . . . . . . . . . . .
Assignment 9, due Wednesday 2015-04-15, at 11:00pm
11
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8.9.1
8.9.2
8.9.3
8.9.4
Finding a cycle in a maze . . . . . . .
Input and output format . . . . . . .
Submitting and testing your program
Sample solution . . . . . . . . . . . . .
9 Common C coding and debugging issues
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Course administration
1.1
Overview
This is the course information for CPSC 223: Data Structures and Programming
Techniques for the Spring 2015 semester. This document is available in two
formats, both of which should contain the same information:
• HTML
• PDF
Code examples can be downloaded from links in the text, or can be found in the
examples directory.
The links above point to www.cs.yale.edu. In case this machine is down,
a backup copy of these files can be found at https://www.dropbox.com/sh/
omg9qcxkxeiam2o/AACRAJOTj8af6V7RC1cXBHjQa?dl=0.
This document is a work in progress, and is likely to change frequently as the
semester progresses.
1.1.1
Resources
• Schedule: list of lectures and events. Includes reading assignments and
pointers to lecture notes.
• Calendar: shows office hours, lectures, and assignment deadlines.
• Assignments: list of homeworks and exams.
• Notes: notes on various topics relevant to the course.
• Syllabus.
• Piazza. This is a web-based question-and-answer system for communicating
with course staff and other students.
• 2012 web pages: web pages for 2012 version of the course.
• Data structure visualizations. Much better than the ASCII art (at best)
illustrations you will find in this document.1
1 I would like to thank David Galles for making this site available and Xiao Shi for pointing
me to it.
12
1.1.2
•
•
•
•
•
•
•
•
1.1.3
Documentation
How to use the Zoo
Zoo help
Coding hints
GNU Online Documentation,
– gcc
– gdb
– ddd
– emacs
Frequently-asked questions (FAQ) for
– C
– C (abridged)
– Unix
– Emacs
Programming in C
Valgrind documentation
UNIXhelp for Users
Questions and comments
Please feel free to send questions or comments on the class or anything connected
to it to james.aspnes@gmail.com.
For questions about individual assignments, you may be able to get a faster
response using Piazza. Note that questions you ask there are visible to other
students if not specifically marked private, so be careful about broadcasting your
draft solutions.
1.2
1.2.1
Lecture schedule
Topics by date
2015-01-12 Introduction. What the course is about. Getting started with C:
running the compiler, the main function, integer data types, a few simple
programs. Readings: Course administration, The Zoo and the Zoo Annex,
The Linux programming environment, Structure of a C program, Basic
integer types; Kernighan and Ritchie §§1.1 and 1.2. 2015-01-14
Arithmetic in C. Readings: Integer constants, Integer operators, Operator
precedence (we didn’t actually do the full operator precedence table in
class, but it’s a nice summary of what operators are available, and if you
don’t like typing parentheses everywhere it’s useful to have an idea of how
precedence and associativity work); K&R §§1.4, 2.2, 2.3, 2.5, 2.6, 2.8, 2.9,
and 2.12. 2015-01-16
13
Local variables and assignment operators. The ++ and -- operators. More
specialized operators: , and ?:. I/O using getchar and putchar. Control
structures: if, switch, while, for. Readings: Variables, Statements
through The for loop; K&R §1.3, 1.5, 2.1, 2.4, and 3.1–3.6. 2015-01-21
Goto-like control structures: break, continue, goto, and return.
Functions.
Readings: Rest of [Statements]{#statements}, Functions{#functions}; K&R §3.7, 3.8, 4.1, and 4.5. Examples from lecture
(with a bit of after-the-fact sprucing up) can be found in the examples
directory. 2015-01-26
Start of pointers and arrays: pointer types and pointer variables. The &
and * operators. Using a pointer to get a value out of a function. Array
declarations. Preventing array modification with const. Storage allocation:
malloc, free, and realloc. Readings: Pointers up through Arrays and
functions; K&R §5.1–5.4. Examples. 2015-01-28
More on pointers and arrays: Multi-dimensional arrays, C99 variablelength arrays. Function pointers. Finding storage allocation bugs using valgrind. Readings: Rest of Pointers, Valgrind; K&R §§5.6–5.9,
5.11. Examples from lecture: original array-of-pointers-to-arrays implementation array2dOriginal.c, valgrind-approved version after removing
printArray2D on uninitialized data array2d.c, buggy version demonstrating how valgrind complains about writing off the end of an array and
doing multiple frees of the same block array2dBad.c. An issue that came
up in lecture is that all of these implementations are a little bit inefficient
in that they allocate a separate block for each row, which means extra overhead for mallocs data and the possibility that the rows may be scattered
throughout memory, causing issues with virtual memory paging. Here
is yet another implementation that gets space for both the array of row
pointers and all rows with a single call to malloc and then does pointer
arithmetic to slice the space up.2 There is also a generic version of the
simple approach in the section on multidimensional arrays. 2015-02-02
Strings in C: null-terminated strings. What goes wrong if you forget to put
on the null. Various library functions from <string.h> and how one might
implement them. The perils of gets and bounded-size buffers, and how
to use malloc and realloc to avoid them. Meaning of argv. Readings:
Strings; K&R §§5.5, 5.10, B2. Examples from lecture. 2015-02-04
Structured data types: structs, unions, and enums. Type aliases using
typedef. Opaque struct definitions and separating interface from implementation. Readings: Structured data types, typedef; K&R Chapter 6,
§2.5 (for enums). Examples from lecture, plus a working version of the
intArray implementation together with the originally stubby intArray.c
and the partial version from the end of lecture. 2015-02-09
More C stuff: Makefiles. Floating-point arithmetic and the math library.
Timing code with a profiler. Basics of file I/O. Readings: Make, Floating
2 Note that because each row is in the same malloc-ed block as its adjacent rows, valgrind
will not detect if you run off the end of a row in this implementation.
14
point types, Performance tuning, Input and output; K&R §§2.7 and 7.5,
Appendix B4. Examples from lecture. 2015-02-11
Start of data structures: efficiency of different data structures, linked
lists. Readings: Asymptotic notation, Linked lists. Examples from lecture
including improved stack.c and working queue.c. 2015-02-16
Invariants and representation functions for abstract data types. The deque
type, with two implementations: a circular doubly-linked list and a ring
buffer. Readings: Abstract data types, deques. Examples from lecture.
2015-02-18
Hash Wednesday: set and map data types, hash tables. Readings: Hash
tables. Examples from lecture. 2015-02-23
Various C preprocessor tricks: macros with arguments, string processing in
macros, non-syntactic macros, conditional compilation. Readings: Macros;
K&R Appendix A12.3. Examples from lecture. 2015-02-25
Polymorphism in C: generic containers and object-oriented programming
(of a sort) using structs full of function pointers. Example of using git
for version control (you will not be tested on version control). Readings:
Generic containers, version control. Examples from lecture. Unlike the
actual example from lecture, this version really works, instead of just
inserting 0 over and over again. 2015-03-02
Recursion. Readings: Recursion. Examples from lecture. 2015-03-04
Exam 1. This took place at the usual class time (1:00–2:15), and was
a closed-book test potentially covering all material discussed in lecture
prior to the exam. Sample exams from previous years: 2005, 2012. Sample
solutions. 2015-03-23
Binary trees and heaps. Readings: Binary trees, Heaps. Examples from
lecture. 2015-03-25
Binary search tree implementation basics: insertion, search, deletion,
various kinds of traversals. Readings: Binary search trees. Examples from
lecture. 2015-03-30
Balanced binary search trees: augmented trees and AVL trees. We also
saw a little bit about red-black trees, but not enough to actually be useful.
Readings: Augmented trees, AVL trees. Example from lecture: see AVL
tree implementation. 2015-04-01
Self-adjusting binary search trees: splay trees, a little bit about scapegoat
trees. Readings: Splay trees. There was no new code in lecture but we
did spend a while looking at a pre-prepared splay tree implementation.
2015-04-06
Graphs: structure of a graph, graph representations, basic ideas of graph
search. Readings: Graphs up to start of graph search. 2015-04-08
More graphs: depth-first and breadth-first search. Minimum spanning trees
and shortest paths. Readings: Rest of graphs. BFS example from lecture.
The program I was using in lecture to make nice pictures of graphs was
dot, specifically dot -Tpng. Thanks to the good folks at ITS, this is now
installed in the Zoo along with the rest of the GraphViz tools. 2015-04-13
Dynamic programming: all-pairs shortest paths, longest increasing subse15
quence, longest common subsequence. Readings: Dynamic programming.
Examples from lecture. 2015-04-15
Randomized data structures. Readings: Randomization. The devurandom.c example from lecture. 2015-04-20
Data structures for strings: tries, TSTs, and variants; radix sort. Readings:
String processing. 2015-04-22
Exam 2. This took place at the usual class time (1:00–2:15), and was a
closed-book test potential covering all material discussed in lecture during
the semester. Sample exams from previous years: 2005, 2012. Sample
solutions.
1.2.2
Topics not covered in 2015
Here are some topics that were not covered specifically this semester but can be
found in the notes.
•
•
•
•
•
•
•
•
1.3
Iterators
Suffix arrays
Testing during development
Algorithm design techniques
Bit manipulation
Persistence
C++
What next
Syllabus
Syllabus for Computer Science 223b, Data Structures and Programming Techniques. Instructor: James Aspnes.
1.3.1
On-line course information
On-line information about the course, including the lecture schedule, lecture
notes, and information about assignments, can be found at http://www.cs.yale.
edu/homes/aspnes/classes/223/notes.html. This document will be updated
frequently during the semester, and is also available in PDF format.
1.3.2
Meeting times
Lectures are MW 13:00–14:15 in WLH 201 (Sudler Hall). The lecture schedule
can be found in the course notes. A calendar is also available.
16
1.3.3
Synopsis of the course
Topics include programming in C; data structures (arrays, stacks, queues, lists,
trees, heaps, graphs); sorting and searching; storage allocation and management;
data abstraction; programming style; testing and debugging; writing efficient
programs.
1.3.4
Prerequisites
CPSC 201, or equivalent background. See me if you aren’t sure.
1.3.5
Textbook
The textbook for this course is:
• The C Programming Language (2nd Edition), by Brian W. Kernighan and
Dennis M. Ritchie. Prentice Hall, 1988. ISBN 0131103628. The definitive
introduction to C. You should memorize this book.
If you are on the Yale campus or are using VPN to get to Yale’s network, you can
access this book at http://proquest.safaribooksonline.com/book/programming/
c/9780133086249. You do not need to buy a physical copy of this book unless
you want to.
1.3.6
Course requirements
Nine weekly homework assignments, and two in-class exams. Assignments will
be weighted equally in computing the final grade. Each exam will count as three
assignments.
1.3.7
Staff
See the calendar for open office hours.
1.3.7.1
Instructor
James Aspnes (james.aspnes@gmail.com, http://www.cs.yale.edu/homes/
aspnes/). Office: AKW 401. If my open office hours don’t work for you, please
send email to make an appointment.
17
1.3.7.2
Teaching Fellows
• Yujia Hu yujia.hu@yale.edu
• Joshua Lockerman joshua.lockerman@yale.edu
• Junaid Nomani junaid.nomani@yale.edu
1.3.7.3
•
•
•
•
Peer tutors
Iulia Tamas iulia.tamas@yale.edu
Dylan Visher dylan.visher@yale.edu
Lining Wang lining.wang@yale.edu
Blake Woodworth blake.woodworth@yale.edu
1.3.8
Use of outside help
Students are free to discuss homework problems and course material with each
other, and to consult with the instructor or a TA. Solutions handed in, however,
should be the student’s own work. If a student benefits substantially from
hints or solutions received from fellow students or from outside sources, then
the student should hand in their solution but acknowledge the outside sources,
and we will apportion credit accordingly. Using outside resources in solving a
problem is acceptable but plagiarism is not.
1.3.9
Clarifications for homework assignments
From time to time, ambiguities and errors may creep into homework assignments.
Questions about the interpretation of homework assignments should be sent
to the instructor at james.aspnes@gmail.com. Clarifications will appear in the
on-line version of the assignment.
1.3.10
Late assignments
Assignments submitted after the deadline without a Dean’s Excuse are automatically assessed a 2%/hour penalty.
1.4
Introduction
There are two purposes to this course: to teach you to program in the C
programming language, and to teach you how to choose, implement, and use
data structures and standard programming techniques.
18
1.4.1
Why should you learn to program in C?
• It is
–
–
–
the de facto substandard of programming languages.
C runs on everything.
C lets you write programs that use very few resources.
C gives you near-total control over the system, down to the level of
pushing around individual bits with your bare hands.
– C imposes very few constraints on programming style: unlike higherlevel languages, C doesn’t have much of an ideology. There are very
few programs you can’t write in C.
– Many of the programming languages people actually use (Visual Basic,
perl, python, ruby, PHP, etc.) are executed by interpreters written in
C (or C++, an extension to C).
• You will learn discipline.
– C makes it easy to shoot yourself in the foot.
– You can learn to avoid this by being careful about where you point it.
– Pain is a powerful teacher of caution.
• You will fail CS323 if you don’t learn C really well in CS223 (CS majors
only).
On the other hand, there are many reasons why you might not want to use C
later in life. It’s missing a lot of features of modern program languages, including:
• A garbage collector.
• Minimal programmer-protection features like array bounds-checking or a
strong type system.
• Non-trivial built-in data structures.
• Language support for exceptions, namespaces, object-oriented programming, etc.
For most problems where minimizing programmer time and maximizing robustness are more important than minimizing runtime, other languages are a better
choice. But for this class, we’ll be using C.
If you want to read a lot of flaming about what C is or is not good for, see
http://c2.com/cgi/wiki?CeeLanguage.
1.4.2
Why should you learn about data structures and programming
techniques?
For small programs, you don’t need much in the way of data structures. But as
soon as you are representing reasonably complicated data, you need some place
to store it. Thinking about how you want to store and organize this data can be
a good framework for organizing the rest of your program.
Many programming environments will give you a rich collection of built-in data
structures as part of their standard library. C does not: unless you use third-
19
party libraries, any data structure you want in C you will have to build yourself.
For most data structures this will require an understanding of pointers and
storage allocation, mechanisms often hidden in other languages. Understanding
these concepts will give you a deeper understanding of how computers actually
work, and will both let you function in minimalist environments where you don’t
have a lot of support and let you understand what more convenient environments
are doing under their abstraction barriers.
The same applies to the various programming techniques we will discuss in this
class. While some of the issues that come up are specific to C and similar lowlevel languages (particular issues involving disciplined management of storage),
some techniques will apply no matter what kinds of programs you are writing
and all will help in understanding what your computer systems are doing even if
some of the details are hidden.
2
The Zoo and the Zoo Annex
The main undergraduate computing facility for Computer Science is the Zoo,
located on the third floor of AKW. The Zoo contains a large number of Linux
workstations.
You don’t need to do your work for this class in the Zoo, but that is where your
assignments will be submitted and tested, so if you do development elsewhere,
you will need to copy your files over and make sure that they work there as well.
The “Zoo Annex” is the informal name for 17HH room 111, which is reserved for
CS student use from 19:00 to 23:59 Sundays through Thursdays. The machines
in 17HH 111 run Windows, but once logged in, you can create a Linux desktop
remotely from a Zoo machine using a program called FastX. You can also
download and use FastX from your own machine to get access to the full Zoo
environment if you are running Windows or OSX.
The best place for information about the Zoo is at http://zoo.cs.yale.edu/. Below
are some points that are of particular relevance for CS223 students.
2.1
Getting an account
To get an account in the Zoo, follow the instructions at http://zoo.cs.yale.edu/
accounts.html. You will need your NetID and password to sign up for an account.
Even if you already have an account, you still need to use this form to register
as a CS 223 student, or you will not be able to submit assignments.
20
2.2
Getting into the room
The Zoo is located on the third floor of Arthur K Watson Hall, toward the
front of the building. If you are a Yale student, your ID should get you into the
building and the room. If you are not a student, you will need to get your ID
validated in AKW 008a to get in after hours.
2.3
Remote use
There are several options for remote use of the Zoo. FastX is the most straightforward if you want to replicate the Zoo experience elsewhere. I personally tend
to connect using ssh.
2.3.1
Access using FastX
These instructions are adapted from Stan Eisenstat’s documentation for CS323.
To use FastX, you will need an installation key. If you are downloading the
Windows or OSX version from the Yale Software Library, this will appear on
the download page. Otherwise, see the instructions in the following section.
2.3.1.1
Getting a license key
In order to use FastX, you will need a license key.
1. Go to the Yale Software Library page for FastX (Windows version).
2. Log in with your NetID and password if required.
3. Copy the installation key that appears below the “Download Now” button.
2.3.1.2
FastX in the Zoo Annex
Using FastX you can turn a window on a Windows machine in the Zoo Annex
(aka, the 17 Hillhouse, Room 111, cluster) into the same Linux desktop that you
see when you log into a Zoo node.
Do the following:
1. If you are not logged in:
• Press CTRL + ALT + DELETE. A new screen with a Usage Agreement will appear.
• Click “OK”. A new screen with a NetID and Password box will appear.
• Enter your NetID and Password and click “Continue”. Windows will
log you in.
2. Click on the Windows icon in the lower left-hand corner of the screen. A
new box will appear.
21
3. Mouse over the “Search programs and files” box in the new window and
type “fastx” (but do not hit return). A list of “Programs” will appear.
4. Click on “FastX” under “Programs” to launch FastX. (If a “Licensing”
window appears asking for an activation key, enter the installation key
from the download page, click “OK”, and click “Close” to dismiss the new
window that appears.)
5. If there is no entry named “Zoo” in the FastX window:
• Click on the green + sign in the upper left-hand corner. A new
window will appear.
• Enter “Zoo” in the Name field.
• Enter “node.zoo.cs.yale.edu” in the Host field.
• Do not change the Port or User field.
• Click “Save” to save the configuration.
6. Double click on the “Zoo” entry. A “Login” box should appear. If a
“Password” box appears instead:
• Click “Cancel”.
• Click on the green pencil in the upper left-hand corner of the FastX
window.
• Delete the contents of the User field. A grayed out “Joe” will appear.
• Click “Save” to save the configuration.
• Double click on the “Zoo” entry again. A “Login” box will appear.
7. Enter your Yale netID in the “Login” box (it should already be there) and
click “Continue”. (If a warning appears asking whether you want to accept
a new key, click on “Accept” to dismiss it.) A “Password” box will appear.
8. Enter your password in the “Password” box and click on “Continue”. A
new “Start new session” entry will appear in the right-hand side of the
FastX window.
9. Click on “Start new session” to produce a pulldown menu and click on
either “Gnome” or “KDE” to choose a window manager (Gnome is the
default in the Zoo) or “Xterm” to open a terminal window.
WARNING: If you click on “Gnome” but the desktop that appears does
not have the usual “Applications” and “Places” in the upper left-hand
corner, then:
1. Type ALT-F2 (that is, press the ALT and F2 keys simultaneously) to
open a command field.
2. Enter the command “gnome-shell –mode=classic -r” (without the
surrounding quotes) and press the “Enter” key.
10. A new window with a Zoo Linux desktop should open. You may resize it
by dragging on the lower left-hand corner if you prefer a different size.
11. When you log out of the Linux desktop, close the “FastX” window by
clicking the red X in the upper right-hand corner.
12. Do not forget to log out of Windows when you are done.
2.3.1.3
Using FastX from Windows
22
Using FastX you can turn a window on Windows into the same Linux desktop
that you see when you log into a Zoo node.
To install the software on your own Windows machine:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Go to the Yale Software Library and click on “Windows”.
Scroll through the list and click on “FastX”.
Copy the installation key that appears below the “Download Now” button.
Click on the “Download Now” button and follow the instructions for
installing the software.
Launch FastX.
Click on the green + sign in the upper left-hand corner of the FastX
window. In the new window that appears:
• Enter “Zoo” in the Name field.
• Enter “node.zoo.cs.yale.edu” in the Host field.
• Do not change the Port field.
• Enter your Yale netID in the User field.
• Click “Save” to save the configuration.
Double click on the “Zoo” entry.
When the “Licensing” window appears asking for an activation key, enter
the number copied above and click “OK”. A new window will appear. Click
“Close” to dismiss it.
Quit FastX.
If you run into difficulties up to this point, seek help from a student tech or one
of the instructional staff.
Once you have installed the software, do the following to run FastX:
1. Launch FastX.
2. Double click on the “Zoo” entry. A “Password” box will appear.
3. Enter the password for your Yale netID and click “OK”. A “Start a new
session” entry will appear in the FastX window.
4. Click on “Start new session” to produce a pulldown menu and click on
either “Gnome” or “KDE” to choose a window manager (Gnome is the
default in the Zoo) or “Xterm” to open a terminal window.
WARNING: If you click on “Gnome” but the desktop that appears does
not have the usual “Applications” and “Places” in the upper left-hand
corner, then:
1. Type ALT-F2 (that is, press the ALT and F2 keys simultaneously) to
open a command field.
2. Enter the command “gnome-shell –mode=classic -r” (without the
surrounding quotes) and press the “Enter” key.
5. A new window with a Zoo Linux desktop should open. You may resize it
by dragging on the lower left-hand corner if you prefer a different size.
6. When you log out of the Linux desktop, close the “FastX” window by
clicking the red X in the upper right-hand corner.
23
2.3.1.4
Using FastX from OSX
Using FastX you can turn a window on a Mac into the same Linux desktop that
you see when you log into a Zoo node.
To install the software on your own Mac:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Go to the Yale Software Library and click on “Mac”.
Scroll through the list and click on “FastX”.
Copy the installation key that appears below the “Download Now” button.
Click on the “Download Now” button and save the downloaded file as
“fastx_1016.dmg” (for version 1.0.16).
Click on the downloaded file “fastx_1016.dmg” to open the disk image. A
“FastX” disk icon will appear on your desktop, and a folder containing a
“FastX” icon will appear.
Drag the “FastX” icon to the folder where you would like to save it, and
launch FastX by clicking on it.
Click on the green + sign in the upper left-hand corner of the FastX
window. In the new window that appears:
• Enter “Zoo” in the Name field.
• Enter “node.zoo.cs.yale.edu” in the Host field.
• Do not change the Port field.
• Enter your Yale netID in the User field.
• Click “Save” to save the configuration.
Double click on the “Zoo” entry.
When the “Licensing” window appears asking for an activation key, enter
the number copied above and click “OK”. A new window will appear. Click
“Close” to dismiss it.
Quit FastX.
Drag the “FastX” disk icon and then the “fastx_1016.dmg” icon to the
Trash to clean up.
If you run into difficulties up to this point, seek help from a student tech or one
of the instructional staff.
Once you have installed the software, do the following to run FastX:
1. Launch FastX from the folder where you saved it.
2. Double click on “Zoo”. A “Password” box will appear, possibly behind the
FastX window. (If it does not, click on the “FastX” icon that is flashing in
your dock.)
3. Enter the password for your Yale netID and click “OK”. A “Start a new
session” entry will appear in the FastX window.
4. Click on “Start new session” to produce a pulldown menu and click on
either “Gnome” or “KDE” to choose a window manager (Gnome is the
default in the Zoo) or “Xterm” to open a terminal window.
WARNING: If you click on “Gnome” but the desktop that appears does
not have the usual “Applications” and “Places” in the upper left-hand
corner, then:
24
1. Type ALT-F2 (that is, press the ALT and F2 keys simultaneously) to
open a command field.
2. Enter the command “gnome-shell –mode=classic -r” (without the
surrounding quotes) and press the “Enter” key.
5. A new window with a Zoo Linux desktop should open behind the FastX
window. You may resize it by dragging on the lower left-hand corner if
you prefer a different size.
6. When you log out of the Linux desktop, close the “FastX” window by
clicking the red dot in the upper left-hand corner.
2.3.2
Terminal access
Date: Mon, 13 Dec 2004 14:34:19 -0500 (EST)
From: Jim Faulkner <james.faulkner@yale.edu>
Subject: Accessing the Zoo
Hello all,
I've been asked to write up a quick guide on how to access the Linux
computers in the Zoo. For those who need this information, please read
on.
There are 2 ways of accessing the Zoo nodes, by walking
logging in on the console (the computers are located on
AKW), or by connecting remotely via SSH. Telnet access
SSH clients for various operating systems are available
up to one and
the 3rd floor of
is not allowed.
here:
http://www.yale.edu/software/
Mac OSX comes with an SSH client by default. A good choice for an SSH
client if you run Microsoft Windows is PuTTY:
http://www.chiark.greenend.org.uk/~sgtatham/putty/
With the exception of a few legacy accounts, the Zoo uses your campus-wide
NetID and password for login access. However, you must sign up for a Zoo
account before access is allowed. To sign up for a Zoo account, go to
this web page:
http://zoo.cs.yale.edu/accounts.html
Then login with your campus-wide NetID and password. You may choose a
different shell, or set up your account to be enrolled in a class if that
is appropriate for you, but neither is necessary. Just click "Submit".
Within an hour, your Zoo account will be created, and you will receive
25
more information via e-mail about how to access the Zoo.
Users cannot log into zoo.cs.yale.edu (the central file server) directly,
they must log into one of the Zoo nodes. Following is the list of Zoo
nodes:
aphid.zoo.cs.yale.edu
bumblebee.zoo.cs.yale.edu
cardinal.zoo.cs.yale.edu
chameleon.zoo.cs.yale.edu
cicada.zoo.cs.yale.edu
cobra.zoo.cs.yale.edu
cricket.zoo.cs.yale.edu
frog.zoo.cs.yale.edu
gator.zoo.cs.yale.edu
giraffe.zoo.cs.yale.edu
grizzly.zoo.cs.yale.edu
hare.zoo.cs.yale.edu
hippo.zoo.cs.yale.edu
hornet.zoo.cs.yale.edu
jaguar.zoo.cs.yale.edu
koala.zoo.cs.yale.edu
ladybug.zoo.cs.yale.edu
leopard.zoo.cs.yale.edu
lion.zoo.cs.yale.edu
macaw.zoo.cs.yale.edu
monkey.zoo.cs.yale.edu
newt.zoo.cs.yale.edu
peacock.zoo.cs.yale.edu
perch.zoo.cs.yale.edu
python.zoo.cs.yale.edu
rattlesnake.zoo.cs.yale.edu
rhino.zoo.cs.yale.edu
scorpion.zoo.cs.yale.edu
swan.zoo.cs.yale.edu
termite.zoo.cs.yale.edu
tick.zoo.cs.yale.edu
tiger.zoo.cs.yale.edu
tucan.zoo.cs.yale.edu
turtle.zoo.cs.yale.edu
viper.zoo.cs.yale.edu
zebra.zoo.cs.yale.edu
If you have already created an account, you can SSH directly to one of
the above computers and log in with your campus-wide NetID and
password. You can also SSH to node.zoo.cs.yale.edu, which will connect
you to a random Zoo node.
Feel free to contact me if you have any questions about the Zoo.
thanks,
Jim Faulkner
Zoo Systems Administrator
2.3.3
GUI access
(These notes from Debayan Gupta.)
For Mac or Linux users, typing “ssh -X netID@node.zoo.cs.yale.edu” into a
terminal and then running “nautilus” will produce an X window interface.
When on Windows, I usually use XMing (I’ve included a step-by-step guide at
the end of this mail).
26
For transferring files, I use CoreFTP (http://www.coreftp.com). FileZilla (https:
//filezilla-project.org/) is another option.
Step-by-step guide to XMIng:
You can download Xming from here: http://sourceforge.net/projects/xming/
Download and install. Do NOT launch Xming at the end of your installation.
Once you’ve installed Xming, go to your start menu and find XLaunch (it should
be in the same folder as Xming).
1. Start XLaunch, and select “Multiple Windows”. Leave “Display Number”
as its default value. Click next.
2. Select “Start a program”. Click next.
3. Type “nautilus” (or “terminal”, if you want a terminal) into the “Start
Program” text area. Select “Using PuTTY (plink.exe)”.
4. Type in the name of the computer (use “node.zoo.cs.yale.edu”) in the
“Connect to computer” text box.
5. Type in your netID in the “Login as user” text box (you can leave the
password blank). Click next.
6. Make sure “Clipboard” is ticked. Leave everything else blank. Click next.
7. Click “Save Configuration”. When saving, make sure your filename ends
with “.xlaunch” - this will let you connect with a click (you won’t need to
do all this every time you connect).
8. Click Finish.
9. You will be prompted for your password - enter it. Ignore any security
warnings.
10. You now have a remote connection to the Zoo.
For more options and information, you can go to: http://www.straightrunning.
com/XmingNotes/
2.4
How to compile and run programs
See the chapter on how to use the Zoo for details of particular commands. The
basic steps are
• Creating the program with a text editor of your choosing. (I like vim for
long programs and cat for very short ones.)
• Compiling it with gcc.
• Running it.
27
If any of these steps fail, the next step is debugging. We’ll talk about debugging
elsewhere.
2.4.1
Creating the program
Use your favorite text editor. The program file should have a name of the form
foo.c; the .c at the end tells the C compiler the contents are C source code.
Here is a typical C program:
#include <stdio.h>
/* print the numbers from 1 to 10 */
int
main(int argc, char **argv)
{
int i;
puts("Now I will count from 1 to 10");
for(i = 1; i <= 10; i++) {
printf("%d\n", i);
}
}
return 0;
examples/count.c
2.4.2
Compiling and running a program
Here’s what happens when I compile and run it on the Zoo:
$ c99 -g3 -o count count.c
$ ./count
Now I will count from 1 to 10
1
2
3
4
5
6
7
8
9
10
$
28
The first line is the command to compile the program. The dollar sign is my
prompt, which is printed by the system to tell me it is waiting for a command.
The command calls gcc as c99 with arguments -g3 (enable maximum debugging
info), -o (specify executable file name, otherwise defaults to a.out), count (the
actual executable file name), and count.c (the source file to compile). This
tells gcc that we should compile count.c to count in C99 mode with maximum
debugging info included in the executable file.
The second line runs the output file count. Calling it ./count is necessary
because by default the shell (the program that interprets what you type) only
looks for programs in certain standard system directories. To make it run a
program in the current directory, we have to include the directory name.
2.4.3
Some notes on what the program does
Noteworthy features of this program include:
• The #include <stdio.h> in line 1. This is standard C boilerplate, and
will appear in any program you see that does input or output. The meaning
is to tell the compiler to include the text of the file /usr/include/stdio.h
in your program as if you had typed it there yourself. This particular file
contains declarations for the standard I/O library functions like puts (put
string) and printf (print formatted), as used in the program. If you don’t
put it in, your program may or may not still compile. Do it anyway.
• Line 3 is a comment; its beginning and end is marked by the /* and */
characters. Comments are ignored by the compiler but can be helpful for
other programmers looking at your code (including yourself, after you’ve
forgotten why you wrote something).
• Lines 5 and 6 declare the main function. Every C program has to have a
main function declared in exactly this way—it’s what the operating system
calls when you execute the program. The int on Line 3 says that main
returns a value of type int (we’ll describe this in more detail later in the
chapter on functions), and that it takes two arguments: argc of type int,
the number of arguments passed to the program from the command line,
and argv, of a pointer type that we will get to eventually, which is an array
of the arguments (essentially all the words on the command line, including
the program name). Note that it would also work to do this as one line
(as K&R typically does); the C compiler doesn’t care about whitespace,
so you can format things however you like, subject to the constraint that
consistency will make it easier for people to read your code.
• Everything inside the curly braces is the body of the main function. This
includes
– The declaration int i;, which says that i will be a variable that
holds an int (see the chapter on Integer Types).
29
– Line 10, which prints an informative message using puts (discussed
in the chapter on input and output.
– The for loop on Lines 11–13, which executes its body for each value
of i from 1 to 10. We’ll explain how for loops work later. Note that
the body of the loop is enclosed in curly braces just like the body
of the main function. The only statement in the body is the call to
printf on Line 12; this includes a format string that specifies that
we want a decimal-formatted integer followed by a newline (the \n).
– The return 0; on Line 15 tells the operating system that the program worked (the convention in Unix is that 0 means success). If
the program didn’t work for some reason, we could have returned
something else to signal an error.
3
The Linux programming environment
The Zoo runs a Unix-like operating system called Linux. Most people run Unix
with a command-line interface provided by a shell. Each line typed to the shell
tells it what program to run (the first word in the line) and what arguments
to give it (remaining words). The interpretation of the arguments is up to the
program.
3.1
The shell
When you sign up for an account in the Zoo, you are offered a choice of possible
shell programs. The examples below assume you have chosen bash, the Bourneagain shell written by the GNU project. Other shells behave similarly for basic
commands.
3.1.1
Getting a shell prompt in the Zoo
When you log in to a Zoo node directly, you may not automatically get a shell
window. If you use the default login environment (which puts you into the KDE
window manager), you need to click on the picture of the display with a shell in
from of it in the toolbar at the bottom of the screen. If you run Gnome instead
(you can change your startup environment using the popup menu in the login
box), you can click on the foot in the middle of the toolbar. Either approach will
pop up a terminal emulator from which you can run emacs, gcc, and so forth.
The default login shell in the Zoo is bash, and all examples of shell command
lines given in these notes will assume bash. You can choose a different login
shell on the account sign-up page if you want to, but you are probably best off
just learning to like bash.
30
3.1.2
The Unix filesystem
Most of what one does with Unix programs is manipulate the filesystem.
Unix files are unstructured blobs of data whose names are given by paths
consisting of a sequence of directory names separated by slashes: for example /home/accts/some-user/cs223/hw1.c. At any time you are in a current
working directory (type pwd to find out what it is and cd new-directory to
change it). You can specify a file below the current working directory by
giving just the last part of the pathname. The special directory names .
and .. can also be used to refer to the current directory and its parent. So
/home/accts/some-user/cs223/hw1.c is just hw1.c or ./hw1.c if your current
working directory is /home/accts/some-user/cs223, cs223/hw1.c if your current working directory is /home/accts/some-user, and ../cs223/hw1.c if your
current working directory is /home/accts/some-user/illegal-downloads.
All Zoo machines share a common filesystem, so any files you create or change
on one Zoo machine will show up in the same place on all the others.
3.1.3
Unix command-line programs
Here are some handy Unix commands:
man man program will show you the on-line documentation (the man page) for
a program (e.g., try man man or man ls). Handy if you want to know what
a program does. On Linux machines like the ones in the Zoo you can also
get information using info program, which has an Emacs-like interface.
You can also use man function to see documentation for standard library
functions. The command man -k string will search for man pages whose
titles contain string.
Sometimes there is more than one man page with the same name. In this
case man -k will distingiush them by different manual section numbers,
e.g., printf (1) (a shell command) vs. printf (3) (a library routine).
To get a man page from a specific section, use man section name, e.g. man
3 printf. ls
ls lists all the files in the current directory. Some useful variants:
• ls /some/other/dir; list files in that directory instead.
• ls -l; long output format showing modification dates and owners.
mkdir mkdir dir will create a new directory in the current directory named
dir. rmdir
rmdir dir deletes a directory. It only works on directories that contain
no files. cd
31
cd dir changes the current working directory. With no arguments, cd
changes back to your home directory. pwd
pwd (“print working directory”) shows what your current directory is. mv
mv old-name new-name changes the name of a file. You can also use this
to move files between directories. cp
cp old-name new-name makes a copy of a file. rm
rm file deletes a file. Deleted files cannot be recovered. Use this command
carefully. chmod
chmod changes the permissions on a file or directory. See the man page for
the full details of how this works. Here are some common chmod’s:
• chmod 644 file; owner can read or write the file, others can only
read it.
• chmod 600 file; owner can read or write the file, others can’t do
anything with it.
• chmod 755 file; owner can read, write, or execute the file, others
can read or execute it. This is typically used for programs or for
directories (where the execute bit has the special meaning of letting
somebody find files in the directory).
• chmod 700 file; owner can read, write, or execute the file, others
can’t do anything with it.
emacs, gcc, make, gdb, git See corresponding sections.
3.1.4
Stopping and interrupting programs
Sometimes you may have a running program that won’t die. Aside from costing
you the use of your terminal window, this may be annoying to other Zoo users,
especially if the process won’t die even if you close the terminal window or log
out.
There are various control-key combinations you can type at a terminal window
to interrupt or stop a running program.
ctrl-C Interrupt the process. Many processes (including any program you write
unless you trap SIGINT using the sigaction system call) will die instantly
when you do this. Some won’t. ctrl-Z
Suspend the process. This will leave a stopped process lying around. Type
jobs to list all your stopped processes, fg to restart the last process (or
fg %1 to start process %1 etc.), bg to keep running the stopped process
in the background, kill %1 to kill process %1 politely, kill -KILL %1 to
kill process %1 whether it wants to die or not. ctrl-D
Send end-of-file to the process. Useful if you are typing test input
to a process that expects to get EOF eventually or writing programs
32
using cat > program.c (not really recommmended). For test input,
you are often better putting it into a file and using input redirection
(./program < test-input-file); this way you can redo the test after
you fix the bugs it reveals. ctrl-\
Quit the process. Sends a SIGQUIT, which asks a process to quit and
dump core. Mostly useful if ctrl-C and ctrl-Z don’t work.
If you have a runaway process that you can’t get rid of otherwise, you can use
ps g to get a list of all your processes and their process ids. The kill command
can then be used on the offending process, e.g. kill -KILL 6666 if your evil
process has process id 6666. Sometimes the killall command can simplify
this procedure, e.g. killall -KILL evil kills all process with command name
evil.
3.1.5
Running your own programs
If you compile your own program, you will need to prefix it with ./ on the
command line to tell the shell that you want to run a program in the current
directory (called ‘.’) instead of one of the standard system directories. So for
example, if I’ve just built a program called count, I can run it by typing
$ ./count
Here the “$ ” is standing in for whatever your prompt looks like; you should
not type it.
Any words after the program name (separated by whitespace—spaces and/or
tabs) are passed in as arguments to the program. Sometimes you may wish to
pass more than one word as a single argument. You can do so by wrapping the
argument in single quotes, as in
$ ./count 'this is the first argument' 'this is the second argument'
3.1.6
Redirecting input and output
Some programs take input from standard input (typically the terminal). If
you are doing a lot of testing, you will quickly become tired of typing test input
at your program. You can tell the shell to redirect standard input from a file
by putting the file name after a < symbol, like this:
$ ./count < huge-input-file
A ‘>’ symbol is used to redirect standard output, in case you don’t want to
read it as it flies by on your screen:
$ ./count < huge-input-file > huger-output-file
A useful file for both input and output is the special file /dev/null. As input,
it looks like an empty file. As output, it eats any characters sent to it:
33
$ ./sensory-deprivation-experiment < /dev/null > /dev/null
You can also pipe programs together, connecting the output of one to the input
of the next. Good programs to put at the end of a pipe are head (eats all but the
first ten lines), tail (eats all but the last ten lines), more (lets you page through
the output by hitting the space bar, and tee (shows you the output but also saves
a copy to a file). A typical command might be something like ./spew | more or
./slow-but-boring | tee boring-output. Pipes can consist of a long train
of programs, each of which processes the output of the previous one and supplies
the input to the next. A typical case might be:
$ ./do-many-experiments | sort | uniq -c | sort -nr
which, if ./do-many-experiments gives the output of one experiment on each
line, produces a list of distinct experimental outputs sorted by decreasing frequency. Pipes like this can often substitute for hours of real programming.
3.2
Text editors
To write your programs, you will need to use a text editor, preferably one that
knows enough about C to provide tools like automatic indentation and syntax
highlighting. There are three reasonable choices for this in the Zoo: kate, emacs,
and vim (which can also be run as vi). Kate is a GUI-style editor that comes
with the KDE window system; it plays nicely with the mouse, but Kate skills will
not translate well into other environements. Emacs and Vi have been the two
contenders for the One True Editor since the 1970s—if you learn one (or both)
you will be able to use the resulting skills everywhere. My personal preference is
to use Vi, but Emacs has the advantage of using the same editing commands as
the shell and gdb command-line interfaces.
3.2.1
Writing C programs with Emacs
To start Emacs, type emacs at the command line. If you are actually sitting at
a Zoo node it should put up a new window. If not, Emacs will take over the
current window. If you have never used Emacs before, you should immediately
type C-h t (this means hold down the Control key, type h, then type t without
holding down the Control key). This will pop you into the Emacs built-in
tutorial.
3.2.1.1
My favorite Emacs commands
General note: C-x means hold down Control and press x; M-x means hold down
Alt (Emacs calls it “Meta”) and press x. For M-x you can also hit Esc and then
x.
34
C-h Get help. Everything you could possibly want to know about Emacs is
available through this command. Some common versions: C-h t puts up
the tutorial, C-h b lists every command available in the current mode,
C-h k tells you what a particular sequence of keystrokes does, and C-h l
tells you what the last 50 or so characters you typed were (handy if Emacs
just garbled your file and you want to know what command to avoid in
the future). C-x u
Undo. Undoes the last change you made to the current buffer. Type it
again to undo more things. A lifesaver. Note that it can only undo back
to the time you first loaded the file into Emacs—if you want to be able to
back out of bigger changes, use git (described below). C-x C-s
Save. Saves changes to the current buffer out to its file on disk. C-x C-f
Edit a different file. C-x C-c
Quit out of Emacs. This will ask you if you want to save any buffers that
have been modified. You probably want to answer yes (y) for each one,
but you can answer no (n) if you changed some file inside Emacs but want
to throw the changes away. C-f
Go forward one character. C-b
Go back one character. C-n
Go to the next line. C-p
Go to the previous line. C-a
Go to the beginning of the line. C-k
Kill the rest of the line starting with the current position. Useful Emacs
idiom: C-a C-k. C-y
“Yank.” Get back what you just killed. TAB
Re-indent the current line. In C mode this will indent the line according
to Emacs’s notion of how C should be indented. M-x compile
Compile a program. This will ask you if you want to save out any unsaved
buffers and then run a compile command of your choice (see the section
on compiling programs below). The exciting thing about M-x compile is
that if your program has errors in it, you can type C-x ‘ to jump to the
next error, or at least where gcc thinks the next error is.
3.2.2
Using Vi instead of Emacs
If you don’t find yourself liking Emacs very much, you might want to try Vim
instead. Vim is a vastly enhanced reimplementation of the classic vi editor,
which I personally find easier to use than Emacs. Type vimtutor to run the
tutorial.
One annoying feature of Vim is that it is hard to figure out how to quit. If you
don’t mind losing all of your changes, you can always get out by hitting the
Escape key a few times and then typing ~ \\\ :qa!\\\ ~
To run Vim, type vim or vim filename from the command line. Or you can use
the graphical version gvim, which pops up its own window.
35
Vim is a modal editor, meaning that at any time you are in one of several modes
(normal mode, insert mode, replace mode, operator-pending mode, etc.), and
the interpretation of keystrokes depends on which mode you are in. So typing
jjjj in normal mode moves the cursor down four lines, while typing jjjj in
insert mode inserts the string jjjj at the current position. Most of the time
you will be in either normal mode or insert mode. There is also a command
mode entered by hitting : that lets you type longer commands, similar to the
Unix command-line or M-x in Emacs.
3.2.2.1
3.2.2.1.1
My favorite Vim commands
Normal mode
:h Get help. (Hit Enter at the end of any command that starts with a colon.)
Escape
Get out of whatever strange mode you are in and go back to normal mode.
You will need to use this whenever you are done typing code and want to
get back to typing commands. i
Enter insert mode. You will need to do this to type anything. The
command a also enters insert mode, but puts new text after the current
cursor position instead of before it. u
Undo. Undoes the last change you made to the current buffer. Type it
again to undo more things. If you undid something by mistake, c-R (control
R) will redo the last undo (and can also be repeated). :w
Write the current file to disk. Use :w filename to write it to filename.
Use :wa to write all files that you have modified. The command ZZ does
the same thing without having to hit Enter at the end. :e filename
Edit a different file. :q
Quit. Vi will refuse to do this if you have unwritten files. See :wa for how
to fix this, or use :q! if you want to throw away your changes and quit
anyway. The shortcuts :x and :wq do a write of the current file followed
by quitting. h, j, k, l
Move the cursor left, down, up, or right. You can also use the arrow keys
(in both normal mode and insert mode). x
Delete the current character. D
Delete to end of line. dd
Delete all of the current line. This is a special case of a more general d
command. If you precede it with a number, you can delete multiple lines:
5dd deletes the next 5 lines. If you replace the second d with a motion
command, you delete until wherever you land: d$ deletes to end of line
(D is faster), dj deletes this line and the line after it, d% deletes the next
matching group of parantheses/braces/brackets and whatever is between
them, dG deletes to end of file—there are many possibilities. All of these
save what you deleted into register "" so you can get them back with p. yy
36
Like dd, but only saves the line to register "" and doesn’t delete it. (Think
copy). All the variants of dd work with yy: 5yy, y$, yj, y%, etc. p
Pull whatever is in register "". (Think paste). << and >>
Outdent or indent the current line one tab stop. :make
Run make in the current directory. You can also give it arguments, e.g.,
:make myprog, :make test. Use :cn to go to the next error if you get
errors. :!
Run a command, e.g., :! echo hello world or :! gdb myprogram. Returns to Vim when the command exits (control-C can sometimes be helpful
if your command isn’t exiting when it should). This works best if you ran
Vim from a shell window; it doesn’t work very well if Vim is running in its
own window.
3.2.2.1.2
Insert mode
control-P and control-N These are completion commands that attempt
to expand a partial word to something it matches elsewhere in the
buffer. So if you are a good person and have named a variable
informativeVariableName instead of ivn, you can avoid having to type
the entire word by typing inf<control-P> if it’s the only word in your
buffer that starts with inf. control-O and control-I
Jump to the last cursor position before a big move / back to the place you
jumped from. ESC
Get out of insert mode!
3.2.2.2
Settings
Unlike Emacs, Vim’s default settings are not very good for editing C programs.
You can fix this by creating a file called .vimrc in your home directory with the
following commands:
set shiftwidth=4
set autoindent
set backup
set cindent
set hlsearch
set incsearch
set showmatch
set number
syntax on
filetype plugin on
filetype indent on
examples/sample.vimrc
(You can download this file by clicking on the link.)
37
In Vim, you can type e.g. :help backup to find out what each setting does.
Note that because .vimrc starts with a ., it won’t be visible to ls unless you
use ls -a or ls -A.
3.3
3.3.1
Compilation tools
The GNU C compiler gcc
A C program will typically consist of one or more files whose names end with .c.
To compile foo.c, you can type gcc foo.c. Assuming foo.c contains no errors
egregious enough to be detected by the extremely forgiving C compiler, this will
produce a file named a.out that you can then execute by typing ./a.out.
If you want to debug your program using gdb or give it a different name,
you will need to use a longer command line. Here’s one that compiles foo.c
to foo (run it using ./foo) and includes the information that gdb needs:
gcc -g3 -o foo foo.c
If you want to use C99 features, you will need to tell gcc to use C99 instead
of its own default dialect of C. You can do this either by adding the argument
-std=c99 as in gcc -std=c99 -o foo foo.c or by calling gcc as c99 as in c99
-o foo foo.c.
By default, gcc doesn’t check everything that might be wrong with your program.
But if you give it a few extra arguments, it will warn you about many (but not
all) potential problems: c99 -g3 -Wall -pedantic -o foo foo.c.
3.3.2
Make
For complicated programs involving multiple source files, you are probably better
off using make than calling gcc directly. Make is a “rule-based expert system”
that figures out how to compile programs given a little bit of information about
their components.
For example, if you have a file called foo.c, try typing make foo and see what
happens.
In general you will probably want to write a Makefile, which is named Makefile
or makefile and tells make how to compile programs in the same directory. Here’s
a typical Makefile:
# Any line that starts with a sharp is a comment and is ignored
# by Make.
# These lines set variables that control make's default rules.
# We STRONGLY recommend putting "-Wall -pedantic -g3" in your CFLAGS.
CC=gcc
38
CFLAGS=-std=c99 -Wall -pedantic -g3
# The next line is a dependency line.
# It says that if somebody types "make all"
# make must first make "hello-world".
# By default the left-hand-side of the first dependency is what you
# get if you just type "make" with no arguments.
all: hello-world
# How do we make hello-world?
# The dependency line says you need to first make hello-world.o
# and hello-library.o
hello-world: hello-world.o hello-library.o
# Subsequent lines starting with a TAB character give
# commands to execute.
# This command uses make built-in variables to avoid
# retyping (and getting things wrong):
# $@ = target hello-world
# $^ = dependencies hello-world.o and hello-library.o
$(CC) $(CFLAGS) -o $@ $^
# You can put whatever commands you want.
echo "I just built hello-world! Hooray!"
# Here we are saying that hello-world.o and hello-library.o
# should be rebuilt whenever their corresponding source file
# or hello-library.h changes.
# There are no commands attached to these dependency lines, so
# make will have to figure out how to do that somewhere else
# (probably from the builtin .c -> .o rule).
hello-world.o: hello-world.c hello-library.h
hello-library.o: hello-library.c hello-library.h
# Command lines can do more than just build things. For example,
# "make test" will rebuild hello-world (if necessary) and then run it.
test: hello-world
./hello-world
# This lets you type "make clean" and get rid of anything you can
# rebuild. The $(RM) variable is predefined to "rm -f"
clean:
$(RM) hello-world *.o
examples/usingMake/Makefile
Given a Makefile, make looks at each dependency line and asks: (a) does the
target on the left hand side exist, and (b) is it older than the files it depends on.
If so, it looks for a set of commands for rebuilding the target, after first rebuilding
39
any of the files it depends on; the commands it runs will be underneath some
dependency line where the target appears on the left-hand side. It has built-in
rules for doing common tasks like building .o files (which contain machine code)
from .c files (which contain C source code). If you have a fake target like all
above, it will try to rebuild everything all depends on because there is no file
named all (one hopes).
3.3.2.1
Make gotchas
Make really really cares that the command lines start with a TAB character.
TAB looks like eight spaces in Emacs and other editors, but it isn’t the same
thing. If you put eight spaces in (or a space and a TAB), Make will get horribly
confused and give you an incomprehensible error message about a “missing
separator”. This misfeature is so scary that I avoided using make for years
because I didn’t understand what was going on. Don’t fall into that trap—make
really is good for you, especially if you ever need to recompile a huge program
when only a few source files have changed.
If you use GNU Make (on a zoo node), note that beginning with version 3.78,
GNU Make prints a message that hints at a possible SPACEs-vs-TAB problem,
like this:
$ make
Makefile:23:*** missing separator (did you mean TAB instead of 8 spaces?).
If you need to repair a Makefile that uses spaces, one way of converting leading
spaces into TABs is to use the unexpand program:
$ mv Makefile Makefile.old
$ unexpand Makefile.old > Makefile
3.4
Debugging tools
The standard debugger on the Zoo is gdb. Also useful is the memory error
checker valgrind. Below are some notes on debugging in general and using
these programs in particular.
3.4.1
Debugging in general
Basic method of all debugging:
1. Know what your program is supposed to do.
2. Detect when it doesn’t.
3. Fix it.
A tempting mistake is to skip step 1, and just try randomly tweaking things
until the program works. Better is to see what the program is doing internally,
40
Stop.
so you can see exactly where and when it is going wrong. A second temptation
is to attempt to intuit where things are going wrong by staring at the code or
the program’s output. Avoid this temptation as well: let the computer tell you
what it is really doing inside your program instead of guessing.
3.4.2
Assertions
Every non-trivial C program should include <assert.h>, which gives you the
assert macro (see Appendix B6 of K&R). The assert macro tests if a condition
is true and halts your program with an error message if it isn’t:
#include <assert.h>
int
main(int argc, char **argv)
{
assert(2+2 == 5);
return 0;
}
examples/debugging/no.c
Compiling and running this program produces the following output:
$ gcc -o no no.c
$ ./no
no: no.c:6: main: Assertion `2+2 == 5' failed.
Line numbers and everything, even if you compile with the optimizer turned
on. Much nicer than a mere segmentation fault, and if you run it under the
debugger, the debugger will stop exactly on the line where the assert failed so
you can poke around and see why.
3.4.3
The GNU debugger gdb
The standard debugger on Linux is called gdb. This lets you run your program
under remote control, so that you can stop it and see what is going on inside.
You can also use ddd, which is a graphical front-end for gdb. There is an extensive
tutorial available for ddd, so we will concentrate on the command-line interface
to gdb here.
We’ll look at a contrived example. Suppose you have the following program
bogus.c:
#include <stdio.h>
/* Print the sum of the integers from 1 to 1000 */
41
int
main(int argc, char **argv)
{
int i;
int sum;
}
sum = 0;
for(i = 0; i -= 1000; i++) {
sum += i;
}
printf("%d\n", sum);
return 0;
examples/debugging/bogus.c
Let’s compile and run it and see what happens. Note that we include the flag
-g3 to tell the compiler to include debugging information. This allows gdb to
translate machine addresses back into identifiers and line numbers in the original
program for us.
$ c99 -g3 -o bogus bogus.c
$ ./bogus
-34394132
$
That doesn’t look like the sum of 1 to 1000. So what went wrong? If we were
clever, we might notice that the test in the for loop is using the mysterious -=
operator instead of the <= operator that we probably want. But let’s suppose
we’re not so clever right now—it’s four in the morning, we’ve been working on
bogus.c for twenty-nine straight hours, and there’s a -= up there because in our
befuddled condition we know in our bones that it’s the right operator to use.
We need somebody else to tell us that we are deluding ourselves, but nobody is
around this time of night. So we’ll have to see what we can get the computer to
tell us.
The first thing to do is fire up gdb, the debugger. This runs our program in
stop-motion, letting us step through it a piece at a time and watch what it is
actually doing. In the example below gdb is run from the command line. You
can also run it directly from Emacs with M-x gdb, which lets Emacs track and
show you where your program is in the source file with a little arrow, or (if you
are logged in directly on a Zoo machine) by running ddd, which wraps gdb in a
graphical user interface.
$ gdb bogus
GNU gdb 4.17.0.4 with Linux/x86 hardware watchpoint and FPU support
Copyright 1998 Free Software Foundation, Inc.
GDB is free software, covered by the GNU General Public License, and you are
welcome to change it and/or distribute copies of it under certain conditions.
42
Type "show copying" to see the conditions.
There is absolutely no warranty for GDB. Type "show warranty" for details.
This GDB was configured as "i386-redhat-linux"...
(gdb) run
Starting program: /home/accts/aspnes/tmp/bogus
-34394132
Program exited normally.
So far we haven’t learned anything. To see our program in action, we need to
slow it down a bit. We’ll stop it as soon as it enters main, and step through it
one line at a time while having it print out the values of the variables.
(gdb) break main
Breakpoint 1 at 0x8048476: file bogus.c, line 9.
(gdb) run
Starting program: /home/accts/aspnes/tmp/bogus
Breakpoint 1, main (argc=1, argv=0xbffff9ac) at bogus.c:9
9
sum = 0;
(gdb) display sum
1: sum = 1
(gdb) n
10
for(i = 0; i -= 1000; i++)
1: sum = 0
(gdb) display i
2: i = 0
(gdb) n
11
sum += i;
2: i = -1000
1: sum = 0
(gdb) n
10
for(i = 0; i -= 1000; i++)
2: i = -1000
1: sum = -1000
(gdb) n
11
sum += i;
2: i = -1999
1: sum = -1000
(gdb) n
10
for(i = 0; i -= 1000; i++)
2: i = -1999
1: sum = -2999
(gdb) quit
The program is running. Exit anyway? (y or n) y
$
43
Here we are using break main to tell the program to stop as soon as it enters
main, display to tell it to show us the value of the variables i and sum whenever
it stops, and n (short for next) to execute the program one line at a time.
When stepping through a program, gdb displays the line it will execute next as
well as any variables you’ve told it to display. This means that any changes you
see in the variables are the result of the previous displayed line. Bearing this in
mind, we see that i drops from 0 to -1000 the very first time we hit the top of
the for loop and drops to -1999 the next time. So something bad is happening
in the top of that for loop, and if we squint at it a while we might begin to
suspect that i -= 1000 is not the nice simple test we might have hoped it was.
3.4.3.1
My favorite gdb commands
help Get a description of gdb’s commands. run
Runs your program. You can give it arguments that get passed in to your
program just as if you had typed them to the shell. Also used to restart
your program from the beginning if it is already running. quit
Leave gdb, killing your program if necessary. break
Set a breakpoint, which is a place where gdb will automatically stop your
program. Some examples: - break somefunction stops before executing
the first line somefunction. - break 117 stops before executing line
number 117. list
Show part of your source file with line numbers (handy for figuring out
where to put breakpoints). Examples: - list somefunc lists all lines of
somefunc. - list 117-123 lists lines 117 through 123.
next Execute the next line of the program, including completing any procedure
calls in that line. step
Execute the next step of the program, which is either the next line if it
contains no procedure calls, or the entry into the called procedure. finish
Continue until you get out of the current procedure (or hit a breakpoint).
Useful for getting out of something you stepped into that you didn’t want
to step into. cont
(Or continue). Continue until (a) the end of the program, (b) a fatal
error like a Segmentation Fault or Bus Error, or (c) a breakpoint. If you
give it a numeric argument (e.g., cont 1000) it will skip over that many
breakpoints before stopping. print
Print the value of some expression, e.g. print i. display
Like print, but runs automatically every time the program stops. Useful
for watching values that change often. set disable-randomization off
Not something you will need every day, but you should try this before
running your program if it is producing segmentation faults outside of
gdb but not inside. Normally the Linux kernel randomizes the position
of bits of your program before running it, to make its response to buffer
overflow attacks less predictable. By default, gdb turns this off so that
the behavior of your program is consistent from one execution to the next.
44
But sometimes this means that a pointer that had been bad with address
randomization (causing a segmentation fault) turns out not to be bad
without. This option will restore the standard behavior outside gdb and
give you some hope of finding what went wrong.
3.4.3.2
Debugging strategies
In general, the idea behind debugging is that a bad program starts out sane, but
after executing for a while it goes bananas. If you can find the exact moment in
its execution where it first starts acting up, you can see exactly what piece of
code is causing the problem and have a reasonably good chance of being able to
fix it. So a typical debugging strategy is to put in a breakpoint (using break)
somewhere before the insanity hits, “instrument” the program (using display)
so that you can watch it going insane, and step through it (using next, step, or
breakpoints and cont) until you find the point of failure. Sometimes this process
requires restarting the program (using run) if you skip over this point without
noticing it immediately.
For large or long-running programs, it often makes sense to do binary search to
find the point of failure. Put in a breakpoint somewhere (say, on a function that
is called many times or at the top of a major loop) and see what the state of the
program is after going through the breakpoint 1000 times (using something like
cont 1000). If it hasn’t gone bonkers yet, try restarting and going through 2000
times. Eventually you bracket the error as occurring (for example) somewhere
between the 4000th and 8000th occurrence of the breakpoint. Now try stepping
through 6000 times; if the program is looking good, you know the error occurs
somewhere between the 6000th and 8000th breakpoint. A dozen or so more
experiments should be enough isolate the bug to a specific line of code.
The key to all debugging is knowing what your code is supposed to do. If you
don’t know this, you can’t tell the lunatic who thinks he’s Napoleon from lunatic
who really is Napoleon. If you’re confused about what your code is supposed
to be doing, you need to figure out what exactly you want it to do. If you can
figure that out, often it will be obvious what is going wrong. If it isn’t obvious,
you can always go back to gdb.
3.4.3.3
Common applications of gdb
Here are some typical classes of bugs and how to squish them with gdb. (The
same instructions usually work for ddd.)
3.4.3.3.1
Watching your program run
1. Compile your program with the -g3 flag. You can still run gdb if you don’t
do this, but it won’t be able to show you variable names or source lines.
2. Run gdb with gdb programname.
45
3. Type break main to stop at the start of the main routine.
4. Run your program with run arguments. The run command stands in for
the program name. You can also redirect input as in the shell with run
arguments < filename.
5. When the program stops, you can display variables in the current function
or expressions involving these variables using display, as in display x,
display a[i], display z+17. In ddd, double-clicking on a variable name
will have the same effect. Use undisplay to get rid of any displays you
don’t want.
6. To step through your program, use next (always goes to next line in the
current function, not dropping down into function calls), step (go to the
next executed line, even if it is inside a called function), finish (run until
the current function returns), and cont (run until the end of the program
or the next breakpoint).
This can be handy if you don’t particularly know what is going on in your
program and want to see.
3.4.3.3.2
Dealing with failed assertions
Run the program as described above. When you hit the bad assert, you will
stop several functions deep from where it actually happened. Use up to get up
to the function that has the call to assert then use print or display to figure
out what is going on.
Example program:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
int
main(int argc, char **argv)
{
int x;
x = 3;
assert(x+x == 4);
}
return 0;
examples/debugging/assertFailed.c
With gdb in action:
$ gcc -g3 -o assertFailed assertFailed.c
46
22:59:39 (Sun Feb 15) zeniba aspnes ~/g/classes/223/notes/examples/debugging
$ gdb assertFailed
GNU gdb (Ubuntu 7.7.1-0ubuntu5~14.04.2) 7.7.1
Copyright (C) 2014 Free Software Foundation, Inc.
License GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>
This is free software: you are free to change and redistribute it.
There is NO WARRANTY, to the extent permitted by law. Type "show copying"
and "show warranty" for details.
This GDB was configured as "i686-linux-gnu".
Type "show configuration" for configuration details.
For bug reporting instructions, please see:
<http://www.gnu.org/software/gdb/bugs/>.
Find the GDB manual and other documentation resources online at:
<http://www.gnu.org/software/gdb/documentation/>.
For help, type "help".
Type "apropos word" to search for commands related to "word"...
Reading symbols from assertFailed...done.
(gdb) run
Starting program: /home/aspnes/g/classes/223/notes/examples/debugging/assertFailed
assertFailed: assertFailed.c:12: main: Assertion `x+x == 4' failed.
Program received signal SIGABRT, Aborted.
0xb7fdd416 in __kernel_vsyscall ()
(gdb) up
#1 0xb7e43577 in __GI_raise (sig=sig@entry=6) at ../nptl/sysdeps/unix/sysv/linux/raise.c:56
56 ../nptl/sysdeps/unix/sysv/linux/raise.c: No such file or directory.
(gdb) up
#2 0xb7e469a3 in __GI_abort () at abort.c:89
89 abort.c: No such file or directory.
(gdb) up
#3 0xb7e3c6c7 in __assert_fail_base (fmt=0xb7f7a8b4 "%s%s%s:%u: %s%sAssertion `%s' failed.\
assertion=assertion@entry=0x804850f "x+x == 4", file=file@entry=0x8048500 "assertFailed.
line=line@entry=12, function=function@entry=0x8048518 <__PRETTY_FUNCTION__.2355> "main")
92 assert.c: No such file or directory.
(gdb) up
#4 0xb7e3c777 in __GI___assert_fail (assertion=0x804850f "x+x == 4", file=0x8048500 "assert
function=0x8048518 <__PRETTY_FUNCTION__.2355> "main") at assert.c:101
101 in assert.c
(gdb) up
#5 0x0804845d in main (argc=1, argv=0xbffff434) at assertFailed.c:12
12
assert(x+x == 4);
(gdb) print x
$1 = 3
Here we see that x has value 3, which may or may not be the right value, but
certainly violates the assertion.
47
3.4.3.3.3
Dealing with segmentation faults
Very much like the previous case. Run gdb until the segmentation fault hits,
then look around for something wrong.
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
int
main(int argc, char **argv)
{
int a[1000];
int i;
i = -1771724;
printf("%d\n", a[i]);
}
return 0;
examples/debugging/segmentationFault.c
$ gcc -g3 -o segmentationFault segmentationFault.c
23:04:18 (Sun Feb 15) zeniba aspnes ~/g/classes/223/notes/examples/debugging
$ gdb segmentationFault
GNU gdb (Ubuntu 7.7.1-0ubuntu5~14.04.2) 7.7.1
[...]
Reading symbols from segmentationFault...done.
(gdb) run
Starting program: /home/aspnes/g/classes/223/notes/examples/debugging/segmentationFault
Program received signal SIGSEGV, Segmentation fault.
0x08048435 in main (argc=1, argv=0xbffff434) at segmentationFault.c:13
13
printf("%d\n", a[i]);
(gdb) print a[i]
$1 = 0
(gdb) print i
$2 = -1771724
Curiously, gdb has no problem coming up with a value for a[i]. But i looks
pretty suspicious.
3.4.3.3.4
Dealing with infinite loops
Run gdb, wait a while, then hit control-C. This will stop gdb wherever it is. If
you have an infinite loop, it’s likely that you will be in it, and that the index
48
variables will be doing something surprising. Use display to keep an eye on
them and do next a few times.
$ gcc -g3 -o infiniteLoop infiniteLoop.c
23:08:05 (Sun Feb 15) zeniba aspnes ~/g/classes/223/notes/examples/debugging
$ gdb infiniteLoop
GNU gdb (Ubuntu 7.7.1-0ubuntu5~14.04.2) 7.7.1
[...]
Reading symbols from infiniteLoop...done.
(gdb) run
Starting program: /home/aspnes/g/classes/223/notes/examples/debugging/infiniteLoop
^C
Program received signal SIGINT, Interrupt.
main (argc=1, argv=0xbffff434) at infiniteLoop.c:11
11
i *= 37;
(gdb) display i
1: i = 0
(gdb) n
10
for(i = 0; i < 10; i += 0) {
1: i = 0
(gdb) n
11
i *= 37;
1: i = 0
(gdb) n
10
for(i = 0; i < 10; i += 0) {
1: i = 0
(gdb) n
11
i *= 37;
1: i = 0
(gdb) n
10
for(i = 0; i < 10; i += 0) {
1: i = 0
(gdb) n
11
i *= 37;
1: i = 0
3.4.3.3.5
Mysterious variable changes
Sometimes pointer botches don’t manifest as good, honest segmentation faults
but instead as mysterious changes to seemingly unrelated variables. You can
catch these in the act using conditional breakpoints. The downside is that you
can only put conditional breakpoints on particular lines.
Here’s a program that violates array bounds (which C doesn’t detect):
#include <stdio.h>
49
#include <stdlib.h>
#include <assert.h>
int
main(int argc, char **argv)
{
int x;
int a[10];
int i;
x = 5;
for(i = -1; i < 11; i++) {
a[i] = 37;
}
assert(x == 5);
}
return 0;
examples/debugging/mysteryChange.c
In the debugging session below, it takes a couple of attempts to catch the change
in x before hitting the failed assertion.
$ gcc -g3 -o mysteryChange mysteryChange.c
23:15:41 (Sun Feb 15) zeniba aspnes ~/g/classes/223/notes/examples/debugging
$ gdb mysteryChange
GNU gdb (Ubuntu 7.7.1-0ubuntu5~14.04.2) 7.7.1
[...]
Reading symbols from mysteryChange...done.
(gdb) run
Starting program: /home/aspnes/g/classes/223/notes/examples/debugging/mysteryChange
mysteryChange: mysteryChange.c:18: main: Assertion `x == 5' failed.
Program received signal SIGABRT, Aborted.
0xb7fdd416 in __kernel_vsyscall ()
(gdb) list main
2
#include <stdlib.h>
3
#include <assert.h>
4
5
int
6
main(int argc, char **argv)
7
{
8
int x;
9
int a[10];
50
10
int i;
11
(gdb) list
12
x = 5;
13
14
for(i = -1; i < 11; i++) {
15
a[i] = 37;
16
}
17
18
assert(x == 5);
19
20
return 0;
21 }
(gdb) break 14 if x != 5
Breakpoint 1 at 0x804842e: file mysteryChange.c, line 14.
(gdb) run
The program being debugged has been started already.
Start it from the beginning? (y or n) y
Starting program: /home/aspnes/g/classes/223/notes/examples/debugging/mysteryChange
mysteryChange: mysteryChange.c:18: main: Assertion `x == 5' failed.
Program received signal SIGABRT, Aborted.
0xb7fdd416 in __kernel_vsyscall ()
(gdb) break 15 if x != 5
Breakpoint 2 at 0x8048438: file mysteryChange.c, line 15.
(gdb) run
The program being debugged has been started already.
Start it from the beginning? (y or n) y
Starting program: /home/aspnes/g/classes/223/notes/examples/debugging/mysteryChange
Breakpoint 2, main (argc=1, argv=0xbffff434) at mysteryChange.c:15
15
a[i] = 37;
(gdb) print i
$1 = 0
(gdb) print a[0]
$2 = 134520832
(gdb) print a[-1]
$3 = 37
(gdb) print x
$4 = 37
One thing to note is that a breakpoint stops before the line it is on executes.
So when we hit the breakpoint on line 15 (gdb having observed that x != 5 is
true), i has the value 0, but the damage happened in the previous interation
51
when i was -1. If we want to see exactly what happened then, we’d need to go
back in time. We can’t do this, but we could set an earlier breakpoint and run
the program again.
3.4.4
Valgrind
The valgrind program can be used to detect some (but not all) common errors
in C programs that use pointers and dynamic storage allocation. On the Zoo,
you can run valgrind on your program by putting valgrind at the start of the
command line:
valgrind ./my-program arg1 arg2 < test-input
This will run your program and produce a report of any allocations and deallocations it did. It will also warn you about common errors like using unitialized
memory, dereferencing pointers to strange places, writing off the end of blocks
allocated using malloc, or failing to free blocks.
You can suppress all of the output except errors using the -q option, like this:
valgrind -q ./my-program arg1 arg2 < test-input
You can also turn on more tests, e.g.
valgrind -q --tool=memcheck --leak-check=yes ./my-program arg1 arg2 < test-input
See valgrind --help for more information about the (many) options, or look
at the documentation at http://valgrind.org/ for detailed information about
what the output means. For some common valgrind messages, see the examples
section below.
If you want to run valgrind on your own machine, you may be able to find a
version that works at http://valgrind.org. Unfortunately, this is only likely to
work if you are running a Unix-like operating system (which includes Linux and
Mac OSX, but not Windows).
3.4.4.1
Compilation flags
You can run valgrind on any program (try valgrind ls); it does not require
special compilation. However, the output of valgrind will be more informative
if you compile your program with debugging information turned on using the -g
or -g3 flags (this is also useful if you plan to watch your program running using
gdb, ).
3.4.4.2
Automated testing
Unless otherwise specified, automated testing of your program will be done
using the script in /c/cs223/bin/vg; this runs /c/cs223/bin/valgrind with
the --tool=memcheck, --leak-check=yes, and -q options, throws away your
52
program’s output, and replaces it with valgrind’s output. If you have a program
named ./prog, running /c/cs223/bin/vg ./prog should produce no output.
3.4.4.3
Examples of some common valgrind errors
Here are some examples of valgrind output. In each case the example program
is compiled with -g3 so that valgrind can report line numbers from the source
code.
3.4.4.3.1
Uninitialized values
Consider this unfortunate program, which attempts to compare two strings, one
of which we forgot to ensure was null-terminated:
#include <stdio.h>
int
main(int argc, char **argv)
{
char a[2];
a[0] = 'a';
if(!strcmp(a, "a")) {
puts("a is \"a\"");
}
}
return 0;
examples/valgrindErrors/uninitialized.c
Run without valgrind, we see no errors, because we got lucky and it turned out
our hand-built string was null-terminated anyway:
$ ./uninitialized
a is "a"
But valgrind is not fooled:
$ valgrind -q ./uninitialized
==4745== Conditional jump or move depends on uninitialised value(s)
==4745==
at 0x4026663: strcmp (mc_replace_strmem.c:426)
==4745==
by 0x8048435: main (uninitialized.c:10)
==4745==
==4745== Conditional jump or move depends on uninitialised value(s)
==4745==
at 0x402666C: strcmp (mc_replace_strmem.c:426)
==4745==
by 0x8048435: main (uninitialized.c:10)
53
==4745==
==4745== Conditional jump or move depends on uninitialised value(s)
==4745==
at 0x8048438: main (uninitialized.c:10)
==4745==
Here we get a lot of errors, but they are all complaining about the same call to
strcmp. Since it’s unlikely that strcmp itself is buggy, we have to assume that
we passed some uninitialized location into it that it is looking at. The fix is to
add an assignment a[1] = '\0' so that no such location exists.
3.4.4.3.2
Bytes definitely lost
Here is a program that calls malloc but not free:
#include <stdio.h>
#include <stdlib.h>
int
main(int argc, char **argv)
{
char *s;
s = malloc(26);
}
return 0;
examples/valgrindErrors/missing_free.c
With no extra arguments, valgrind will not look for this error. But if we turn
on --leak-check=yes, it will complain:
$ valgrind -q --leak-check=yes ./missing_free
==4776== 26 bytes in 1 blocks are definitely lost in loss record 1 of 1
==4776==
at 0x4024F20: malloc (vg_replace_malloc.c:236)
==4776==
by 0x80483F8: main (missing_free.c:9)
==4776==
Here the stack trace in the output shows where the bad block was allocated: inside
malloc (specifically the paranoid replacement malloc supplied by valgrind),
which was in turn called by main in line 9 of missing_free.c. This lets us go
back and look at what block was allocated in that line and try to trace forward
to see why it wasn’t freed. Sometimes this is as simple as forgetting to include
a free statement anywhere, but in more complicated cases it may be because
I somehow lose the pointer to the block by overwriting the last variable that
points to it or by embedding it in some larger structure whose components I
forget to free individually.
54
3.4.4.3.3
Invalid write or read operations
These are usually operations that you do off the end of a block from malloc or
on a block that has already been freed.
An example of the first case:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
int
main(int argc, char **argv)
{
char *s;
s = malloc(1);
s[0] = 'a';
s[1] = '\0';
puts(s);
}
return 0;
examples/valgrindErrors/invalid_operations.c
==7141== Invalid write of size 1
==7141==
at 0x804843B: main (invalid_operations.c:12)
==7141== Address 0x419a029 is 0 bytes after a block of size 1 alloc'd
==7141==
at 0x4024F20: malloc (vg_replace_malloc.c:236)
==7141==
by 0x8048428: main (invalid_operations.c:10)
==7141==
==7141== Invalid read of size 1
==7141==
at 0x4026063: __GI_strlen (mc_replace_strmem.c:284)
==7141==
by 0x409BCE4: puts (ioputs.c:37)
==7141==
by 0x8048449: main (invalid_operations.c:14)
==7141== Address 0x419a029 is 0 bytes after a block of size 1 alloc'd
==7141==
at 0x4024F20: malloc (vg_replace_malloc.c:236)
==7141==
by 0x8048428: main (invalid_operations.c:10)
==7141==
An example of the second:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
int
55
main(int argc, char **argv)
{
char *s;
s = malloc(2);
free(s);
s[0] = 'a';
s[1] = '\0';
puts(s);
}
return 0;
examples/valgrindErrors/freed_block.c
==7144== Invalid write of size 1
==7144==
at 0x804846D: main (freed_block.c:13)
==7144== Address 0x419a028 is 0 bytes inside a block of size 2 free'd
==7144==
at 0x4024B3A: free (vg_replace_malloc.c:366)
==7144==
by 0x8048468: main (freed_block.c:11)
==7144==
==7144== Invalid write of size 1
==7144==
at 0x8048477: main (freed_block.c:14)
==7144== Address 0x419a029 is 1 bytes inside a block of size 2 free'd
==7144==
at 0x4024B3A: free (vg_replace_malloc.c:366)
==7144==
by 0x8048468: main (freed_block.c:11)
==7144==
==7144== Invalid read of size 1
==7144==
at 0x4026058: __GI_strlen (mc_replace_strmem.c:284)
==7144==
by 0x409BCE4: puts (ioputs.c:37)
==7144==
by 0x8048485: main (freed_block.c:16)
[... more lines of errors deleted ...]
In both cases the problem is that we are operating on memory that is not
guaranteed to be allocated to us. For short programs like these, we might get
lucky and have the program work anyway. But we still want to avoid bugs like
this because we might not get lucky.
How do we know which case is which? If I write off the end of an existing block, I’ll
see something like Address 0x419a029 is 0 bytes after a block of size 1 alloc'd,
telling me that I am working on an address after a block that is still allocated. When I try to write to a freed block, the message changes
to
Address 0x419a029 is 1 bytes inside a block of size 2 free'd,
where the free'd part tells me I freed something I probably shouldn’t have.
Fixing the first class of bugs is usually just a matter of allocating a bigger block
(but don’t just do this without figuring out why you need a bigger block, or
56
you’ll just be introducing random mutations into your code that may cause
other problems elsewhere). Fixing the second class of bugs usually involves
figuring out why you freed this block prematurely. In some cases you may need
to re-order what you are doing so that you don’t free a block until you are
completely done with it.
3.4.5
Not recommended: debugging output
A tempting but usually bad approach to debugging is to put lots of printf
statements in your code to show what is going on. The problem with this
compared to using assert is that there is no built-in test to see if the output
is actually what you’d expect. The problem compared to gdb is that it’s not
flexible: you can’t change your mind about what is getting printed out without
editing the code. A third problem is that the output can be misleading: in
particular, printf output is usually buffered, which means that if your program
dies suddenly there may be output still in the buffer that is never flushed to
stdout. This can be very confusing, and can lead you to believe that your
program fails earlier than it actually does.
If you really need to use printf or something like it for debugging output, here
are a few rules of thumb to follow to mitigate the worst effects:
1. Use fprintf(stderr, ...) instead of printf(...); this allows you to
redirect your program’s regular output somewhere that keeps it separate
from the debugging output (but beware of misleading interleaving of the
two streams—buffering may mean that output to stdout and stderr
appears to arrive out of order). It also helps that output to stderr is
usually unbuffered, avoiding the problem of lost output.
2. If you must output to stdout, put fflush(stdout) after any output
operation you suspect is getting lost in the buffer. The fflush function
forces any buffered output to be emitted immediately.
3. Keep all arguments passed to printf as simple as possible and
beware of faults in your debugging code itself.
If you write
printf("a[key] == %d\n", a[key]) and key is some bizarre value,
you will never see the result of this printf because your program will
segfault while evaluating a[key]. Naturally, this is more likely to occur if
the argument is a[key]->size[LEFTOVERS].cleanupFunction(a[key])
than if it’s just a[key], and if it happens it will be harder to figure out
where in this complex chain of array indexing and pointer dereferencing
the disaster happened. Better is to wait for your program to break in
gdb, and use the print statement on increasingly large fragments of the
offending expression to see where the bogus array index or surprising null
pointer is hiding.
4. Wrap your debugging output in an #ifdef so you can turn it on and off
easily.
57
Bearing in mind that this is a bad idea, here is an example of how one might do
it as well as possible:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
/* initialize the application */
void
init(void)
{
int x;
}
x = *((int *) 0xbad1dea); /* if we are lucky, maybe the optimizer will remove it? */
int
main(int argc, char **argv)
{
init();
#ifdef DEBUGGING_OUTPUT
/*
* this type of debugging output is not recommended
* but if you do it anyway:
*
* 1. Use stderr, which flushes automatically.
* 2. Be wary of buffered data on stdout.
* 3. Wrap your debugging statement in an #ifdef,
*
so it is not active by default.
*/
fputs("Returned from init() in main()\n", stderr);
#endif
}
return 0;
examples/debugging/usingPrintf.c
Note that we get much more useful information if we run this under gdb (which
will stop exactly on the bad line in init), but not seeing the result of the fputs
at least tells us something.
58
3.5
Performance tuning
Chapter 7 of Kernighan and Pike, The Practice of Programming (Addison-Wesley,
1998) gives an excellent overview of performance tuning. This page will be limited
to some Linux-specific details and an example.
3.5.1
Timing under Linux
Use time, e.g.
$ time wc /usr/share/dict/words
45378 45378 408865 /usr/share/dict/words
real
user
sys
0m0.010s
0m0.006s
0m0.004s
This measures “real time” (what it sounds like), “user time” (the amount of time
the program runs), and “system time” (the amount of time the operating system
spends supporting your program, e.g. by loading it from disk and doing I/O).
Real time need not be equal to the sum of user time and system time, since the
operating system may be simultaneously running other programs.
Particularly for fast programs, times can vary from one execution to the next,
e.g.
$ time wc /usr/share/dict/words
45378 45378 408865 /usr/share/dict/words
real
0m0.009s
user
0m0.008s
sys
0m0.001s
$ time wc /usr/share/dict/words
45378 45378 408865 /usr/share/dict/words
real
user
sys
0m0.009s
0m0.007s
0m0.002s
This arises because of measurement errors and variation in how long different
operations take. But usually the variation will not be much.
Note also that time is often a builtin operation of your shell, so the output
format may vary depending on what shell you use.
59
3.5.2
Profiling with gprof
The problem with time is that it only tells you how much time your whole
program took, but not where it spent its time. This is similar to looking at a
program without a debugger: you can’t see what’s happening inside. If you want
to see where your program is spending its time, you need to use a profiler.
For example, here’s a short but slow program for calculating the number of
primes less than some limit passed as argv[1]:
#include <stdio.h>
#include <stdlib.h>
/* return 1 if n is prime, 0 otherwise */
int
isPrime(int n)
{
int factor;
}
if(n < 2) return 0;
/* else */
for(factor = 2; factor < n; factor++) {
if(n % factor == 0) return 0;
}
/* else */
return 1;
/* return number of primes < n */
int
countPrimes(int n)
{
int i;
int count;
count = 0;
for(i = 0; i < n; i++) {
if(isPrime(i)) count++;
}
}
return count;
int
main(int argc, char **argv)
60
{
if(argc != 2) {
fprintf(stderr, "Usage: %s n\n", argv[0]);
return 1;
}
printf("%d\n", countPrimes(atoi(argv[1])));
}
return 0;
examples/profiling/countPrimes.c
And now we’ll time countPrimes 100000:
$ c99 -g3 -o countPrimes countPrimes.c
$ time ./countPrimes 100000
9592
real
0m4.711s
user
0m4.608s
sys 0m0.004s
This shows that the program took just under five seconds of real time, of which
most was spent in user mode and a very small fraction was spent in kernel (sys)
mode. The user-mode part corresponds to the code we wrote and any library
routines we call that don’t require special privileges from the operation system.
The kernel-mode part will mostly be I/O (not much in this case). Real time is
generally less useful than CPU time, because it depends on how loaded the CPU
is. Also, none of these times are especially precise, because the program only
gets charged for time on a context switch (when it switches between user and
kernel mode or some other program takes over the CPU for a bit) or when the
kernel decides to see what it is up to (typically every 10 milliseconds).
The overall cost is not too bad, but the reason I picked 100000 and not some
bigger number was that it didn’t terminate fast enough for larger inputs. We’d
like to see why it is taking so long, to have some idea what to try to speed up.
So we’ll compile it with the -pg option to gcc, which inserts profiling code that
counts how many times each function is called and how long (on average) each
call takes.
Because the profile is not very smart about shared libraries, we also including
the --static option to force the resulting program to be statically linked. This
means that all the code that is used by the program is baked into the executable
instead of being linked in at run-time. (Normally we don’t do this because
it makes for big executables and big running programs, since statically-linked
libraries can’t be shared between more than one running program.)
$ c99 -pg --static -g3 -o countPrimes countPrimes.c
61
$ time ./countPrimes 100000
9592
real
0m4.723s
user
0m4.668s
sys 0m0.000s
Hooray! We’ve made the program slightly slower. But we also just produced
a file gmon.out that we can read with gprof. Note that we have to pass the
name of the program so that gprof can figure out which executable generated
gmon.out.
$ gprof countPrimes
Flat profile:
Each sample counts as 0.01 seconds.
%
cumulative
self
time
seconds
seconds
calls
100.00
4.66
4.66
100000
0.00
4.66
0.00
1
0.00
4.66
0.00
1
self
s/call
0.00
0.00
0.00
total
s/call
0.00
4.66
4.66
name
isPrime
countPrimes
main
[...much explanatory text deleted]
It looks like we are spending all of our time in isPrime, at least if we read
the columns on the left. The per-call columns are not too helpful because of
granularity: isPrime is too fast for the profiler to wake up and detect how long
it runs for. The total columns are less suspicious because they are obtained
by sampling: from time to time, the profiler looks and sees what function it’s
in, and charges each function a fraction of the total CPU time proportional to
how often it gets sampled. So we probable aren’t really spending zero time in
countPrimes and main, but the amount of time we do spend is small enough
not to be detected.
This is handy because it means we don’t need to bother trying to speed up the
rest of the program. We have two things we can try:
1. Call isPrime less.
2. Make isPrime faster.
Let’s start by seeing if we can make isPrime faster.
What isPrime is doing is testing if a number n is prime by the most direct way
possible: dividing by all numbers less than n until it finds a factor. That’s a lot
of divisions: if n is indeed prime, it’s linear in n. Since division is a relatively
expensive operation, the first thing to try is to get rid of some.
Here’s a revised version of isPrime:
/* return 1 if n is prime, 0 otherwise */
62
int
isPrime(int n)
{
int factor;
}
if(n < 2) { return 0; }
if(n % 2 == 0) {
/* special case for the only even prime */
return n == 2;
}
/* else */
for(factor = 3; factor < n; factor+=2) {
if(n % factor == 0) return 0;
}
/* else */
return 1;
examples/profiling/countPrimesSkipEvenFactors.c
The trick is to check first if n is divisible by 2, and only test odd potential factors
thereafter. This requires some extra work to handle 2, but maybe the extra code
complexity will be worth it.
Let’s see how the timing goes:
$ c99 -pg --static -g3 -o countPrimes ./countPrimesSkipEvenFactors.c
$ time ./countPrimes 100000
9592
real
0m2.608s
user
0m2.400s
sys 0m0.004s
$ gprof countPrimes
Flat profile:
Each sample counts as 0.01 seconds.
%
cumulative
self
time
seconds
seconds
calls
100.00
2.29
2.29
100000
0.00
2.29
0.00
1
0.00
2.29
0.00
1
self
s/call
0.00
0.00
0.00
total
s/call
0.00
2.29
2.29
name
isPrime
countPrimes
main
[...]
Twice as fast! And the answer is still the same, too—this is important.
Can we test even fewer factors? Suppose n has a non-trivial factor x. Then n
equals x*y for some y which is also nontrivial. One of x or y will be no bigger
63
than the square root of n. So perhaps we can stop when we reach the square
root of n,
Let’s try it:
#include <math.h>
/* return 1 if n is prime, 0 otherwise */
int
isPrime(int n)
{
int factor;
}
}
if(n < 2) { return 0; }
if(n % 2 == 0) {
/* special case for the only even prime */
return n == 2;
}
/* else */
for(factor = 3; factor < sqrt(n)+1; factor+=2) {
if(n % factor == 0) return 0;
}
/* else */
return 1;
examples/profiling/countPrimesSqrt.c
I added +1 to the return value of sqrt both to allow for factor to be equal
to the square root of n, and because the output of sqrt is not exact, and it
would be embarrassing if I announced that 25 was prime because I stopped at
4.9999999997.
Using the math library not only requires including <math.h> but also requires
compiling with the -lm flag after all .c or .o files, to link in the library routines:
$ c99 -pg --static -g3 -o countPrimes ./countPrimesSqrt.c -lm
$ time ./countPrimes 1000000
78498
real
0m1.008s
user
0m0.976s
sys 0m0.000s
$ gprof countPrimes
Flat profile:
Each sample counts as 0.01 seconds.
%
cumulative
self
64
self
total
time
50.00
50.00
0.00
0.00
seconds
0.02
0.04
0.04
0.04
seconds
0.02
0.02
0.00
0.00
calls
100000
ms/call
0.00
ms/call
0.00
1
1
0.00
0.00
20.00
20.00
name
isPrime
__sqrt_finite
countPrimes
main
[...]
Whoosh!
Can we optimize further? Let’s see what happens on a bigger input:
$ time ./countPrimes 1000000
78498
real
0m0.987s
user
0m0.960s
sys 0m0.000s
$ gprof countPrimes
Flat profile:
Each sample counts as 0.01 seconds.
%
cumulative
self
self
time
seconds
seconds
calls ms/call
51.04
0.49
0.49
44.79
0.92
0.43 1000000
0.00
3.65
0.96
0.04
0.52
0.96
0.01
1
5.00
0.00
0.96
0.00
1
0.00
total
ms/call
0.00
435.00
430.00
name
__sqrt_finite
isPrime
sqrt
main
countPrimes
[...]
This is still very good, although we’re spending a lot of time in sqrt (more
specifically, its internal helper routine __sqrt_finite). Can we do better?
Maybe moving the sqrt out of the loop in isPrime will make a difference:
/* return 1 if n is prime, 0 otherwise */
int
isPrime(int n)
{
int factor;
int sqrtValue;
if(n < 2) { return 0; }
if(n % 2 == 0) {
/* special case for the only even prime */
return n == 2;
}
65
}
/* else */
sqrtValue = sqrt(n) + 1;
for(factor = 3; factor < sqrtValue; factor+=2) {
if(n % factor == 0) return 0;
}
/* else */
return 1;
examples/profiling/countPrimesSqrtOutsideLoop.c
$ c99 -pg --static -g3 -o countPrimes ./countPrimesSqrtOutsideLoop.c -lm
$ time ./countPrimes 1000000
78498
real
0m0.413s
user
0m0.392s
sys 0m0.000s
$ gprof countPrimes
Flat profile:
Each sample counts as 0.01 seconds.
%
cumulative
self
self
time
seconds
seconds
calls ms/call
97.44
0.38
0.38 1000000
0.00
2.56
0.39
0.01
1
10.00
0.00
0.39
0.00
1
0.00
total
ms/call
0.00
390.00
390.00
name
isPrime
countPrimes
main
[...]
This worked! We are now spending almost so little time in sqrt that the profiler
doesn’t notice it.
What if we get rid of the call to sqrt and test if factor * factor <= n instead?
This way we could dump the math library:
/* return 1 if n is prime, 0 otherwise */
int
isPrime(int n)
{
int factor;
if(n < 2) { return 0; }
if(n % 2 == 0) {
/* special case for the only even prime */
return n == 2;
}
/* else */
66
}
for(factor = 3; factor*factor <= n; factor+=2) {
if(n % factor == 0) return 0;
}
/* else */
return 1;
examples/profiling/countPrimesSquaring.c
$ c99 -pg --static -g3 -o countPrimes ./countPrimesSquaring.c
$ time ./countPrimes 1000000
78498
real
0m0.450s
user
0m0.428s
sys 0m0.000s
This is slower, but not much slower. We might need to decide how much we care
about avoiding floating-point computation in our program.
At this point we could decide that countPrimes is fast enough, or maybe we
could look for further improvements, say, by testing out many small primes at
the beginning instead of just 2, calling isPrime only on odd values of i, or
reading a computational number theory textbook to find out how we ought
to be doing this. A reasonable strategy for code for your own use is often to
start running one version and make improvements on a separate copy while it’s
running. If the first version terminates before you are done writing new code,
it’s probably fast enough.
3.5.2.1
Effect of optimization during compilation
We didn’t use any optimization flags for this example, because the optimizer
can do a lot of rewriting that can make the output of the profiler confusing.
For example, at high optimization levels, the compiler will often avoid functioncall overhead by inserting the body of a helper function directly into its caller.
But this can make a big difference in performance, so in real life you will
want to compile with optimization turned on. Here’s how the performance of
countPrimes 100000 is affected by optimization level:
Version
No optimization
With -O1
With -O2
With -O3
4.600
2.260
0.036
0.012
0.012
4.060
1.948
0.028
0.012
0.012
3.928
1.964
0.028
0.008
0.008
3.944
1.984
0.028
0.008
0.012
countPrimes.c
countPrimesSkipEvenFactors.c
countPrimesSqrt.c
countPrimesSqrtOutsideLoop.c
countPrimesSquaring.c
67
In each case, the reported time is the sum of user and system time in seconds.3
For the smarter routines, more optimization doesn’t necessarily help, although
some of this may be experimental error since I was too lazy to get a lot of
samples by running each program more than once, and the times for the faster
programs are so small that granularity is going to be an issue.
Here’s the same table using countPrimes 10000000 on the three fastest programs:
Version
No optimization
With -O1
With -O2
With -O3
24.236
9.388
9.748
18.840
9.364
9.248
18.720
9.368
9.236
18.564
9.360
9.160
countPrimesSqrt.c
countPrimesSqrtOutsideLoop.c
countPrimesSquaring.c
Again there are the usual caveats that I am a lazy person and should probably be
doing more do deal with sampling and granularity issues, but if you believe these
numbers, we actually win by going to countPrimesSquaring once the optimizer
is turned on. I suspect that it is benefiting from strength reduction, which would
generate the product factor*factor in isPrime incrementally using addition
rather than multiplying from scratch each time.
It’s also worth noting that the optimizer works better if we leave a lot of easy
optimization lying around. For countPrimesSqrt.c, my guess is that most of
the initial gains are from avoiding function call overhead on sqrt by compiling
it in-line. But even the optimizer is not smart enough to recognize that we are
computing the same value over and over again, so we still win by pulling sqrt
out of the loop in countPrimesSqrtOutsideLoop.c.
If I wanted to see if my guesses about the optimizer were correct, there I could
use gcc -S and look at the assembler code. But see earlier comments about
laziness.
3.6
Version control
When you are programming, you will make mistakes. If you program long enough,
these will eventually include true acts of boneheadedness like accidentally deleting
all of your source files. You are also likely to spend some of your time trying
out things that don’t work, at the end of which you’d like to go back to the last
version of your program that did work. All these problems can be solved by
using a version control system.
3 The compiler is GCC version 4.8.2-19ubuntu1 running on a Linux 3.13.0-44-generic kernel
running inside VirtualBox on a Windows 8.1 machine with a 3.30-Ghz AMD FX-6100 CPU,
so don’t be surprised if you get different numbers on a real machine.
68
There are six respectable version control systems installed on the Zoo: rcs, cvs,
svn, bzr, hg, and git. If you are familiar with any of them, you should use
that. If you have to pick one from scratch, I recommend using git. A brief
summary of git is given below. For more details, see the tutorials available at
http://git-scm.com.
3.6.1
Setting up Git
Typically you run git inside a directory that holds some project you are working
on (say, hw1). Before you can do anything with git, you will need to create the
repository, which is a hidden directory .git that records changes to your files:
$ mkdir git-demo
$ cd git-demo
$ git init
Initialized empty Git repository in /home/classes/cs223/class/aspnes.james.ja54/git-demo/.gi
Now let’s create a file and add it to the repository:
$ echo 'int main(int argc, char **argv) { return 0; }' > tiny.c
$ git add tiny.c
The git status command will tell us that Git knows about tiny.c, but hasn’t
commited the changes to the repository yet:
$
#
#
#
#
#
#
#
#
#
git status
On branch master
Initial commit
Changes to be committed:
(use "git rm --cached <file>..." to unstage)
new file:
tiny.c
The git commit command will commit the actual changes, along with a message
saying what you did. For short messages, the easiest way to do this is to include
the message on the command line:
$ git commit -a -m"add very short C program"
[master (root-commit) 5393616] add very short C program
Committer: James Aspnes <ja54@tick.zoo.cs.yale.edu>
Your name and email address were configured automatically based
on your username and hostname. Please check that they are accurate.
You can suppress this message by setting them explicitly:
git config --global user.name "Your Name"
69
git config --global user.email you@example.com
If the identity used for this commit is wrong, you can fix it with:
git commit --amend --author='Your Name <you@example.com>'
1 files changed, 1 insertions(+), 0 deletions(-)
create mode 100644 tiny.c
The -a argument tells Git to include any changes I made to files it already knows
about. The -m argument sets the commit message.
Because this is the first time I ever did a commit, and because I didn’t tell Git
who I was before, it complains that its guess for my name and email address may
be wrong. It also tells me what to do to get it to shut up about this next time:
$ git config --global user.name "James Aspnes"
$ git config --global user.email "aspnes@cs.yale.edu"
$ git commit --amend --author="James Aspnes <aspnes@cs.yale.edu>" -m"add a very short C prog
[master a44e1e1] add a very short C program
1 files changed, 1 insertions(+), 0 deletions(-)
create mode 100644 tiny.c
Note that I repeated the -m business to git commit --amend; if I hadn’t, it would
have run the default editor (vim) to let me edit my commit message. If I don’t
like vim, I can change the default using git config --global core.editor,
e.g.:
$ git config --global core.editor "emacs -nw"
I can see what commits I’ve done so far using git log:
$ git log
commit a44e1e195de4ce785cd95cae3b93c817d598a9ee
Author: James Aspnes <aspnes@cs.yale.edu>
Date:
Thu Dec 29 20:21:21 2011 -0500
add a very short C program
3.6.2
Editing files
Suppose I edit tiny.c using my favorite editor to turn it into the classic helloworld program:
#include <stdio.h>
int
main(int argc, char **argv)
{
70
}
puts("hello, world");
return 0;
I can see what files have changed using git status:
$ git status
# On branch master
# Changed but not updated:
#
(use "git add <file>..." to update what will be committed)
#
(use "git checkout -- <file>..." to discard changes in working directory)
#
#
modified:
tiny.c
#
no changes added to commit (use "git add" and/or "git commit -a")
Notice how Git reminds me to use git commit -a to include these changes
in my next commit. I can also do git add tiny.c if I just want include the
changes to tiny.c (maybe I made changes to a different file that I want to
commit separately), but usually that’s too much work.
If I want to know the details of the changes since my last commit, I can do
git diff:
$ git diff
diff --git a/tiny.c b/tiny.c
index 0314ff1..f8d9dcd 100644
--- a/tiny.c
+++ b/tiny.c
@@ -1 +1,8 @@
-int main(int argc, char **argv) { return 0; }
+#include <stdio.h>
+
+int
+main(int argc, char **argv)
+{
+
puts("hello, world");
+
return 0;
+}
Since I like these changes, I do a commit:
$ git commit -a -m"expand previous program to hello world"
[master 13a73be] expand previous program to hello world
1 files changed, 8 insertions(+), 1 deletions(-)
Now there are two commits in my log:
$ git log | tee /dev/null
commit 13a73bedd3a48c173898d1afec05bd6fa0d7079a
71
Author: James Aspnes <aspnes@cs.yale.edu>
Date:
Thu Dec 29 20:34:06 2011 -0500
expand previous program to hello world
commit a44e1e195de4ce785cd95cae3b93c817d598a9ee
Author: James Aspnes <aspnes@cs.yale.edu>
Date:
Thu Dec 29 20:21:21 2011 -0500
add a very short C program
3.6.3
Renaming files
You can rename a file with git mv. This is just like regular mv, except that it
tells Git what you are doing.
$
$
#
#
#
#
#
#
git mv tiny.c hello.c
git status
On branch master
Changes to be committed:
(use "git reset HEAD <file>..." to unstage)
renamed:
tiny.c -> hello.c
These changes don’t get written to the repository unless you do another
git commit:
$ git commit -a -m"give better name to hello program"
[master 6d2116c] give better name to hello program
1 files changed, 0 insertions(+), 0 deletions(-)
rename tiny.c => hello.c (100%)
3.6.4
Adding and removing files
To add a file, create it and then call git add:
$ cp hello.c goodbye.c
$ git status
# On branch master
# Untracked files:
#
(use "git add <file>..." to include in what will be committed)
#
#
goodbye.c
nothing added to commit but untracked files present (use "git add" to track)
$ git add goodbye.c
72
$ git commit -a -m"we need a second program to say goodbye"
[master 454b24c] we need a second program to say goodbye
1 files changed, 8 insertions(+), 0 deletions(-)
create mode 100644 goodbye.c
To remove a file, use git rm:
$ git rm goodbye.c
$ git status
# On branch master
# Changed but not updated:
#
(use "git add/rm <file>..." to update what will be committed)
#
(use "git checkout -- <file>..." to discard changes in working directory)
#
#
deleted:
goodbye.c
#
no changes added to commit (use "git add" and/or "git commit -a")
$ git commit -a -m"no, goodbye.c was a bad idea"
[master defa0e0] no, goodbye.c was a bad idea
1 files changed, 0 insertions(+), 8 deletions(-)
delete mode 100644 goodbye.c
3.6.5
Recovering files from the repository
If you make a mistake, you can back out using the repository. Here I will delete
my hello.c file and then get it back using git checkout -- hello.c:
$ rm hello.c
$ ls
$ git checkout -- hello.c
$ ls
hello.c
I can also get back old versions of files by putting the commit id before the --:
$ git checkout a44e1 -- tiny.c
$ ls
hello.c tiny.c
The commit id can be any unique prefix of the ridiculously long hex name shown
by git log.
Having recovered tiny.c, I will keep it around by adding it to a new commit:
$ git commit -a -m"keep tiny.c around"
[master 23d6219] keep tiny.c around
1 files changed, 1 insertions(+), 0 deletions(-)
create mode 100644 tiny.c
73
3.6.6
Undoing bad commits
Suppose I commit a change that I didn’t want to make. For example, let’s
suppose I decide to add some punctuation to the greeting in hello.c but botch
my edit:
$ vim hello.c
$ git commit -a -m"add exclamation point"
[master f40d8d3] add exclamation point
1 files changed, 1 insertions(+), 1 deletions(-)
Only now does it occur to me to test my program:
$ c99 -o hello hello.c
$ ./hello
hello, wolrd!
Disaster!
I can use git diff to see what went wrong. The command below compares the
current working directory to HEADˆ, the commit before the most recent commit:4
$ git diff HEAD^ | tee /dev/null
diff --git a/hello.c b/hello.c
index f8d9dcd..dc227a8 100644
--- a/hello.c
+++ b/hello.c
@@ -3,6 +3,6 @@
int
main(int argc, char **argv)
{
puts("hello, world");
+
puts("hello, wolrd!");
return 0;
}
And I see my mistake leaping out at me on the new line I added (which git diff
puts a + in front of). But now what do I do? I already commited the change,
which means that I can’t get it out of the history.5
Instead, I use git revert on HEAD, the most recent commit:
$ git revert HEAD
[master fca3166] Revert "add exclamation point"
1 files changed, 1 insertions(+), 1 deletions(-)
4 The pattern here is that HEAD is the most recent commit, HEADˆ the one before it, HEADˆˆ
the one before that, and so on. This is sometimes nicer than having to pull hex gibberish out
of the output of git log.
5 Technically I can use git reset to get rid of the commit, but git reset can be dangerous,
since it throws away information.
74
(Not shown here is where it popped up a vim session to let me edit the commit
message; I just hit :x<ENTER> to get out of it without changing the default.)
Now everything is back to the way it was before the bad commit:
$ ./hello
hello, world
3.6.7
Looking at old versions
Running git log will now show me the entire history of my project, newest
commits first:
fca3166a697c6d72fb9e8aec913bb8e36fb5fe4e
f40d8d386890103abacd0bf4142ecad62eed5aeb
23d6219c9380ba03d9be0672f0a7b25d18417731
defa0e0430293ca910f077d5dd19fccc47ab0521
454b24c307121b5a597375a99a37a825b0dc7e81
6d2116c4c72a6ff92b8b276eb88ddb556d1b8fdd
13a73bedd3a48c173898d1afec05bd6fa0d7079a
a44e1e195de4ce785cd95cae3b93c817d598a9ee
Revert "add exclamation point"
add exclamation point
keep tiny.c around
no, goodbye.c was a bad idea
we need a second program to say goodbye
give better name to hello program
expand previous program to hello world
add a very short C program
If I want to look at an old version (say, after I created goodbye.c), I can go
back to it using git checkout:
$ git checkout 454b2
Note: checking out '454b2'.
You are in 'detached HEAD' state. You can look around, make experimental
changes and commit them, and you can discard any commits you make in this
state without impacting any branches by performing another checkout.
If you want to create a new branch to retain commits you create, you may
do so (now or later) by using -b with the checkout command again. Example:
git checkout -b new_branch_name
HEAD is now at 454b24c... we need a second program to say goodbye
$ ls
goodbye.c hello hello.c
Now I have both goodbye.c and hello.c, as well as my compiled program
hello, since I didn’t tell Git about it. Note that I also got lots of horrendous
warnings about the fact that I am living in the past and shouldn’t expect to
make any permanent changes here.
To go back to the last commit, use git checkout master:
$ git checkout master
75
Previous HEAD position was 454b24c... we need a second program to say goodbye
Switched to branch 'master'
$ ls
hello hello.c tiny.c
3.6.8
More information about Git
All Git commands take a --help argument that brings up their manual page.
There is also extensive documentation at http://git-scm.com.
3.7
Submitting assignments
The submit command is is found in /c/cs223/bin on the Zoo. Here is the
documentation (adapted from comments in the script):
submit
unsubmit
check
makeit
protect
unprotect
retrieve
testit
assignment-number
assignment-number
assignment-number
assignment-number
assignment-number
assignment-number
assignment-number
assignment-number
file(s)
file(s)
[file]
file(s)
file(s)
file[s]
test
The submit program can be invoked in eight different ways:
/c/cs223/bin/submit
1
Makefile tokenize.c unique.c time.log
submits the named source files as your solution to Homework #1;
/c/cs223/bin/check
2
lists the files that you have submitted for Homework #2;
/c/cs223/bin/unsubmit
3
error.submit bogus.solution
deletes the named files that you had submitted previously for Homework #3
(i.e., withdraws them from submission, which is useful if you accidentally
submit the wrong file);
/c/cs223/bin/makeit
4
tokenize unique
runs "make" on the files that you submitted previously for Homework #4;
/c/cs223/bin/protect
5
tokenize.c time.log
76
protects the named files that you submitted previously for Homework #5 (so
they cannot be deleted accidentally); and
/c/cs223/bin/unprotect
6
unique.c time.log
unprotects the named files that you submitted previously for Homework #6
(so they can be deleted); and
/c/cs223/bin/retrieve
7
Csquash.c
retrieves copies of the named files that you submitted previously for Homework #7
/c/cs223/bin/testit
8
BigTest
runs the test script /c/cs223/Hwk8/test.BigTest.
The submit program will only work if there is a directory with your name and
login on it under /c/cs223/class. If there is no such directory, you need to
make sure that you have correctly signed up for CS223 using the web form. Note
that it may take up to an hour for this directory to appear after you sign up.
4
The C programming language
The C programming language was developed at Bell Laboratories in the early
1970s as the system programming language for Unix, based on the earlier and
even more primitive languages BCPL and B. When originally developed, it was
targeted at machines that were extremely limited by modern standards: the
first Unix implementation (and the B compiler that supported it) ran on a
DEC PDP-7 with only 8192 18-bit words of memory (Dennis M. Ritchie, The
development of the C language, in Thomas J. Bergin, Jr., and Richard G. Gibson,
Jr., History of Programming Languages-II ed., ACM Press, 1996). So using as
few resources as possible, both in the compiler and in the resulting code, was a
priority.
This priority is reflected in the features (and lack of features) of C, and is partly
responsible for its success. Programs written in C place almost no demands on
the system they run on and give the programmer nearly complete control over
their execution: this allows programs that were previously written in assembly
language, like operating system kernels and device drivers, to be implemented in
C. So C is often the first language ported to any new architecture, and many
higher-level languages are either executed using interpreters written in C or use
C as in intermediate language in the compilation process.
Since its initial development, C has gone through four major versions:
77
• The original K&R C defined in the 1978 first edition of Kernighan and
Ritchie’s book The C Programming Language;
• ANSI C, from 1988, which fixed some oddities in the syntax and which is
documented in the 1988 second edition of The C Programming Language;
• C99, from 1999, the ISO/IEC 9899:1999 standard for C, which added
some features from C++ and many new features for high-performance
numerical computing;
• C11, from 2011, the ISO/IEC 9899:2011 standard for C, which relaxed
some of the requirements of C99 that most compilers hadn’t bothered
implementing and which added a few extra features.
Unfortunately, C99 and C11 both exemplify the uselessness of standards committees in general and the ISO in particular. Because the ISO has no power to
enforce standards on compiler writers, and because they will charge you CHF
198 just to look at the C11 standard, many compiler writers have ignored much
of C99 and C11. In particular, Microsoft pretty much gave up on adding any
features after ANSI C, and support for C99 and C11 is spotty in gcc and clang,
the two dominant open source C compilers. So if you want to write portable C
code, it is safest to limit yourself to features in ANSI C.
For this class, we will permit you to use any feature of C99 that gcc supports,
which also includes all features of ANSI C. You can compile with C99 support by
using gcc --std=c99 or by calling gcc as c99, as in c99 -o hello hello.c.
Compiling with straight gcc will give you GNU’s own peculiar dialect of C, which
is basically ANSI C with some extras. For maximum portability when using
gcc, it is safest to use gcc -ansi -pedantic, which expects straight ANSI C
and will complain about any extensions.
4.1
Structure of a C program
A C program consists of one or more files (which act a little bit like modules
in more structured programming languages, each of which typically contains
definitions of functions, each of which consists of statements, which are
either compound statements like if, while, etc. or expressions that typically perform some sort of arithmetic or call other functions. Files may also
include declarations of global variables (not recommended), and functions will
often contain declarations of local variables that can only be used inside that
function.
Here is a typical small C program that sums a range of integers. Since this is
our first real program, it’s a little heavy on the comments (shown between /*
and */).
#include <stdio.h>
#include <stdlib.h>
/* This is needed to get the declarations of fprintf and printf */
/* This is needed to get the declaration of atoi */
/* Return the sum of all integers i
78
* such that start <= i and i < end. */
int
sumRange(int start, int end)
{
int i;
/* loop variable */
int sum; /* sum of all values so far */
/* a mathematician would use a formula for this,
* but we are computer programmers! */
sum = 0;
/* The three parts of the header for
* 1. Set i to start initially.
* 2. Keep doing the loop as long as
* 3. After each iteration, add 1 to
*/
for(i = start; i < end; i++) {
sum += i; /* This adds i to sum
}
}
this loop mean:
i is less than end.
i.
*/
/* This exits the function immediately,
* sending the value of sum back to the caller. */
return sum;
int
main(int argc, char **argv)
{
int start;
/* initial value in range */
int end;
/* one past the last value in the range */
/* This tests for the wrong number of arguments.
* The != operator returns true (1) if its arguments are not equal,
* and false (0) otherwise.
* Note that the program name itself counts as an argument
* (which is why we want the argument count to be 3)
* and appears in position 0 in the argument vector
* (which means we can get it using argv[0]). */
if(argc != 3) {
fprintf(stderr, "Usage: %s\n start end", argv[0]);
return 1;
}
/* Convert start and end positions from strings to ints */
start = atoi(argv[1]);
end = atoi(argv[2]);
79
/* Call sumRange and print the result */
printf("sumRange(%d, %d) = %d\n", start, end, sumRange(start, end));
}
return 0;
examples/sumRange.c
This is what the program does if we compile and run it:
$ c99 -g -Wall -pedantic -o sumRange sumRange.c
$ ./sumRange 1 100
sumRange(1, 100) = 4950
The sumRange.c program contains two functions, sumRange and main. The
sumRange function does the actual work, while main is the main routine of the
program that gets called with the command-line arguments when the program
is run. Every C program must have a routine named main with these particular
arguments.
In addition, main may call three library functions, fprintf (which in this case
is used to generate error messages), printf (which generates ordinary output),
and atoi (which is used to translate the command-line arguments into numerical
values). These functions must all be declared before they can be used. In the case
of sumRange, putting the definition of sumRange before the definition of main
is enough. For the library routines, the include files stdio.h and stdlib.h
contain declarations of the functions that contain enough information about
there return types and arguments that the compiler knows how to generate
machine code to call them. These files are included in sumRange.c by the C
preprocessor, which pastes in the contents of any file specified by the #include
command, strips out any comments (delimited by /* and */, or by // and the
end of the line if you are using C99), and does some other tricks that allow you
to muck with the source code before the actual compiler sees it (see Macros).
You can see what the output of the preprocessor looks like by calling the C
compiler with the -E option, as in c99 -E sumRange.c.
The body of each function consists of some variable declarations followed
by a sequence of statements that tell the computer what to do. Unlike some
languages, every variable used in a C program must be declared. A declaration
specifies the type of a variable, which tells the compiler how much space to
allocate for it and how to interpret some operations on its value. Statements
may be compound statements like the if statement in main that executes
its body only if the program is called with the wrong number of command-line
arguments or the for loop in sumRange that executes its body as long as the
test in its header remains true; or they may be simple statements that consist
of a single expression followed by a semicolon.
An expression is usually either a bare function call whose value is discarded (for
80
example, the calls to fprintf and printf in main), or an arithmetic expression
(which may include function calls, like the calls to atoi or in main) whose value
is assigned to some variable using the assignment operator = or sometimes
variants like += (which is shorthand for adding a value to an existing variable: x
+= y is equivalent to x = x+y).
When you compile a C program, after running the preprocessor, the compiler
generates assembly language code that is a human-readable description of the
ultimate machine code for your target CPU. Assembly language strips out all the
human-friendly features of your program and reduces it to simple instructions
usually involving moving things from one place to the other or performing a
single arithmetic operation. For example, the C line
x = y + 1;
/* add 1 to y, store result in x */
gets translated into x86 assembly as
movl
-24(%rbp), %edi
addl
$1, %edi
movl
%edi, -28(%rbp)
These three operations copy the value of y into the CPU register %edi, add 1
to the %edi register, and then copy the value back into x. This corresponds
directly to what you would have to do to evaluate x = y + 1 if you could only
do one very basic operation at a time and couldn’t do arithmetic operations on
memory locations: fetch y, add 1, store x. Note that the CPU doesn’t know
about the names y and x; instead, it computes their addresses by adding -24
and -28 respectively to the base pointer register %rbp. This is why it can be
hard to debug compiled code unless you tell the compiler to keep around extra
information.
For an arbitrary C program, if you are using gcc, you can see what your code
looks like in assembly language using the -S option. For example, c99 -S
sumRange.c will create a file sumRange.s that looks like this:
.file
"sumRange.c"
.text
.globl
sumRange
.type
sumRange, @function
sumRange:
.LFB0:
.cfi_startproc
pushl
%ebp
.cfi_def_cfa_offset 8
.cfi_offset 5, -8
movl
%esp, %ebp
.cfi_def_cfa_register 5
subl
$16, %esp
movl
$0, -4(%ebp)
81
.L3:
.L2:
.LFE0:
movl
movl
jmp
8(%ebp), %eax
%eax, -8(%ebp)
.L2
movl
addl
addl
-8(%ebp), %eax
%eax, -4(%ebp)
$1, -8(%ebp)
movl
-8(%ebp), %eax
cmpl
12(%ebp), %eax
jl
.L3
movl
-4(%ebp), %eax
leave
.cfi_restore 5
.cfi_def_cfa 4, 4
ret
.cfi_endproc
.size
.section
.LC0:
.string
"Usage: %s\n start end"
.string
.text
.globl
.type
"sumRange(%d, %d) = %d\n"
.LC1:
main:
.LFB1:
sumRange, .-sumRange
.rodata
main
main, @function
.cfi_startproc
pushl
%ebp
.cfi_def_cfa_offset 8
.cfi_offset 5, -8
movl
%esp, %ebp
.cfi_def_cfa_register 5
andl
$-16, %esp
subl
$32, %esp
cmpl
$3, 8(%ebp)
je
.L6
movl
12(%ebp), %eax
movl
(%eax), %edx
movl
stderr, %eax
movl
%edx, 8(%esp)
movl
$.LC0, 4(%esp)
movl
%eax, (%esp)
call
fprintf
82
.L6:
.L7:
.LFE1:
movl
jmp
movl
addl
movl
movl
call
movl
movl
addl
movl
movl
call
movl
movl
movl
movl
movl
call
movl
movl
movl
movl
movl
movl
call
movl
$1, %eax
.L7
12(%ebp), %eax
$4, %eax
(%eax), %eax
%eax, (%esp)
atoi
%eax, 24(%esp)
12(%ebp), %eax
$8, %eax
(%eax), %eax
%eax, (%esp)
atoi
%eax, 28(%esp)
28(%esp), %eax
%eax, 4(%esp)
24(%esp), %eax
%eax, (%esp)
sumRange
%eax, 12(%esp)
28(%esp), %eax
%eax, 8(%esp)
24(%esp), %eax
%eax, 4(%esp)
$.LC1, (%esp)
printf
$0, %eax
leave
.cfi_restore 5
.cfi_def_cfa 4, 4
ret
.cfi_endproc
.size
.ident
.section
main, .-main
"GCC: (Ubuntu 4.8.2-19ubuntu1) 4.8.2"
.note.GNU-stack,"",@progbits
examples/sumRange.s
You usually don’t need to look at assembly language, but it can sometimes be
enlightening to see what the compiler is doing with your code. One thing that I
find interesting about this particular code (which is for the x86 architecture) is
that most of the instructions are movl, the x86 instruction for copying a 32-bit
quantity from one location to another: most of what this program is doing is
copying data into the places expected by the library functions it is calling. Also
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noteworthy is that the beautiful compound statements like if and for that so
eloquently express the intent of the programmer get turned into a pile of jump
(jmp) and conditional jump (jl, je) instructions, the machine code versions of
the often dangerous and confusing goto statement. This is because CPUs are
dumb: they don’t know how to carry out an if branch or a loop, and all they
can do instead is be told to replace the value of their program counter register
with some new value instead of just incrementing it as they usually do.
Assembly language is not the last stage in this process. The assembler (as) is a
program that translates the assembly language in sumRange.s into machine code
(which will be store in sumRange.o if we aren’t compiling a single program all
at once). Machine code is not human-readable, and is close to the raw stream of
bytes that gets stored in the computer’s memory to represent a running program.
The missing parts are that the addresses of each function and global variables are
generally left unspecified, so that they can be moved around to make room for
other functions and variables coming from other files and from system libraries.
The job of stitching all of these pieces together, putting everything in the right
place, filling in any placeholder addresses, and generating the executable file
sumRange that we can actually run is given to the linker ld.
The whole process looks like this:
sumRange.c (source code)
|
v
[preprocessor (cpp)]
|
v
preprocessed version of sumRange.c
|
v
[compiler (gcc)]
|
v
sumRange.s (assembly code)
|
v
[assembler (as)]
|
v
sumRange.o (machine code)
|
v
[linker (ld)] <- system library (glibc.a)
|
v
sumRange (executable)
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The good news is, you don’t actually have to run all of these steps yourself;
instead, gcc (which you may be calling as c99) will take care of everything for
you, particularly for simple programs like sumRange.c that fit in a single file.
4.2
Numeric data types
All data stored inside a computer is ultimately represented as a sequence of bits,
0 or 1 values, typically organized into words consisting of several 8-bit bytes.6
A typical desktop computer might have enough RAM to store 23 2 bytes (4
gigabytes); the Zoo machines store 23 5 bytes (32 gigabytes). However, the
address space of a process might be much larger: on a 64-bit machine, the
address space is 26 4 bytes. There’s no way to store 26 4 different addresses in
23 5 bytes of RAM; instead, a memory mapper, typically built in to the CPU,
translates the large addresses of the parts of the address space that are actually
used into smaller addresses corresponding to actual RAM locations. In some
cases, regions of memory that have not been used in a while will be swapped
out to disk, leaving more RAM free for other parts of the process (or other
processes). This technique is known as virtual memory and is usually invisible
to the programmer. The use of virtual memory can increase the available space
beyond the size of the RAM a little bit, but if you try to run a process that is
actively using significantly more space that can be stored in RAM, it will slow
down dramatically, because disk drives are roughly ten million times slower than
memory.
The most basic kind of data represents integer values from some bounded range.
C supports several integer data types, varying in their size (and thus range),
and whether or not they are considered to be signed. These are described in
more detail below.
For numerical computation, integer data types can be inconvenient. So C also
supports floating-point types that consist of a fixed-size mantissa, which
is essentially an integer, together with an exponent that is used to multiply
the mantissa by 2x for some x. These allow very small or very large values to
be represented with small relative error, but do not allow exact computation
because of the limited precision of the mantissa. Floating-point types are also
described below.
All other data is represented by converting it to either integer or floating-point
numbers. For example, text characters in C are represented as small integer
values, so that the character constant 'z' representation a lower-case “z” is
exactly the same as the integer constant 122 (which is the ASCII code for “z”).
A string like "hi there" is represented by a sequence of 8-bit ASCII characters,
with a special 0 character to mark the end of the string. Strings that go beyond
6 This convention was not always followed in the early days of computing. For example, the
PDP-7 on which UNIX was first developed used 18-bit words, which conveniently translated
into six octal digits back in the pre-hexadecimal era.
85
the English characters available in the ASCII encoding are typically represented
using Unicode and encoded as sequences of bytes using a particular representation
called UTF-8. The color of a pixel in an image might be represented as three
8-bit integers representing the intensity of red, green, and blue in the color, while
an image itself might be a long sequence of such 3-byte RGB values. At the
bottom, every operation applied to these more complex data types translates
into a whole lot of copies and arithmetic operations on individual bytes and
words.
From the CPU’s point of view, even much of this manipulation consists of
operating on integers that happen to represent addresses instead of data. So
when a C program writes a zero to the 19th entry in a sequence of 4-byte integers,
somewhere in the implementation of this operation the CPU will be adding 4 · 19
to a base address for the sequence to computer where to write this value. Unlike
many higher-level languages, C allows the program direct access to address
computations via pointer types, which are tricky enough to get their own
chapter. Indeed, most of the structured types that C provides for representing
more complicated data can best be understood as a thin layer of abstraction on
top of pointers. We will see examples of these in later chapters as well.
For now, we concentrate on integer and floating-point types, and on the operations
that can be applied to them.
4.2.1
Integer types in C
Most variables in C programs tend to hold integer values, and indeed most
variables in C programs tend to be the default-width integer type int. Declaring
a variable to have a particular integer type controls how much space is used to
store the variable (any values too big to fit will be truncated) and specifies that
the arithmetic on the variable is done using integer operations.
4.2.1.1
Basic integer types
The standard C integer types are:
Name
Typical size
Signed by default?
char
short
int
long
long long
8 bits
16 bits
32 bits
32 bits
64 bits
unspecified
signed
signed
signed
signed
The typical size is for 32-bit architectures like the Intel i386. Some 64-bit
machines might have 64-bit ints and longs, and some microcontrollers have
86
16-bit ints. Particularly bizarre architectures might have even wilder sizes, but
you are not likely to see this unless you program vintage 1970s supercomputers.
The general convention is that int is the most convenient size for whatever
computer you are using and should be used by default.
Many compilers also support a long long type that is usually twice the length
of a long (e.g. 64 bits on i386 machines). This type was not officially added to
the C standard prior to C99, so it may or may not be available if you insist on
following the ANSI specification strictly.
Each of these types comes in signed and unsigned variants.
This controls the interpretation of some operations (mostly comparisons and
shifts) and determines the range of the type: for example, an unsigned char
holds values in the range 0 through 255 while a signed char holds values in
the range -128 through 127, and in general an unsigned n-bit type runs from
0 through 2n − 1 while the signed version runs from −2n−1 through 2n−1 − 1.
The representation of signed integers uses two’s-complement notation, which
means that a positive value x is represented as the unsigned value x while a
negative value −x is represented as the unsigned value 2n − x. For example, if
we had a peculiar implementation of C that used 3-bit ints, the binary values
and their interpretation as int or unsigned int would look like this:
bits
as unsigned int
as int
000
001
010
011
100
101
110
111
0
1
2
3
4
5
6
7
0
1
2
3
-4
-3
-2
-1
The reason we get one extra negative value for an unsigned integer type is this
allows us to interpret the first bit as the sign, which makes life a little easier for
whoever is implementing our CPU. Two useful features of this representation
are:
1. We can convert freely between signed and unsigned values as long as we
are in the common range of both, and
2. Addition and subtraction work exactly the same we for both signed and
unsigned values. For example, on our hypothetical 3-bit machine, 1 + 5
represented as 001 + 101 = 110 gives the same answer as 1 + (−3) =
001 + 101 = 110. In the first case we interpret 110 as 6, while in the
second we interpret it as −2, but both answers are right in their respective
contexts.
87
Note that in order to make this work, we can’t detect overflow: when the CPU
adds two 3-bit integers, it doesn’t know if we are adding 7 + 6 = 111 + 110 =
1101 = 13 or (−1) + (−2) = 111 + 110 = 101 = (−3). In both cases the result
is truncated to 101, which gives the incorrect answer 5 when we are adding
unsigned values.
This can often lead to surprising uncaught errors in C programs, although using
more than 3 bits will make overflow less likely. It is usually a good idea to
pick a size for a variable that is substantially larger than the largest value you
expect the variable to hold (although most people just default to int), unless
you are very short on space or time (larger values take longer to read and write
to memory, and may make some arithmetic operations take longer).
Taking into account signed and unsigned versions, the full collection of integer
types looks like this:
char
short
int
long
long long
signed char
unsigned
unsigned
unsigned
unsigned
unsigned
char
short
int
long
long long
So these are all examples of declarations of integer variables:
int i;
char c;
signed char temperature; /* degrees Celsius, only valid for Earth's surface */
long netWorthInPennies;
long long billGatesNetWorthInPennies;
unsigned short shaveAndAHaircutTwoBytes;
For chars, whether the character is signed (−128 . . . 127) or unsigned (0 . . . 255) is
at the whim of the compiler. If it matters, declare your variables as signed char
or unsigned char. For storing actual 8-bit characters that you aren’t doing
arithmetic on, it shouldn’t matter.
There is a slight gotcha for character processing with input function like getchar
and getc. These return the special value EOF (defined in stdio.h to be −1) to
indicate end of file. But 255, which represents 'ÿ' in the ISO Latin-1 alphabet
and in Unicode and which may also appear quite often in binary files, will map
to −1 if you put it in a character. So you should store the output of these
functions in an int if you need to test for end of file. Once you have done this
test, it’s OK to put the non-end-of-file character back in a char.
/* right */
int c;
while((c = getchar()) != EOF) {
88
}
putchar(c);
/* WRONG */
char c;
while((c = getchar()) != EOF) {
putchar(c);
}
4.2.1.2
/* <- DON'T DO THIS! */
C99 fixed-width types
C99 provides a stdint.h header file that defines integer types with known size
independent of the machine architecture. So in C99, you can use int8_t instead
of signed char to guarantee a signed type that holds exactly 8 bits, or uint64_t
instead of unsigned long long to get a 64-bit unsigned integer type. The full
set of types typically defined are int8_t, int16_t, int32_t, and int64_t for
signed integers and uint8_t, uint16_t, uint32_t, and uint64_t for unsigned
integers. There are also types for integers that contain the fewest number of bits
greater than some minimum (e.g., int_least16_t is a signed type with at least
16 bits, chosen to minimize space) or that are the fastest type with at least the
given number of bits (e.g., int_fast16_t is a signed type with at least 16 bits,
chosen to minimize time). The stdint.h file also defines constants giving the
minimum and maximum values of these and standard integer types; for example,
INT_MIN and INT_MAX give the smallest and largest values that can be stored in
an int.
All of these types are defined as aliases for standard integer types using typedef;
the main advantage of using stdint.h over defining them yourself is that if
somebody ports your code to a new architecture, stdint.h should take care
of choosing the right types automatically. The main disadvantage is that, like
many C99 features, stdint.h is not universally available on all C compilers.
Also, because these fixed-width types are a late addition to the language, the
built-in routines for printing and parsing integers, as well as the mechanisms for
specifying the size of an integer constant, are not adapted to deal with them.
But if you do need to print or parse types defined in stdint.h, the larger
inttypes.h header defines macros that give the corresponding format strings
for printf and scanf. The inttypes.h file includes stdint.h, so you do not
need to include both. Below is an example of a program that uses the various
features provided by inttypes.h and stdint.h.
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
#include <inttypes.h>
89
/* example of using fixed-width types */
/* largest value we can apply 3x+1 to without overflow */
#define MAX_VALUE ((UINT64_MAX - 1) / 3)
int
main(int argc, char **argv)
{
uint64_t big;
if(argc != 2) {
fprintf(stderr, "Usage: %s number\n", argv[0]);
return 1;
}
/* parse argv[1] as a uint64_t */
/* SCNu64 expands to the format string for scanning uint64_t (without the %) */
/* We then rely on C concatenating adjacent string constants. */
sscanf(argv[1], "%" SCNu64, &big);
/* do some arithmetic on big */
while(big != 1) {
/* PRIu64 expands to the format string for printing uint64_t */
printf("%" PRIu64 "\n", big);
}
}
if(big % 2 == 0) {
big /= 2;
} else if(big <= MAX_VALUE) {
big = 3*big + 1;
} else {
/* overflow! */
puts("overflow");
return 1;
}
puts("Reached 1");
return 0;
examples/integerTypes/fixedWidth.c
90
4.2.2
size_t and ptrdiff_t
The type aliases size_t and ptrdiff_t are provided in stddef.h to represent
the return types of the sizeof operator and pointer subtraction. On a 32bit architecture, size_t will be equivalent to the unsigned 32-bit integer type
uint32_t (or just unsigned int) and ptrdiff_t will be equivalent to the signed
32-bit integer type int32_t (int). On a 64-bit architecture, size_t will be
equivalent to uint64_t and ptrdiff_t will be equivalent to uint64_t.
The place where you will most often see size_t is as an argument to malloc,
where it gives the number of bytes to allocate.
Because stdlib.h includes stddef.h, it is often not necessary to include
stddef.h explicitly.
4.2.2.1
Integer constants
Constant integer values in C can be written in any of four different ways:
• In the usual decimal notation, e.g. 0, 1, -127, 9919291, 97.
• In octal or base 8, when the leading digit is 0, e.g. 01 for 1, 010 for 8,
0777 for 511, 0141 for 97. Octal is not used much any more, but it is still
conventional for representing Unix file permissions.
• In hexadecimal or base 16, when prefixed with 0x. The letters a through
f are used for the digits 10 through 15. For example, 0x61 is another way
to write 97.
• Using a character constant, which is a single ASCII character or an
escape sequence inside single quotes. The value is the ASCII value of
the character: 'a' is 97.7 Unlike languages with separate character types,
C characters are identical to integers; you can (but shouldn’t) calculate
972 by writing 'a'*'a'. You can also store a character anywhere.
Except for character constants, you can insist that an integer constant is unsigned
or long by putting a u or l after it. So 1ul is an unsigned long version of 1.
By default integer constants are (signed) ints. For long long constants, use
ll, e.g., the unsigned long long constant 0xdeadbeef01234567ull. It is also
permitted to write the l as L, which can be less confusing if the l looks too
much like a 1.
Some examples:
‘a’
97
97u
0xbea00diful
int
int
unsigned int
unsigned long, written in hexadecimal
7 Certain ancient versions of C ran on machines with a different character set encoding, like
EBCDIC. The C standard does not guarantee ASCII encoding.
91
0777s
short, written in octal
A curious omission is that there is no way to write a binary integer directly in C.
So if you want to write the bit pattern 00101101, you will need to encode it in
hexadecimal as 0x2d (or octal as 055). Another potential trap is that leading
zeros matter: 012 is an octal value corresponding to what normal people call 10.
4.2.2.1.1
Naming constants
Having a lot of numeric constants in your program—particularly if the same
constant shows up in more than one place—is usually a sign of bad programming.
There are a few constants, like 0 and 1, that make sense on their own, but many
constant values are either mostly arbitrary, or might change if the needs of the
program change. It’s helpful to assign these constants names that explain their
meaning, instead of requiring the user to guess why there is a 37 here or an
0x1badd00d there. This is particularly important if the constants might change
in later versions of the program, since even though you could change every 37
in your program into a 38, this might catch other 37 values that have different
intended meanings.
For example, suppose that you have a function (call it getchar) that needs to
signal that sometimes it didn’t work. The usual way is to return a value that
the function won’t normally return. Now, you could just tell the user what value
that is:
/* get a character (as an `int` ASCII code) from `stdin` */
/* return -1 on end of file */
int getchar(void);
and now the user can write
while((c = getchar()) != -1) {
...
}
But then somebody reading the code has to remember that -1 means “end
of file” and not “signed version of 0xff” or “computer room on fire, evacuate
immediately.” It’s much better to define a constant EOF that happens to equal -1,
because among other things if you change the special return value from getchar
later then this code will still work (assuming you fixed the definition of EOF):
while((c = getchar()) != EOF) {
...
}
So how do you declare a constant in C? The traditional approach is to use the
C preprocessor, the same tool that gets run before the compiler to expand out
92
#include directives. To define EOF, the file /usr/include/stdio.h includes
the text
#define EOF (-1)
What this means is that whenever the characters EOF appear in a C program as
a separate word (e.g. in 1+EOF*3 but not in appurtenancesTherEOF), then the
preprocessor will replace them with the characters (-1). The parentheses around
the -1 are customary to ensure that the -1 gets treated as a separate constant
and not as part of some larger expression. So from the compiler’s perspective,
EOF really is -1, but from the programmer’s perspective, it’s end-of-file. This is
a special case of the C preprocessor’s macro mechanism.
In general, any time you have a non-trivial constant in a program, it should be
#defined. Examples are things like array dimensions, special tags or return
values from functions, maximum or minimum values for some quantity, or
standard mathematical constants (e.g., /usr/include/math.h defines M_PI as
pi to umpteen digits). This allows you to write
char buffer[MAX_FILENAME_LENGTH+1];
area = M_PI*r*r;
if(status == COMPUTER_ROOM_ON_FIRE) {
evacuate();
}
instead of
char buffer[513];
area = 3.141592319*r*r;
if(status == 136) {
evacuate();
}
which is just an invitation to errors (including the one in the area computation).
Like typedefs, #defines that are intended to be globally visible are best done in
header files; in large programs you will want to #include them in many source
files. The usual convention is to write #defined names in all-caps to remind the
user that they are macros and not real variables.
4.2.2.2
4.2.2.2.1
Integer operators
Arithmetic operators
93
The usual + (addition), - (negation or subtraction), and * (multiplication)
operators work on integers pretty much the way you’d expect. The only caveat
is that if the result lies outside of the range of whatever variable you are storing
it in, it will be truncated instead of causing an error:
unsigned char c;
c = -1;
c = 255 + 255;
c = 256 * 1772717;
/* sets c = 255 */
/* sets c = 254 */
/* sets c = 0 */
This can be a source of subtle bugs if you aren’t careful. The usual giveaway is
that values you thought should be large positive integers come back as randomlooking negative integers.
Division (/) of two integers also truncates: 2/3 is 0, 5/3 is 1, etc. For positive
integers it will always round down.
Prior to C99, if either the numerator or denominator is negative, the behavior
was unpredictable and depended on what your processor chose to do. In practice
this meant you should never use / if one or both arguments might be negative.
The C99 standard specified that integer division always removes the fractional
part, effectively rounding toward 0; so (-3)/2 is -1, 3/-2 is -1, and (-3)/-2 is
1.
There is also a remainder operator % with e.g. 2%3 = 2, 5%3 = 2, 27 % 2 = 1, etc.
The sign of the modulus is ignored, so 2%-3 is also 2. The sign of the dividend
carries over to the remainder: (-3)%2 and (-3)%(-2) are both -1. The reason
for this rule is that it guarantees that y == x*(y/x) + y%x is always true.
4.2.2.2.2
Bitwise operators
In addition to the arithmetic operators, integer types support bitwise logical
operators that apply some Boolean operation to all the bits of their arguments
in parallel. What this means is that the i-th bit of the output is equal to some
operation applied to the i-th bit(s) of the input(s). The bitwise logical operators
are ~ (bitwise negation: used with one argument as in ~0 for the all-1’s binary
value), & (bitwise AND), ‘|’ (bitwise OR), and ‘ˆ’ (bitwise XOR, i.e. sum mod 2).
These are mostly used for manipulating individual bits or small groups of bits
inside larger words, as in the expression x & 0x0f, which strips off the bottom
four bits stored in x.
Examples:
x
y
expression
value
0011
0011
0011
0101
0101
0101
x&y
x|y
xˆy
0001
0111
0110
94
x
y
expression
value
0011
0101
~x
1100
The shift operators << and >> shift the bit sequence left or right: x << y produces
the value x · 2y (ignoring overflow); this is equivalent to shifting every bit in x y
positions to the left and filling in y zeros for the missing positions. In the other
direction, x >> y produces the value bx · 2− yc by shifting x y positions to the
right. The behavior of the right shift operator depends on whether x is unsigned
or signed; for unsigned values, it shifts in zeros from the left end always; for
signed values, it shifts in additional copies of the leftmost bit (the sign bit). This
makes x >> y have the same sign as x if x is signed.
If y is negative, it reverses the direction of the shift; so x << -2 is equivalent to
x >> 2.
Examples (unsigned char x):
x
y
x << y
x >> y
00000001
11111111
10111001
1
3
-2
00000010
11111000
00101110
00000000
00011111
11100100
Examples (signed char x):
x
y
x << y
x >> y
00000001
11111111
10111001
1
3
-2
00000010
11111000
11101110
00000000
11111111
11100100
Shift operators are often used with bitwise logical operators to set or extract
individual bits in an integer value. The trick is that (1 << i) contains a 1 in the
i-th least significant bit and zeros everywhere else. So x & (1<<i) is nonzero if
and only if x has a 1 in the i-th place. This can be used to print out an integer
in binary format (which standard printf won’t do).
The following program gives an example of this technique. For example, when
called as ./testPrintBinary 123, it will print 111010 followed by a newline.
#include <stdio.h>
#include <stdlib.h>
/* print out all bits of n */
void
95
print_binary(unsigned int n)
{
unsigned int mask = 0;
/* this grotesque hack creates a bit pattern 1000... */
/* regardless of the size of an unsigned int */
mask = ~mask ^ (~mask >> 1);
}
for(; mask != 0; mask >>= 1) {
putchar((n & mask) ? '1' : '0');
}
int
main(int argc, char **argv)
{
if(argc != 2) {
fprintf(stderr, "Usage: %s n\n", argv[0]);
return 1;
}
print_binary(atoi(argv[1]));
putchar('\n');
}
return 0;
examples/integerTypes/testPrintBinary.c
In the other direction, we can set the i-th bit of x to 1 by doing x | (1 << i)
or to 0 by doing x & ~(1 << i). See the section on bit manipulation. for
applications of this to build arbitrarily-large bit vectors.
4.2.2.2.3
Logical operators
To add to the confusion, there are also three logical operators that work on
the truth-values of integers, where 0 is defined to be false and anything else is
defined by be true. These are && (logical AND), ||, (logical OR), and ! (logical
NOT). The result of any of these operators is always 0 or 1 (so !!x, for example,
is 0 if x is 0 and 1 if x is anything else). The && and || operators evaluate their
arguments left-to-right and ignore the second argument if the first determines the
answer (this is the only place in C where argument evaluation order is specified);
so
0 && executeProgrammer();
1 || executeProgrammer();
96
is in a very weak sense perfectly safe code to run.
Watch out for confusing & with &&. The expression 1 & 2 evaluates to 0, but
1 && 2 evaluates to 1. The statement 0 & executeProgrammer(); is also
unlikely to do what you want.
Yet another logical operator is the ternary operator ?:, where x ? y : z
equals the value of y if x is nonzero and z if x is zero. Like && and ||, it only
evaluates the arguments it needs to:
fileExists(badFile) ? deleteFile(badFile) : createFile(badFile);
Most uses of ?: are better done using an if-then-else statement.
4.2.2.2.4
Relational operators
Logical operators usually operate on the results of relational operators or
comparisons: these are == (equality), != (inequality), < (less than), > (greater
than), <= (less than or equal to) and >= (greater than or equal to). So, for
example,
if(size >= MIN_SIZE && size <= MAX_SIZE) {
puts("just right");
}
tests if size is in the (inclusive) range [MIN_SIZE..MAX_SIZE].
Beware of confusing == with =. The code
/* DANGER! DANGER! DANGER! */
if(x = 5) {
...
is perfectly legal C, and will set x to 5 rather than testing if it’s equal to 5.
Because 5 happens to be nonzero, the body of the if statement will always
be executed. This error is so common and so dangerous that gcc will warn
you about any tests that look like this if you use the -Wall option. Some
programmers will go so far as to write the test as 5 == x just so that if their
finger slips, they will get a syntax error on 5 = x even without special compiler
support.
4.2.2.3
Converting to and from strings
To input or output integer values, you will need to convert them from or to
strings. Converting from a string is easy using the atoi or atol functions
declared in stdlib.h; these take a string as an argument and return an int or
long, respectively. C99 also provides atoll for converting to long long. These
routines have no ability to signal an error other than returning 0, so if you do
atoi("Sweden"), that’s what you’ll get.
97
Output is usually done using printf (or sprintf if you want to write to a
string without producing output). Use the %d format specifier for ints, shorts,
and chars that you want the numeric value of, %ld for longs, and %lld for
long longs.
A contrived program that uses all of these features is given below:
#include <stdio.h>
#include <stdlib.h>
/* This program can be used to show how atoi etc. handle overflow. */
/* For example, try "overflow 1000000000000". */
int
main(int argc, char **argv)
{
char c;
int i;
long l;
long long ll;
if(argc != 2) {
fprintf(stderr, "Usage: %s n\n", argv[0]);
return 1;
}
c = atoi(argv[1]);
i = atoi(argv[1]);
l = atol(argv[1]);
ll = atoll(argv[1]);
printf("char: %d
}
int: %d
long: %ld
long long: %lld", c, i, l, ll);
return 0;
examples/integerTypes/overflow.c
4.2.3
Floating-point types
Real numbers are represented in C by the floating point types float, double,
and long double. Just as the integer types can’t represent all integers because
they fit in a bounded number of bytes, so also the floating-point types can’t
represent all real numbers. The difference is that the integer types can represent
values within their range exactly, while floating-point types almost always give
only an approximation to the correct value, albeit across a much larger range.
The three floating point types differ in how much space they use (32, 64, or
98
80 bits on x86 CPUs; possibly different amounts on other machines), and thus
how much precision they provide. Most math library routines expect and return
doubles (e.g., sin is declared as double sin(double), but there are usually
float versions as well (float sinf(float)).
4.2.3.1
Floating point basics
The core idea of floating-point representations (as opposed to fixed point
representations as used by, say, ints), is that a number x is written as m · be
where m is a mantissa or fractional part, b is a base, and e is an exponent.
On modern computers the base is almost always 2, and for most floating-point
representations the mantissa will be scaled to be between 1 and b. This is done
by adjusting the exponent, e.g.
1 = 1 · 20
2 = 1 · 21
0.375 = 1.5 · 2−2
etc.
The mantissa is usually represented in base 2, as a binary fraction. So (in a
very low-precision format), $1 would be 1.000 · 20 , 2 would be 1.000 · 21 , and
0.375 = 3/8 would be 1.100 · 2−2 , where the first 1 after the decimal point counts
as 1/2, the second as 1/4, etc. Note that for a properly-scaled (or normalized)
floating-point number in base 2 the digit before the decimal point is always 1.
For this reason it is usually dropped to save space (although this requires a
special representation for 0).
Negative values are typically handled by adding a sign bit that is 0 for positive
numbers and 1 for negative numbers.
4.2.3.2
Floating-point constants
Any number that has a decimal point in it will be interpreted by the compiler as
a floating-point number. Note that you have to put at least one digit after the
decimal point: 2.0, 3.75, -12.6112. You can specific a floating point number
in scientific notation using e for the exponent: 6.022e23.
4.2.3.3
Operators
Floating-point types in C support most of the same arithmetic and relational
operators as integer types; x > y, x / y, x + y all make sense when x and y are
floats. If you mix two different floating-point types together, the less-precise
one will be extended to match the precision of the more-precise one; this also
works if you mix integer and floating point types as in 2 / 3.0. Unlike integer
99
division, floating-point division does not discard the fractional part (although
it may produce round-off error: 2.0/3.0 gives 0.66666666666666663, which
is not quite exact). Be careful about accidentally using integer division when
you mean to use floating-point division: 2/3 is 0. Casts can be used to force
floating-point division (see below).
Some operators that work on integers will not work on floating-point types. These
are % (use modf from the math library if you really need to get a floating-point
remainder) and all of the bitwise operators ~, <<, >>, &, ˆ, and |.
4.2.3.4
Conversion to and from integer types
Mixed uses of floating-point and integer types will convert the integers to floatingpoint.
You can convert floating-point numbers to and from integer types explicitly using
casts. A typical use might be:
/* return the average of a list */
double
average(int n, int a[])
{
int sum = 0;
int i;
for(i = 0; i < n; i++) {
sum += a[i];
}
}
return (double) sum / n;
If we didn’t put in the (double) to convert sum to a double, we’d end up doing
integer division, which would truncate the fractional part of our average. Note
that casts bind tighter than arithmetic operations, so the (double) applies to
just sum, and not the whole expression sum / n.
In the other direction, we can write:
i = (int) f;
to convert a float f to int i. This conversion loses information by throwing
away the fractional part of f: if f was 3.2, i will end up being just 3.
The math library contains a pile of functions for converting values of type double
to integer values of type double that give more control over the rounding: see
for example the descriptions of floor, ceil, round, trunc, and nearbyint in
the GNU libc reference manual.
100
4.2.3.5
The IEEE-754 floating-point standard
The IEEE-754 floating-point standard is a standard for representing and manipulating floating-point quantities that is followed by all modern computer systems.
It defines several standard representations of floating-point numbers, all of which
have the following basic pattern (the specific layout here is for 32-bit floats):
bit
31 30
23 22
0
S EEEEEEEE MMMMMMMMMMMMMMMMMMMMMMM
The bit numbers are counting from the least-significant bit. The first bit is the
sign (0 for positive, 1 for negative). The following 8 bits are the exponent in
excess-127 binary notation; this means that the binary pattern 01111111 =
127 represents an exponent of 0, 1000000 = 128, represents 1, 01111110 = 126
represents -1, and so forth. The mantissa fits in the remaining 24 bits, with its
leading 1 stripped off as described above.
Certain numbers have a special representation. Because 0 cannot be represented
in the standard form (there is no 1 before the decimal point), it is given the
special representation 0 00000000 00000000000000000000000. (There is also
a -0 = 1 00000000 00000000000000000000000, which looks equal to +0 but
prints differently.) Numbers with exponents of 11111111 = 255 = 2128 represent
non-numeric quantities such as “not a number” (NaN), returned by operations like
(0.0/0.0) and positive or negative infinity. A table of some typical floating-point
numbers (generated by the program float.c) is given below:
0
-0
0.125
0.25
0.5
1
2
4
8
0.375
0.75
1.5
3
6
0.1
0.2
0.4
0.8
1e+12
1e+24
1e+36
inf
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
0
-0
0.125
0.25
0.5
1
2
4
8
0.375
0.75
1.5
3
6
0.10000000149011612
0.20000000298023224
0.40000000596046448
0.80000001192092896
999999995904
1.0000000138484279e+24
9.9999996169031625e+35
inf
101
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
00000000
00000000
01111100
01111101
01111110
01111111
10000000
10000001
10000010
01111101
01111110
01111111
10000000
10000001
01111011
01111100
01111101
01111110
10100110
11001110
11110110
11111111
00000000000000000000000
00000000000000000000000
00000000000000000000000
00000000000000000000000
00000000000000000000000
00000000000000000000000
00000000000000000000000
00000000000000000000000
00000000000000000000000
10000000000000000000000
10000000000000000000000
10000000000000000000000
10000000000000000000000
10000000000000000000000
10011001100110011001101
10011001100110011001101
10011001100110011001101
10011001100110011001101
11010001101010010100101
10100111100001000011100
10000001001011111001110
00000000000000000000000
-inf =
nan =
-inf = 1 11111111 00000000000000000000000
nan = 0 11111111 10000000000000000000000
What this means in practice is that a 32-bit floating-point value (e.g. a float) can
represent any number between 1.17549435e-38 and 3.40282347e+38, where
the e separates the (base 10) exponent. Operations that would create a smaller
value will underflow to 0 (slowly—IEEE 754 allows “denormalized” floating point
numbers with reduced precision for very small values) and operations that would
create a larger value will produce inf or -inf instead.
For a 64-bit double, the size of both the exponent and mantissa are larger; this
gives a range from 1.7976931348623157e+308 to 2.2250738585072014e-308,
with similar behavior on underflow and overflow.
Intel processors internally use an even larger 80-bit floating-point format for
all operations. Unless you declare your variables as long double, this should
not be visible to you from C except that some operations that might otherwise
produce overflow errors will not do so, provided all the variables involved sit in
registers (typically the case only for local variables and function parameters).
4.2.3.6
Error
In general, floating-point numbers are not exact: they are likely to contain
round-off error because of the truncation of the mantissa to a fixed number
of bits. This is particularly noticeable for large values (e.g. 1e+12 in the table
above), but can also be seen in fractions with values that aren’t powers of 2 in the
denominator (e.g. 0.1). Round-off error is often invisible with the default float
output formats, since they produce fewer digits than are stored internally, but
can accumulate over time, particularly if you subtract floating-point quantities
with values that are close (this wipes out the mantissa without wiping out the
error, making the error much larger relative to the number that remains).
The easiest way to avoid accumulating error is to use high-precision floating-point
numbers (this means using double instead of float). On modern CPUs there
is little or no time penalty for doing so, although storing doubles instead of
floats will take twice as much space in memory.
Note that a consequence of the internal structure of IEEE 754 floating-point
numbers is that small integers and fractions with small numerators and
power-of-2 denominators can be represented exactly—indeed, the IEEE 754
standard carefully defines floating-point operations so that arithmetic on such
exact integers will give the same answers as integer arithmetic would (except,
of course, for division that produces a remainder). This fact can sometimes
be exploited to get higher precision on integer values than is available from
the standard integer types; for example, a double can represent any integer
between -253 and 253 exactly, which is a much wider range than the values from
2ˆ-31ˆ to 2ˆ31ˆ-1 that fit in a 32-bit int or long. (A 64-bit long
long does better.) So double‘ should be considered for applications where
102
large precise integers are needed (such as calculating the net worth in pennies of
a billionaire.)
One consequence of round-off error is that it is very difficult to test floating-point
numbers for equality, unless you are sure you have an exact value as described
above. It is generally not the case, for example, that (0.1+0.1+0.1) == 0.3
in C. This can produce odd results if you try writing something like
for(f = 0.0; f <= 0.3; f += 0.1): it will be hard to predict in advance
whether the loop body will be executed with f = 0.3 or not. (Even more
hilarity ensues if you write for(f = 0.0; f != 0.3; f += 0.1), which
after not quite hitting 0.3 exactly keeps looping for much longer than I am
willing to wait to see it stop, but which I suspect will eventually converge to
some constant value of f large enough that adding 0.1 to it has no effect.)
Most of the time when you are tempted to test floats for equality, you are
better off testing if one lies within a small distance from the other, e.g. by
testing fabs(x-y) <= fabs(EPSILON * y), where EPSILON is usually some
application-dependent tolerance. This isn’t quite the same as equality (for
example, it isn’t transitive), but it usually closer to what you want.
4.2.3.7
Reading and writing floating-point numbers
Any numeric constant in a C program that contains a decimal point is treated
as a double by default. You can also use e or E to add a base-10 exponent (see
the table for some examples of this.) If you want to insist that a constant value
is a float for some reason, you can append F on the end, as in 1.0F.
For I/O, floating-point values are most easily read and written using scanf (and
its relatives fscanf and sscanf) and printf. For printf, there is an elaborate
variety of floating-point format codes; the easiest way to find out what these
do is experiment with them. For scanf, pretty much the only two codes you
need are "%lf", which reads a double value into a double *, and "%f", which
reads a float value into a float *. Both these formats are exactly the same
in printf, since a float is promoted to a double before being passed as an
argument to printf (or any other function that doesn’t declare the type of its
arguments). But you have to be careful with the arguments to scanf or you
will get odd results as only 4 bytes of your 8-byte double are filled in, or—even
worse—8 bytes of your 4-byte float are.
4.2.3.8
Non-finite numbers in C
The values nan, inf, and -inf can’t be written in this form as floating-point
constants in a C program, but printf will generate them and scanf seems to
recognize them. With some machines and compilers you may be able to use the
macros INFINITY and NAN from <math.h> to generate infinite quantities. The
macros isinf and isnan can be used to detect such quantities if they occur.
103
4.2.3.9
The math library
(See also K&R Appendix B4.)
Many mathematical functions on floating-point values are not linked into C programs by default, but can be obtained by linking in the math library. Examples
would be the trigonometric functions sin, cos, and tan (plus more exotic ones),
sqrt for taking square roots, pow for exponentiation, log and exp for base-e
logs and exponents, and fmod for when you really want to write x%y but one
or both variables is a double. The standard math library functions all take
doubles as arguments and return double values; most implementations also
provide some extra functions with similar names (e.g., sinf) that use floats
instead, for applications where space or speed is more important than accuracy.
There are two parts to using the math library. The first is to include the line
#include <math.h>
somewhere at the top of your source file. This tells the preprocessor to paste in
the declarations of the math library functions found in /usr/include/math.h.
The second step is to link to the math library when you compile. This is done
by passing the flag -lm to gcc after your C program source file(s). A typical
command might be:
c99 -o program program.c -lm
If you don’t do this, you will get errors from the compiler about missing functions.
The reason is that the math library is not linked in by default, since for many
system programs it’s not needed.
4.3
Operator precedence
Operator precedence in C controls the interpretation of ambiguous expressions
like 2+3*4, which could in principle be parsed either as 2+(3*4) (the right
way) or as (2+3)*4 (the cheap calculator way). For the most part, C parses
unparenthesized expressions the right way, but if you are not sure what it will
do with an expression, you can always put in parentheses to force it to do the
right thing.
There is a table on page 53 of Kernighan and Ritchie that shows the precedence
of all operators in C, which we reproduce below.
The interpretation of this table is that higher entries bind tighter than lower
ones; so the fact that * has higher precedence that + and both have higher
precedence that > means that 2+3*4 > 5 gets parsed as (2+(3*4)) > 5.
Associativity controls how an expression with multiple operators of the same
precedence is interpreted. The fact that + and - associate left-to-right means
that the expression 2+3-4-5 is interpreted as (((2+3)-4)-5): the leftmost
104
operation is done first. Unary operators, ternary ?: and assignment operators
are the only ones that associate right-to-left. For assignment operators, this is so
x = y = 0 is interpreted as x = (y = 0) (assigning 0 to both x and y) and not
(x = y) = 0 (which would give an error because (x = y) isn’t something you
can assign to). For unary operators, this mostly affects expressions like *p++,
which is equivalent to *(p++) (increment the pointer first then dereference it)
rather than (*p)++ (increment the thing that p points to).
() [] -> .
! ~ - (unary) * (unary) &(unary) ++ -- (type)
sizeof
* (binary) / %
+ (binary) - (binary)
<< >>
< <= >= >
== !=
& (binary)
ˆ
|
&&
||
?:
= += -= *= /= %= &= ˆ= |= <<= >>=
,
4.4
function calls and indexing
unary operators (associate
right-to-left)
multiplication and division
addition and subtraction
shifts
inequalities
equality
bitwise AND
bitwise XOR
bitwise OR
logical AND
logical OR
ternary if (associates
right-to-left)
assignment (associate
right-to-left)
comma
Programming style
The C programming language imposes very few constraints on how programs
are formatted and organized. Both of the following are legitimate C programs,
which compile to exactly the same machine code using gcc with a high enough
optimization level:
/*
* Count down from COUNTDOWN_START (defined below) to 0.
* Prints all numbers in the range including both endpoints.
*/
#include <stdio.h>
#define COUNTDOWN_START (10)
int
main(int argc, char **argv)
105
{
for(int i = COUNTDOWN_START; i >= 0; i--) {
printf("%d\n", i);
}
return 0;
}
examples/style/countdown.c
#include <stdio.h>
int main(int _,char**xb){_=0xb;while(_--)printf("%d\n",_);return ++_;}
examples/style/badCountdown.c
The difference between these programs is that the first is designed to be easy to
read and understand while the second is not. Though computer can’t tell the
difference between them, the second will be much harder to debug or modify to
accomplish some new task.
Certain formatting and programming conventions have evolved over the years to
make C code as comprehensible as possible, and as we introduce various features
of C, we will talk about how best to use them to make your programs understood
by both computers and humans.
Submitted assignments may be graded for style in addition to correctness. Below
is a checklist that has been used in past versions of the course to identify some
of the more egregious violations of reasonable coding practice. For more extreme
examples of what not to do, see the International Obfuscated C Code Contest.
Style grading checklist
Score is 20 points minus 1 for each box checked (but never less than 0)
Comments
[
[
[
[
[
[
[
[
[
]
]
]
]
]
]
]
]
]
Undocumented module.
Undocumented function other than main.
Underdocumented function: return value or args not described.
Undocumented program input and output (when main is provided).
Undocumented struct or union components.
Undocumented #define.
Failure to cite code taken from other sources.
Insufficient comments.
Excessive comments.
Naming
[ ] Meaningless function name.
[ ] Confusing variable name.
106
[ ] Inconsistent variable naming style (UgLyName, ugly_name, NAME___UGLY_1).
[ ] Inconsistent use of capitalization to distinguish constants.
Whitespace
[ ] Inconsistent or misleading indentation.
[ ] Spaces not used or used misleadingly to break up complicated expressions.
[ ] Blank lines not used or used misleadingly to break up long function bodies.
Macros
[
[
[
[
]
]
]
]
Non-trivial constant with no symbolic name.
Failure to parenthesize expression in macro definition.
Dependent constant not written as expression of earlier constant.
Underdocumented parameterized macro.
Global variables
[ ] Inappropriate use of a global variable.
Functions
[ ] Kitchen-sink function that performs multiple unrelated tasks.
[ ] Non-void function that returns no useful value.
[ ] Function with too many arguments.
Code organization
[
[
[
[
[
[
]
]
]
]
]
]
Lack of modularity.
Function used in multiple source files but not declared in header file.
Internal-use-only function not declared static.
Full struct definition in header files when components should be hidden.
#include "file.c"
Substantial repetition of code.
Miscellaneous
[ ] Other obstacle to readability not mentioned above.
4.5
Variables
Variables in C are a direct abstraction of physical memory locations. To understand how variables work, it helps to start by understanding how computer
memory works.
107
4.5.1
Memory
Basic model: memory consists of many bytes of storage, each of which has an
address which is itself a sequence of bits. Though the actual memory architecture
of a modern computer is complex, from the point of view of a C program we
can think of as simply a large address space that the CPU can store things
in (and load things from), provided it can supply an address to the memory.
Because we don’t want to have to type long strings of bits all the time, the C
compiler lets us give names to particular regions of the address space, and will
even find free space for us to use.
4.5.2
Variables as names
A variable is a name given in a program for some region of memory. Each
variable has a type, which tells the compiler how big the region of memory
corresponding to it is and how to treat the bits stored in that region when performing various kinds of operations (e.g. integer variables are added together by
very different circuitry than floating-point variables, even though both represent
numbers as bits). In modern programming languages, a variable also has a
scope (a limit on where the name is meaningful, which allows the same name to
be used for different variables in different parts of the program) and an extent
(the duration of the variable’s existence, controlling when the program allocates
and deallocates space for it).
4.5.2.1
Variable declarations
Before you can use a variable in C, you must declare it. Variable declarations
show up in three places:
• Outside a function. These declarations declare global variables that are
visible throughout the program (i.e. they have global scope). Use of
global variables is almost always a mistake.
• In the argument list in the header of a function. These variables are
parameters to the function. They are only visible inside the function
body (local scope), exist only from when the function is called to when
the function returns (bounded extent—note that this is different from
what happens in some garbage-collected languages like Scheme), and get
their initial values from the arguments to the function when it is called.
• At the start of any block delimited by curly braces. Such variables are
visible only within the block (local scope again) and exist only when the
containing function is active (bounded extent). The convention in C is has
generally been to declare all such local variables at the top of a function;
this is different from the convention in C++ or Java, which encourage
variables to be declared when they are first used. This convention may
be less strong in C99 code, since C99 adopts the C++ rule of allowing
108
variables to be declared anywhere (which can be particularly useful for
index variables in for loops).
Another feature of function parameters and local variables is that if a function is
called more than once (even if the function calls itself), each copy of the function
gets its own local variables.
Variable declarations consist of a type name followed by one or more variable
names separated by commas and terminated by a semicolon (except in argument
lists, where each declaration is terminated by a comma). I personally find it
easiest to declare variables one per line, to simplify documenting them. It is also
possible for global and local variables (but not function arguments) to assign an
initial value to a variable by putting in something like = 0 after the variable name.
It is good practice to put a comment after each variable declaration that explains
what the variable does (with a possible exception for conventionally-named loop
variables like i or j in short functions). Below is an example of a program with
some variable declarations in it:
#include <stdio.h>
#include <ctype.h>
/* This program counts the number of digits in its input. */
/*
*This global variable is not used; it is here only to demonstrate
* what a global variable declaration looks like.
*/
unsigned long SpuriousGlobalVariable = 127;
int
main(int argc, char **argv)
{
int c;
/* character read */
int count = 0;
/* number of digits found */
while((c = getchar()) != EOF) {
if(isdigit(c)) {
count++;
}
}
printf("%d\n", count);
}
return 0;
examples/variables/countDigits.c
109
4.5.2.2
Variable names
The evolution of variable names in different programming languages:
11101001001001 Physical addresses represented as bits. #FC27
Typical assembly language address represented in hexadecimal to save
typing (and because it’s easier for humans to distinguish #A7 from #B6
than to distinguish 10100111 from 10110110.) A1$
A string variable in BASIC, back in the old days where BASIC variables
were one uppercase letter, optionally followed by a number, optionally
followed by $ for a string variable and % for an integer variable. These
type tags were used because BASIC interpreters didn’t have a mechanism
for declaring variable types. IFNXG7
A typical FORTRAN variable name, back in the days of 6-character allcaps variable names. The I at the start means it’s an integer variable.
The rest of the letters probably abbreviate some much longer description
of what the variable means. The default type based on the first letter
was used because FORTRAN programmers were lazy, but it could be
overridden by an explicit declaration. i, j, c, count, top_of_stack,
accumulatedTimeInFlight
Typical names from modern C programs. There is no type information contained in the name; the type is specified in the declaration and remembered
by the compiler elsewhere. Note that there are two different conventions
for representing multi-word names: the first is to replace spaces with
underscores, and the second is to capitalize the first letter of each word
(possibly excluding the first letter), a style called camel case. You should
pick one of these two conventions and stick to it. prgcGradeDatabase
An example of Hungarian notation, a style of variable naming in which
the type of the variable is encoded in the first few character. The type
is now back in the variable name again. This is not enforced by the
compiler: even though iNumberOfStudents is supposed to be an int,
there is nothing to prevent you from declaring float iNumberOfStudents
if you are teaching a class on improper chainsaw handling and want to
allow for the possibility of fractional students. See this MSDN page for a
much more detailed explanation of the system.
Not clearly an improvement on standard naming conventions, but it is
popular in some programming shops.
In C, variable names are called identifiers. These are also used to identify
things that are not variables, like functions and user-defined types.
An identifier in C must start with a lower or uppercase letter or the underscore
character _. Typically variables starting with underscores are used internally
by system libraries, so it’s dangerous to name your own variables this way.
Subsequent characters in an identifier can be letters, digits, or underscores.
110
So for example a, ____a___a_a_11727_a, AlbertEinstein, aAaAaAaAaAAAAAa,
and ______ are all legal identifiers in C, but $foo and 01 are not.
The basic principle of variable naming is that a variable name is a substitute for
the programmer’s memory. It is generally best to give identifiers names that
are easy to read and describe what the variable is used for. Such variables are
called self-documenting. None of the variable names in the preceding list are
any good by this standard. Better names would be total_input_characters,
dialedWrongNumber, or stepsRemaining. Non-descriptive single-character
names are acceptable for certain conventional uses, such as the use of i and j
for loop iteration variables, or c for an input character. Such names should only
be used when the scope of the variable is small, so that it’s easy to see all the
places where it is used at the same time.
C identifiers are case-sensitive, so aardvark, AArDvARK, and AARDVARK are all
different variables. Because it is hard to remember how you capitalized something
before, it is important to pick a standard convention and stick to it. The
traditional convention in C goes like this:
• Ordinary variables and functions are lowercased or camel-cased, e.g. count,
countOfInputBits.
• User-defined types (and in some conventions global variables) are capitalized, e.g. Stack, TotalBytesAllocated.
• Constants created with #define or enum are put in all-caps:
MAXIMUM_STACK_SIZE, BUFFER_LIMIT.
4.5.3
Using variables
Ignoring pointers for the moment, there are essentially two things you can do to
a variable. You can assign a value to it using the = operator, as in:
x = 2;
y = 3;
/* assign 2 to x */
/* assign 3 to y */
or you can use its value in an expression:
x = y+1;
/* assign y+1 to x */
The assignment operator is an ordinary operator, and assignment expressions
can be used in larger expressions:
x = (y=2)*3; /* sets y to 2 and x to 6 */
This feature is usually only used in certain standard idioms, since it’s confusing
otherwise.
There are also shorthand operators for expressions of the form variable = variable
operator expression. For example, writing x += y is equivalent to writing
x = x + y, x /= y is the same as x = x / y, etc.
111
For the special case of adding or subtracting 1, you can abbreviate still further
with the ++ and -- operators. These come in two versions, depending on whether
you want the result of the expression (if used in a larger expression) to be the
value of the variable before or after the variable is incremented:
x
y
y
y
y
=
=
=
=
=
0;
x++;
++x;
x--;
--x;
/*
/*
/*
/*
sets
sets
sets
sets
x
x
x
x
to
to
to
to
1
2
1
0
and
and
and
and
y
y
y
y
to
to
to
to
0
2
2
0
(the
(the
(the
(the
old
new
old
new
value)
value)
value)
value)
*/
*/
*/
*/
The intuition is that if the ++ comes before the variable, the increment happens
before the value of the variable is read (a preincrement; if it comes after, it
happens after the value is read (a postincrement). This is confusing enough
that it is best not to use the value of preincrement or postincrement operations
except in certain standard idioms. But using x++ or ++x by itself as a substitute
for x = x+1 is perfectly acceptable style.8
4.5.4
Initialization
It is a serious error to use the value of a variable that has never been assigned to,
because you will get whatever junk is sitting in memory at the address allocated
to the variable, and this might be some arbitrary leftover value from a previous
function call that doesn’t even represent the same type.9
Fortunately, C provides a way to guarantee that a variable is initialized as soon
as it is declared. Many of the examples in the notes do not use this mechanism,
because of bad habits learned by the instructor using early versions of C that
imposed tighter constraints on initialization. But initializing variables is a good
habit to get in the practice of doing.
For variables with simple types (that is, not arrays, structs, or unions), an
initializer looks like an assignment:
int sum = 0;
int n = 100;
int nSquared = n*n;
double gradeSchoolPi = 3.14;
const char * const greeting = "Hi!";
const int greetingLength = strlen(greeting);
For ordinary local variables, the initializer value can be any expression, including
expressions that call other functions. There is an exception for variables allocated
8 C++ programmers will prefer ++x if they are not otherwise using the return value, because
if x is some very complicated type with overloaded ++, using preincrement avoids having to
save a copy of the old value.
9 Exception: Global variables and static local variables are guaranteed to be initialized to
an all-0 pattern, which will give the value 0 for most types.
112
when the program starts (which includes global variables outside functions and
static variables inside functions), which can only be initialized to constant
expressions.
The last two examples show how initializers can set the values of variables that
are declared to be const (the variable greeting is both constant itself, because
of const greeting, and points to data that is also constant, because it is of
type const char). This is the only way to set the values of such variables
without cheating, because the compiler will complain if you try to do an ordinary
assignment to a variable declared to be constant.
For fixed-size arrays and structs, it is possible to supply an initializer for each
component, by enclosing the initializer values in braces, separated by commas.
For example:
int threeNumbers[3] = { 1, 2, 3 };
struct numericTitle {
int number;
const char *name;
};
struct numericTitle s = { 7, "Samurai" };
struct numericTitle n = { 3, "Ninjas" };
4.5.5
Storage class qualifiers
It is possible to specify additional information about how a variable can be used
using storage class qualifiers, which usually go before the type of a variable
in a declaration.
4.5.5.1
Scope and extent
Most variables that you will use in C are either parameters to functions or local
variables inside functions. These have local scope, meaning the variable names
can only be used in the function in which they are declared, and automatic
extent, meaning the space for the variable is allocated, typically on the stack,
when the function is called, and reclaimed when the function exits. (If the
function calls itself, you get another copy of all the local variables; see recursion.)
On very rare occasions you might want to have a variable that survives the entire
execution of a program (has static extent) or that is visible throughout the
program (has global scope). C provides a mechanism for doing this that you
shold never use under normal circumstances. Pretty much the only time you are
going to want to have a variable with static extent is if you are keeping track of
some piece of information that (a) you only need one instance of, (b) you need
to survive between function calls, and (c) it would be annoying to pass around
113
as an extra argument to any function that uses it. An example would be the
internal data structures used by malloc, or the count variable in the function
below:
/* returns the number of times this function has previously been called */
/* this can be used to generate unique numerical identifiers */
unsigned long long
ticketMachine(void)
{
static unsigned long long count = 0;
}
return count++;
To declare a local variable with static extent, use the static qualifier as in the
above example. To declare a global variable with static extent, declare it outside
a function. In both cases you should provide an initializer for the variable.
4.5.5.1.1
Additional qualifiers for global variables
It is possible to put some additional constraints on the visibility of global
variables. By default, a global variable will be visible everywhere, but functions
files other than the one in which it is defined won’t necessarily know what type
it has. This latter problem can be fixed using an extern declaration, which says
that there is a variable somewhere else of a particular type that we are declaring
(but not defining, so no space is allocated). In contrast, the static keyword (on
a global variable) specifies that it will only be visible in the current file, even if
some other file includes a declaration of a global variable of the same name.
Here are three variable declarations that illustrate how this works:
unsigned short Global = 5;
/* global variable, can be used anywhere */
extern float GlobalFloat;
/* this global variable, defined somewhere else, has type
static char Character = 'c';
/* global variable, can only be used by functions in this
(Note the convention of putting capital letters on global variables to distinguish
them from local variables.)
Typically, an extern definition would appear in a header file so that it can be
included in any function that uses the variable, while an ordinary global variable
definition would appear in a C file so it only occurs once.
4.5.6
Marking variables as constant
The const qualifier declares a variable to be constant:
114
const int three = 3;
/* this will always be free */
It is an error to apply any sort of assignment (=, +=,++, etc.) to a variable
qualified asconst‘.
4.5.6.1
Pointers to const
A pointer to a region that should not be modified should be declared with const
type:
const char *string = "You cannot modify this string.";
The const in the declaration above applies to the characters that string points
to: string is not const itself, but is instead a pointer to const. It is still
possible to make string point somewhere else, say by doing an assignment:
string = "You cannot modify this string either."
If you want to make it so that you can’t assign to string, put const right before
the variable name:
/* prevent assigning to string as well */
const char * const string = "You cannot modify this string.";
Now string is a const pointer to const: you can neither modify string nor
the values it points to.
Note that const only restricts what you can do using this particular variable
name. If you can get at the memory that something points to by some other
means, say through another pointer, you may be able to change the values in
these memory locations anyway:
int x = 5;
const int *p = &x;
int *q;
*p = 1; /* will cause an error at compile time */
x = 3; /* also changes *p, but will not cause an error */
4.6
Input and output
Input and output from C programs is typically done through theconst standard I/O library, whose functions etc. are declared in stdio.h. A detailed
descriptions of the functions in this library is given in Appendix B of Kernighan
and Ritchie. We’ll talk about some of the more useful functions and about how
input-output (I/O) works on Unix-like operating systems in general.
115
4.6.1
Character streams
The standard I/O library works on character streams, objects that act like
long sequences of incoming or outgoing characters. What a stream is connected
to is often not apparent to a program that uses it; an output stream might go to
a terminal, to a file, or even to another program (appearing there as an input
stream).
Three standard streams are available to all programs: these are stdin (standard
input), stdout (standard output), and stderr (standard error). Standard I/O
functions that do not take a stream as an argument will generally either read
from stdin or write to stdout. The stderr stream is used for error messages.
It is kept separate from stdout so that you can see these messages even if you
redirect output to a file:
$ ls no-such-file > /tmp/dummy-output
ls: no-such-file: No such file or directory
4.6.2
Reading and writing single characters
To read a single character from stdin, use getchar:
int c;
c = getchar();
The getchar routine will return the special value EOF (usually -1; short for end
of file) if there are no more characters to read, which can happen when you
hit the end of a file or when the user types the end-of-file key control-D to the
terminal. Note that the return value of getchar is declared to be an int since
EOF lies outside the normal character range.
To write a single character to stdout, use putchar:
putchar('!');
Even though putchar can only write single bytes, it takes an int as an argument.
Any value outside the range 0..255 will be truncated to its last byte, as in the
usual conversion from int to unsigned char.
Both getchar and putchar are wrappers for more general routines getc and
putc that allow you to specify which stream you are using. To illustrate getc
and putc, here’s how we might define getchar and putchar if they didn’t exist
already:
int
getchar2(void)
{
return getc(stdin);
116
}
int
putchar2(int c)
{
return putc(c, stdout);
}
Note that putc, putchar2 as defined above, and the original putchar all return
an int rather than void; this is so that they can signal whether the write
succeeded. If the write succeeded, putchar or putc will return the value written.
If the write failed (say because the disk was full), then putc or putchar will
return EOF.
Here’s another example of using putc to make a new function putcerr that
writes a character to stderr:
int
putcerr(int c)
{
return putc(c, stderr);
}
A rather odd feature of the C standard I/O library is that if you don’t like the
character you just got, you can put it back using the ungetc function. The
limitations on ungetc are that (a) you can only push one character back, and (b)
that character can’t be EOF. The ungetc function is provided because it makes
certain high-level input tasks easier; for example, if you want to parse a number
written as a sequence of digits, you need to be able to read characters until you
hit the first non-digit. But if the non-digit is going to be used elsewhere in your
program, you don’t want to eat it. The solution is to put it back using ungetc.
Here’s a function that uses ungetc to peek at the next character on stdin
without consuming it:
/* return the next character from stdin without consuming it */
int
peekchar(void)
{
int c;
c = getchar();
if(c != EOF) ungetc(c, stdin);
}
return c;
117
/* puts it back */
4.6.3
Formatted I/O
Reading and writing data one character at a time can be painful. The C standard
I/O library provides several convenient routines for reading and writing formatted
data. The most commonly used one is printf, which takes as arguments a
format string followed by zero or more values that are filled in to the format
string according to patterns appearing in it.
Here are some typical printf statements:
printf("Hello\n");
printf("%c", c);
printf("%d", n);
printf("%u", n);
printf("%o", n);
printf("%x", n);
printf("%f", x);
/*
/*
/*
/*
/*
/*
/*
print "Hello" followed by a newline */
equivalent to putchar(c) */
print n (an int) formatted in decimal */
print n (an unsigned int) formatted in decimal */
print n (an unsigned int) formatted in octal */
print n (an unsigned int) formatted in hexadecimal */
print x (a float or double) */
/* print total (an int) and average (a double) on two lines with labels */
printf("Total: %d\nAverage: %f\n", total, average);
For a full list of formatting codes see Table B-1 in Kernighan and Ritchie, or
run man 3 printf.
The inverse of printf is scanf. The scanf function reads formatted data from
stdin according to the format string passed as its first argument and stuffs the
results into variables whose addresses are given by the later arguments. This
requires prefixing each such argument with the & operator, which takes the
address of a variable.
Format strings for scanf are close enough to format strings for printf that you
can usually copy them over directly. However, because scanf arguments don’t go
through argument promotion (where all small integer types are converted to int
and floats are converted to double), you have to be much more careful about
specifying the type of the argument correctly. For example, while printf("%f",
x) will work whetherx is a float or a double, scanf("%f", &x) will work only
if x is a float, which means that scanf("%lf", &x) is needed if x is in fact a
double.
Some examples:
scanf("%c", &c);
scanf("%d", &n);
scanf("%u", &n);
scanf("%o", &n);
scanf("%x", &n);
scanf("%f", &x);
scanf("%lf", &x);
/*
/*
/*
/*
/*
/*
/*
118
like
read
read
read
read
read
read
c = getchar(); c must be a char; will NOT put EOF in
an int formatted in decimal */
an unsigned int formatted in decimal */
an unsigned int formatted in octal */
an unsigned int formatted in hexadecimal */
a float */
a double */
/* read total (an int) and average (a float) on two lines with labels */
/* (will also work if input is missing newlines or uses other whitespace, see below) */
scanf("Total: %d\nAverage: %f\n", &total, &average);
For a full list of formatting codes, run man 3 scanf.
The scanf routine usually eats whitespace (spaces, tabs, newlines, etc.) in its
input whenever it sees a conversion specification or a whitespace character in its
format string. The one exception is that a %c conversion specifier will not eat
whitespace and will instead return the next character whether it is whitespace
or not. Non-whitespace characters that are not part of conversion specifications
must match exactly. To detect if scanf parsed everything successfully, look at
its return value; it returns the number of values it filled in, or EOF if it hits
end-of-file before filling in any values.
The printf and scanf routines are wrappers for fprintf and fscanf, which
take a stream as their first argument, e.g.:
fprintf(stderr, "BUILDING ON FIRE, %d%% BURNT!!!\n", percentage);
This sends the output the the standard error output handle stderr. Note the
use of “%%” to print a single percent in the output.
4.6.4
Rolling your own I/O routines
Since we can write our own functions in C, if we don’t like what the standard
routines do, we can build our own on top of them. For example, here’s a function
that reads in integer values without leading minus signs and returns the result.
It uses the peekchar routine we defined above, as well as the isdigit routine
declared in ctype.h.
/* read an integer written in decimal notation from stdin until the first
* non-digit and return it. Returns 0 if there are no digits. */
int
readNumber(void)
{
int accumulator;
/* the number so far */
int c;
/* next character */
accumulator = 0;
while((c = peekchar()) != EOF && isdigit(c)) {
c = getchar();
/* consume it */
accumulator *= 10;
/* shift previous digits over */
accumulator += (c - '0');
/* add decimal value of new digit */
}
119
}
return accumulator;
Here’s another implementation that does almost the same thing:
int
readNumber2(void)
{
int n;
}
if(scanf("%u", &n) == 1) {
return n;
} else {
return 0;
}
The difference is that readNumber2 will consume any whitespace before the first
digit, which may or may not be what we want.
More complex routines can be used to parse more complex input. For example,
here’s a routine that uses readNumber to parse simple arithmetic expressions,
where each expression is either a number or of the form (expression+expression)
or (expression*expression). The return value is the value of the expression after
adding together or multiplying all of its subexpressions. (A complete program
including this routine and the others defined earlier that it uses can be found
examples/IO/calc.c.
#define EXPRESSION_ERROR (-1)
/* read an expression from stdin and return its value */
/* returns EXPRESSION_ERROR on error */
int
readExpression(void)
{
int e1;
/* value of first sub-expression */
int e2;
/* value of second sub-expression */
int c;
int op;
/* operation: '+' or '*' */
c = peekchar();
if(c == '(') {
c = getchar();
e1 = readExpression();
op = getchar();
e2 = readExpression();
120
c = getchar(); /* this had better be ')' */
if(c != ')') return EXPRESSION_ERROR;
}
/* else */
switch(op) {
case '*':
return e1*e2;
break;
case '+':
return e1+e2;
break;
default:
return EXPRESSION_ERROR;
break;
}
} else if(isdigit(c)) {
return readNumber();
} else {
return EXPRESSION_ERROR;
}
Because this routine calls itself recursively as it works its way down through
the input, it is an example of a recursive descent parser. Parsers for more
complicated languages like C are usually not written by hand like this, but are
instead constructed mechanically using a Parser generator.
4.6.5
File I/O
Reading and writing files is done by creating new streams attached to the files.
The function that does this is fopen. It takes two arguments: a filename, and a
flag that controls whether the file is opened for reading or writing. The return
value of fopen has type FILE * and can be used in putc, getc, fprintf, etc.
just like stdin, stdout, or stderr. When you are done using a stream, you
should close it using fclose.
Here’s a program that reads a list of numbers from a file whose name is given as
argv[1] and prints their sum:
#include <stdio.h>
#include <stdlib.h>
int
main(int argc, char **argv)
{
121
FILE *f;
int x;
int sum;
if(argc < 2) {
fprintf(stderr, "Usage: %s filename\n", argv[0]);
exit(1);
}
f = fopen(argv[1], "r");
if(f == 0) {
/* perror is a standard C library routine */
/* that prints a message about the last failed library routine */
/* prepended by its argument */
perror(filename);
exit(2);
}
/* else everything is ok */
sum = 0;
while(fscanf("%d", &x) == 1) {
sum += x;
}
printf("%d\n", sum);
/* not strictly necessary but it's polite */
fclose(f);
}
return 0;
examples/IO/sum.c
To write to a file, open it with fopen(filename, "w"). Note that as soon as
you call fopen with the "w" flag, any previous contents of the file are erased.
If you want to append to the end of an existing file, use "a" instead. You can
also add + onto the flag if you want to read and write the same file (this will
probably involve using fseek).
Some operating systems (Windows) make a distinction between text and binary
files. For text files, use the same arguments as above. For binary files, add a b,
e.g. fopen(filename, "wb") to write a binary file.
/* leave a greeting in the current directory */
#include <stdio.h>
#include <stdlib.h>
122
#define FILENAME "hello.txt"
#define MESSAGE "hello world"
int
main(int argc, char **argv)
{
FILE *f;
f = fopen(FILENAME, "w");
if(f == 0) {
perror(FILENAME);
exit(1);
}
/* unlike puts, fputs doesn't add a newline */
fputs(MESSAGE, f);
putc('\n', f);
fclose(f);
}
return 0;
examples/IO/helloFile.c
4.7
Statements and control structures
The bodies of C functions (including the main function) are made up of statements. These can either be simple statements that do not contain other
statements, or compound statements that have other statements inside them.
Control structures are compound statements like if/then/else, while, for, and
do..while that control how or whether their component statements are executed.
4.7.1
Simple statements
The simplest kind of statement in C is an expression (followed by a semicolon,
the terminator for all simple statements). Its value is computed and discarded.
Examples:
x = 2;
x = 2+3;
2+3;
puts("hi");
root2 = sqrt(2);
/*
/*
/*
/*
/*
an assignment statement */
another assignment statement */
has no effect---will be discarded by smart compilers */
a statement containing a function call */
an assignment statement with a function call */
123
Most statements in a typical C program are simple statements of this form.
Other examples of simple statements are the jump statements return, break,
continue, and goto. A return statement specifies the return value for a function
(if there is one), and when executed it causes the function to exit immediately.
The break and continue statements jump immediately to the end of a loop (or
switch; see below) or the next iteration of a loop; we’ll talk about these more
when we talk about loops. The goto statement jumps to another location in the
same function, and exists for the rare occasions when it is needed. Using it in
most circumstances is a sin.
4.7.2
Compound statements
Compound statements come in two varieties: conditionals and loops.
4.7.2.1
Conditionals
These are compound statements that test some condition and execute one or
another block depending on the outcome of the condition. The simplest is the
if statement:
if(houseIsOnFire) {
/* ouch! */
scream();
runAway();
}
The body of the if statement is executed only if the expression in parentheses
at the top evaluates to true (which in C means any value that is not 0).
The braces are not strictly required, and are used only to group one or more
statements into a single statement. If there is only one statement in the body,
the braces can be omitted:
if(programmerIsLazy) omitBraces();
This style is recommended only for very simple bodies. Omitting the braces
makes it harder to add more statements later without errors.
if(underAttack)
launchCounterAttack();
hideInBunker();
/* executed only when attacked */
/* ### DO NOT INDENT LIKE THIS ### executed always */
In the example above, the lack of braces means that the hideInBunker()
statement is not part of the if statement, despite the misleading indentation.
This sort of thing is why I generally always put in braces in an if.
An if statement may have an else clause, whose body is executed if the test is
false (i.e. equal to 0).
124
if(happy) {
smile();
} else {
frown();
}
A common idiom is to have a chain of if and else if branches that test several
conditions:
if(temperature < 0) {
puts("brrr");
} else if(temperature < 100) {
puts("hooray");
} else {
puts("ouch!");
}
This can be inefficient if there are a lot of cases, since the tests are applied
sequentially. For tests of the form <expression> == <small constant>, the
switch statement may provide a faster alternative. Here’s a typical switch
statement:
/* print plural of cow, maybe using the obsolete dual number construction */
switch(numberOfCows) {
case 1:
puts("cow");
break;
case 2:
puts("cowen");
break;
default:
puts("cows");
break;
}
This prints the string “cow” if there is one cow, “cowen” if there are two cowen,
and “cows” if there are any other number of cows. The switch statement
evaluates its argument and jumps to the matching case label, or to the default
label if none of the cases match. Cases must be constant integer values.
The break statements inside the block jump to the end of the block. Without
them, executing the switch with numberOfCows equal to 1 would print all three
lines. This can be useful in some circumstances where the same code should be
used for more than one case:
switch(c) {
case 'a':
case 'e':
case 'i':
125
case 'o':
case 'u':
type = VOWEL;
break;
default:
type = CONSONANT;
break;
}
or when a case “falls through” to the next:
switch(countdownStart) {
case 3:
puts("3");
case 2:
puts("2");
case 1:
puts("1")
case 0:
puts("KABLOOIE!");
break;
default:
puts("I can't count that high!");
break;
}
Note that it is customary to include a break on the last case even though it has
no effect; this avoids problems later if a new case is added. It is also customary
to include a default case even if the other cases supposedly exhaust all the
possible values, as a check against bad or unanticipated inputs.
switch(oliveSize) {
case JUMBO:
eatOlives(SLOWLY);
break;
case COLLOSSAL:
eatOlives(QUICKLY);
break;
case SUPER_COLLOSSAL:
eatOlives(ABSURDLY);
break;
default:
/* unknown size! */
abort();
break;
}
Though switch statements are better than deeply nested if/else-if constructions,
126
it is often even better to organize the different cases as data rather than code.
We’ll see examples of this when we talk about function pointers.
Nothing in the C standards prevents the case labels from being buried inside
other compound statements. One rather hideous application of this fact is Duff’s
device.
4.7.2.2
Loops
There are three kinds of loops in C.
4.7.2.2.1
The while loop
A while loop tests if a condition is true, and if so, executes its body. It then
tests the condition is true again, and keeps executing the body as long as it is.
Here’s a program that deletes every occurence of the letter e from its input.
#include <stdio.h>
int
main(int argc, char **argv)
{
int c;
while((c = getchar()) != EOF) {
switch(c) {
case 'e':
case 'E':
break;
default:
putchar(c);
break;
}
}
}
return 0;
Note that the expression inside the while argument both assigns the return
value of getchar to c and tests to see if it is equal to EOF (which is returned
when no more input characters are available). This is a very common idiom in C
programs. Note also that even though c holds a single character, it is declared
as an int. The reason is that EOF (a constant defined in stdio.h) is outside the
normal character range, and if you assign it to a variable of type char it will be
quietly truncated into something else. Because C doesn’t provide any sort of
exception mechanism for signalling unusual outcomes of function calls, designers
of library functions often have to resort to extending the output of a function to
127
include an extra value or two to signal failure; we’ll see this a lot when the null
pointer shows up in the chapter on pointers.
4.7.2.2.2
The do..while loop
The do..while statement is like the while statement except the test is done at
the end of the loop instead of the beginning. This means that the body of the
loop is always executed at least once.
Here’s a loop that does a random walk until it gets back to 0 (if ever). If we
changed the do..while loop to a while loop, it would never take the first step,
because pos starts at 0.
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
int
main(int argc, char **argv)
{
int pos = 0;
/* position of random walk */
srandom(time(0));
/* initialize random number generator */
do {
pos += random() & 0x1 ? +1 : -1;
printf("%d\n", pos);
} while(pos != 0);
}
return 0;
examples/statements/randomWalk.c
The do..while loop is used much less often in practice than the while loop.
Note that it is always possible to convert a do..while loop to a while loop by
making an extra copy of the body in front of the loop.
4.7.2.2.3
The for loop
The for loop is a form of syntactic sugar that is used when a loop iterates over
a sequence of values stored in some variable (or variables). Its argument consists
of three expressions: the first initializes the variable and is called once when the
statement is first reached. The second is the test to see if the body of the loop
should be executed; it has the same function as the test in a while loop. The
third sets the variable to its next value. Some examples:
128
/* count from 0 to 9 */
for(i = 0; i < 10; i++) {
printf("%d\n", i);
}
/* and back from 10 to 0 */
for(i = 10; i >= 0; i--) {
printf("%d\n", i);
}
/* this loop uses some functions to move around */
for(c = firstCustomer(); c != END_OF_CUSTOMERS; c = customerAfter(c)) {
helpCustomer(c);
}
/* this loop prints powers of 2 that are less than n*/
for(i = 1; i < n; i *= 2) {
printf("%d\n", i);
}
/* this loop does the same thing with two variables by using the comma operator */
for(i = 0, power = 1; power < n; i++, power *= 2) {
printf("2^%d = %d\n", i, power);
}
/* Here are some nested loops that print a times table */
for(i = 0; i < n; i++) {
for(j = 0; j < n; j++) {
printf("%d*%d=%d ", i, j, i*j);
}
putchar('\n');
}
A for loop can always be rewritten as a while loop.
for(i = 0; i < 10; i++) {
printf("%d\n", i);
}
/* is exactly the same as */
i = 0;
while(i < 10) {
printf("%d\n", i);
i++;
}
129
4.7.2.2.4
Loops with break, continue, and goto
The break statement immediately exits the innermmost enclosing loop or switch
statement.
for(i = 0; i < n; i++) {
openDoorNumber(i);
if(boobyTrapped()) {
break;
}
}
The continue statement skips to the next iteration. Here is a program with a
loop that iterates through all the integers from -10 through 10, skipping 0:
#include <stdio.h>
/* print a table of inverses */
#define MAXN (10)
int
main(int argc, char **argv)
{
int n;
for(n = -MAXN; n <= MAXN; n++) {
if(n == 0) continue;
printf("1.0/%3d = %+f\n", n, 1.0/n);
}
}
return 0;
examples/statements/inverses.c
Occasionally, one would like to break out of more than one nested loop. The
way to do this is with a goto statement.
for(i = 0; i < n; i++) {
for(j = 0; j < n; j++) {
doSomethingTimeConsumingWith(i, j);
if(checkWatch() == OUT_OF_TIME) {
goto giveUp;
}
}
}
giveUp:
puts("done");
130
The target for the goto is a label, which is just an identifier followed by a colon
and a statement (the empty statement ; is ok).
The goto statement can be used to jump anywhere within the same function
body, but breaking out of nested loops is widely considered to be its only
genuinely acceptable use in normal code.
4.7.2.3
Choosing where to put a loop exit
Choosing where to put a loop exit is usually pretty obvious: you want it after
any code that you want to execute at least once, and before any code that you
want to execute only if the termination test fails.
If you know in advance what values you are going to be iterating over, you will
most likely be using a for loop:
for(i = 0; i < n; i++) {
a[i] = 0;
}
Most of the rest of the time, you will want a while loop:
while(!done()) {
doSomething();
}
The do..while loop comes up mostly when you want to try something, then try
again if it failed:
do {
result = fetchWebPage(url);
} while(result == 0);
Finally, leaving a loop in the middle using break can be handy if you have
something extra to do before trying again:
for(;;) {
result = fetchWebPage(url);
if(result != 0) {
break;
}
/* else */
fprintf(stderr, "fetchWebPage failed with error code %03d\n", result);
sleep(retryDelay); /* wait before trying again */
}
(Note the empty for loop header means to loop forever; while(1) also works.)
131
4.8
Functions
A function, procedure, or subroutine encapsulates some complex computation as a single operation. Typically, when we call a function, we pass as
arguments all the information this function needs, and any effect it has will
be reflected in either its return value or (in some cases) in changes to values
pointed to by the arguments. Inside the function, the arguments are copied into
local variables, which can be used just like any other local variable—they can
even be assigned to without affecting the original argument.
4.8.1
Function definitions
A typical function definition looks like this:
/* Returns the square of the distance between two points separated by
dx in the x direction and dy in the y direction. */
int
distSquared(int dx, int dy)
{
return dx*dx + dy*dy;
}
examples/functions/distSquaredNoHeader.c
The part outside the braces is called the function declaration; the braces and
their contents is the function body.
Like most complex declarations in C, once you delete the type names the
declaration looks like how the function is used: the name of the function comes
before the parentheses and the arguments inside. The ints scattered about
specify the type of the return value of the function (before the function name)
and of the parameters (inside the parentheses after the function name); these are
used by the compiler to determine how to pass values in and out of the function
and (usually for more complex types, since numerical types will often convert
automatically) to detect type mismatches.
If you want to define a function that doesn’t return anything, declare its return
type as void. You should also declare a parameter list of void if the function
takes no arguments.
/* Prints "hi" to stdout */
void
helloWorld(void)
{
puts("hi");
}
examples/functions/helloWorld.c
132
It is not strictly speaking an error to omit the second void here. Putting void
in for the parameters tells the compiler to enforce that no arguments are passed
in. If we had instead declared helloWorld as
/* Prints "hi" to stdout */
void
helloWorld()
/* DANGER! */
{
puts("hi");
}
it would be possible to call it as
helloWorld("this is a bogus argument");
without causing an error. The reason is that a function declaration with no
arguments means that the function can take an unspecified number of arguments,
and it’s up to the user to make sure they pass in the right ones. There are good
historical reasons for what may seem like obvious lack of sense in the design
of the language here, and fixing this bug would break most C code written
before 1989. But you shouldn’t ever write a function declaration with an empty
argument list, since you want the compiler to know when something goes wrong.
4.8.2
When to write a function
As with any kind of abstraction, there are two goals to making a function:
• Encapsulation: If you have some task to carry out that is simple do describe from the outside but messy to understand from the inside, wrapping
it in a function lets somebody carry out this task without having to know
the details. This is also useful if you want to change the implementation
later.
• Code re-use: If you find yourself writing the same lines of code in several
places (or worse, are tempted to copy a block of code to several places),
you should probably put this code in a function (or perhaps more than one
function, if there is no succinct way to describe what this block of code is
doing).
Both of these goals may be trumped by the goal of making your code understandable. If you can’t describe what a function is doing in a single, simple
sentence, this is a sign that maybe you need to restructure your code. Having a
function that does more than one thing (or does different thing depending on
its arguments) is likely to lead to confusion. So, for example, this is not a good
function definition:
/*** ### UGLY CODE AHEAD ### ***/
/*
133
* If getMaximum is true, return maximum of x and y,
* else return minimum.
*/
int
computeMaximumOrMinimum(int x, int y, int getMaximum)
{
if(x > y) {
if(getMaximum) {
return x;
} else {
return y;
}
} else {
if(getMaximum) {
return y;
} else {
return x;
}
}
}
Better would be to write two functions:
/* return the maximum of x and y */
int
maximum(int x, int y)
{
if(x > y) {
return x;
} else {
return y;
}
}
/* return the minimum of x and y */
int
minimum(int x, int y)
{
if(x < y) {
return x;
} else {
return y;
}
}
At the same time, it’s possible for a function to be too simple. Suppose I write
the function
134
/* print x to stdout followed by a newline */
void
printIntWithNewline(int x)
{
printf("%d\n", x);
}
It’s pretty clear from the name what this function does. But since anybody
who has been using C for a while has seen printf("%d\n", ...) over and over
again, it’s usually more clear to expand out the definition:
printIntWithNewline(2+5);
printf("%d\n", 2+7);
/* this could do anything */
/* this does exactly what it says */
As with all caveats, this caveat comes with its own caveat: what might justify
a function like this is if you want to be able to do some kind of specialized
formatting that should be consistent for all values of a particular form. So you
might write a printDistance function like the above as a stub for a fancier
function that might use different units at different scales or something.
A similar issue will come up with non-syntactic macros, which also tend to fail
the “does this make my code more or less understandable” test. Usually it is a
bad idea to try to replace common C idioms.
4.8.3
Calling a function
A function call consists of the function followed by its arguments (if any) inside
parentheses, separated by comments. For a function with no arguments, call it
with nothing between the parentheses. A function call that returns a value can
be used in an expression just like a variable. A call to a void function can only
be used as an expression by itself:
totalDistance += distSquared(x1 - x2, y1 - y2);
helloWorld();
greetings += helloWorld(); /* ERROR */
4.8.4
The return statement
To return a value from a function, write a return statement, e.g.
return 172;
The argument to return can be any expression. Unlike the expression in, say,
an if statement, you do not need to wrap it in parentheses. If a function is
declared void, you can do a return with no expression, or just let control reach
the end of the function.
135
Executing a return statement immediately terminates the function. This can
be used like break to get out of loops early.
/* returns 1 if n is prime, 0 otherwise */
int
isPrime(int n)
{
int i;
if (n < 2) return 0;
/* special case for 0, 1, negative n */
for(i = 2; i < n; i++) {
if (n % i == 0) {
/* found a factor */
return 0;
}
}
}
/* no factors */
return 1;
examples/functions/isPrime.c
4.8.5
Function declarations and modules
By default, functions have global scope: they can be used anywhere in your
program, even in other files. If a file doesn’t contain a declaration for a function
someFunc before it is used, the compiler will assume that it is declared like
int someFunc() (i.e., return type int and unknown arguments). This can
produce infuriating complaints later when the compiler hits the real declaration
and insists that your function someFunc should be returning an int and you are
a bonehead for declaring it otherwise.
The solution to such insulting compiler behavior errors is to either (a) move the
function declaration before any functions that use it; or (b) put in a declaration
without a body before any functions that use it, in addition to the declaration
that appears in the function definition. (Note that this violates the no separate
but equal rule, but the compiler should tell you when you make a mistake.)
Option (b) is generally preferred, and is the only option when the function is
used in a different file.
To make sure that all declarations of a function are consistent, the usual practice
is to put them in an include file. For example, if distSquared is used in a lot
of places, we might put it in its own file distSquared.c:
#include "distSquared.h"
136
int
distSquared(int dx, int dy)
{
return dx*dx + dy*dy;
}
examples/functions/distSquared.c
The file distSquared.c above uses #include to include a copy of the following
header file distSquared.h:
/* Returns the square of the distance between two points separated by
dx in the x direction and dy in the y direction. */
int distSquared(int dx, int dy);
examples/functions/distSquared.h
Note that the declaration in distSquared.h doesn’t have a body. Instead, it’s
terminated by a semicolon, like a variable declaration. It’s also worth noting
that we moved the documenting comment to distSquared.h: the idea is that
distSquared.h is the public face of this (very small one-function) module, and
so the explanation of how to use the function should be there.
The reason distSquared.c includes distSquared.h is to get the compiler to
verify that the declarations in the two files match. But to use the distSquared
function, we also put #include "distSquared.h" at the top of the file that
uses it:
#include "distSquared.h"
#define THRESHOLD (100)
int
tooClose(int x1, int y1, int x2, int y2)
{
return distSquared(x1 - x2, y1 - y2) < THRESHOLD;
}
examples/functions/tooClose.c
The #include on line 1 uses double quotes instead of angle brackets; this tells
the compiler to look for distSquared.h in the current directory instead of the
system include directory (typically /usr/include).
4.8.6
Static functions
By default, all functions are global; they can be used in any file of your program
whether or not a declaration appears in a header file. To restrict access to the
137
current file, declare a function static, like this:
static void
helloHelper(void)
{
puts("hi!");
}
void
hello(int repetitions)
{
int i;
}
for(i = 0; i < repetitions; i++) {
helloHelper();
}
examples/functions/staticHello.c
The function hello will be visible everywhere. The function helloHelper will
only be visible in the current file.
It’s generally good practice to declare a function static unless you intend to
make it available, since not doing so can cause namespace conflicts, where
the presence of two functions with the same name either prevent the program
from linking or—even worse—cause the wrong function to be called. The latter
can happen with library functions, since C allows the programmer to override
library functions by defining a new function with the same name. Early on
in my career as a C programmer, I once had a program fail in a spectacularly
incomprehensible way because I’d written a select function without realizing
that select is a core library function in Unix.
4.8.7
Local variables
A function may contain definitions of local variables, which are visible only
inside the function and which survive only until the function returns. These
may be declared at the start of any block (group of statements enclosed by
braces), but it is conventional to declare all of them at the outermost block of
the function.
/* Given n, compute n! = 1*2*...*n */
/* Warning: will overflow on 32-bit machines if n > 12 */
int
factorial(int n)
{
int i;
138
int product;
if(n < 2) return n;
/* else */
product = 1;
for(i = 2; i <= n; i++) {
product *= i;
}
}
return product;
examples/functions/factorial.c
4.8.8
Mechanics of function calls
Several things happen under the hood when a function is called. Since a function
can be called from several different places, the CPU needs to store its previous
state to know where to go back. It also needs to allocate space for function
arguments and local variables.
Some of this information will be stored in registers, memory locations built
into the CPU itself, but most will go on the stack, a region of memory that on
typical machines grows downward, even though the most recent additions to the
stack are called the “top” of the stack. The location of the top of the stack is
stored in the CPU in a special register called the stack pointer.
So a typical function call looks like this internally:
1. The current instruction pointer or program counter value, which
gives the address of the next line of machine code to be executed, is pushed
onto the stack.
2. Any arguments to the function are copied either into specially designated
registers or onto new locations on the stack. The exact rules for how to do
this vary from one CPU architecture to the next, but a typical convention
might be that the first few arguments are copied into registers and the rest
(if any) go on the stack.
3. The instruction pointer is set to the first instruction in the code for the
function.
4. The code for the function allocates additional space on the stack to hold
its local variables (if any) and to save copies of the values of any registers
it wants to use (so that it can restore their contents before returning to its
caller).
5. The function body is executed until it hits a return statement.
139
6. Returning from the function is the reverse of invoking it: any saved registers
are restored from the stack, the return value is copied to a standard register,
and the values of the instruction pointer and stack pointer are restored to
what they were before the function call.
From the programmer’s perspective, the important point is that both the arguments and the local variables inside a function are stored in freshly-allocated
locations that are thrown away after the function exits. So after a function call
the state of the CPU is restored to its previous state, except for the return value.
Any arguments that are passed to a function are passed as copies, so changing
the values of the function arguments inside the function has no effect on the
caller. Any information stored in local variables is lost.
Under very rare circumstances, it may be useful to have a variable local to a
function that persists from one function call to the next. You can do so by
declaring the variable static. For example, here is a function that counts how
many times it has been called:
/* return the number of times the function has been called */
int
counter(void)
{
static count = 0;
}
return ++count;
examples/functions/staticCounter.c
Static local variables are stored outside the stack with global variables, and have
unbounded extent. But they are only visible inside the function that declares
them. This makes them slightly less dangerous than global variables—there is no
fear that some foolish bit of code elsewhere will quietly change their value—but
it is still the case that they usually aren’t what you want. It is also likely that
operations on static variables will be slightly slower than operations on ordinary
(“automatic”) variables, since making them persistent means that they have to
be stored in (slow) main memory instead of (fast) registers.
4.9
4.9.1
Pointers
Memory and addresses
Memory in a typical modern computer is divided into two classes: a small number
of registers, which live on the CPU chip and perform specialized functions like
keeping track of the location of the next machine code instruction to execute
or the current stack frame, and main memory, which (mostly) lives outside
the CPU chip and which stores the code and data of a running program. When
140
the CPU wants to fetch a value from a particular location in main memory, it
must supply an address: a 32-bit or 64-bit unsigned integer on typical current
architectures, referring to one of up to 232 or 264 distinct 8-bit locations in the
memory. These integers can be manipulated like any other integer; in C, they
appear as pointers, a family of types that can be passed as arguments, stored
in variables, returned from functions, etc.
4.9.2
Pointer variables
A pointer variable is a variable that holds a pointer, just like an int variable
is a variable that holds an int.
4.9.2.1
Declaring a pointer variable
The convention is C is that the declaration of a complex type looks like its use.
To declare a pointer-valued variable, write a declaration for the thing that it
points to, but include a * before the variable name:
int *pointerToInt;
double *pointerToDouble;
char *pointerToChar;
char **pointerToPointerToChar;
These declarations create four pointer variables, named pointerToInt,
pointerToDouble, pointerToChar, and pointerToPointerToChar. On a
typical 64-bit machine, each will be allocated 8 bytes, enough to represent an
address in memory.
The contents of these variables are initially arbitrary: to use them, you will need
to compute the address of something and assign it to the variable.
4.9.2.2
Assigning to pointer variables
Declaring a pointer-valued variable allocates space to hold the pointer but not
to hold anything it points to. Like any other variable in C, a pointer-valued
variable will initially contain garbage—in this case, the address of a location
that might or might not contain something important. To initialize a pointer
variable, you have to assign to it the address of something that already exists.
Typically this is done using the & (address-of ) operator:
int n;
int *p;
/* an int variable */
/* a pointer to an int */
p = &n;
/* p now points to n */
141
4.9.2.3
Using a pointer
Pointer variables can be used in two ways. The simplest way is to get their value
as with any other variable. This value will be an address, which can be stored
in another pointer variable of the same type.
int n;
int *p;
int *q;
/* an int variable */
/* a pointer to an int */
/* another pointer to an int */
p = &n;
q = p;
/* p now points to n */
/* q now points to n as well */
But more often you will want to work on the value stored at the location pointed
to. You can do this by using the * (dereference) operator, which acts as an
inverse of the address-of operator:
int n;
int *p;
/* an int variable */
/* a pointer to an int */
p = &n;
/* p now points to n */
*p = 2;
*p = *p + *p;
/* sets n to 2 */
/* sets n to 4 */
The * operator binds very tightly, so you can usually use *p anywhere you could
use the variable it points to without worrying about parentheses. However, a
few operators, such as the -- and ++ operators and the . operator used to
unpack structs, bind tighter. These require parentheses if you want the * to take
precedence.
(*p)++;
*p++;
4.9.2.4
/* increment the value pointed to by p */
/* WARNING: increments p itself */
Printing pointers
You can print a pointer value using printf with the %p format specifier. To do
so, you should convert the pointer to type void * first using a cast (see below for
void * pointers), although on machines that don’t have different representations
for different pointer types, this may not be necessary.
Here is a short program that prints out some pointer values:
#include <stdio.h>
#include <stdlib.h>
int G = 0;
/* a global variable, stored in BSS segment */
int
142
main(int argc, char **argv)
{
static int s; /* static local variable, stored in BSS segment */
int a;
/* automatic variable, stored on stack */
int *p;
/* pointer variable for malloc below */
/* obtain a block big enough for one int from the heap */
p = malloc(sizeof(int));
printf("&G
printf("&s
printf("&a
printf("&p
printf("p
printf("main
=
=
=
=
=
=
%p\n",
%p\n",
%p\n",
%p\n",
%p\n",
%p\n",
(void
(void
(void
(void
(void
(void
*)
*)
*)
*)
*)
*)
&G);
&s);
&a);
&p);
p);
main);
free(p);
}
return 0;
examples/pointers/lookingAtPointers.c
When I run this on a Mac OS X 10.6 machine after compiling with gcc, the
output is:
&G
&s
&a
&p
p
main
=
=
=
=
=
=
0x100001078
0x10000107c
0x7fff5fbff2bc
0x7fff5fbff2b0
0x100100080
0x100000e18
The interesting thing here is that we can see how the compiler chooses to allocate
space for variables based on their storage classes. The global variable G and the
static local variable s both persist between function calls, so they get placed in
the BSS segment (see .bss) that starts somewhere around 0x100000000, typically
after the code segment containing the actual code of the program. Local variables
a and p are allocated on the stack, which grows down from somewhere near the
top of the address space. The block returned from malloc that p points to is
allocated off the heap, a region of memory that may also grow over time and
starts after the BSS segment. Finally, main appears at 0x100000e18; this is in
the code segment, which is a bit lower in memory than all the global variables.
143
4.9.3
The null pointer
The special value 0, known as the null pointer, may be assigned to a pointer
of any type. It may or may not be represented by the actual address 0, but
it will act like 0 in all contexts (e.g., it has the value false in an if or while
statement). Null pointers are often used to indicate missing data or failed
functions. Attempting to dereference a null pointer can have catastrophic effects,
so it’s important to be aware of when you might be supplied with one.
4.9.4
Pointers and functions
A simple application of pointers is to get around C’s limit on having only one
return value from a function. Because C arguments are copied, assigning a value
to an argument inside a function has no effect on the outside. So the doubler
function below doesn’t do much:
#include <stdio.h>
/* doesn't work */
void
doubler(int x)
{
x *= 2;
}
int
main(int argc, char **argv)
{
int y;
y = 1;
}
doubler(y);
/* no effect on y */
printf("%d\n", y);
/* prints 1 */
return 0;
examples/pointers/badDoubler.c
However, if instead of passing the value of y into doubler we pass a pointer to
y, then the doubler function can reach out of its own stack frame to manipulate
y itself:
#include <stdio.h>
144
void
doubler(int *x)
{
*x *= 2;
}
int
main(int argc, char **argv)
{
int y;
y = 1;
}
doubler(&y);
/* sets y to 2 */
printf("%d\n", y);
/* prints 2 */
return 0;
examples/pointers/goodDoubler.c
Generally, if you pass the value of a variable into a function (with no &), you
can be assured that the function can’t modify your original variable. When you
pass a pointer, you should assume that the function can and will change the
variable’s value. If you want to write a function that takes a pointer argument
but promises not to modify the target of the pointer, use const, like this:
void
printPointerTarget(const int *p)
{
printf("%d\n", *p);
}
The const qualifier tells the compiler that the target of the pointer shouldn’t be
modified. This will cause it to return an error if you try to assign to it anyway:
void
printPointerTarget(const int *p)
{
*p = 5; /* produces compile-time error */
printf("%d\n", *p);
}
Passing const pointers is mostly used when passing large structures to functions,
where copying a 32-bit pointer is cheaper than copying the thing it points to.
If you really want to modify the target anyway, C lets you “cast away const”:
void
145
printPointerTarget(const int *p)
{
*((int *) p) = 5; /* no compile-time error */
printf("%d\n", *p);
}
There is usually no good reason to do this. The one exception might be if the
target of the pointer represents an abstract data type, and you want to modify
its representation during some operation to optimize things somehow in a way
that will not be visible outside the abstraction barrier, making it appear to leave
the target constant.
Note that while it is safe to pass pointers down into functions, it is very dangerous
to pass pointers up. The reason is that the space used to hold any local variable
of the function will be reclaimed when the function exits, but the pointer will
still point to the same location, even though something else may now be stored
there. So this function is very dangerous:
int *
dangerous(void)
{
int n;
}
return &n;
/* NO! */
*dangerous() = 12;
/* writes 12 to some unknown location */
...
An exception is when you can guarantee that the location pointed to will survive
even after the function exits, e.g. when the location is dynamically allocated
using malloc (see below) or when the local variable is declared static:
int *
returnStatic(void)
{
static int n;
}
return &n;
...
*returnStatic() = 12;
/* writes 12 to the hidden static variable */
Usually returning a pointer to a static local variable is not good practice, since
the point of making a variable local is to keep outsiders from getting at it. If
you find yourself tempted to do this, a better approach is to allocate a new block
146
using malloc (see below) and return a pointer to that. The downside of the
malloc method is that the caller has to promise to call free on the block later,
or you will get a storage leak.
4.9.5
Pointer arithmetic and arrays
Because pointers are just numerical values, one can do arithmetic on them.
Specifically, it is permitted to
• Add an integer to a pointer or subtract an integer from a pointer. The
effect of p+n where p is a pointer and n is an integer is to compute the
address equal to p plus n times the size of whatever p points to (this is
why int * pointers and char * pointers aren’t the same).
• Subtract one pointer from another. The two pointers must have the same
type (e.g. both int * or both char *). The result is a signed integer value
of type ptrdiff_t, equal to the numerical difference between the addresses
divided by the size of the objects pointed to.
• Compare two pointers using ==, !=, <, >, <=, or >=.
• Increment or decrement a pointer using ++ or --.
4.9.5.1
Arrays
The main application of pointer arithmetic in C is in arrays. An array is a
block of memory that holds one or more objects of a given type. It is declared
by giving the type of object the array holds followed by the array name and the
size in square brackets:
int a[50];
char *cp[100];
/* array of 50 ints */
/* array of 100 pointers to char */
Declaring an array allocates enough space to hold the specified number of objects
(e.g. 200 bytes for a above and 400 for cp—note that a char * is an address, so
it is much bigger than a char). The number inside the square brackets must be
a constant whose value can be determined at compile time.
The array name acts like a constant pointer to the zeroth element of the array.
It is thus possible to set or read the zeroth element using *a. But because the
array name is constant, you can’t assign to it:
1
2
3
*a = 12;
/* sets zeroth element to 12 */
a = &n;
/* #### DOESN'T WORK #### */
More common is to use square brackets to refer to a particular element of the
array. The expression a[n] is defined to be equivalent to *(a+n); the index n
(an integer) is added to the base of the array (a pointer), to get to the location
of the n-th element of a. The implicit * then dereferences this location so that
you can read its value (in a normal expression) or assign to it (on the left-hand
147
side of an assignment operator). The effect is to allow you to use a[n] just as
you would any other variable of type int (or whatever type a was declared as).
Note that C doesn’t do any sort of bounds checking. Given the declaration
int a[50];, only indices from a[0] to a[49] can be used safely. However, the
compiler will not blink at a[-12] or a[10000]. If you read from such a location
you will get garbage data; if you write to it, you will overwrite god-knows-what,
possibly trashing some other variable somewhere else in your program or some
critical part of the stack (like the location to jump to when you return from
a function). It is up to you as a programmer to avoid such buffer overruns,
which can lead to very mysterious (and in the case of code that gets input from
a network, security-damaging) bugs. The valgrind program can help detect such
overruns in some cases.
Another curious feature of the definition of a[n] as identical to *(a+n) is that
it doesn’t actually matter which of the array name or the index goes inside the
braces. So all of a[0], *a, and 0[a] refer to the zeroth entry in a. Unless you
are deliberately trying to obfuscate your code, it’s best to write what you mean.
4.9.5.2
Arrays and functions
Because array names act like pointers, they can be passed into functions that
expect pointers as their arguments. For example, here is a function that computes
the sum of all the values in an array a of size n:
/* compute the sum of the first n elements of array a */
int
sumArray(int n, const int *a)
{
int i;
int sum;
sum = 0;
for(i = 0; i < n; i++) {
sum += a[i];
}
}
return sum;
examples/pointers/sumArray.c
Note the use of const to promise that sumArray won’t modify the contents of a.
Another way to write the function header is to declare a as an array of unknown
size:
/* return the sum of the values in a, an array of size n */
int
148
sumArray(int n, const int a[])
{
...
}
This has exactly the same meaning to the compiler as the previous definition.
Even though normally the declarations int a[10] and int *a mean very different things (the first one allocates space to hold 10 ints, and prevents assigning
a new value to a), in a function argument int a[] is just syntactic sugar for
int *a. You can even modify what a points to inside sumArray by assigning to
it. This will allow you to do things that you usually don’t want to do, like write
this hideous routine:
/* return the sum of the first n values in a */
int
sumArray(int n, const int a[])
{
const int *an;
/* pointer to first element not in a */
int sum;
sum = 0;
an = a+n;
while(a < an) {
sum += *a++;
}
}
return sum;
4.9.5.3
Multidimensional arrays
Arrays can themselves be members of arrays. The result is a multidimensional
array, where a value in row i and column j is accessed by a[i][j].
Declaration is similar to one-dimensional arrays:
int a[6][4];
/* declares an array of 6 rows of 4 ints each */
This declaration produces an array of 24 int values, packed contiguously in
memory. The interpretation is that a is an array of 6 objects, each of which is
an array of 4 ints.
If we imagine the array to contain increasing values like this:
0 1 2 3 4 5
6 7 8 9 10 11
12 13 14 15 16 17
149
the actual positions in memory will look like this:
0 1
^
a[0]
2
3
4
5
6 7
^
a[1]
8
9 10 11 12 13 14 15 16 17
^
a[2]
To look up a value, we do the usual array-indexing magic. Suppose we want
to find a[1][4]. The name a acts as a pointer to the base of the array.The
name a[1] says to skip ahead 1 times the size of the things pointed to by a,
which are arrays of 6 ints each, for a total size of 24 bytes assuming 4-byte
ints. For a[1][4], we start at a[1] and move forward 4 times the size of the
thing pointed to by a[1], which is an int; this puts us 24+16 bytes from a, the
position of 10 in the picture above.
Like other array declarations, the size must be specified at compile time in
pre-C99 C. If this is not desirable, a similar effect can be obtained by allocating
each row separately using malloc and building a master list of pointers to
rows, of type int **. The downside of this approach is that the array is no
longer contiguous (which may affect cache performance) and it requires reading
a pointer to find the location of a particular value, instead of just doing address
arithmetic starting from the base address of the array. But elements can still be
accessed using the a[i][j] syntax. An example of this approach is given below:
/* Demo program for malloc'd two-dimensional arrays */
#include <stdio.h>
#include <stdlib.h>
/* frees a 2d array created by malloc2d */
void
free2d(void **a)
{
void **row;
/* first free rows */
for(row = a; *row != 0; row++) {
free(*row);
}
}
/* then free array of rows */
free(a);
/* returns a two-dimensional array with numRows rows and
* rowSize bytes per row, or 0 on allocation failure.
* The caller is responsible for freeing the result with free2d. */
void **
malloc2d(size_t numRows, size_t rowSize)
150
{
void **a;
size_t i;
/* a is an array of void * pointers that point to the rows */
/* The last element is 0, so free2d can detect the last row */
a = malloc(sizeof(void *) * (numRows + 1));
/* one extra for sentinel */
if(a == 0) {
/* malloc failed */
return 0;
}
/* now allocate the actual rows */
for(i = 0; i < numRows; i++) {
a[i] = malloc(rowSize);
if(a[i] == 0) {
/* note that 0 in a[i] will stop freed2d after it frees previous rows */
free2d(a);
return 0;
}
}
/* initialize the sentinel value */
a[numRows] = 0;
}
return a;
int
main(int argc, char **argv)
{
int rows;
int cols;
int **a;
int i;
int j;
if(argc != 3) {
fprintf(stderr, "Usage: %s rows cols\n", argv[0]);
return 1;
}
/* else */
rows = atoi(argv[1]);
cols = atoi(argv[2]);
151
/* note that void ** is not converted automatically,
* so we need an explicit cast */
a = (int **) malloc2d(rows, cols * sizeof(int));
if(a == 0) {
fprintf(stderr, "malloc2d failed, exiting\n");
return 2;
}
for(i = 0; i < rows; i++) {
for(j = 0; j < cols; j++) {
a[i][j] = i - j;
}
}
for(i = 0; i < rows; i++) {
for(j = 0; j < cols; j++) {
printf("%4d", a[i][j]);
}
putchar('\n');
}
free2d((void **) a);
}
/* always clean up */
return 0;
examples/pointers/malloc2d.c
4.9.5.4
Variable-length arrays
C99 adds the feature of variable-length arrays, where the size of the array is
determined at run-time. These can only appear as local variables in procedures
(automatic variables) or in argument lists. In the case of variable-length arrays
in argument lists, it is also necessary that the length of the array be computable
from previous arguments.
For example, we could make the length of the array explicit in our sumArray
function:
/* return the sum of the values in a, an array of size n */
int
sumArray(int n, const int a[n])
{
int i;
int sum;
sum = 0;
152
for(i = 0; i < n; i++) {
sum += a[i];
}
}
return sum;
This doesn’t accomplish much, because the length of the array is not used.
However, it does become useful if we have a two-dimensional array, as otherwise
there is no way to compute the length of each row:
int
sumMatrix(int rows, int cols, const int m[rows][cols])
{
int i;
int j;
int sum;
sum = 0;
for(i = 0; i < rows; i++) {
for(j = 0; j < cols; j++) {
sum += a[i][j];
}
}
}
return sum;
Here the fact that each row of m is known to be an array of cols many ints makes
the implicit pointer computation in a[i][j] actually work. It is considerably
more difficult to to this in ANSI C; the simplest approach is to pack m into a
one-dimensional array and do the address computation explicitly:
int
sumMatrix(int rows, int cols, const int a[])
{
int i;
int j;
int sum;
sum = 0;
for(i = 0; i < rows; i++) {
for(j = 0; j < cols; j++) {
sum += a[i*cols + j];
}
}
return sum;
153
}
Variable-length arrays can sometimes be used for run-time storage allocation, as
an alternative to malloc and free (see below). A variable-length array allocated
as a local variable will be deallocated when the containing scope (usually a
function body, but maybe just a compound statement marked off by braces)
exits. One consequence of this is that you can’t return a variable-length array
from a function.
Here is an example of code using this feature:
/* reverse an array in place */
void
reverseArray(int n, int a[n])
{
/* algorithm: copy to a new array in reverse order */
/* then copy back */
int i;
int copy[n];
for(i = 0; i < n; i++) {
/* the -1 is needed to that a[0] goes to a[n-1] etc. */
copy[n-i-1] = a[i];
}
}
for(i = 0; i < n; i++) {
a[i] = copy[i];
}
While using variable-length arrays for this purpose can simplify code in some cases,
as a general programming practice it is extremely dangerous. The reason is
that, unlike allocations through malloc, variable-length array allocations are
typically allocated on the stack (which is often more constrainted than the heap)
and have no way of reporting failure. So if there isn’t enough room for your
variable-length array, odds are you won’t find out until a segmentation fault
occurs somewhere later in your code when you try to use it.
(As an additional annoyance, gdb is confused by two-dimensional variable-length
arrays.)
Here’s a safer version of the above routine, using malloc and free.
/* reverse an array in place */
void
reverseArray(int n, int a[n])
{
/* algorithm: copy to a new array in reverse order */
154
/* then copy back */
int i;
int *copy;
copy = (int *) malloc(n * sizeof(int));
assert(copy); /* or some other error check */
for(i = 0; i < n; i++) {
/* the -1 is needed to that a[0] goes to a[n-1] etc. */
copy[n-i-1] = a[i];
}
for(i = 0; i < n; i++) {
a[i] = copy[i];
}
}
free(copy);
4.9.6
Void pointers
A special pointer type is void *, a “pointer to void”. Such pointers are declared
in the usual way:
void *nothing;
/* pointer to nothing */
Unlike ordinary pointers, you can’t dereference a void * pointer or do arithmetic
on it, because the compiler doesn’t know what type it points to. However, you
are allowed to use a void * as a kind of “raw address” pointer value that you
can store arbitrary pointers in. It is permitted to assign to a void * variable
from an expression of any pointer type; conversely, a void * pointer value
can be assigned to a pointer variable of any type. An example is the return
value of malloc or the argument to free, both of which are declared as void
*. (Note that K&R suggests using an explicit cast for the return value of
malloc. This is now acknowledged by the authors to be an error, which arose
from the need for a cast prior to the standardization of void * in ANSI C. See
http://cm.bell-labs.com/cm/cs/cbook/2ediffs.html.)
int *block;
block = malloc(sizoef(int) * 12);
free(block);
/* void * converted to int * before assignment */
/* int * converted to void * before passing to free *
If you need to use a void * pointer as a pointer of a particular type in an
expression, you can cast it to the appropriate type by prefixing it with a type
name in parentheses, like this:
155
int a[50];
void *p;
/* typical array of ints */
/* dangerous void pointer */
a[12] = 17;
p = a;
/* save that valuable 17 */
/* p now holds base address of a */
printf("%d\n", ((int *) p)[12]);
/* get 17 back */
Usually if you have to start writing casts, it’s a sign that you are doing something
wrong, and you run the danger of violating the type system—say, by tricking
the compiler into treating a block of bits that are supposed to be an int as
four chars. But violating the type system like this will be necessary for some
applications, because even the weak type system in C turns out to be too
restrictive for writing certain kinds of “generic” code that work on values of
arbitrary types.
4.9.6.1
Alignment
One issue with casting pointers to and from void * is that you may violate the
alignment restrictions for a particular kind of pointer on some architectures.
Back in the 8-bit era of the 1970s, a single load or store operation would access
a single byte of memory, and because some data (chars) are still only one byte
wide, C pointers retain the ability to address individual bytes. But present-day
memory architectures typically have a wider data path, and the CPU may load
or store as much as 8 bytes (64 bits) in a single operation. This makes it natural
to organize memory into 4-byte or 8-byte words even though addresses still refer
to individual bytes. The effect of the memory architecture is that the address of
memory words must be aligned to a multiple of the word size: so with 4-byte
words, the address 0x1037ef44 (a multiple of 4) could refer to a full word, but
0x1037ef45 (one more than a multiple of 4) could only be used to refer to a
byte within a word.
What this means for a C program depends on your particular CPU and compiler.
If you try to use someting like 0x1037ef45 as an int *, one of three things
might happen:
1. The CPU might load the 4 bytes starting at this address, using two accesses
to memory to piece together the full int out of fragments of words. This
is done on Intel architectures, but costs performance.
2. The CPU might quietly zero out the last two bits of the address, loading
from 0x1037ef44 even though you asked for 0x1037ef45. This happens
on some other architectures, notably ARM.
3. The CPU might issue a run-time exception.
All of these outcomes are bad, and the C standard does not specify what happens
if you try to dereference a pointer value that does not satisfy the alignment
restrictions of its target type. Fortunately, unless you are doing very nasty things
156
with casts, this is unlikely to come up, because any pointer value you will see in
a typical program is likely to arise in one of three ways:
1. By taking the address of some variable. This pointer will be appropriately
aligned, because the compiler allocates space for each variable (including
fields within structs) with appropriate alignment.
2. By computing an offset address using pointer arithmetic either explicitly
(p + n) or implicilty (p[n]). In either case, as long as the base pointer is
correctly aligned, the computed pointer will also be correctly aligned.
3. By obtaining a pointer to an allocated block of memory using malloc or a
similar function. Here malloc is designed to always return blocks with the
maximum possible required alignment, just to avoid problems when you
use the results elsewhere.
On many compilers, you can use __alignof(type) to get the alignment restriction
for a particular type. This was formalized in C11 (with a different name!).
The other place where alignment can create issues is that if you make a struct
with components with different alignment restrictions, you may end up with
some empty space. For example, on a machine that enforces 4-byte alignment for
ints, building a struct that contains a char and an int will give you something
bigger than you might expect:
#include <stdio.h>
struct ci {
char c;
};
int i;
struct ic {
int i;
char c;
};
/* offset 0 */
/* 3 unused bytes go here */
/* offset 4 */
/* offset 0 */
/* offset 4 */
/* 3 unused bytes go here */
int
main(int argc, char **argv)
{
printf("sizeof(struct ci) == %lu\n", sizeof(struct ci));
printf("sizeof(struct ic) == %lu\n", sizeof(struct ic));
}
return 0;
examples/alignment/structPacking.c
$ c99 -Wall -o structPacking structPacking.c
157
$ ./structPacking
sizeof(struct ci) == 8
sizeof(struct ic) == 8
In both cases, the compiler packs in an extra 3 bytes to make the size of the
struct a multiple of the worst alignment of any of its components. If it didn’t
do this, you would have trouble as soon as you tried to make an array of these
things.
4.9.7
Run-time storage allocation using malloc
C does not generally permit arrays to be declared with variable sizes. C also
doesn’t let local variables outlive the function they are declared in. Both features
can be awkward if you want to build data structures at run time that have
unpredictable (perhaps even changing) sizes and that are intended to persist
longer than the functions that create them. To build such structures, the
standard C library provides the malloc routine, which asks the operating system
for a block of space of a given size (in bytes). With a bit of pushing and shoving,
this can be used to obtain a block of space that for all practical purposes acts
just like an array.
To use malloc, you must include stdlib.h at the top of your program. The
declaration for malloc is
void *malloc(size_t);
where size_t is an integer type (often unsigned long). Calling malloc with
an argument of n allocates and returns a pointer to the start of a block of n
bytes if possible. If the system can’t give you the space you asked for (maybe
you asked for more space than it has), malloc returns a null pointer. It is good
practice to test the return value of malloc whenever you call it.
Because the return type of malloc is void *, its return value can be assigned
to any variable with a pointer type. Computing the size of the block you need is
your responsibility—and you will be punished for any mistakes with difficultto-diagnose buffer overrun errors—but this task is made slightly easier by the
built-in sizeof operator that allows you to compute the size in bytes of any
particular data type. A typical call to malloc might thus look something like
this:
#include <stdlib.h>
/* allocate and return a new integer array with n elements */
/* calls abort() if there isn't enough space */
int *
makeIntArray(int n)
{
int *a;
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a = malloc(sizeof(int) * n);
if(a == 0) abort();
}
/* die on failure */
return a;
examples/pointers/makeIntArray.c
When you are done with a malloc’d region, you should return the space to
the system using the free routine, also defined in stdlib.h. If you don’t do
this, your program will quickly run out of space. The free routine takes a
void * as its argument and returns nothing. It is good practice to write a
matching destructor that de-allocates an object for each constructor (like
makeIntArray) that makes one.
void
destroyIntArray(int *a)
{
free(a);
}
It is a serious error to do anything at all with a block after it has been freed.
This is not necessarily because free modifies the contents of the block (although
it might), but because when you free a block you are granting the storage
allocator permission to hand the same block out in response to a future call to
malloc, and you don’t want to step on whatever other part of your program is
now trying to use that space.
It is also possible to grow or shrink a previously allocated block. This is done
using the realloc function, which is declared as
void *realloc(void *oldBlock, size_t newSize);
The realloc function returns a pointer to the resized block. It may or may not
allocate a new block. If there is room, it may leave the old block in place and
return its argument. But it may allocate a new block and copy the contents of
the old block, so you should assume that the old pointer has been freed.
Here’s a typical use of realloc to build an array that grows as large as it needs
to be:
/* read numbers from stdin until there aren't any more */
/* returns an array of all numbers read, or null on error */
/* returns the count of numbers read in *count */
int *
readNumbers(int *count /* RETVAL */ )
{
int mycount;
/* number of numbers read */
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int size;
int *a;
int n;
/* size of block allocated so far */
/* block */
/* number read */
mycount = 0;
size = 1;
a = malloc(sizeof(int) * size);
if(a == 0) return 0;
/* allocating zero bytes is tricky */
while(scanf("%d", &n) == 1) {
/* is there room? */
while(mycount >= size) {
/* double the size to avoid calling realloc for every number read */
size *= 2;
a = realloc(a, sizeof(int) * size);
if(a == 0) return 0;
}
}
/* put the new number in */
a[mycount++] = n;
/* now trim off any excess space */
a = realloc(a, sizeof(int) * mycount);
/* note: if a == 0 at this point we'll just return it anyway */
/* save out mycount */
*count = mycount;
}
return a;
examples/pointers/readNumbers.c
Because errors involving malloc and its friends can be very difficult to spot, it
is recommended to test any program that uses malloc using valgrind.
4.9.8
Function pointers
A function pointer, internally, is just the numerical address for the code for a
function. When a function name is used by itself without parentheses, the value
is a pointer to the function, just as the name of an array by itself is a pointer
to its zeroth element. Function pointers can be stored in variables, structs,
unions, and arrays and passed to and from functions just like any other pointer
type. They can also be called: a variable of type function pointer can be used in
160
place of a function name.
Function pointers are not used as much in C as in functional languages, but
there are many common uses even in C code.
4.9.8.1
Function pointer declarations
A function pointer declaration looks like a function declaration, except that
the function name is wrapped in parentheses and preceded by an asterisk. For
example:
/* a function taking two int arguments and returning an int */
int function(int x, int y);
/* a pointer to such a function */
int (*pointer)(int x, int y);
As with function declarations, the names of the arguments can be omitted.
Here’s a short program that uses function pointers:
/* Functional "hello world" program */
#include <stdio.h>
int
main(int argc, char **argv)
{
/* function for emitting text */
int (*say)(const char *);
say = puts;
say("hello world");
}
return 0;
4.9.8.2
Callbacks
A callback is when we pass a function pointer into a function so that that
function can call our function when some event happens or it needs to compute
something.
A classic example is the comparison argument to qsort, from the standard
library:
/* defined in stdlib.h */
void
161
qsort(
void *base,
size_t n,
size_t size,
int (*cmp)(const void *key1, const void *key2)
);
This is a generic sorting routine that will sort any array in place. It needs to
know (a) the base address of the array; (b) how many elements there are; (c)
how big each element is; and (d) how to compare two elements. The only tricky
part is supplying the comparison, which could involve arbitrarily-complex code.
So we supply this code as a function with an interface similar to strcmp.
static int
compare_ints(void *key1, void *key2)
{
return *((int *) key1) - *((int *) key2);
}
int
sort_int_array(int *a, int n)
{
qsort(a, n, sizeof(*a), compare_ints);
}
Other examples might include things like registering an error handler for a
library, instead of just having it call abort() or something equally catastrophic,
or providing a cleanup function for freeing data passed into a data structure.
4.9.8.3
Dispatch tables
Alternative to gigantic if/else if or switch statements. The idea is to build
an array of function pointers (or, more generally, some sort of dictionary data
structure), and use the value we might otherwise be feeding to switch as an index
into this array. Here is a simple example, which echoes most of the characters in
its input intact, except for echoing every lowercase vowel twice:
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<assert.h>
<limits.h>
/*
* Demonstrate use of dispatch tables.
*/
/* print a character twice */
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/* like putchar, returns character if successful or EOF on error */
int
putcharTwice(int c)
{
if(putchar(c) == EOF || putchar(c) == EOF) {
return EOF;
} else {
return c;
}
}
#define NUM_CHARS (UCHAR_MAX + 1) /* UCHAR_MAX is in limits.h */
int
main(int argc, char **argv)
{
/* this declares table as an array of function pointers */
int (*table[UCHAR_MAX+1])(int);
int i;
int c;
for(i = 0; i < UCHAR_MAX; i++) {
/* default is to call putchar */
table[i] = putchar;
}
/* but lower-case vowels show up twice */
table['a'] = putcharTwice;
table['e'] = putcharTwice;
table['i'] = putcharTwice;
table['o'] = putcharTwice;
table['u'] = putcharTwice;
while((c = getchar()) != EOF) {
table[c](c);
}
}
return 0;
examples/pointers/dispatchTable.c
And here is the program translating Shakespeare into mock-Swedish:
$ c99 -Wall -pedantic -g3 -o dispatchTable dispatchTable.c
$ echo Now is the winter of our discontent made glorious summer by this sun of York. | ./dis
Noow iis thee wiinteer oof oouur diiscoonteent maadee glooriioouus suummeer by thiis suun oo
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In this particular case, we did a lot of work to avoid just writing a switch
statement. But being able to build a dispatch table dynamically can be very
useful sometimes. An example might be a graphical user interface where each
button has an associated function. If buttons can be added by different parts of
the program, using a table mapping buttons to functions allows a single dispatch
routine to figure out where to route button presses.
(For some applications, we might want to pass additional information in to the
function to change its behavior. This can be done by replacing the function
pointers with closures.)
4.9.9
The restrict keyword
In C99, it is possible to declare that a pointer variable is the only way to reach
its target as long as it is in scope. This is not enforced by the compiler; instead,
it is a promise from the programmer to the compiler that any data reached
through this point will not be changed by other parts of the code, which allows
the compiler to optimize code in ways that are not possible if pointers might
point to the same place (a phenomenon called pointer aliasing). For example,
consider the following short function:
// write 1 + *src to *dst and return *src
int
copyPlusOne(int * restrict dst, int * restrict src)
{
*dst = *src + 1;
return *src;
}
For this function, the output of c99 -O3 -S includes one more instruction if
the restrict qualifiers are removed. The reason is that if dst and src may
point to the same location, src needs to be re-read for the return statement, in
case it changed. But if they are guaranteed to point to different locations, the
compiler can re-use the previous value it already has in one of the CPU registers.
For most code, this feature is useless, and potentially dangerous if someone calls
your routine with aliased pointers. However, it may sometimes be possible to
increase performance of time-critical code by adding a restrict keyword. The
cost is that the code might no longer work if called with aliased pointers.
Curiously, C assumes that two pointers are never aliases if you have two arguments
with different pointer types, neither of which is char * or void *.10 This is
known as the strict aliasing rule and cannot be overridden from within the
program source code: there is no unrestrict keyword. You probably only need
10 The reason for excluding char * and void * is that these are often used to represent
pointers to objects with arbitrary types.
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to worry about this if you are casting pointers to different types and then passing
the cast pointers around in the same context as the original pointers.
4.10
Strings
Processing strings of characters is one of the oldest application of mechanical
computers, arguably predating numerical computation by at least fifty years.
Assuming you’ve already solved the problem of how to represent characters in
memory (e.g. as the C char type encoded in ASCII), there are two standard
ways to represent strings:
• As a delimited string, where the end of a string is marked by a special
character. The advantages of this method are that only one extra byte
is needed to indicate the length of an arbitrarily long string, that strings
can be manipulated by simple pointer operations, and in some cases that
common string operations that involve processing the entire string can
be performed very quickly. The disadvantage is that the delimiter can’t
appear inside any string, which limits what kind of data you can store in a
string.
• As a counted string, where the string data is prefixed or supplemented
with an explicit count of the number of characters in the string. The
advantage of this representation is that a string can hold arbitrary data
(including delimiter characters) and that one can quickly jump to the end
of the string without having to scan its entire length. The disadvantage
is that maintaining a separate count typically requires more space than
adding a one-byte delimiter (unless you limit your string length to 255
characters) and that more care needs to be taken to make sure that the
count is correct.
4.10.1
C strings
Because delimited strings are simpler and take less space, C went for delimited
strings. A string is a sequence of characters terminated by a null character '\0'.
Looking back from almost half a century later, this choice may have been a
mistake in the long run, but we are pretty much stuck with it.
Note that the null character is not the same as a null pointer, although both
appear to have the value 0 when used in integer contexts. A string is represented
by a variable of type char *, which points to the zeroth character of the string.
The programmer is responsible for allocating and managing space to store strings,
except for explicit string constants, which are stored in a special non-writable
string space by the compiler.
If you want to use counted strings instead, you can build your own
using a struct.
Most scripting languages written in C (e.g.
Perl,
165
Python_programming_language, PHP, etc.)
use this approach internally. (Tcl is an exception, which is one of many good reasons not to use
Tcl).
4.10.2
String constants
A string constant in C is represented by a sequence of characters within double
quotes. Standard C character escape sequences like \n (newline), \r (carriage
return), \a (bell), \0x17 (character with hexadecimal code 0x17), \\ (backslash),
and \" (double quote) can all be used inside string constants. The value of a
string constant has type const char *, and can be assigned to variables and
passed as function arguments or return values of this type.
Two string constants separated only by whitespace will be concatenated by the
compiler as a single constant: "foo" "bar" is the same as "foobar". This
feature is not much used in normal code, but shows up sometimes in macros.
4.10.3
String buffers
The problem with string constants is that you can’t modify them. If you want to
build strings on the fly, you will need to allocate space for them. The traditional
approach is to use a buffer, an array of chars. Here is a particularly painful
hello-world program that builds a string by hand:
#include <stdio.h>
int
main(int argc, char **argv)
{
char hi[3];
hi[0] = 'h';
hi[1] = 'i';
hi[2] = '\0';
puts(hi);
}
return 0;
examples/strings/hi.c
Note that the buffer needs to have size at least 3 in order to hold all three
characters. A common error in programming with C strings is to forget to leave
space for the null at the end (or to forget to add the null, which can have comical
results depending on what you are using your surprisingly long string for).
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4.10.3.1
String buffers and the perils of gets
Fixed-size buffers are a common source of errors in older C programs, particularly
ones written with the library routine gets. The problem is that if you do
something like
strcpy(smallBuffer, bigString);
the strcpy function will happily keep copying characters across memory long
after it has passed the end of smallBuffer. While you can avoid this to a
certain extent when you control where bigString is coming from, the situation
becomes particularly fraught if the string you are trying to store comes from
the input, where it might be supplied by anybody, including somebody who is
trying to execute a buffer overrun attack to seize control of your program.
If you do need to read a string from the input, you should allocate the receiving
buffer using malloc and expand it using realloc as needed. Below is a program
that shows how to do this, with some bad alternatives commented out:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
#define NAME_LENGTH (2)
#define INITIAL_LINE_LENGTH (2)
/* return a freshly-malloc'd line with next line of input from stdin */
char *
getLine(void)
{
char *line;
int size;
/* how much space do I have in line? */
int length; /* how many characters have I used */
int c;
size = INITIAL_LINE_LENGTH;
line = malloc(size);
assert(line);
length = 0;
while((c = getchar()) != EOF && c != '\n') {
if(length >= size-1) {
/* need more space! */
size *= 2;
/* make length equal to new size */
167
}
}
/* copy contents if necessary */
line = realloc(line, size);
line[length++] = c;
line[length] = '\0';
}
return line;
int
main(int argc, char **argv)
{
int x = 12;
/* char name[NAME_LENGTH]; */
char *line;
int y = 17;
puts("What is your name?");
/* gets(name); */
/* scanf("%s\n", name); */
/* fgets(name, NAME_LENGTH, stdin); */
line = getLine();
printf("Hi %s!
free(line);
}
/*
/*
/*
/*
may
may
may
has
overrun buffer */
overrun buffer */
truncate input */
none of these problems */
Did you know that x == %d and y == %d?\n", line, x, y);
/* but we do have to free line when we are done with it */
return 0;
examples/strings/getLine.c
4.10.4
Operations on strings
Unlike many programming languages, C provides only a rudimentary stringprocessing library. The reason is that many common string-processing tasks in
C can be done very quickly by hand.
For example, suppose we want to copy a string from one buffer to another. The
library function strcpy declared in string.h will do this for us (and is usually
the right thing to use), but if it didn’t exist we could write something very close
to it using a famous C idiom.
168
void
strcpy2(char *dest, const char *src)
{
/* This line copies characters one at a time from *src to *dest. */
/* The postincrements increment the pointers (++ binds tighter than *) */
/* to get to the next locations on the next iteration through the loop. */
/* The loop terminates when *src == '\0' == 0. */
/* There is no loop body because there is nothing to do there. */
while(*dest++ = *src++);
}
The externally visible difference between strcpy2 and the original strcpy is
that strcpy returns a char * equal to its first argument. It is also likely that
any implementation of strcpy found in a recent C library takes advantage of
the width of the memory data path to copy more than one character at a time.
Most C programmers will recognize the while(*dest++ = *src++); from having seen it before, although experienced C programmers will generally be able to
figure out what such highly abbreviated constructions mean. Exposure to such
constructions is arguably a form of hazing.
Because C pointers act exactly like array names, you can also write strcpy2
using explicit array indices. The result is longer but may be more readable if
you aren’t a C fanatic.
char *
strcpy2a(char *dest, const char *src)
{
int ;
i = 0;
for(i = 0; src[i] != '\0'; i++) {
dest[i] = src[i];
}
/* note that the final null in src is not copied by the loop */
dest[i] = '\0';
}
return dest;
An advantage of using a separate index in strcpy2a is that we don’t trash dest,
so we can return it just like strcpy does. (In fairness, strcpy2 could have saved
a copy of the original location of dest and done the same thing.)
Note that nothing in strcpy2, strcpy2a, or the original strcpy will save you if
dest points to a region of memory that isn’t big enough to hold the string at
src, or if somebody forget to tack a null on the end of src (in which case strcpy
will just keep going until it finds a null character somewhere). As elsewhere, it’s
169
your job as a programmer to make sure there is enough room. Since the compiler
has no idea what dest points to, this means that you have to remember how
much room is available there yourself.
If you are worried about overrunning dest, you could use strncpy instead. The
strncpy function takes a third argument that gives the maximum number of
characters to copy; however, if src doesn’t contain a null character in this range,
the resulting string in dest won’t either. Usually the only practical application
to strncpy is to extract the first k characters of a string, as in
/* copy the substring of src consisting of characters at positions
start..end-1 (inclusive) into dest */
/* If end-1 is past the end of src, copies only as many characters as
available. */
/* If start is past the end of src, the results are unpredictable. */
/* Returns a pointer to dest */
char *
copySubstring(char *dest, const char *src, int start, int end)
{
/* copy the substring */
strncpy(dest, src + start, end - start);
/* add null since strncpy probably didn't */
dest[end - start] = '\0';
}
return dest;
Another quick and dirty way to extract a substring of a string you don’t care
about (and can write to) is to just drop a null character in the middle of the
sacrificial string. This is generally a bad idea unless you are certain you aren’t
going to need the original string again, but it’s a surprisingly common practice
among C programmers of a certain age.
A similar operation to strcpy is strcat. The difference is that strcat concatenates src on to the end of dest; so that if dest previous pointed to "abc" and
src to "def", dest will now point to "abcdef". Like strcpy, strcat returns
its first argument. A no-return-value version of strcat is given below.
void
strcat2(char *dest, const char *src)
{
while(*dest) dest++;
while(*dest++ = *src++);
}
Decoding this abomination is left as an exercise for the reader. There is also a
function strncat which has the same relationship to strcat that strncpy has
to strcpy.
170
As with strcpy, the actual implementation of strcat may be much more subtle,
and is likely to be faster than rolling your own.
4.10.5
Finding the length of a string
Because the length of a string is of fundamental importance in C (e.g., when
deciding if you can safely copy it somewhere else), the standard C library provides
a function strlen that counts the number of non-null characters in a string.
Note that if you are allocating space for a copy of a string, you will need to add
one to the value returned by strlen to account for the null.
Here’s a possible implementation:
int
strlen(const char *s)
{
int i;
for(i = 0; *s; i++, s++);
}
return i;
Note the use of the comma operator in the increment step. The comma operator
applied to two expressions evaluates both of them and discards the value of the
first; it is usually used only in for loops where you want to initialize or advance
more than one variable at once.
Like the other string routines, using strlen requires including string.h.
4.10.5.1
The strlen tarpit
A common mistake is to put a call to strlen in the header of a loop; for example:
/* like strcpy, but only copies characters at indices 0, 2, 4, ...
from src to dest */
char *
copyEvenCharactersBadVersion(char *dest, const char *src)
{
int i;
int j;
/* BAD: Calls strlen on every pass through the loop */
for(i = 0, j = 0; i < strlen(src); i += 2, j++) {
dest[j] = src[i];
}
171
dest[j] = '\0';
}
return dest;
The problem is that strlen has to scan all of src every time the test is done,
which adds time proportional to the length of src to each iteration of the loop.
So copyEvenCharactersBadVersion takes time proportional to the square of
the length of src.
Here’s a faster version:
/* like strcpy, but only copies characters at indices 0, 2, 4, ...
from src to dest */
char *
copyEvenCharacters(char *dest, const char *src)
{
int i;
int j;
int len;
/* length of src */
len = strlen(src);
/* GOOD: uses cached value of strlen(src) */
for(i = 0, j = 0; i < len; i += 2, j++) {
dest[j] = src[i];
}
dest[j] = '\0';
}
return dest;
Because it doesn’t call strlen all the time, this version of copyEvenCharacters
will run much faster than the original even on small strings, and several million
times faster if src is megabytes long.
4.10.6
Comparing strings
If you want to test if strings s1 and s2 contain the same characters, writing
s1 == s2 won’t work, since this tests instead whether s1 and s2 point to the
same address. Instead, you should use strcmp, declared in string.h. The
strcmp function walks along both of its arguments until it either hits a null
on both and returns 0, or hits two different characters, and returns a positive
integer if the first string’s character is bigger and a negative integer if the second
string’s character is bigger (a typical implementation will just subtract the two
characters). A straightforward implementation might look like this:
172
int
strcmp(const char *s1, const char *s2)
{
while(*s1 && *s2 && *s1 == *s2) {
s1++;
s2++;
}
}
return *s1 - *s2;
To use strcmp to test equality, test if the return value is 0:
if(strcmp(s1, s2) == 0) {
/* strings are equal */
...
}
You may sometimes see this idiom instead:
if(!strcmp(s1, s2)) {
/* strings are equal */
...
}
My own feeling is that the first version is more clear, since !strcmp always
suggested to me that you were testing for the negation of some property (e.g. not
equal). But if you think of strcmp as telling you when two strings are different
rather than when they are equal, this may not be so confusing.
4.10.7
Formatted output to strings
You can write formatted output to a string buffer with sprintf just like you
can write it to stdout with printf or to a file with fprintf. Make sure when
you do so that there is enough room in the buffer you are writing to, or the
usual bad things will happen.
4.10.8
Dynamic allocation of strings
When allocating space for a copy of a string s using malloc, the required space
is strlen(s)+1. Don’t forget the +1, or bad things may happen.11
11 In this case you will get lucky most of the time, since the odds are that malloc will give you
a block that is slightly bigger than strlen(s) anyway. But bugs that only manifest themselves
occasionally are even worse than bugs that kill your program every time, because they are
much harder to track down.
173
Because allocating space for a copy of a string is such a common operation, many
C libraries provide a strdup function that does exactly this. If you don’t have
one (it’s not required by the C standard), you can write your own like this:
/* return a freshly-malloc'd copy of s */
/* or 0 if malloc fails */
/* It is the caller's responsibility to free the returned string when done. */
char *
strdup(const char *s)
{
char *s2;
s2 = malloc(strlen(s)+1);
if(s2 != 0) {
strcpy(s2, s);
}
}
return s2;
Exercise: Write a function strcatAlloc that returns a freshly-malloc’d string
that concatenates its two arguments. Exactly how many bytes do you need to
allocate?
4.10.9
Command-line arguments
Now that we know about strings, we can finally do something with argc and
argv.
Recall that argv in main is declared as char **; this means that it is a pointer
to a pointer to a char, or in this case the base address of an array of pointers
to char, where each such pointer references a string. These strings correspond
to the command-line arguments to your program, with the program name itself
appearing in argv[0]12
The count argc counts all arguments including argv[0]; it is 1 if your program
is called with no arguments and larger otherwise.
Here is a program that prints its arguments. If you get confused about what
argc and argv do, feel free to compile this and play with it:
#include <stdio.h>
int
main(int argc, char **argv)
12 Some programs (e.g. /c/cs223/bin/submit) will use this to change their behavior depending on what name you call them with.
174
{
int i;
printf("argc = %d\n\n", argc);
for(i = 0; i < argc; i++) {
printf("argv[%d] = %s\n", i, argv[i]);
}
}
return 0;
examples/strings/printArgs.c
Like strings, C terminates argv with a null: the value of argv[argc] is always
0 (a null pointer to char). In principle this allows you to recover argc if you
lose it.
4.11
Structured data types
C has two kinds of structured data types: structs and unions. A struct holds
multiple values in consecutive memory locations, called fields, and implements
what in type theory is called a product type: the set of possible values is the
Cartesian product of the sets of possible values for its fields. In contrast, a union
has multiple fields but they are all stored in the same location: effectively, this
means that only one field at a time can hold a value, making a union a sum
type whose set of possible values is the union of the sets of possible values for
each of its fields. Unlike what happens in more sensible programming languages,
unions are not tagged: unless you keep track of this somewhere else, you can’t
tell which field in a union is being used, and you can store a value of one type in
a union and try to read it back as a different type, and C won’t complain.13
4.11.1
Structs
A struct is a way to define a type that consists of one or more other types
pasted together. Here’s a typical struct definition:
struct string {
int length;
char *data;
};
This defines a new type struct string that can be used anywhere you would
use a simple type like int or float. When you declare a variable with type
13 There are various ways to work around this. The simplest is to put a union inside a larger
struct that includes an explicit type tag.
175
struct string, the compiler allocates enough space to hold both an int and
a char * (8 bytes on a typical 32-bit machine). You can get at the individual
components using the . operator, like this:
struct string {
int length;
char *data;
};
int
main(int argc, char **argv)
{
struct string s;
s.length = 4;
s.data = "this string is a lot longer than you think";
puts(s.data);
}
return 0;
examples/structs/structExample.c
Variables of type struct can be assigned to, passed into functions, returned
from functions, and tested for equality, just like any other type. Each such
operation is applied componentwise; for example, s1 = s2; is equivalent to
s1.length = s2.length; s1.data = s2.data; and s1 == s2 is equivalent
to s1.length == s2.length && s1.data = s2.data.
These operations are not used as often as you might think: typically, instead of
copying around entire structures, C programs pass around pointers, as is done
with arrays. Pointers to structs are common enough in C that a special syntax
is provided for dereferencing them.14 Suppose we have:
struct string s;
struct string *sp;
/* a struct */
/* a pointer to a struct */
s.length = 4;
s.data = "another overly long string";
sp = &s;
/* sp now points to s */
We can then refer to elements of the struct string that sp points to (i.e. s)
in either of two ways:
14 Arguably, this is a bug in the design of the language: if the compiler knows that sp
has type struct string *, there is no particular reason why it can’t interpret sp.length as
sp->length. But it doesn’t do this, so you will have to remember to write sp->length instead.
176
puts((*sp).data);
puts(sp->data);
The second is more common, since it involves typing fewer parentheses. It is an
error to write *sp.data in this case; since . binds tighter than *, the compiler
will attempt to evaluate sp.data first and generate an error, since sp doesn’t
have a data field.
Pointers to structs are commonly used in defining abstract data data, since it
is possible to declare that a function returns e.g. a struct string * without
specifying the components of a struct string. (All pointers to structs in C
have the same size and structure, so the compiler doesn’t need to know the
components to pass around the address.) Hiding the components discourages
code that shouldn’t look at them from doing so, and can be used, for example,
to enforce consistency between fields.
For example, suppose we wanted to define a struct string * type that held
counted strings that could only be accessed through a restricted interface that
prevented (for example) the user from changing the string or its length. We
might create a file myString.h that contained the declarations:
/* make a struct string * that holds a copy of s */
/* returns 0 if malloc fails */
struct string *makeString(const char *s);
/* destroy a struct string * */
void destroyString(struct string *);
/* return the length of a struct string * */
int stringLength(struct string *);
/* return the character at position index in the struct string * */
/* or returns -1 if index is out of bounds */
int stringCharAt(struct string *s, int index);
examples/myString/myString.h
and then the actual implementation in myString.c would be the only place
where the components of a struct string were defined:
#include <stdlib.h>
#include <string.h>
#include "myString.h"
struct string {
int length;
char *data;
};
177
struct string *
makeString(const char *s)
{
struct string *s2;
s2 = malloc(sizeof(struct string));
if(s2 == 0) { return 0; } /* let caller worry about malloc failures */
s2->length = strlen(s);
s2->data = malloc(s2->length);
if(s2->data == 0) {
free(s2);
return 0;
}
strncpy(s2->data, s, s2->length);
}
return s2;
void
destroyString(struct string *s)
{
free(s->data);
free(s);
}
int
stringLength(struct string *s)
{
return s->length;
}
int
stringCharAt(struct string *s, int index)
{
if(index < 0 || index >= s->length) {
return -1;
} else {
return s->data[index];
}
}
examples/myString/myString.c
178
In practice, we would probably go even further and replace all the
struct string * types with a new name declared with typedef.
4.11.1.1
Operations on structs
What you can do to structs is pretty limited: you can look up or set individual
components in a struct, you can pass structs to functions or as return values
from functions (which makes a copy of the original struct), and you can assign
the contents of one struct to another using s1 = s2 (which is equivalent to
copying each component separately).
One thing that you can’t do is test two structs for equality using ==; this is
because structs may contain extra space holding junk data. If you want to test
for equality, you will need to do it componenti by component.
4.11.1.2
Layout in memory
The C99 standard guarantees that the components of a struct are stored in
memory in the same order that they are defined in: that is, later components are
placed at higher address. This allows sneaky tricks like truncating a structure
if you don’t use all of its components. Because of alignment restrictions, the
compiler may add padding between components to put each component on its
prefered alignment boundary.
You can find the position of a component within a struct using the offsetof
macro, which is defined in stddef.h. This returns the number of bytes from the
base of the struct that the component starts at, and can be used to do various
terrifying non-semantic things with pointers.
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<stddef.h>
<assert.h>
int
main(int argc, char **argv)
{
struct foo {
int i;
char c;
double d;
float f;
char *s;
};
printf("i is at %lu\n", offsetof(struct foo, i));
printf("c is at %lu\n", offsetof(struct foo, c));
179
printf("d is at %lu\n", offsetof(struct foo, d));
printf("f is at %lu\n", offsetof(struct foo, f));
printf("s is at %lu\n", offsetof(struct foo, s));
}
return 0;
examples/structs/offsetof.c
4.11.1.3
Bit fields
It is possible to specify the exact number of bits taken up by a member of a
struct of integer type. This is seldom useful, but may in principle let you pack
more information in less space. Bit fields are sometimes used to unpack data
from an external source that uses this trick, but this is dangerous, because there
is no guarantee that the compiler will order the bit fields in your struct in any
particular order (at the very least, you will need to worry about endianness.
Example:
struct color
unsigned
unsigned
unsigned
unsigned
};
{
int
int
int
int
red
green
blue
alpha
:
:
:
:
2;
2;
2;
2;
This defines a struct that (probably) occupies only one byte, and supplies four
2-bit fields, each of which can hold values in the range 0-3.
4.11.2
Unions
A union is just like a struct, except that instead of allocating space to store
all the components, the compiler only allocates space to store the largest one,
and makes all the components refer to the same address. This can be used to
save space if you know that only one of several components will be meaningful
for a particular object. An example might be a type representing an object in a
LISP-like language like Scheme:
struct lispObject {
int type;
/* type code */
union {
int
intVal;
double floatVal;
char * stringVal;
struct {
struct lispObject *car;
180
};
struct lispObject *cdr;
} consVal;
} u;
Now if you wanted to make a struct lispObject that held an integer value,
you might write
lispObject o;
o.type = TYPE_INT;
o.u.intVal = 27;
Here TYPE_INT has presumably been defined somewhere. Note that nothing
then prevents you from writing
x = 2.7 * o.u.floatVal;
/* BAD */
The effects of this will be strange, since it’s likely that the bit pattern representing
27 as an int represents something very different as a double. Avoiding such
mistakes is your responsibility, which is why most uses of union occur inside
larger structs that contain enough information to figure out which variant of
the union applies.
4.11.3
Enums
C provides the enum construction for the special case where you want to have a
sequence of named constants of type int, but you don’t care what their actual
values are, as in
enum color { RED, BLUE, GREEN, MAUVE, TURQUOISE };
This will assign the value 0 to RED, 1 to BLUE, and so on. These values are
effectively of type int, although you can declare variables, arguments, and return
values as type enum color to indicate their intended interpretation.
Despite declaring a variable enum color c (say), the compiler will still allow c
to hold arbitrary values of type int.
So the following ridiculous code works just fine:
#include <stdio.h>
#include <stdlib.h>
enum foo { FOO };
enum apple { MACINTOSH, CORTLAND, RED_DELICIOUS };
enum orange { NAVEL, CLEMENTINE, TANGERINE };
int
main(int argc, char **argv)
181
{
enum foo x;
if(argc != 1) {
fprintf(stderr, "Usage: %s\n", argv[0]);
return 1;
}
printf("FOO = %d\n", FOO);
printf("sizeof(enum foo) = %d\n", sizeof(enum foo));
x = 127;
printf("x = %d\n", x);
/* note we can add apples and oranges */
printf("%d\n", RED_DELICIOUS + TANGERINE);
}
return 0;
examples/definitions/enumsAreInts.c
4.11.3.1
Specifying particular values
It is also possible to specify particular values for particular enumerated constants,
as in
enum color { RED = 37, BLUE = 12, GREEN = 66, MAUVE = 5, TURQUOISE };
Anything that doesn’t get a value starts with one plus the previous value; so
the above definition would set TURQUOISE to 6. This may result in two names
mapping to the same value.
4.11.3.2
What most people do
In practice, enums are seldom used, and you will more commonly see a stack of
#defines:
#define
#define
#define
#define
#define
RED
(0)
BLUE
(1)
GREEN
(2)
MAUVE
(3)
TURQUOISE (4)
The reason for this is partly historical—enum arrived late in the evolution of
C—but partly practical: a table of #defines makes it much easier to figure out
which color is represented by 3, without having to count through a list. But if
182
you never plan to use the numerical values, enum may be a better choice, because
it guarantees that all the values will be distinct.
4.11.3.3
Using enum with union
A natural place to use an enum is to tag a union with the type being used. For
example, a Lisp-like language might implement the following multi-purpose data
type:
enum TypeCode { TYPE_INT, TYPE_DOUBLE, TYPE_STRING };
struct LispValue {
enum TypeCode typeCode;
union {
int i;
double d;
char *s;
} value;
};
Here we don’t care what the numeric values of TYPE_INT, TYPE_DOUBLE, and
TYPE_STRING are, as long as we can apply switch to typeCode to figure out
what to do with one of these things.
4.12
Type aliases using typedef
Suppose that you want to represent character strings as
struct string {
int length;
char *data;
};
/* malloc'd block */
int stringLength(const struct string *s);
If you later change the representation to, say, traditional null-terminated char *
strings or some even more complicated type (union string **some_string[2];),
you will need to go back and replace ever occurrence of struct string * in
every program that uses it with the new type. Even if you don’t expect to
change the type, you may still get tired of typing struct string * all the time,
especially if your fingers slip and give you struct string sometimes.
The solution is to use a typedef, which defines a new type name:
typedef struct string *String;
int stringLength(String s);
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The syntax for typedef looks like a variable declaration preceded by typedef,
except that the variable is replaced by the new type name that acts like whatever
type the defined variable would have had. You can use a name defined with
typedef anywhere you could use a normal type name, as long as it is later in
the source file than the typedef definition. Typically typedefs are placed in a
header file (.h file) that is then included anywhere that needs them.
You are not limited to using typedefs only for complex types. For example, if
you were writing numerical code and wanted to declare overtly that a certain
quantity was not just any double but actually a length in meters, you could
write
typedef double LengthInMeters;
typedef double AreaInSquareMeters;
AreaInSquareMeters rectangleArea(LengthInMeters height, LengthInMeters width);
Unfortunately, C does not do type enforcement on typedef’d types: it is perfectly
acceptable to the compiler if you pass a value of type AreaInSquareMeters
as the first argument to rectangleArea, since by the time it checks it has
replaced by AreaInSquareMeters and LengthInMeters by double. So this
feature is not as useful as it might be, although it does mean that you can write
rectangleArea(2.0, 3.0) without having to do anything to convert 2.0 and
3.0 to type LengthInMeters.
4.12.1
Opaque structs
There are certain cases where the compiler needs to know the definition of a
struct:
1. When the program accesses its components.
2. When the compiler needs to know its size. This may be because you
are building an array of these structs, because they appear in a larger
struct, when you are passing the struct as an argument or assigning it
to a variable, or just because you applied sizeof to the struct.
But the compile does not need to know the definition of a struct to know how
create a pointer to it. This is because all struct pointers have the same size
and structure.
This allows a trick called an opaque struct, which can be used for information
hiding, where one part of your program is allowed to see the definition of a
struct but other parts are not.
The idea is to create a header file that defines all the functions that might be used
to access the struct, but does not define the struct itself. For example, suppose
we want to create a counter, where the user can call a function increment that
acts like ++ in the sense that it increments the counter and returns the new
184
value, but we don’t want to allow the user to change the value of the counter in
any other way. This header file defines the interface to the counter.
Here is the header file:
/* Create a new counter, initialized to 0.
struct counter * counterCreate(void);
Call counterDestroy to get rid of it. */
/* Free space used by a counter. */
void counterDestroy(struct counter *);
/* Increment a counter and return new value. */
int counterIncrement(struct counter *);
examples/structs/opaqueStructs/counter.h
We can now write code that uses the struct counter * type without knowing
what it is actually pointing to:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
#include "counter.h"
int
main(int argc, char **argv)
{
struct counter *c;
int value;
c = counterCreate();
while((value = counterIncrement(c)) < 10) {
printf("%d\n", value);
}
counterDestroy(c);
}
return 0;
examples/structs/opaqueStructs/testCounter.c
To make this work, we do have to provide an implementation. The obvious
way to do it is have a struct counter store the counter value in an int, but
one could imagine other (probably bad) implementations that did other things,
as long as from the outside they acted like we expect.
We only put the definition of a struct counter in this file. This means that
185
only functions in this file can access a counter’s components, compute the size of
a counter, and so forth. While we can’t absolutely prevent some other function
from extracting or modifying the contents of a counter (C doesn’t provide
that kind of memory protection), we can at least hint very strongly that the
programmer shouldn’t be doing this.
#include <stdlib.h>
#include <assert.h>
#include "counter.h"
struct counter {
int value;
};
struct counter *
counterCreate(void)
{
struct counter *c;
c = malloc(sizeof(struct counter));
assert(c);
c->value = 0;
}
return c;
void
counterDestroy(struct counter *c)
{
free(c);
}
int
counterIncrement(struct counter *c)
{
return ++(c->value);
}
examples/structs/opaqueStructs/counter.c
We will see this trick used over and over again when we build abstract data
types.
186
4.13
Macros
See K&R Appendix A12.3 for full details on macro expansion in ANSI C and
http://gcc.gnu.org/onlinedocs/cpp/Macros.html for documentation on what gcc
supports.
The short version: the command
#define FOO (12)
causes any occurrence of the word FOO in your source file to be replaced by
(12) by the preprocessor. To count as a word, FOO can’t be adjacent to other
alphanumeric characters, so for example FOOD will not expand to (12)D.
4.13.1
Macros with arguments
To create a macro with arguments, put them in parentheses separated by commas
after the macro name, e.g.
#define Square(x) ((x)*(x))
Now if you write Square(foo) it will expand as ((foo)*(foo)). Note the heavy
use of parentheses inside the macro definition to avoid trouble with operator
precedence; if instead we had written
#define BadSquare(x) x*x
then BadSquare(3+4) would give 3+4*3+4, which evaluates to 19, which is
probably not what we intended. The general rule is that macro arguments
should always be put in parentheses if you are using them in an expression where
precedence might be an issue.
4.13.1.1
Multiple arguments
You can have multiple arguments to a macro, e.g.
#define Average(x,y) (((x)+(y))/2.0)
The usual caveats about using lots of parentheses apply.
4.13.1.2
Perils of repeating arguments
Macros can have odd effects if their arguments perform side-effects. For example, Square(++x) expands to ((++x)*(++x)); if x starts out equal to 1, this
expression may evaluate to any of 2, 6, or 9 depending on when the ++ operators
are evaluated, and will definitely leave 3 in x instead of the 2 the programmer
probably expects. For this reason it is generally best to avoid side-effects in
macro arguments, and to mark macro names (e.g. by capitalization) to clearly
distinguish them from function names, where this issue doesn’t come up.
187
4.13.1.3
Variable-length argument lists
C99 added variadic macros that may have a variable number of arguments;
these are mostly useful for dealing with variadic functions (like printf) that
also take a variable number of arguments.
To define a variadic macro, define a macro with arguments where the last
argument is three periods: ... . The macro __VA_ARGS__ then expands to
whatever arguments matched this ellipsis in the macro call.
For example:
#include <stdio.h>
#define Warning(...) fprintf(stderr, __VA_ARGS__)
int
main(int argc, char **argv)
{
Warning("%s: this program contains no useful code\n", argv[0]);
}
return 1;
It is possible to mix regular arguments with ..., as long as ... comes last:
#define Useless(format, ...) printf(format, __VA_ARGS__)
4.13.1.4
Macros vs. inline functions
It is sometimes tempting to use a macro to avoid having to retype some small
piece of code that does not seem big enough to justify a full-blown function,
especially if the cost of the body of the function is small relative to the cost of a
function call. Inline functions are a mechanism that is standard in C99 (and
found in some compilers for older variants of C) that give you the ability to
write a function that will never pay this function call overhead; instead, any call
to an inline function is effectively replaced by the body of the function. Unlike
parameterized macros, inline functions do not suffer from issues with duplicated
parameters or weird text-substitution oddities.
To take a simple example, the distSquared function that we used to illustrate
function definitions doesn’t do very much: just two multiplications and an
addition. If we are doing a lot of distSquared computations, we could easily
double the cost of each computation with function call overhead. One alternative
might be to use a macro:
#define DistSquared(x,y) ((x)*(x)+(y)*(y))
but this suffers from the parameter-duplication problem, which could be particularly unfortunate if we compute DistSquared(expensiveFunctionWithManySideEffects(),
188
12). A better alternative is to use an inline function.
Like macros, inline functions should be defined in header files. Ordinary functions
always go in C files because (a) we only want to compile them once, and (b)
the linker will find them in whatever .o file they end up in anyway. But inline
functions generally don’t get compiled independently, so this doesn’t apply.
Here is a header file for an inline version of distSquared:
/* Returns the square of the distance between two points separated by
dx in the x direction and dy in the y direction. */
static inline int
distSquared(int dx, int dy)
{
return dx*dx + dy*dy;
}
examples/functions/distSquaredInline.h
This looks exactly like the original distSquared, except that we added static
inline. We want this function to be declared static because otherwise some
compilers will try to emit a non-inline definition for it in ever C file this header
is included in, which could have bad results.15
The nice thing about this approach is that if we do decide to make distSquared
an ordinary function (maybe it will make debugging easier, or we realize we
want to be able to take its address), then we can just move the definition into
a .c file and take the static inline off. Indeed, this is probably the safest
thing to start with, since we can also do the reverse if we find that function call
overhead on this particular function really does account for a non-trivial part of
our running time (see profiling).
4.13.2
Macros that include other macros
One macro can expand to another; for example, after defining
#define FOO BAR
#define BAR (12)
it will be the case that FOO will expand to BAR which will then expand to (12).
For obvious reasons, it is a bad idea to have a macro expansion contain the
original macro name.
15 This is also the simplest way to deal with the inconsistencies between different compilers
in how they handle inline functions. For an extensive discussion of the terrifying portability
issues that arise in pre-C99 C compilers, see http://www.greenend.org.uk/rjk/tech/inline.html.
189
4.13.3
More specialized macros
Some standard idioms have evolved over the years to deal with issues that come
up in defining complex macros. Usually, having a complex macro is a sign of
bad design, but these tools can be useful in some situations.
4.13.3.1
Multiple expressions in a macro
Use the comma operator, e.g.
#define NoisyInc(x) (puts("incrementing"), (x)++)
The comma operator evaluates both of its operands and returns the value of the
one on the right-hand side.
You can also choose between alternatives using the ternary ?: operator, as in
#define Max(a,b) ((a) > (b) ? (a) : (b))
(but see the warning about repeated parameters above).
4.13.3.2
Non-syntactic macros
Suppose you get tired of writing
for(i = 0; i < n; i++) ...
all the time. In principle, you can write a macro
#define UpTo(i, n) for((i) = 0; (i) < (n); (i)++)
and then write
UpTo(i, 10) ...
in place of your former for loop headers. This is generally a good way to
make your code completely unreadable. Such macros are called non-syntactic
because they allow code that doesn’t look like syntactically correct C.
Sometimes, however, it makes sense to use non-syntactic macros when you want
something that writes to a variable without having to pass it to a function as a
pointer. An example might be something like this malloc wrapper:
#define TestMalloc(x) ((x) = malloc(sizeof(*x)), assert(x))
(Strictly speaking, this is probably more of a “non-semantic” macro.)
Whether the confusion of having a non-syntactic macro is worth the gain in
safety or code-writing speed is a judgment call that can only be made after long
and painful experience. If in doubt, it’s probably best not to do it.
190
4.13.3.3
Multiple statements in one macro
If you want to write a macro that looks like a function call but contains multiple
statements, the correct way to do it is like
#define HiHi() do { puts("hi"); puts("hi"); } while(0)
This can safely be used in place of single statements, like this:16
if(friendly)
HiHi();
else
snarl();
Note that no construct except do..while will work here. Just using braces
will cause trouble with the semicolon before the else, and no other compound
statement besides do..while expects to be followed by a semicolon in this way.
4.13.3.4
String expansion
Let’s rewrite NoisyInc to include the variable name:
#define BadNoisyInc2(x) (puts("Incrementing x"), x++)
Will this do what we want? No. The C preprocessor is smart enough not to
expand macro parameters inside strings, so BadNoisyInc2(y) will expand to
(puts("Incrementing x"), y++). Instead, we have to write
#define NoisyInc2(x) (puts("Incrementing " #x), x++)
Here #x expands to whatever the value of x is wrapped in double quotes. The
resulting string constant is then concatenated with the adjacent string constant
according to standard C string constant concatenation rules.
To concatenate things that aren’t strings, use the ## operator, as in
#define FakeArray(n) fakeArrayVariableNumber ## n
This lets you write FakeArray(12) instead of fakeArrayVariableNumber12.
Note that there is generally no good reason to ever do this.
Where this feature does become useful is if you want to be able to refer to part
of the source code of your program. For example, here is short program that
includes a macro that prints the source code and value of an expression:
#include <stdio.h>
#define PrintExpr(x) (printf("%s = %d\n", #x, (x)))
int
16 To make the example work, we are violating our usual rule of always using braces in if
statements.
191
main(int argc, char **argv)
{
PrintExpr(2+2);
return 0;
}
examples/macros/printExpr.c
When run, this program prints
2+2 = 4
Without using a macro, there is no way to capture the text string "2+2" so we
can print it.
This sort of trickery is mostly used in debugging. The assert macro is a more
sophisticated version, which uses the built-in macros __FILE__ (which expands
to the current source file as a quoted string) and __LINE__ (which expands to
the current source line number, not quoted) to not only print out an offending
expression, but also the location of it in the source.
4.13.3.5
Big macros
Nothing restricts a macro expansion to a single line, although you must put a
backslash at the end of each line to keep it going. Here is a macro that declares
a specialized sorting routine for any type that supports <:
#define DeclareSort(prefix, type) \
static int \
_DeclareSort_ ## prefix ## _Compare(const void *a, const void *b) \
{ \
const type *aa; const type *bb; \
aa = a; bb = b; \
if(*aa < *bb) return -1; \
else if(*bb < *aa) return 1; \
else return 0; \
} \
\
void \
prefix ## _sort(type *a, int n)\
{ \
qsort(a, n, sizeof(type), _DeclareSort_ ## prefix ## _Compare); \
}
examples/macros/declareSort.h
A typical use might be
#include <stdio.h>
#include <stdlib.h>
192
#include "declareSort.h"
/* note: must appear outside of any function, and has no trailing semicolon */
DeclareSort(int, int)
#define N (50)
int
main(int argc, char **argv)
{
int a[N];
int i;
for(i=0; i < N; i++) {
a[i] = N-i;
}
int_sort(a, N);
for(i=0; i < N; i++) {
printf("%d ", a[i]);
}
putchar('\n');
}
return 0;
examples/macros/useDeclareSort.c
Do this too much and you will end up reinventing C++ templates, which are a
more or less equivalent mechanism for generating polymorphic code that improve
on C macros like the one above by letting you omit the backslashes.
4.13.4
Conditional compilation
In addition to generating code, macros can be used for conditional compiliation, where a section of the source code is included only if a particular macro is
defined. This is done using the #ifdef and #ifndef preprocessor directives. In
its simplest form, writing #ifdef NAME includes all code up to the next #endif
if and only if NAME is defined. Similarly, #ifndef NAME includes all code up to
the next #endif if and only if NAME is not defined.
Like regular C if statements, #ifdef and #ifndef directives can be nested, and
can include else cases, which are separated by an #else directive.
#include <stdio.h>
193
#include <assert.h>
int
main(int argc, char **argv)
{
#ifdef SAY_HI
puts("Hi.");
#else /* matches #ifdef SAY_HI */
#ifndef BE_POLITE
puts("Go away!");
#else /* matches #ifndef BE_POLITE */
puts("I'm sorry, I don't feel like talking today.");
#endif /* matches #ifndef BE_POLITE */
#endif /* matches #ifdfe SAY_HI */
#ifdef DEBUG_ARITHMETIC
assert(2+2 == 5);
#endif
}
return 0;
examples/macros/ifdef.c
4.13.5
Defining macros on the command line
You can turn these conditional compilation directives on and off at compile
time by passing the -D flag to gcc. Here is the program above, running after
compiling with different choices of options:
$ gcc -DSAY_HI -o ifdef ifdef.c
$ ./ifdef
Hi.
$ gcc -DBE_POLITE -DDEBUG_ARITHMETIC -o ifdef ifdef.c
$ ./ifdef
I'm sorry, I don't feel like talking today.
ifdef: ifdef.c:18: main: Assertion `2+2 == 5' failed.
Aborted
An example of how this mechanism can be useful is the NDEBUG macro: if you
define this before including assert.h, it turns every assert in your code into
a no-op. This can be handy if you are pretty sure your code works and you
want to speed it up in its final shipped version, or if you are pretty sure your
code doesn’t work but you want to hide the evidence. (It also means you should
not perform side-effects inside an assert unless you are happy with them not
happening.)
194
Using the flag -DNAME defines NAME to be 1. If you want something else, use
-DNAME=VALUE. This can be used to bake useful information into your program at
compile time, and is often used to specify filenames. Below is a simple example.
#include <stdio.h>
int
main(int argc, char **argv)
{
#ifdef MESSAGE
puts(MESSAGE);
#endif
}
return 0;
examples/macros/message.c
$ gcc -DMESSAGE='"Hi there!"' -o message message.c
$ ./message
Hi there!
Note that we had to put an extra layer of single quotes in the command line to
keep the shell from stripping off the double quotes. This is unavoidable: had we
written puts("MESSAGE") in the code, the preprocessor would have recognized
that MESSAGE appeared inside a string and would not have replaced it.17
4.13.6
The #if directive
The preprocessor also includes a more general #if directive that evaluates simple
arithmetic expressions. The limitations are that it can only do integer arithmetic
(using the widest signed integer type available to the compiler) and can only do
it to integer and character constants and the special operator defined(NAME),
which evaluates to 1 if NAME is defined and 0 otherwise. The most common use
of this is to combine several #ifdef-like tests into one:
#include <stdio.h>
int
main(int argc, char **argv)
{
#if VERBOSITY >= 3 && defined(SAY_HI)
puts("Hi!");
17 The # operator looks like it ought to be useful here, but it only works for expanding
arguments to macros and not for expanding macros themselves. Attempting to get around
this by wrapping MESSAGE in a macro that applies the # operator to its first argument will end
in tears if MESSAGE contains any special characters like commas or right parentheses. The C
preprocessor has many unfortunate limitations.
195
#endif
}
return 0;
examples/macros/if.c
4.13.7
Debugging macro expansions
One problem with using a lot of macros is that you can end up with no idea what
input is actually fed to the compiler after the preprocessor is done with it. You
can tell gcc to tell you how everything expands using gcc -E source_file.c.
If your source file contains any #include statements it is probably a good idea
to send the output of gcc -E to a file so you can scroll down past the thousands
of lines of text they may generate.
4.13.8
Can a macro call a preprocessor command?
E.g., can you write something like
#define DefinePlus1(x, y)
#define IncludeLib(x)
#define x ((y)+1)
#include "lib/" #x
The answer is no. C preprocessor commands are only recognized in unexpanded
text. If you want self-modifying macros you will need to use a fancier macro
processor like m4.
5
Data structures and programming techniques
Up until this point we have mostly concentrated on the details of the C programming language. In this part of the notes, we will be looking more at how to
construct data structures and how to organize a program. In principle, these
techniques can be applied to any programming language that supports the appropriate low-level data types, but we will continue to emphasize issues involved
with implementation in C.
5.1
Asymptotic notation
Asymptotic notation is a tool for measuring the growth rate of functions,
which for program design usually means the way in which the time or space
costs of a program scale with the size of the input. We’ll start with an example
of why this is important.
196
5.1.1
Two sorting algorithms
Suppose we want to sort in increasing order a deck of n cards, numbered 1
through n. Here are two algorithms for doing this.
In the ‘”mergesort”’ algorithm, we start with n piles of one card each. We then
take pairs of piles and merge them together, by repeatedly pulling the smaller
of the two smallest cards off the top of the pile and putting it on the bottom
of our output pile. After the first round of this, we have n/2 piles of two cards
each. After another round, n/4 piles of four cards each, and so on until we get
one pile with n cards after roughly log2 n rounds of merging.
Here’s a picture of this algorithm in action on 8 cards:
5 7 1 2 3 4 8 6
57 12 34 68
1257 3468
12345678
Suppose that we want to estimate the cost of this algorithm without actually
coding it up. We might observe that each time a card is merged into a new pile,
we need to do some small, fixed number of operations to decide that it’s the
smaller card, and then do an additional small, fixed number of operations to
physically move it to a new place. If we are really clever, we might notice that
since the size of the pile a card is in doubles with each round, there can be at
most dlog2 ne rounds until all cards are in the same pile. So the cost of getting
a single card in the right place will be at most c log n where c counts the “small,
fixed” number of operations that we keep mentioning, and the cost of getting
every card in the right place will be at most cn log n.
In the ‘”selection sort”’ algorithm, we look through all the cards to find the
smallest one, swap it to the beginning of the list, then look through the remaining
cards for the second smallest, swap it to the next position, and so on.
Here’s a picture of this algorithm in action on 8 cards:
57123486
17523486
12573486
12375486
12345786
197
12345786
12345687
12345678
This is a simpler algorithm to implement that mergesort, but it is usually slower
on large inputs. We can formalize this by arguing that each time we scan k cards
to find the smallest, it’s going to take some small, fixed number of operations
to test each card against the best one we found so far, and an additional small,
fixed number of operations to swap the smallest card to the right place. To
compute the total cost we have to add these costs for all cards, which will give
us a total cost that looks something like (c1 n + c2 ) + (c1 (n − 1) + c2 ) + (c1 (n −
2) + c2 ) + . . . + (c11̇ + c2 ) = c1 n(n + 1)/2 + c2 n.
For large n, it looks like this is going to cost more than mergesort. But how can
we make this claim cleanly, particularly if we don’t know the exact values of c,
c1 , and c2 ?
5.1.2
Big-O to the rescue
The idea is to replace complex running time formulae like cn log n or c1 n(n +
1)/2 + c2 n with an asymptotic growth rate O(n log n) or O(n2 ). These asymptotic growth rates omit the specific details of exactly how fast our algorithms
run (which we don’t necessarily know without actually coding them up) and
concentrate solely on how the cost scales as the size of the input n becomes
large.
This avoids two issues:
1. Different computers run at different speeds, and we’d like to be able to say
that one algorithm is better than another without having to measure its
running time on specific hardware.
2. Performance on large inputs is more important than performance on small
inputs, since programs running on small inputs are usually pretty fast.
The idea of ‘”asymptotic notation”’ is to consider the shape of the worst-case
cost T (n) to process an input of size n. Here, worst-case means we consider
the input that gives the greatest cost, where cost is usually time, but may be
something else like space. To formalize the notion of shape, we define classes of
functions that behave like particular interesting functions for large inputs. The
definition looks much like a limit in calculus:
O(n) A function f (n) is in the class O(g(n)) if there exist constants N and c
such that f (n) < c · g(n) when n > N .
If f (n) is in O(g(n)) we say f (n) is ‘”big-O”’ of g(n) or just f (n) = O(g(n)).18
18 This
is an abuse of notation, where the equals sign is really acting like set membership.
198
Unpacked, this definition says that f (n) is less than a constant times g(n) when
n is large enough.
Some examples:
• Let f (n) = 3n + 12, and let g(n) = n. To show that f (n) is in O(g(n)) =
O(n), we can pick whatever constants we want for c and N (as long as
they work). So let’s make N be 100 and c be 4. Then we need to show
that if n > 100, 3n + 12 < 4n. But 3n + 12 < 4n holds precisely when
12 < n, which is implied by our assumption that n > 100.
• Let f (n) = 4n2 + 23n + 15, and let g(n) = n2 . Now let N be 100 again
and c be 5. So we need 4n2 + 23n + 15 < 5n2 , or 23n + 15 < n2 . But
n > 100 means that n2 > 100n = 50n + 50n > 50n + 5000 > 23n + 15,
which proves that f (n) is in O(n2 ).
• Let f (n) < 146 for all n, and let g(n) = 1. Then for N = 0 and c = 146,
f (n) < 146 = 146g(n), and f (n) is in O(1).
Writing proofs like this over and over again is a nuisance, so we can use some
basic rules of thumb to reduce messy functions f (n) to their asymptotic forms:
• If c is a constant (doesn’t depend on n), then c · f (n) = O(f (n)). This
follows immediately from being able to pick c in the definition. So we can
always get rid of constant factors: 137n5 = O(n5 ).
• If f (n) = g(n) + h(n), then the bigger of g(n) or h(n) wins. This is because
if g(n) ≤ h(n), then g(n) + h(n) ≤ 2g(n), and then big-O eats the 2. So
12n2 + 52n + 3 = O(n2 ) because n2 dominates all the other terms.
• To figure out which of two terms dominates, the rule is
– Bigger exponents win: If a < b, then O(na ) + O(nb ) = O(nb ).
– Polynomials beat logarithms: For any a and any b > 0, O(loga n) +
O(nb ) = O(nb ).
– Exponentials beat polynomials: For any a and any b > 1, O(na ) +
O(bn ) = O(bn ).
– The distributive law works: Because O(log n) dominates O(1),
O(n log n) dominates O(n).
This means that almost any asymptotic bound can be reduced down to one of a
very small list of common bounds. Ones that you will typically see in practical
algorithms, listed in increasing order, are O(1), O(log n), O(n), O(n log n), or
O(n2 ).
Applying these rules to mergesort and selection sort gives us asymptotic bounds
of cn log n = O(n log n) (the constant vanishes) and c1 n(n + 1)/2 + c2 n =
c1 n2 /2 + c1 n/2 + c2 n = O(n2 ) + O(n) + O(n) = O(n2 ) (the constants vanish
and then O(n2 ) dominates). Here we see that no matter how fast our machine
is at different low-level operations, for large enough inputs mergesort will beat
selection sort.
The general rule is that an expression O(f (n)) = O(g(n)) is true if for any choice of a function
in O(f (n)), that function is in O(g(n)). This relation is transitive and symmetric, but unlike
real equality it’s not symmetric.
199
5.1.3
Asymptotic cost of programs
To compute the asymptotic cost of a program, the rule of thumb is that any
simple statement costs O(1) time to evaluate, and larger costs are the result of
loops or calls to expensive functions, where a loop multiplies the cost by the
number of iterations in the loop. When adding costs together, the biggest cost
wins:
So this function takes O(1) time:
/* return the sum of the integers i with 0 <= i
int
sumTo(int n)
{
return n*(n-1)/2;
}
and i < n */
But this function, which computes exactly the same value, takes O(n) time:
/* return the sum of the integers i with 0 <= i
int
sumTo(int n)
{
int i;
int sum = 0;
and i < n */
for(i = 0; i < n; i++) {
sum += i;
}
}
return sum;
The reason it takes so long is that each iteration of the loop takes only O(1)
time, but we execute the loop n times, and n · O(1) = O(n).
Here’s an even worse version that takes O(n2 ) time:
/* return the sum of the integers i with 0 <= i
int
sumTo(int n)
{
int i;
int j;
int sum = 0;
for(i = 0; i < n; i++) {
for(j = 0; j < i; j++) {
sum++;
200
and i < n */
}
}
}
return sum;
Here we have two nested loops. The outer loop iterates exactly n times, and
for each iteration the inner loop iterates at most n times, and the innermost
iteration costs O(1) each time, so the total is at most O(n2 ). (In fact, it’s no
better than this, because at least n/2 times we execute the inner loop, we do at
least n/2 iterations.)
So even if we knew that the constant on the first implementation was really large
(maybe our CPU is bad at dividing by 2?), for big values of n it’s still likely to
be faster than the other two.
(This example is a little misleading, because n is not the size of the input but the
actual input value. More typical might be a statement that the cost of strlen
is O(n) where n is the length of the string.)
5.1.4
Other variants of asymptotic notation
Big-O notation is good for upper bounds, but the inequality in the definition
means that it can’t be used for anything else: it is the case that 12 = O(n67 )
just because 12 < n67 when n is large enough. There is an alternative definition,
called ‘”big-Omega”’, that works in the other direction:
Ω(n) A function f (n) is in the class Ω(g(n)) if there exist constants N and c
such that f (n) > c · g(n) when n > N .
This is exactly the same as the definition of O(g(n)) except that the inequality
goes in the other direction. So if we want to express that some algorithm is very
expensive, we might write that it’s Ω(n2 ), which says that once the size of the
input is big enough, then the cost grows at least as fast as n2 .
If you want to claim that your bound is tight—both an upper and a lower
bound—use big-Theta: f (n) is Θ(g(n)) if it is both O(f (n)) and Ω(g(n)).
Mostly we will just use big-O, with the understanding that when we say that a
particular algorithm is O(n), that’s the best bound we could come up with.
5.2
Linked lists
Linked lists are about the simplest data structure beyond arrays. They aren’t
very efficient for many purposes, but have very good performance for certain
specialized applications.
201
The basic idea is that instead of storing n items in one big array, we store each
item in its own struct, and each of these structs includes a pointer to the
next struct in the list (with a null pointer to indicate that there are no more
elements). If we follow the pointers we can eventually reach all of the elements.
For example, if we declare the struct holding each element like this:
struct elt {
struct elt *next;
int contents;
};
/* pointer to next element in the list */
/* contents of this element */
We can build a structure like this:
The box on the far left is not a struct elt, but a struct elt *; in order to
keep track of the list we need a pointer to the first element. As usual in C, we
will have to do all the work of allocating these elements and assigning the right
pointer values in the right places ourselves.
5.2.1
Stacks
The selling point of linked lists in comparison to arrays is that inserting or
removing elements can be cheap: at the front of the list, inserting a new element
just requires allocating another struct and hooking up a few pointers, while
removing an element just requires moving the pointer to the first element to
point to the second element instead, and then freeing the first element.
For example here’s what happens the linked list above looks like after we insert
a new element at the front:
To make this work, we need to change two pointers: the head pointer and the
next pointer in the new element holding 0. These operations aren’t affected by
the size of the rest of the list and so take O(1) time.
202
Removal is the reverse of installation: We patch out the first element by shifting
the head pointer to the second element, then deallocate it with free. (We do
have to be careful to get any data we need out of it before calling free). This is
also an O(1) operation.
The fact that we can add and remove elements at the start of linked lists for
cheap makes them particularly useful for implementing a stack, an abstract
data type that supports operations push (insert a new element on the top of the
stack) and pop (remove and return the element at the top of the stack. Here
is an example of a simple linked-list implementation of a stack, together with
some test code:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
struct elt {
struct elt *next;
int value;
};
/*
* We could make a struct for this,
* but it would have only one component,
* so this is quicker.
*/
typedef struct elt *Stack;
#define STACK_EMPTY (0)
/* push a new value onto top of stack */
void
stackPush(Stack *s, int value)
{
struct elt *e;
e = malloc(sizeof(struct elt));
assert(e);
}
e->value = value;
e->next = *s;
*s = e;
int
stackEmpty(const Stack *s)
{
203
}
return (*s == 0);
int
stackPop(Stack *s)
{
int ret;
struct elt *e;
assert(!stackEmpty(s));
ret = (*s)->value;
/* patch out first element */
e = *s;
*s = e->next;
free(e);
}
return ret;
/* print contents of stack on a single line */
void
stackPrint(const Stack *s)
{
struct elt *e;
for(e = *s; e != 0; e = e->next) {
printf("%d ", e->value);
}
}
putchar('\n');
int
main(int argc, char **argv)
{
int i;
Stack s;
s = STACK_EMPTY;
for(i = 0; i < 5; i++) {
printf("push %d\n", i);
stackPush(&s, i);
204
}
stackPrint(&s);
while(!stackEmpty(&s)) {
printf("pop gets %d\n", stackPop(&s));
stackPrint(&s);
}
}
return 0;
examples/linkedLists/stack.c
Unlike most of our abstract data types, we do not include a struct representing
the linked list itself. This is because the only thing we need to keep track of a
linked list is the head pointer, and it feels a little silly to have a struct with
just one component. But we might choose to do this if we wanted to make the
linked list implementation opaque or allow for including more information later.
struct stack {
struct elt *head;
};
5.2.1.1
Building a stack out of an array
When the elements of a stack are small, or when a maximum number of elements
is known in advance, it often makes sense to build a stack from an array (with
a variable storing the index of the top element) instead of a linked list. The
reason is that pushes and pops only require updating the stack pointer instead
of calling malloc or free to allocate space, and pre-allocating is almost always
faster than allocating as needed. This is the strategy used to store the function
call stack in almost all programs (the exception is in languages like Scheme,
where the call stack is allocated on the heap because stack frames may outlive
the function call that creates them).
5.2.2
Queues
Stacks are last-in-first-out (LIFO) data structures: when we pop, we get the
last item we pushed. What if we want a first-in-first-out (FIFO) data structure?
Such a data structure is called a queue and can also be implemented by a linked
list. The difference is that if we want O(1) time for both the enqueue (push)
and dequeue (pop) operations, we must keep around pointers to both ends of
the linked list.
So now we get something that looks like this:
205
Enqueuing a new element typically requires (a) allocating a new struct to hold
it; (b) making the old tail struct point at the new struct; and (c) updating
the tail pointer to also point to the new struct. There is a minor complication
when the stack is empty; in this case instead of updating tail->next we must put
a pointer to the new struct in head. Dequeuing an element involves updating
the head pointer and freeing the removed struct, exactly like a stack pop.
Here is the queue above after enqueuing a new element 6. The updated pointers
are indicated by dotted lines:
Because we are only changing two pointers, each of which we can reach by
following a constant number of pointers from the main struct, we can do this
in O(1) time.
There is a slight complication when we enqueue the very first element, because
we need to update the head pointer instead of the pointer in the previous tail
(which doesn’t yet exist). This requires testing for an empty queue in the enqueue
routine, which we’ll do in the sample code below.
Dequeuing is easier because it requires updating only one pointer:
If we adopt the convention that a null in head means an empty queue, and use
this property to check if the queue is empty when enqueuing, we don’t even have
to clear out tail when we dequeue the last element.
Here is a simple implementation of a queue holding ints, together with some
test code showing how its behavior differs from a stack:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
206
struct elt {
struct elt *next;
int value;
};
struct queue {
struct elt *head;
struct elt *tail;
};
typedef struct queue *q;
struct queue *
queueCreate(void)
{
struct queue *q;
q = malloc(sizeof(struct queue));
q->head = q->tail = 0;
}
return q;
/* push a new value onto top of Queue */
void
enq(struct queue *q, int value)
{
struct elt *e;
e = malloc(sizeof(struct elt));
assert(e);
e->value = value;
/* Because I will be the tail, nobody is behind me */
e->next = 0;
if(q->head == 0) {
/* If the queue was empty, I become the head */
q->head = e;
} else {
/* Otherwise I get in line after the old tail */
q->tail->next = e;
207
}
}
/* I become the new tail */
q->tail = e;
int
queueEmpty(const struct queue *q)
{
return (q->head == 0);
}
int
deq(struct queue *q)
{
int ret;
struct elt *e;
assert(!queueEmpty(q));
ret = q->head->value;
/* patch out first element */
e = q->head;
q->head = e->next;
free(e);
}
return ret;
/* print contents of queue on a single line, head first */
void
queuePrint(struct queue *q)
{
struct elt *e;
for(e = q->head; e != 0; e = e->next) {
printf("%d ", e->value);
}
}
putchar('\n');
/* free a queue and all of its elements */
void
208
queueDestroy(struct queue *q)
{
while(!queueEmpty(q)) {
deq(q);
}
}
free(q);
int
main(int argc, char **argv)
{
int i;
struct queue *q;
q = queueCreate();
for(i = 0; i < 5; i++) {
printf("enq %d\n", i);
enq(q, i);
queuePrint(q);
}
while(!queueEmpty(q)) {
printf("deq gets %d\n", deq(q));
queuePrint(q);
}
queueDestroy(q);
}
return 0;
examples/linkedLists/queue.c
It is a bit trickier to build a queue out of an array than to build a stack. The
difference is that while a stack pointer can move up and down, leaving the base of
the stack in the same place, a naive implementation of a queue would have head
and tail pointers both marching ever onward across the array leaving nothing
but empty cells in their wake. While it is possible to have the pointers wrap
around to the beginning of the array when they hit the end, if the queue size is
unbounded the tail pointer will eventually catch up to the head pointer. At this
point (as in a stack that overflows), it is necessary to allocate more space and
copy the old elements over. See the section on ring buffers for an example of
how to do this.
209
5.2.3
Looping over a linked list
Looping over a linked list is not hard if you have access to the next pointers.
(For a more abstract way to do this see iterators.)
Let’s imagine somebody gave us a pointer to the first struct stack in a list; call
this pointer first. Then we can write a loop like this that prints the contents
of the stack:
void
stackPrint(struct stack *first)
{
struct stack *elt;
}
for(elt = first; elt != 0; elt = elt->next) {
puts(elt->book);
}
There’s not a whole lot to notice here except that for is perfectly happy to
iterate over something that isn’t a range of integers. The running time is linear
in the length of the list (O(n)).
5.2.4
Looping over a linked list backwards
What if we want to loop over a linked list backwards? The next pointers all go
the wrong way, so we have to save a trail of breadcrumbs to get back. The safest
way to do this is to reverse the original list into an auxiliary list:
void
stackPrintReversed(struct stack *first)
{
struct stack *elt;
Stack s2;
/* uses imperative implementation */
s2 = stackCreate();
for(elt = first; elt != 0; elt = elt->next) {
s2 = stackPush(s2, elt->book);
}
}
stackPrint(s2);
stackDestroy(s2);
Pushing all the elements from the first list onto s2 puts the first element on the
bottom, so when we print s2 out, it’s in the reverse order of the original stack.
210
We can also write a recursive function that prints the elements backwards. This
function effectively uses the function call stack in place of the extra stack s2
above.
void
stackPrintReversedRecursive(struct stack *first)
{
if(first != 0) {
/* print the rest of the stack */
stackPrintReversedRecursive(first->next);
}
}
/* then print the first element */
puts(first->book);
The code in stackPrintReversedRecursive is shorter than the code in
stackPrintReversed, and it is likely to be faster since it doesn’t require
allocating a second stack and copying all the elements. But it will only work for
small stacks: because the function call stack is really a fixed-size array, if the
input to stackPrintReversedRecursive is too big the recursion will go too
deep cause a stack overflow.
If we want to do this sort of thing a lot, we should build a doubly-linked list,
with a pointer in each element both to the next element and the previous element
instead of a singly-linked list (see below for more).
5.2.5
Deques and doubly-linked lists
Suppose we want a data structure that represents a line of elements where we
can push or pop elements at either end. Such a data structure is known as a
deque (pronounced like “deck”), and can be implemented with all operations
taking O(1) time by a doubly-linked list, where each element has a pointer to
both its successor and its predecessor.
An ordinary singly-linked list is not good enough. The reason is that even if we
keep a pointer to both ends as in a queue, when it comes time to pop an element
off the tail, we have no pointer to its predecessor ready to hand; the best we can
do is scan from the head until we get to an element whose successor is the tail,
which takes O(n) time.
So instead we need a doubly-linked list, where each node points to both its
successor and predecessor. The most straightforward way to build this is to
make it circular, and use a dummy node to represent the head of the list. The
resulting data structure might look something like this:
211
Below is an implementation of this structure. We have separated the interface in
deque.h from the implementation in deque.c. This will allow us to change the
implementation if we decide we don’t like it, without affecting any other code in
the system.
A nice feature of this data structure is that we don’t need to use null pointers to
mark the ends of the deque. Instead, each end is marked by a pointer to the
dummy head element. For an empty deque, this just means that the head points
to itself. The cost of this is that to detect an empty deque we have to test for
equality with the head (which might be slightly more expensive that just testing
for null) and the head may contain some wasted space for its missing value if we
allocate it like any other element.19
To keep things symmetric, we implement the pointers as an array, indexed by
the directions DEQUE_FRONT and DEQUE_BACK (defined in deque.h). This means
we can use the same code to push or pop on either end of the deque.
typedef struct deque Deque;
#define DEQUE_FRONT (0)
#define DEQUE_BACK (1)
#define DEQUE_EMPTY (-1)
/* returned by dequePop if deque is empty */
/* return a new empty deque */
Deque *dequeCreate(void);
/* push new value onto direction side of deque d */
void dequePush(Deque *d, int direction, int value);
/* pop and return first value on direction side of deque d */
19 The example below uses the offsetof macro, defined in stddef.h, to allocate a truncated
head that doesn’t include this extra space. This is probably more trouble than it is worth in
this case, but might be useful if we were creating a lot of dummy heads and the contents were
more than 4 bytes long.
212
/* returns DEQUE_EMPTY if deque is empty */
int dequePop(Deque *d, int direction);
/* return 1 if deque contains no elements, 0 otherwise */
int dequeIsEmpty(const Deque *d);
/* free space used by a deque */
void dequeDestroy(Deque *d);
examples/linkedLists/deque/deque.h
#include <stdlib.h>
#include <assert.h>
#include <stddef.h>
/* for offsetof */
#include "deque.h"
#define NUM_DIRECTIONS (2)
struct deque {
struct deque *next[NUM_DIRECTIONS];
int value;
};
Deque *
dequeCreate(void)
{
Deque *d;
/*
* We don't allocate the full space for this object
* because we don't use the value field in the dummy head.
*
* Saving these 4 bytes doesn't make a lot of sense here,
* but it might be more significant if value where larger.
*/
d = malloc(offsetof(struct deque, value));
/* test is to deal with malloc failure */
if(d) {
d->next[DEQUE_FRONT] = d->next[DEQUE_BACK] = d;
}
}
return d;
void
213
dequePush(Deque *d, int direction, int value)
{
struct deque *e; /* new element */
assert(direction == DEQUE_FRONT || direction == DEQUE_BACK);
e = malloc(sizeof(struct deque));
assert(e);
e->next[direction] = d->next[direction];
e->next[!direction] = d;
e->value = value;
}
d->next[direction] = e;
e->next[direction]->next[!direction] = e;
/* preserves invariant */
int
dequePop(Deque *d, int direction)
{
struct deque *e;
int retval;
assert(direction == DEQUE_FRONT || direction == DEQUE_BACK);
e = d->next[direction];
if(e == d) {
return DEQUE_EMPTY;
}
/* else remove it */
d->next[direction] = e->next[direction];
e->next[direction]->next[!direction] = d;
retval = e->value;
free(e);
}
return retval;
int
dequeIsEmpty(const Deque *d)
{
return d->next[DEQUE_FRONT] == d;
214
}
void
dequeDestroy(Deque *d)
{
while(!dequeIsEmpty(d)) {
dequePop(d, DEQUE_FRONT);
}
}
free(d);
examples/linkedLists/deque/deque.c
And here is some test code:
testDeque.c.
5.2.5.1
Alternate implementation using a ring buffer
The deque.h file carefully avoids revealing any details of the implementation
of a deque. This allows us to replace the implementation with a different
implementation that is more efficient in its use of both time and space, at the
cost of additional code complexity. Below is a replacement for deque.c that
uses a ring buffer in place of the circular linked list.
The idea of a ring buffer is to store the deque elements in an array, with a pointer
to the first element and a length field that says how many elements are in the
deque. The information needed to manage the array (which is allocated using
malloc) is stored in a struct.
The sequence of elements wraps around the endpoints of the array, leaving a gap
somewhere in the middle. Deque pushes extend the sequence into this gap from
one side or another, while pops increase the size of the gap. If the user wants
to do a push and the array is full, we build a new, larger deque, move all the
elements there, and then transplant all the bits of the new struct deque into
the old one. This transplant trick avoids changing the address of the struct
deque that the user needs to access it.
#include <stdlib.h>
#include <assert.h>
#include "deque.h"
/*
* Alternative implementation of a deque using a ring buffer.
*
* Conceptually, this is an array whose indices wrap around at
215
* the endpoints.
*
* The region in use is specified by a base index pointing
* to the first element, and a length count giving the number
* of elements. A size field specifies the number of slots
* in the block.
*
* Picture:
*
* --------------------------------------------------* |7|8|9| | | | | | | | | | | | | | | | |1|2|3|4|5|6|
* --------------------------------------------------*
^
^
*
|
|
*
base + length - 1
base
*
*/
struct deque {
size_t base;
size_t length;
size_t size;
int *contents;
};
/* location of front element */
/* length of region in use */
/* total number of positions in contents */
#define INITIAL_SIZE (8)
/* create a new deque of the given size */
static Deque *
dequeCreateInternal(size_t size)
{
struct deque *d;
d = malloc(sizeof(struct deque));
assert(d);
d->base = 0;
d->length = 0;
d->size = size;
d->contents = malloc(sizeof(int) * d->size);
assert(d->contents);
}
return d;
216
/* return a new empty deque */
Deque *
dequeCreate(void)
{
return dequeCreateInternal(INITIAL_SIZE);
}
void
dequePush(Deque *d, int direction, int value)
{
struct deque *d2;
/* replacement deque if we grow */
int *oldContents;
/* old contents of d */
/*
* First make sure we have space.
*/
if(d->length == d->size) {
/* nope */
d2 = dequeCreateInternal(d->size * 2);
/* evacuate d */
while(!dequeIsEmpty(d)) {
dequePush(d2, DEQUE_BACK, dequePop(d, DEQUE_FRONT));
}
/* do a transplant from d2 to d */
/* but save old contents so we can free them */
oldContents = d->contents;
*d = *d2;
/* this is equivalent to copying the components one by one */
}
/* these are the pieces we don't need any more */
free(oldContents);
free(d2);
/*
* This requires completely different code
* depending on the direction, which is
* annoying.
*/
if(direction == DEQUE_FRONT) {
/* d->base is unsigned, so we have to check for zero first */
if(d->base == 0) {
d->base = d->size - 1;
} else {
217
}
d->base--;
d->length++;
}
d->contents[d->base] = value;
} else {
d->contents[(d->base + d->length++) % d->size] = value;
}
/* pop and return first value on direction side of deque d */
/* returns DEQUE_EMPTY if deque is empty */
int
dequePop(Deque *d, int direction)
{
int retval;
if(dequeIsEmpty(d)) {
return DEQUE_EMPTY;
}
/* else */
if(direction == DEQUE_FRONT) {
/* base goes up by one, length goes down by one */
retval = d->contents[d->base];
d->base = (d->base+1) % d->size;
d->length--;
}
return retval;
} else {
/* length goes down by one */
return d->contents[(d->base + --d->length) % d->size];
}
int
dequeIsEmpty(const Deque *d)
{
return d->length == 0;
}
void
dequeDestroy(Deque *d)
{
218
}
free(d->contents);
free(d);
examples/linkedLists/deque/ringBuffer.c
Here is a Makefile that compiles testDeque.c against both the linked list and
the ring buffer implementations. You can do make time to race them against
each other.
CC=gcc
CFLAGS=-std=c99 -Wall -pedantic -O3 -g3
# how many iterations for test
ITERATIONS=10000000
VALGRIND_ITERATIONS=100
all: testDeque testRingBuffer
test: all
./testDeque $(ITERATIONS)
valgrind -q --leak-check=yes ./testDeque $(VALGRIND_ITERATIONS)
./testRingBuffer $(ITERATIONS)
valgrind -q --leak-check=yes ./testRingBuffer $(VALGRIND_ITERATIONS)
time: all
time ./testDeque $(ITERATIONS)
time ./testRingBuffer $(ITERATIONS)
testDeque: testDeque.o deque.o
$(CC) $(CFLAGS) -o $@ $^
testRingBuffer: testDeque.o ringBuffer.o
$(CC) $(CFLAGS) -o $@ $^
clean:
$(RM) testDeque testRingBuffer *.o
examples/linkedLists/deque/Makefile
5.2.6
Circular linked lists
For some applications, there is no obvious starting or ending point to a list, and a
circular list (where the last element points back to the first) may be appropriate.
Circular doubly-linked lists can also be used to build deques; a single pointer
into the list tracks the head of the deque, with some convention adopted for
219
whether the head is an actual element of the list (at the front, say, with its left
neighbor at the back) or a dummy element that is not considered to be part of
the list.
The selling point of circular doubly-linked lists as a concrete data structure is
that insertions and deletions can be done anywhere in the list with only local
information. For example, here are some routines for manipulating a doublylinked list directly. We’ll make our lives easy and assume (for the moment) that
the list has no actual contents to keep track of.
#include <stdlib.h>
/* directions for doubly-linked list next pointers */
#define RIGHT (0)
#define LEFT (1)
struct elt {
struct elt *next[2];
};
typedef struct elt *Elt;
/* create a new circular doubly-linked list with 1 element */
/* returns 0 on allocation error */
Elt
listCreate(void)
{
Elt e;
e = malloc(sizeof(*e));
if(e) {
e->next[LEFT] = e->next[RIGHT] = e;
}
}
return e;
/* remove an element from a list */
/* Make sure you keep a pointer to some other element! */
/* does not free the removed element */
void
listRemove(Elt e)
{
/* splice e out */
e->next[RIGHT]->next[LEFT] = e->next[LEFT];
e->next[LEFT]->next[RIGHT] = e->next[RIGHT];
}
220
/* insert an element e into list in direction dir from head */
void
listInsert(Elt head, int dir, Elt e)
{
/* fill in e's new neighbors */
e->next[dir] = head->next[dir];
e->next[!dir] = head;
}
/* make neigbhors point back at e */
e->next[dir]->next[!dir] = e;
e->next[!dir]->next[dir] = e;
/* split a list, removing all elements between e1 and e2 */
/* e1 is the leftmost node of the removed subsequence, e2 rightmost */
/* the removed elements are formed into their own linked list */
/* comment: listRemove could be implemented as listSplit(e,e) */
void
listSplit(Elt e1, Elt e2)
{
/* splice out the new list */
e2->next[RIGHT]->next[LEFT] = e1->next[LEFT];
e1->next[LEFT]->next[RIGHT] = e2->next[RIGHT];
}
/* fix up the ends */
e2->next[RIGHT] = e1;
e1->next[LEFT] = e2;
/* splice a list starting at e2 after e1 */
/* e2 becomes e1's right neighbor */
/* e2's left neighbor becomes left neighbor of e1's old right neighbor */
void
listSplice(Elt e1, Elt e2)
{
/* fix up tail end */
e2->next[LEFT]->next[RIGHT] = e1->next[RIGHT];
e1->next[RIGHT]->next[LEFT] = e2->next[LEFT];
}
/* fix up e1 and e2 */
e1->next[RIGHT] = e2;
e2->next[LEFT] = e1;
/* free all elements of the list containing e */
221
void
listDestroy(Elt e)
{
Elt target;
Elt next;
/* we'll free elements until we get back to e, then free e */
/* note use of pointer address comparison to detect end of loop */
for(target = e->next[RIGHT]; target != e; target = next) {
next = target->next[RIGHT];
free(target);
}
}
free(e);
examples/linkedLists/circular.c
The above code might or might not actually work. What if it doesn’t? It may
make sense to include some sanity-checking code that we can run to see if our
pointers are all going to the right place:
/* assert many things about correctness of the list */
/* Amazingly, this is guaranteed to abort or return no matter
how badly screwed up the list is. */
void
listSanityCheck(Elt e)
{
Elt check;
assert(e != 0);
check = e;
do {
/* are our pointers consistent with our neighbors? */
assert(check->next[RIGHT]->next[LEFT] == check);
assert(check->next[LEFT]->next[RIGHT] == check);
/* on to the next */
check = check->next[RIGHT];
}
} while(check != e);
What if we want to store something in this list? The simplest approach is to
extend the definition of struct elt:
222
struct elt {
struct elt *next[2];
char *name;
int socialSecurityNumber;
int gullibility;
};
But then we can only use the code for one particular type of data. An alternative
approach is to define a new Elt-plus struct:
struct fancyElt {
struct elt *next[2];
char *name;
int socialSecurityNumber;
int gullibility;
};
and then use pointer casts to convert the fancy structs into Elts:
struct fancyElt *e;
e = malloc(sizeof(*e));
/* fill in fields on e */
listInsert(someList, (Elt) e);
The trick here is that as long as the initial part of the struct fancyElt looks
like a struct elt, any code that expects a struct elt will happily work with
it and ignore the fields that happen to be sitting later in memory. (This trick is
how C++ inheritance works.)
The downside is that if something needs to be done with the other fields (e.g
freeing e->name if e is freed), then the Elt functions won’t know to do this. So
if you use this trick you should be careful.
A similar technique using void * pointers can be used to implement generic
containers.
5.2.7
What linked lists are and are not good for
Linked lists are good for any task that involves inserting or deleting elements
next to an element you already have a pointer to; such operations can usually
be done in O(1) time. They generally beat arrays (even resizeable arrays) if you
need to insert or delete in the middle of a list, since an array has to copy any
elements above the insertion point to make room; if inserts or deletes always
happen at the end, an array may be better.
223
Linked lists are not good for any operation that requires random access, since
reaching an arbitrary element of a linked list takes as much as O(n) time. For
such applications, arrays are better if you don’t need to insert in the middle; if
you do, you should use some sort of tree.
5.2.8
Further reading
A description of many different kinds of linked lists with pictures can be found
in the WikiPedia article on the subject.
Animated versions can be found at http://www.cs.usfca.edu/~galles/
visualization/Algorithms.html.
5.3
Abstract data types
One of the hard parts about computer programming is that, in general, programs
are bigger than brains. Unless you have an unusally capacious brain, it is unlikely
that you will be able to understand even a modestly large program in its entirety.
So in order to be able to write and debug large programs, it is important to be
able to break it up into pieces, where each piece can be treated as a tool whose
use and description is simpler (and therefor fits in your brain better) than its
actual code. Then you can forget about what is happening inside that piece,
and just treat it as an easily-understood black box from the outside.
This process of wrapping functionality up in a box and forgetting about its
internals is called abstraction, and it is the single most important concept in
computer science. In these notes we will describe a particular kind of abstraction,
the construction of abstract data types or ADTs. Abstract data types are
data types whose implementation is not visible to their user; from the outside,
all the user knows about an ADT is what operations can be performed on it and
what those operations are supposed to do.
ADTs have an outside and an inside. The outside is called the interface; it
consists of the minimal set of type and function declarations needed to use the
ADT. The inside is called the implementation; it consists of type and function
definitions, and sometime auxiliary data or helper functions, that are not visible
to users of the ADT. This separation between interface and implementation
is called the abstraction barrier, and allows the implementation to change
without affecting the rest of the program.
What joins the implementation to the interface is an abstraction function.
This is a function (in the mathematical sense) that takes any state of the
implementation and trims off any irrelevant details to leave behind an idealized
pictures of what the data type is doing. For example, a linked list implementation
translates to a sequence abstract data type by forgetting about the pointers used
to hook up the elements and just keeping the sequence of elements themselves.
224
To exclude bad states of the implementation (for example, a singly-linked list that
loops back on itself instead of having a terminating null pointer), we may have a
representation invariant, which is just some property of the implementation
that is always true. Representation invariants are also useful for detecting
when we’ve bungled our implementation, and a good debugging strategy for
misbehaving abstract data type implementations is often to look for the first
point at which they violated some property that we thought was an invariant.
Some programming language include very strong mechanisms for enforcing
abstraction barriers. C relies somewhat more on politeness, and as a programmer
you violate an abstraction barrier (by using details of an implementation that
are supposed to be hidden) at your peril. In C, the interface will typically consist
of function and type declarations contained in a header file, with implementation
made up of the corresponding function definitions (and possibly a few extra
static functions) in one or more .c files. The opaque struct technique can be
used to hide implementation details of the type.
5.3.1
A sequence type
Too much abstraction at once can be hard to take, so let’s look at a concrete
example of an abstract data type. This ADT will represent an infinite sequence of
ints. Each instance of the Sequence type supports a single operation seq_next
that returns the next int in the sequence. We will also need to provide one
or more constructor functions to generate new Sequences, and a destructor
function to tear them down.
Here is an example of a typical use of a Sequence:
void
seq_print(Sequence s, int limit)
{
int i;
}
for(i = seq_next(s); i < limit; i = seq_next(s)) {
printf("%d\n", i);
}
Note that seq_print doesn’t need to know anything at all about what a
Sequence is or how seq_next works in order to print out all the values in
the sequence until it hits one greater than or equal to limit. This is a good
thing— it means that we can use with any implementation of Sequence we like,
and we don’t have to change it if Sequence or seq_next changes.
5.3.1.1
Interface
225
In C, the interface of an abstract data type will usually be declared in a header
file, which is included both in the file that implements the ADT (so that the
compiler can check that the declarations match up with the actual definitions in
the implementation. Here’s a header file for sequences:
/* opaque struct: hides actual components of struct sequence,
* which are defined in sequence.c */
typedef struct sequence *Sequence;
/* constructors */
/* all our constructors return a null pointer on allocation failure */
/* returns a Sequence representing init, init+1, init+2, ... */
Sequence seq_create(int init);
/* returns a Sequence representing init, init+step, init+2*step, ... */
Sequence seq_create_step(int init, int step);
/* destructor */
/* destroys a Sequence, recovering all interally-allocated data */
void seq_destroy(Sequence);
/* accessor */
/* returns the first element in a sequence not previously returned */
int seq_next(Sequence);
examples/ADT/sequence/sequence.h
Here we have defined two different constructors for Sequences, one of which
gives slightly more control over the sequence than the other. If we were willing
to put more work into the implementation, we could imagine building a very
complicated Sequence type that supported a much wider variety of sequences
(for example, sequences generated by functions or sequences read from files); but
we’ll try to keep things simple for now. We can always add more functionality
later, since the users won’t notice if the Sequence type changes internally.
5.3.1.2
Implementation
The implementation of an ADT in C is typically contained in one (or sometimes
more than one) .c file. This file can be compiled and linked into any program
that needs to use the ADT. Here is our implementation of Sequence:
#include <stdlib.h>
#include "sequence.h"
struct sequence {
226
};
int next;
int step;
/* next value to return */
/* how much to increment next by */
Sequence
seq_create(int init)
{
return seq_create_step(init, 1);
}
Sequence
seq_create_step(int init, int step)
{
Sequence s;
}
s = malloc(sizeof(*s));
if(s == 0) return 0;
s->next = init;
s->step = step;
return s;
void
seq_destroy(Sequence s)
{
free(s);
}
int
seq_next(Sequence s)
{
int ret;
/* saves the old value before we increment it */
ret = s->next;
s->next += s->step;
}
return ret;
examples/ADT/sequence/sequence.c
Things to note here: the definition of struct sequence appears only in this
file; this means that only the functions defined here can (easily) access the
next and step components. This protects Sequences to a limited extent from
outside interference, and defends against users who might try to “violate the
abstraction boundary” by examining the components of a Sequence directly. It
also means that if we change the components or meaning of the components in
227
struct sequence, we only have to fix the functions defined in sequence.c.
5.3.1.3
Compiling and linking
Now that we have sequence.h and sequence.c, how do we use them? Let’s
suppose we have a simple main program:
#include <stdio.h>
#include "sequence.h"
void
seq_print(Sequence s, int limit)
{
int i;
}
for(i = seq_next(s); i < limit; i = seq_next(s)) {
printf("%d\n", i);
}
int
main(int argc, char **argv)
{
Sequence s;
Sequence s2;
puts("Stepping by 1:");
s = seq_create(0);
seq_print(s, 5);
seq_destroy(s);
puts("Now stepping by 3:");
s2 = seq_create_step(1, 3);
seq_print(s2, 20);
seq_destroy(s2);
}
return 0;
examples/ADT/sequence/main.c
We can compile main.c and sequence.c together into a single binary with the
228
command c99 main.c sequence.c. Or we can build a Makefile which will
compile the two files separately and then link them. Using make may be more
efficient, especially for large programs consisting of many components, since if
we make any changes make will only recompile those files we have changed. So
here is our Makefile:
CC=c99
CFLAGS=-g3 -pedantic -Wall
all: seqprinter
seqprinter: main.o sequence.o
$(CC) $(CFLAGS) -o $@ $^
test: seqprinter
./seqprinter
# these rules say to rebuild main.o and sequence.o if sequence.h changes
main.o: main.c sequence.h
sequence.o: sequence.c sequence.h
clean:
$(RM) -f seqprinter *.o
examples/ADT/sequence/Makefile
And now running make test produces this output. Notice how the built-in make
variables $@ and $ˆ expand out to the left-hand side and right-hand side of the
dependency line for building seqprinter.
$ make test
c99 -g3 -pedantic -Wall
-c -o main.o main.c
c99 -g3 -pedantic -Wall
-c -o sequence.o sequence.c
c99 -g3 -pedantic -Wall -o seqprinter main.o sequence.o
./seqprinter
Stepping by 1:
0
1
2
3
4
Now stepping by 3:
1
4
7
10
13
16
229
19
5.3.2
Designing abstract data types
Now we’ve seen how to implement an abstract data type. How do we choose
when to use when, and what operations to give it? Let’s try answering the
second question first.
5.3.2.1
Parnas’s Principle
Parnas’s Principle is a statement of the fundamental idea of information hiding,
which says that abstraction boundaries should be as narrow as possible:
• The developer of a software component must provide the intended user with
all the information needed to make effective use of the services provided
by the component, and should provide no other information.
• The developer of a software component must be provided with all the
information necessary to carry out the given responsibilities assigned to
the component, and should be provided with no other information.
(David Parnas, “On the Criteria to Be Used in Decomposing Systems into
Modules,” Communications of the ACM, 15(12): 1059–1062, 1972.)
For ADTs, this means we should provide as few functions for accessing and
modifying the ADT as we can get away with. The Sequence type we defined
early has a particularly narrow interface; the developer of Sequence (whoever is
writing sequence.c) needs to know nothing about what its user wants except for
the arguments passed in to seq_create or seq_create_step, and the user only
needs to be able to call seq_next. More complicated ADTs might provide larger
sets of operations, but in general we know that an ADT provides a successful
abstraction when the operations are all “natural” ones given our high-level
description. If we find ourselves writing a lot of extra operations to let users
tinker with the guts of our implementation, that may be a sign that either we
aren’t taking our abstraction barrier seriously enough, or that we need to put
the abstraction barrier in a different place.
5.3.2.2
When to build an abstract data type
The short answer: Whenever you can.
A better answer: The best heuristic I know for deciding what ADTs to include
in a program is to write down a description of how your program is going to
work. For each noun or noun phrase in the description, either identify a built-in
data type to implement it or design an abstract data type.
For example: a grade database maintains a list of students, and for each student
it keeps a list of grades. So here we might want data types to represent:
230
•
•
•
•
A
A
A
A
list of students,
student,
list of grades,
grade.
If grades are simple, we might be able to make them just be ints (or maybe
doubles); to be on the safe side, we should probably create a Grade type with a
typedef. The other types are likely to be more complicated. Each student might
have in addition to his or her grades a long list of other attributes, such as a name,
an email address, etc. By wrapping students up as abstract data types we can
extend these attributes if we need to, or allow for very general implementations
(say, by allowing a student to have an arbitrary list of keyword-attribute pairs).
The two kinds of lists are likely to be examples of sequence types; we’ll be seeing
a lot of ways to implement these as the course progresses. If we want to perform
the same kinds of operations on both lists, we might want to try to implement
them as a single list data type, which then is specialized to hold either students
or grades; this is not always easy to do in C, but we’ll see examples of how to
do this, too.
Whether or not this set of four types is the set we will finally use, writing it down
gives us a place to start writing our program. We can start writing interface files
for each of the data types, and then evolve their implementations and the main
program in parallel, adjusting the interfaces as we find that we have provided
too little (or too much) data for each component to do what it must.
5.4
Hash tables
A hash table is a randomized data structure that supports the INSERT,
DELETE, and FIND operations in expected O(1) time. The core idea behind
hash tables is to use a hash function that maps a large keyspace to a smaller
domain of array indices, and then use constant-time array operations to store
and retrieve the data.
5.4.1
Dictionary data types
A hash table is typically used to implement a dictionary data type, where
keys are mapped to values, but unlike an array, the keys are not conveniently
arranged as integers 0, 1, 2, . . .. Dictionary data types are a fundamental data
structure often found in scripting languages like AWK, Perl, Python, PHP, Lua,
or Ruby. For example, here is some Python code that demonstrates use of a
dictionary accessed using an array-like syntax:
title = {}
# empty dictionary
title["Barack"] = "President"
user = "Barack"
print("Welcome" + title[user] + " " + user)
231
In C, we don’t have the convenience of reusing [] for dictionary lookups (we’d
need C++ for that), but we can still get the same effect with more typing using
functions. For example, using an abstract dictionary in C might look like this:
Dict *title;
const char *user;
title = dictCreate();
dictSet(title, "Barack", "President");
user = "Barack";
printf("Welcome %s %s\n", dictGet(title, user), user);
As with other abstract data types, the idea is that the user of the dictionary type
doesn’t need to know how it is implemented. For example, we could implement
the dictionary as an array of structs that we search through, but that would
be expensive: O(n) time to find a key in the worst case.
Closely related to a dictionary is a set, which has keys but no values. It’s usually
pretty straightforward to turn an implementation of a dictionary into a set (leave
out the values) or vice versa (add values to the end of keys but don’t use them
in searching).
5.4.2
Basics of hashing
If our keys were conveniently named 0, 1, 2, . . . , n−1, we could simply use an array,
and be able to find a record given a key in constant time. Unfortunately, naming
conventions for most objects are not so convenient, and even enumerations like
Social Security numbers are likely to span a larger range than we want to allocate.
But we would like to get the constant-time performance of an array anyway.
The solution is to feed the keys through some hash function H, which maps them
down to array indices. So in a database of people, to find “Smith, Wayland”,
we would first compute H(“Smith, Wayland”)$ = 137$ (say), and then look in
position 137 in the array. Because we are always using the same function H, we
will always be directed to the same position 137.
5.4.3
Resolving collisions
But what if H(“Smith, Wayland”) and H(“Hephaestos”) both equal 137? Now
we have a collision, and we have to resolve it by finding some way to either (a)
effectively store both records in a single array location, or (b) move one of the
records to a new location that we can still find later. Let’s consider these two
approaches separately.
5.4.3.1
Chaining
232
We can’t really store more than one record in an array location, but we can fake
it by making each array location be a pointer to a linked list. Every time we
insert a new element in a particular location, we simply add it to this list.
Since the cost of scanning a linked list is linear in its size, this means that the
worst-case cost of searching for a particular key will be linear in the number of
keys in the table that hash to the same location. Under the assumption that
the hash function is a random function (which does not mean that it returns
random values every time you call it but instead means that we picked one of
the many possible hash functions uniformly at random), on average we get n/m
elements in each list.
So on average a failed search takes O(n/m) time.
This quantity n/m is called the load factor of the hash table and is often
written as α. If we want our hash table to be efficient, we will need to keep
this load factor down. If we can guarantee that it’s a constant, then we get
constant-time searches.
5.4.3.2
Open addressing
With open addressing, we store only one element per location, and handle
collisions by storing the extra elements in other unused locations in the array.
To find these other locations, we fix some probe sequence that tells us where
to look if A[H(x)] contains an element that is not x. A typical probe sequence
(called linear probing) is just H(x), H(x) + 1, H(x) + 2, . . . , wrapping around at
the end of the array. The idea is that if we can’t put an element in a particular
place, we just keep walking up through the array until we find an empty slot. As
long as we follow the same probe sequence when looking for an element, we will
be able to find the element again. If we are looking for an element and reach an
empty location, then we know that the element is not present in the table.
For open addressing, we always have that α = n/m is less than or equal to 1,
since we can’t store more elements in the table than we have locations. In fact,
we must ensure that the load factor is strictly less than 1, or some searches will
never terminate because they never reach an empty location. Assuming α < 1
and that the hash function is uniform, it is possible to calculate the worst-case
expected cost of a FIND operation, which as before will occur when we have
an unsuccessful FIND. Though we won’t do this calculation here, the result is
bounded by 1/(1 − n/m), which gets pretty bad if n/m is very close to 1, but is
a constant as long as n/m is bounded by a constant (say 3/4, which makes the
expected number of probes at most 4).
5.4.4
Choosing a hash function
Here we will describe three methods for generating hash functions. The first
two are typical methods used in practice. The last has additional desirable
233
theoretical properties.
5.4.4.1
Division method
We want our hash function to look as close as it can to a random function,
but random functions are (provably) expensive to store. So in practice we do
something simpler and hope for the best. If the keys are large integers, a typical
approach is to just compute the remainder mod m. This can cause problems if
m is, say, a power of 2, since it may be that the low-order bits of all the keys
are similar, which will produce lots of collisions. So in practice with this method
m is typically chosen to be a large prime.
What if we want to hash strings instead of integers? The trick P
is to treat the
strings as integers. Given a string a1 a2 a3 . . . ak , we represent it as i ai bi , where
b is a base chosen to be larger than the number of characters. We can then feed
this resulting huge integer to our hash function. Typically we do not actually
compute the huge integer directly, but instead compute its remainder mod m,
as in this short C function:
/* treat strings as base-256 integers */
/* with digits in the range 1 to 255 */
#define BASE (256)
size_t
hash(const char *s, size_t m)
{
size_t h;
unsigned const char *us;
/* cast s to unsigned const char * */
/* this ensures that elements of s will be treated as having values >= 0 */
us = (unsigned const char *) s;
h = 0;
while(*us != '\0') {
h = (h * BASE + *us) % m;
us++;
}
}
return h;
The division method works best when m is a prime, as otherwise regularities in
the keys can produce clustering in the hash values. (Consider, for example, what
happens if m is 256). But this can be awkward for computing hash functions
quickly, because computing remainders is a relatively slow operation.
234
5.4.4.2
Multiplication method
For this reason, the most commonly-used hash functions replace the modulus m
with something like 23 2 and replace the base with some small prime, relying on
the multiplier to break up patterns in the input. This yields the multiplication
method. Typical code might look something like this:
#define MULTIPLIER (37)
size_t
hash(const char *s)
{
size_t h;
unsigned const char *us;
/* cast s to unsigned const char * */
/* this ensures that elements of s will be treated as having values >= 0 */
us = (unsigned const char *) s;
h = 0;
while(*us != '\0') {
h = h * MULTIPLIER + *us;
us++;
}
}
return h;
The only difference between this code and the division method code is that we’ve
renamed BASE to MULTIPLIER and dropped m. There is still some remaindertaking happening: since C truncates the result of any operation that exceeds
the size of the integer type that holds it, the h = h * MULTIPLIER + *us; line
effectively has a hidden mod 23 2 or 26 4 at the end of it (depending on how big
your size_t is). Now we can’t use, say, 256, as the multiplier, because then the
hash value h would be determined by just the last four characters of s.
The choice of 37 is based on folklore. I like 97 myself, and 31 also has supporters.
Almost any medium-sized prime should work.
5.4.4.3
Universal hashing
The preceding hash functions offer no guarantees that the adversary can’t find a
set of n keys that all hash to the same location; indeed, since they’re deterministic,
as long as the keyspace contains at least nm keys the adversary can always do
so. Universal families of hash functions avoid this problem by choosing the hash
function randomly, from some set of possible functions that is small enough that
we can write our random choice down.
235
The property that makes a family of hash functions {Hr } universal is that, for
any distinct keys x and y, the probability that r is chosen so that Hr (x) = Hr (y)
is exactly 1/m.
Why is this important? Recall that for chaining, the expected number of collisions
between an element x and other elements was just the sum over all particular
elements y of the probability that x collides with that particular element. If Hr
is drawn from a universal family, this probability is 1/m for each y, and we get
the same n/m expected collisions as if Hr were completely random.
Several universal families of hash functions are known. Here is a simple one
that works when the size of the keyspace and the size of the output space are
both powers of 2. Let the keyspace consist of n-bit strings and let m = 2k .
Then the random index r consists of nk independent random bits organized as
n m-bit strings a1 a2 . . . an . To compute the hash function of a particular input
x, compute the bitwise exclusive or of ai for each position i where the i-th bit of
x is 1.
We can implement this in C as
/* implements universal hashing using random bit-vectors in x */
/* assumes number of elements in x is at least BITS_PER_CHAR * MAX_STRING_SIZE */
#define BITS_PER_CHAR (8)
/* not true on all machines! */
#define MAX_STRING_SIZE (128)
/* we'll stop hashing after this many */
#define MAX_BITS (BITS_PER_CHAR * MAX_STRING_SIZE)
size_t
hash(const char *s, size_t x[])
{
size_t h;
unsigned const char *us;
int i;
unsigned char c;
int shift;
/* cast s to unsigned const char * */
/* this ensures that elements of s will be treated as having values >= 0 */
us = (unsigned const char *) s;
h = 0;
for(i = 0; *us != 0 && i < MAX_BITS; us++) {
c = *us;
for(shift = 0; shift < BITS_PER_CHAR; shift++, i++) {
/* is low bit of c set? */
if(c & 0x1) {
h ^= x[i];
}
236
}
}
}
/* shift c to get new bit in lowest position */
c >>= 1;
return h;
As you can see, this requires a lot of bit-fiddling. It also fails if we get a lot of
strings that are identical for the first MAX_STRING_SIZE characters. Conceivably,
the latter problem could be dealt with by growing x dynamically as needed. But
we also haven’t addressed the question of where we get these random values
from—see the chapter on randomization for some possibilities.
In practice, universal families of hash functions are seldom used, since a reasonable
fixed hash function is unlikely to be correlated with any patterns in the actual
input. But they are useful for demonstrating provably good performance.
5.4.5
Maintaining a constant load factor
All of the running time results for hash tables depend on keeping the load factor
α small. But as more elements are inserted into a fixed-size table, the load factor
grows without bound. The usual solution to this problem is rehashing: when
the load factor crosses some threshold, we create a new hash table of size 2n or
thereabouts and migrate all the elements to it.
This approach raises the worst-case cost of an insertion to O(n). However, we
can bring the expected cost down to O(1) by rehashing only with probability
O(1/n) for each insert after the threshold is crossed. Or we can apply amortized
analysis to argue that the amortized cost (total cost divided by number of
operations) is O(1) assuming we double the table size on each rehash. Neither the
expected-cost nor the amortized-cost approaches actually change the worst-case
cost, but they make it look better by demonstrating that we at least don’t incur
that cost every time.
With enough machinery, it may be possible to deamortize the cost of rehashing
by doing a little bit of it with every insertion. The idea is to build the new hash
table incrementally, and start moving elements to it once it is fully initialized.
This requires keeping around two copies of the hash table and searching both,
and for most purposes is more trouble than it’s worth. But a mechanism like
this is often used for real-time garbage collection, where it’s important not to
have the garbage collector lock up the entire system while it does its work.
237
5.4.6
5.4.6.1
Examples
A low-overhead hash table using open addressing
Here is a very low-overhead hash table based on open addressing. The application
is rapidly verifying ID numbers in the range 000000000 to 999999999 by checking
them against a list of known good IDs. Since the quantity of valid ID numbers
may be very large, a goal of the mechanism is to keep the amount of extra
storage used as small as possible. This implementation uses a tunable overhead
parameter. Setting the parameter to a high value makes lookups fast but requires
more space per ID number in the list. Setting it to a low value can reduce the
storage cost arbitrarily close to 4 bytes per ID, at the cost of increasing search
times.
Here is the header file giving the interface:
typedef struct idList *IDList;
#define MIN_ID (0)
#define MAX_ID (999999999)
/* build an IDList out of an unsorted array of n good ids */
/* returns 0 on allocation failure */
IDList IDListCreate(int n, int unsortedIdList[]);
/* destroy an IDList */
void IDListDestroy(IDList list);
/* check an id against the list */
/* returns nonzero if id is in the list */
int IDListContains(IDList list, int id);
examples/hashTables/idList/idList.h
And here is the implementation:
#include <stdlib.h>
#include <assert.h>
#include "idList.h"
/* overhead parameter that determines both space and search costs */
/* must be strictly greater than 1 */
#define OVERHEAD (1.1)
#define NULL_ID (-1)
struct idList {
238
};
int size;
int ids[1];
/* we'll actually malloc more space than this */
IDList
IDListCreate(int n, int unsortedIdList[])
{
IDList list;
int size;
int i;
int probe;
size = (int) (n * OVERHEAD + 1);
list = malloc(sizeof(*list) + sizeof(int) * (size-1));
if(list == 0) return 0;
/* else */
list->size = size;
/* clear the hash table */
for(i = 0; i < size; i++) {
list->ids[i] = NULL_ID;
}
/* load it up */
for(i = 0; i < n; i++) {
assert(unsortedIdList[i] >= MIN_ID);
assert(unsortedIdList[i] <= MAX_ID);
/* hashing with open addressing by division */
/* this MUST be the same pattern as in IDListContains */
for(probe = unsortedIdList[i] % list->size;
list->ids[probe] != NULL_ID;
probe = (probe + 1) % list->size);
assert(list->ids[probe] == NULL_ID);
}
}
list->ids[probe] = unsortedIdList[i];
return list;
void
239
IDListDestroy(IDList list)
{
free(list);
}
int
IDListContains(IDList list, int id)
{
int probe;
/* this MUST be the same pattern as in IDListCreate */
for(probe = id % size;
list->ids[probe] != NULL_ID;
probe = (probe + 1) % size) {
if(list->ids[probe] == id) {
return 1;
}
}
}
return 0;
examples/hashTables/idList/idList.c
5.4.6.2
A string to string dictionary using chaining
Here is a more complicated string to string dictionary based on chaining.
typedef struct dict *Dict;
/* create a new empty dictionary */
Dict DictCreate(void);
/* destroy a dictionary */
void DictDestroy(Dict);
/* insert a new key-value pair into an existing dictionary */
void DictInsert(Dict, const char *key, const char *value);
/* return the most recently inserted value associated with a key */
/* or 0 if no matching key is present */
const char *DictSearch(Dict, const char *key);
/* delete the most recently inserted record with the given key */
/* if there is no such record, has no effect */
void DictDelete(Dict, const char *key);
240
examples/hashTables/dict/dict.h
#include <stdlib.h>
#include <assert.h>
#include <string.h>
#include "dict.h"
struct elt {
struct elt *next;
char *key;
char *value;
};
struct dict {
int size;
/* size of the pointer table */
int n;
/* number of elements stored */
struct elt **table;
};
#define INITIAL_SIZE (1024)
#define GROWTH_FACTOR (2)
#define MAX_LOAD_FACTOR (1)
/* dictionary initialization code used in both DictCreate and grow */
Dict
internalDictCreate(int size)
{
Dict d;
int i;
d = malloc(sizeof(*d));
assert(d != 0);
d->size = size;
d->n = 0;
d->table = malloc(sizeof(struct elt *) * d->size);
assert(d->table != 0);
for(i = 0; i < d->size; i++) d->table[i] = 0;
}
return d;
241
Dict
DictCreate(void)
{
return internalDictCreate(INITIAL_SIZE);
}
void
DictDestroy(Dict d)
{
int i;
struct elt *e;
struct elt *next;
for(i = 0; i < d->size; i++) {
for(e = d->table[i]; e != 0; e = next) {
next = e->next;
}
}
}
free(e->key);
free(e->value);
free(e);
free(d->table);
free(d);
#define MULTIPLIER (97)
static unsigned long
hash_function(const char *s)
{
unsigned const char *us;
unsigned long h;
h = 0;
for(us = (unsigned const char *) s; *us; us++) {
h = h * MULTIPLIER + *us;
}
}
return h;
static void
grow(Dict d)
242
{
Dict d2;
struct dict swap;
int i;
struct elt *e;
/* new dictionary we'll create */
/* temporary structure for brain transplant */
d2 = internalDictCreate(d->size * GROWTH_FACTOR);
for(i = 0; i < d->size; i++) {
for(e = d->table[i]; e != 0; e = e->next) {
/* note: this recopies everything */
/* a more efficient implementation would
* patch out the strdups inside DictInsert
* to avoid this problem */
DictInsert(d2, e->key, e->value);
}
}
/* the hideous part */
/* We'll swap the guts of d and d2 */
/* then call DictDestroy on d2 */
swap = *d;
*d = *d2;
*d2 = swap;
}
DictDestroy(d2);
/* insert a new key-value pair into an existing dictionary */
void
DictInsert(Dict d, const char *key, const char *value)
{
struct elt *e;
unsigned long h;
assert(key);
assert(value);
e = malloc(sizeof(*e));
assert(e);
e->key = strdup(key);
e->value = strdup(value);
h = hash_function(key) % d->size;
243
e->next = d->table[h];
d->table[h] = e;
d->n++;
}
/* grow table if there is not enough room */
if(d->n >= d->size * MAX_LOAD_FACTOR) {
grow(d);
}
/* return the most recently inserted value associated with a key */
/* or 0 if no matching key is present */
const char *
DictSearch(Dict d, const char *key)
{
struct elt *e;
for(e = d->table[hash_function(key) % d->size]; e != 0; e = e->next) {
if(!strcmp(e->key, key)) {
/* got it */
return e->value;
}
}
}
return 0;
/* delete the most recently inserted record with the given key */
/* if there is no such record, has no effect */
void
DictDelete(Dict d, const char *key)
{
struct elt **prev;
/* what to change when elt is deleted */
struct elt *e;
/* what to delete */
for(prev = &(d->table[hash_function(key) % d->size]);
*prev != 0;
prev = &((*prev)->next)) {
if(!strcmp((*prev)->key, key)) {
/* got it */
e = *prev;
*prev = e->next;
free(e->key);
244
free(e->value);
free(e);
return;
}
}
}
examples/hashTables/dict/dict.c
And here is some (very minimal) test code.
{.c include= examples/hashTables/dict/test_dict.c}
5.5
Generic containers
The first rule of programming is that you should never write the same code
twice. Suppose that you happen to have lying around a dictionary type whose
keys are ints and whose values are strings. Tomorrow you realize that what
you really want is a dictionary type whose keys are strings and whose values
are ints, or one whose keys are ints but whose values are stacks. If you have
n different types that may appear as keys or values, can you avoid writing n2
different dictionary implementations to get every possible combination?
Many languages provide special mechanisms to support generic types, ones
for which part of the type is not specified. It’s as if you could declare an array
in C to be an array of some type to be specified later, and then write functions
that operate on any such array without knowing what the missing type is going
to be (templates in C++ are an example of such a mechanism). Unfortunately,
C does not provide generic types. But by aggressive use of function pointers and
void *, it is possible to fake them.
5.5.1
Generic dictionary: interface
Below is an example of an interface to a generic dictionary type for storing
maps from constant values to constant values. The void * pointers are used to
avoid having to declare exactly what kinds of keys and values the dictionary will
contain.
/* Set dict[key] = value. */
/* Both key and value are copied internally. */
/* If data is the null pointer, remove dict[key]. */
void dictSet(Dict d, const void *key, const void *value);
/* Return dict[key], or null if dict[key] has not been set. */
const void *dictGet(Dict d, const void *key);
245
We’ll also need a constructor and destructor, but we’ll get to those in a moment.
First we need to think about what dictSet and dictGet are supposed to do,
and how we might possibly be able to implement them. Suppose we want to
build a dictionary with strings as both keys and values. Internally, this might be
represented as some sort of hash table or tree. Suppose it’s a hash table. Now,
given some void *key, we’d like to be able to compute its hash value. But we
don’t know what type key points to, and if we guess wrong we are likely to end
up with segmentation faults or worse. So we need some way to register a hash
function for our keys, whatever type they might really be behind that void *.
Similarly, we will want to be able to compare keys for equality (since not all keys
that hash together will necessarily be the same), and we may want to be able
to copy keys and values so that the data inside the dictionary is not modified
if somebody changes a value passed in from the outside. So we need a fair bit
of information about keys and values. We’ll organize all of this information in
a struct made up of function pointers. (This includes a few extra components
that came up while writing the implementation.)
/* Provides operations for working with keys or values */
struct dictContentsOperations {
/* hash function */
unsigned long (*hash)(const void *datum, void *arg);
/* returns nonzero if *datum1 == *datum2 */
int (*equal)(const void *datum1, const void *datum2, void *arg);
/* make a copy of datum that will survive changes to original */
void *(*copy)(const void *datum, void *arg);
/* free a copy */
void (*delete)(void *datum, void *arg);
};
/* extra argument, to allow further specialization */
void *arg;
We could write a similar but smaller struct for values, but to save a little bit of
effort in the short run we’ll use the same struct for both keys and values. We
can now write a constructor for our generic dictionary that consumes two such
structs that provide operations for working on keys and values, respectively:
/* create a new dictionary with given key and
/* Note: valueOps.hash and valueOps.equal are
Dict dictCreate(struct dictContentsOperations
struct dictContentsOperations
value operations */
not used. */
keyOps,
valueOps);
So now to create a dict, we just need to fill in two dictContentsOperations
structures. For convenience, it might be nice if dict.c provided some preloaded
246
structures for common types like ints and strings. We can also use the arg field
in struct dictContentsOperations to make the keys and values themselves
be parameterized types, for example a type of byte-vectors of given length.
We can declare these various convenience structures in dict.h as
/* Some predefined dictContentsOperations structures */
/*
* DictIntOps supports int's that have been cast to (void *), e.g.:
*
d = dictCreate(DictIntOps, DictIntOps);
*
dictSet(d, (void *) 1, (void * 2));
*
x = (int) dictGet(d, (void * 1));
*/
struct dictContentsOperations DictIntOps;
/*
* Supports null-terminated strings, e.g.:
*
d = dictCreate(DictStringOps, DictStringOps);
*
dictSet(d, "foo", "bar");
*
s = dictGet(d, "foo");
* Note: no casts are needed since C automatically converts
* between (void *) and other pointer types.
*/
struct dictContentsOperations DictStringOps;
/*
* Supports fixed-size blocks of memory, e.g.:
*
int x = 1;
*
int y = 2;
*
d = dictCreate(dictMemOps(sizeof(int)), dictMemOps(sizeof(int));
*
dictSet(d, &x, &y);
*
printf("%d", *dictGet(d, &x);
*/
struct dictContentsOperations dictMemOps(int size);
We’ll define the operations in DictIntOps to expect ints cast directly to void *,
the operations in DictStringOps to expect char * cast to void *, and the
operations in dictMemOps(size) will expect void * arguments pointing to
blocks of the given size. There is a subtle difference between a dictionary using
DictIntOps and dictMemOps(sizeof(int)); in the former case, keys and values
are the ints themselves (after being case), which in the latter, keys and values
are pointers to ints.
Implementations of these structures can be found below.
To make a dictionary that maps strings to ints, we just call:
d = dictCreate(DictStringOps, DictIntOps);
247
and then we can do things like:
dictSet(d, "foo", (void *) 2);
v = (int) dictGet(d, "foo');
If we find ourselves working with an integer-valued dictionary a lot, we might
want to define a few macros or inline functions to avoid having to type casts all
the time.
5.5.2
Generic dictionary: implementation
To implement our generic dictionary, we just take our favorite non-generic hash
table, and replace any calls to fixed hash functions, copier, free, etc. with calls
to elements of the appropriate structure. The result is shown below.
typedef struct dict *Dict;
/* Provides operations for working with keys or values */
struct dictContentsOperations {
/* hash function */
unsigned long (*hash)(const void *datum, void *arg);
/* returns nonzero if *datum1 == *datum2 */
int (*equal)(const void *datum1, const void *datum2, void *arg);
/* make a copy of datum that will survive changes to original */
void *(*copy)(const void *datum, void *arg);
/* free a copy */
void (*delete)(void *datum, void *arg);
};
/* extra argument, to allow further specialization */
void *arg;
/* create a new dictionary with given key and value operations */
/* Note: valueOps.hash and valueOps.equal are not used. */
Dict dictCreate(struct dictContentsOperations keyOps,
struct dictContentsOperations valueOps);
/* free a dictionary and all the space it contains */
/* This will call the appropriate delete function for all keys and */
/* values. */
void dictDestroy(Dict d);
/* Set dict[key] = value. */
248
/* Both key and value are copied internally. */
/* If data is the null pointer, remove dict[key]. */
void dictSet(Dict d, const void *key, const void *value);
/* Return dict[key], or null if dict[key] has not been set. */
const void *dictGet(Dict d, const void *key);
/* Some predefined dictContentsOperations structures */
/*
* DictIntOps supports int's that have been cast to (void *), e.g.:
*
d = dictCreate(DictIntOps, DictIntOps);
*
dictSet(d, (void *) 1, (void * 2));
*
x = (int) dictGet(d, (void * 1));
*/
struct dictContentsOperations DictIntOps;
/*
* Supports null-terminated strings, e.g.:
*
d = dictCreate(DictStringOps, DictStringOps);
*
dictSet(d, "foo", "bar");
*
s = dictGet(d, "foo");
* Note: no casts are needed since C automatically converts
* between (void *) and other pointer types.
*/
struct dictContentsOperations DictStringOps;
/*
* Supports fixed-size blocks of memory, e.g.:
*
int x = 1;
*
int y = 2;
*
d = dictCreate(dictMemOps(sizeof(int)), dictMemOps(sizeof(int));
*
dictSet(d, &x, &y);
*
printf("%d", *dictGet(d, &x);
*/
struct dictContentsOperations dictMemOps(int size);
examples/generic/dict.h
#include <stdlib.h>
#include <string.h>
#include "dict.h"
struct dictElt {
unsigned long hash;
void *key;
void *value;
/* full hash of key */
249
};
struct dictElt *next;
struct dict {
int tableSize;
/* number of slots in table */
int numElements;
/* number of elements */
struct dictElt **table; /* linked list heads */
/* these save arguments passed at creation */
struct dictContentsOperations keyOps;
struct dictContentsOperations valueOps;
};
#define INITIAL_TABLESIZE (16)
#define TABLESIZE_MULTIPLIER (2)
#define TABLE_GROW_DENSITY (1)
Dict
dictCreate(struct dictContentsOperations keyOps,
struct dictContentsOperations valueOps)
{
Dict d;
int i;
d = malloc(sizeof(*d));
if(d == 0) return 0;
d->tableSize = INITIAL_TABLESIZE;
d->numElements = 0;
d->keyOps = keyOps;
d->valueOps = valueOps;
d->table = malloc(sizeof(*(d->table)) * d->tableSize);
if(d->table == 0) {
free(d);
return 0;
}
for(i = 0; i < d->tableSize; i++) d->table[i] = 0;
}
return d;
void
dictDestroy(Dict d)
{
int i;
struct dictElt *e;
250
struct dictElt *next;
}
for(i = 0; i < d->tableSize; i++) {
for(e = d->table[i]; e != 0; e = next) {
next = e->next;
d->keyOps.delete(e->key, d->keyOps.arg);
d->valueOps.delete(e->value, d->valueOps.arg);
free(e);
}
}
free(d->table);
free(d);
/* return pointer to element with given key, if any */
static struct dictElt *
dictFetch(Dict d, const void *key)
{
unsigned long h;
int i;
struct dictElt *e;
}
h = d->keyOps.hash(key, d->keyOps.arg);
i = h % d->tableSize;
for(e = d->table[i]; e != 0; e = e->next) {
if(e->hash == h && d->keyOps.equal(key, e->key, d->keyOps.arg)) {
/* found it */
return e;
}
}
/* didn't find it */
return 0;
/* increase the size of the dictionary, rehashing all table elements */
static void
dictGrow(Dict d)
{
struct dictElt **old_table;
int old_size;
int i;
struct dictElt *e;
struct dictElt *next;
int new_pos;
/* save old table */
251
old_table = d->table;
old_size = d->tableSize;
/* make new table */
d->tableSize *= TABLESIZE_MULTIPLIER;
d->table = malloc(sizeof(*(d->table)) * d->tableSize);
if(d->table == 0) {
/* put the old one back */
d->table = old_table;
d->tableSize = old_size;
return;
}
/* else */
/* clear new table */
for(i = 0; i < d->tableSize; i++) d->table[i] = 0;
/* move all elements of old table to new table */
for(i = 0; i < old_size; i++) {
for(e = old_table[i]; e != 0; e = next) {
next = e->next;
/* find the position in the new table */
new_pos = e->hash % d->tableSize;
e->next = d->table[new_pos];
d->table[new_pos] = e;
}
}
}
/* don't need this any more */
free(old_table);
void
dictSet(Dict d, const void *key, const void *value)
{
int tablePosition;
struct dictElt *e;
e = dictFetch(d, key);
if(e != 0) {
/* change existing setting */
d->valueOps.delete(e->value, d->valueOps.arg);
e->value = value ? d->valueOps.copy(value, d->valueOps.arg) : 0;
} else {
/* create new element */
e = malloc(sizeof(*e));
if(e == 0) abort();
252
e->hash = d->keyOps.hash(key, d->keyOps.arg);
e->key = d->keyOps.copy(key, d->keyOps.arg);
e->value = value ? d->valueOps.copy(value, d->valueOps.arg) : 0;
/* link it in */
tablePosition = e->hash % d->tableSize;
e->next = d->table[tablePosition];
d->table[tablePosition] = e;
d->numElements++;
}
}
if(d->numElements > d->tableSize * TABLE_GROW_DENSITY) {
/* grow and rehash */
dictGrow(d);
}
const void *
dictGet(Dict d, const void *key)
{
struct dictElt *e;
}
e = dictFetch(d, key);
if(e != 0) {
return e->value;
} else {
return 0;
}
/* int functions */
/* We assume that int can be cast to void * and back without damage */
static unsigned long dictIntHash(const void *x, void *arg) { return (int) x; }
static int dictIntEqual(const void *x, const void *y, void *arg)
{
return ((int) x) == ((int) y);
}
static void *dictIntCopy(const void *x, void *arg) { return (void *) x; }
static void dictIntDelete(void *x, void *arg) { ; }
struct dictContentsOperations DictIntOps = {
dictIntHash,
dictIntEqual,
dictIntCopy,
253
};
dictIntDelete,
0
/* common utilities for string and mem */
static unsigned long hashMem(const unsigned char *s, int len)
{
unsigned long h;
int i;
}
h = 0;
for(i = 0; i < len; i++) {
h = (h << 13) + (h >> 7) + h + s[i];
}
return h;
static void dictDeleteFree(void *x, void *arg) { free(x); }
/* string functions */
static unsigned long dictStringHash(const void *x, void *arg)
{
return hashMem(x, strlen(x));
}
static int dictStringEqual(const void *x, const void *y, void *arg)
{
return !strcmp((const char *) x, (const char *) y);
}
static void *dictStringCopy(const void *x, void *arg)
{
const char *s;
char *s2;
}
s = x;
s2 = malloc(sizeof(*s2) * (strlen(s)+1));
strcpy(s2, s);
return s2;
struct dictContentsOperations DictStringOps = {
dictStringHash,
dictStringEqual,
dictStringCopy,
dictDeleteFree,
254
};
0
/* mem functions */
static unsigned long dictMemHash(const void *x, void *arg)
{
return hashMem(x, (int) arg);
}
static int dictMemEqual(const void *x, const void *y, void *arg)
{
return !memcmp(x, y, (size_t) arg);
}
static void *dictMemCopy(const void *x, void *arg)
{
void *x2;
}
x2 = malloc((size_t) arg);
memcpy(x2, x, (size_t) arg);
return x2;
struct dictContentsOperations
dictMemOps(int len)
{
struct dictContentsOperations memOps;
memOps.hash = dictMemHash;
memOps.equal = dictMemEqual;
memOps.copy = dictMemCopy;
memOps.delete = dictDeleteFree;
memOps.arg = (void *) len;
}
return memOps;
examples/generic/dict.c
And here is some test code and a Makefile: test-dict.c, tester.h, tester.c, Makefile.
5.6
Recursion
Recursion is when a function calls itself. Some programming languages (particularly functional programming languages like Scheme, ML, or Haskell use
recursion as a basic tool for implementing algorithms that in other languages
255
would typically be expressed using iteration (loops). Procedural languages like
C tend to emphasize iteration over recursion, but can support recursion as well.
5.6.1
Example of recursion in C
Here are a bunch of routines that print the numbers from 0 to 9:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
/* all of these routines print numbers i where start <= i < stop */
void
printRangeIterative(int start, int stop)
{
int i;
}
for(i = start; i < stop; i++) {
printf("%d\n", i);
}
void
printRangeRecursive(int start, int stop)
{
if(start < stop) {
printf("%d\n", start);
printRangeRecursive(start+1, stop);
}
}
void
printRangeRecursiveReversed(int start, int stop)
{
if(start < stop) {
printRangeRecursiveReversed(start+1, stop);
printf("%d\n", start);
}
}
void
printRangeRecursiveSplit(int start, int stop)
{
int mid;
256
if(start < stop) {
mid = (start + stop) / 2;
}
}
printRangeRecursiveSplit(start, mid);
printf("%d\n", mid);
printRangeRecursiveSplit(mid+1, stop);
#define Noisy(x) (puts(#x), x)
int
main(int argc, char **argv)
{
if(argc != 1) {
fprintf(stderr, "Usage: %s\n", argv[0]);
return 1;
}
Noisy(printRangeIterative(0, 10));
Noisy(printRangeRecursive(0, 10));
Noisy(printRangeRecursiveReversed(0, 10));
Noisy(printRangeRecursiveSplit(0, 10));
}
return 0;
examples/recursion/recursion.c
And here is the output:
printRangeIterative(0, 10)
0
1
2
3
4
5
6
7
8
9
printRangeRecursive(0, 10)
0
1
257
2
3
4
5
6
7
8
9
printRangeRecursiveReversed(0, 10)
9
8
7
6
5
4
3
2
1
0
printRangeRecursiveSplit(0, 10)
0
1
2
3
4
5
6
7
8
9
The first function printRangeIterative is simple and direct: it’s what we’ve
been doing to get loops forever. The others are a bit more mysterious.
The function printRangeRecursive is an example of solving a problem using
a divide and conquer approach. If we don’t know how to print a range of
numbers 0 through 9, maybe we can start by solving a simpler problem of
printing the first number 0. Having done that, we have a new, smaller problem:
print the numbers 1 through 9. But then we notice we already have a function
printRangeRecursive that will do that for us. So we’ll call it.
If you aren’t used to this, it has the feeling of trying to make yourself fly by
pulling very hard on your shoelaces.20 But in fact the computer will happily
generate the eleven nested instances of printRangeRecursive to make this
happen. When we hit the bottom, the call stack will look something like this:
20 A small child of my acquaintance once explained that this wouldn’t work, because you
would hit your head on the ceiling.
258
printRangeRecursive(0, 10)
printRangeRecursive(1, 10)
printRangeRecursive(2, 10)
printRangeRecursive(3, 10)
printRangeRecursive(4, 10)
printRangeRecursive(5, 10)
printRangeRecursive(6, 10)
printRangeRecursive(7, 10)
printRangeRecursive(8, 10)
printRangeRecursive(9, 10)
printRangeRecursive(10, 10)
This works because each call to printRangeRecursive gets its own parameters
and its own variables separate from the others, even the ones that are still in
progress. So each will print out start and then call another copy in to print
start+1 etc. In the last call, we finally fail the test start < stop, so the
function exits, then its parent exits, and so on until we unwind all the calls on
the stack back to the first one.
In printRangeRecursiveReversed, the calling pattern is exactly the same,
but now instead of printing start on the way down, we print start on
the way back up, after making the recursive call. This means that in
printRangeRecursiveReversed(0, 10), 0 is printed only after the results of
printRangeRecursiveReversed(1, 10), which gives us the countdown effect.
So far these procedures all behave very much like ordinary loops, with increasing values on the stack standing in for the loop variable. More exciting is
printRangeRecursiveSplit. This function takes a much more aggressive approach to dividing up the problem: it splits a range [0, 10) as two ranges [0, 5)
and [6, 10) separated by a midpoint 5.ˆ[The notation [x, y) means all numbers z
such that x ≤ z < y.] We want to print the midpoint in the middle, of course,
and we can use printRangeRecursiveSplit recursively to print the two ranges.
Following the execution of this procedure is more complicated, with the start of
the sequence of calls looking something like this:
printRangeRecursiveSplit(0, 10)
printRangeRecursiveSplit(0, 5)
printRangeRecursiveSplit(0, 2)
printRangeRecursiveSplit(0, 1)
printRangeRecursiveSplit(0, 0)
printRangeRecursiveSplit(1, 1)
printRangeRecursiveSplit(2, 2)
printRangeRecursiveSplit(3, 5)
printRangeRecursiveSplit(3, 4)
printRangeRecursiveSplit(3, 3)
printRangeRecursiveSplit(4, 4)
printRangeRecursiveSplit(5, 5)
printRangeRecursiveSplit(6, 10)
259
... etc.
Here the computation has the structure of a tree instead of a list, so it is not so
obvious how one might rewrite this procedure as a loop.
5.6.2
Common problems with recursion
Like iteration, recursion is a powerful tool that can cause your program to do
much more than expected. While it may seem that errors in recursive functions
would be harder to track down than errors in loops, most of the time there are a
few basic causes.
5.6.2.1
Omitting the base case
Suppose we leave out the if statement in printRangeRecursive:
void
printRangeRecursiveBad(int start, int stop)
{
printf("%d\n", start);
printRangeRecursiveBad(start+1, stop);
}
This will still work, in a sense. When called as printRangeRecursiveBad(0, 10),
it will print 0, call itself with printRangeRecursiveBad(1, 10), print 1, 2, 3,
etc., but there is nothing to stop it at 10 (or anywhere else). So our output will
be a long string of numbers followed by a segmentation fault, when we blow out
the stack.
This is the recursive version of an infinite loop: the same thing happens if we
forget a loop test and write
void
printRangeIterativeBad(int start, int stop)
{
for(i = 0; ; i++) {
printf("%d\n", i);
}
}
except that now the program just runs forever, since it never runs out of resources.
This is an example of how iteration is more efficient than recursion, at least in C.
5.6.2.2
Blowing out the stack
Blowing out the stack is what happens when a recursion is too deep. Typically,
the operating system puts a hard limit on how big the stack can grow, on the
assumption that any program that grows the stack too much has gone insane
260
and needs to be killed before it does more damage. One of the ways this can
happen is if we forget the base case as above, but it can also happen if we
just try to use a recursive function to do too much. For example, if we call
printRangeRecursive(0, 1000000), we will probably get a segmentation fault
after the first 100,000 numbers or so.
For this reason, it’s best to try to avoid linear recursions like the one in
printRangeRecursive, where the depth of the recursion is proportional
to the number of things we are doing. Much safer are even splits like
printRangeRecursiveSplit, since the depth of the stack will now be only
logarithmic in the number of things we are doing. Fortunately, linear recursions
are often tail-recursive, where the recursive call is the last thing the recursive
function does; in this case, we can use a standard transformation (see below) to
convert the tail-recursive function into an iterative function.
5.6.2.3
Failure to make progress
Sometimes we end up blowing out the stack because we thought we were recursing
on a smaller instance of the problem, but in fact we weren’t. Consider this
broken version of printRangeRecursiveSplit:
void
printRangeRecursiveSplitBad(int start, int stop)
{
int mid;
if(start == stop) {
printf("%d\n", start);
} else {
mid = (start + stop) / 2;
}
}
printRangeRecursiveSplitBad(start, mid);
printRangeRecursiveSplitBad(mid, stop);
This will get stuck on as simple a call as printRangeRecursiveSplitBad(0, 1);
it will set mid to 0, and while the recursive call to printRangeRecursiveSplitBad(0, 0)
will work just fine, the recursive call to printRangeRecursiveSplitBad(0, 1)
will put us back where we started, giving an infinite recursion.
Detecting these errors is usually not too hard (segmentation faults that produce
huge piles of stack frames when you type where in gdb are a dead give-away).
Figuring out how to make sure that you do in fact always make progress can be
trickier.
261
5.6.3
Tail-recursion and iteration
Tail recursion is when a recursive function calls itself only once, and as the
last thing it does. The printRangeRecursive function is an example of a
tail-recursive function:
void
printRangeRecursive(int start, int stop)
{
if(start < stop) {
printf("%d\n", start);
printRangeRecursive(start+1, stop);
}
}
The nice thing about tail-recursive functions is that we can always translate
them directly into iterative functions. The reason is that when we do the tail
call, we are effectively replacing the current copy of the function with a new
copy with new arguments. So rather than keeping around the old zombie parent
copy—which has no purpose other than to wait for the child to return and then
return itself—we can reuse it by assigning new values to its arguments and
jumping back to the top of the function.
Done literally, this produces this goto-considered-harmful monstrosity:
void
printRangeRecursiveGoto(int start, int stop)
{
topOfFunction:
if(start < stop) {
printf("%d\n", start);
}
}
start = start+1;
goto topOfFunction;
But we can almost always remove goto statements using less dangerous control
structures. In this particular case, the pattern of jumping back to just before an
if matches up exactly with what we get from a while loop:
void
printRangeRecursiveNoMore(int start, int stop)
{
while(start < stop) {
printf("%d\n", start);
262
}
}
start = start+1;
In functional programming languages, this transformation is usually done in
the other direction, to unroll loops into recursive functions. Since C doesn’t
like recursive functions so much (they blow out the stack!), we usually do this
transformation got get rid of recursion instead of adding it.
5.6.3.1
Binary search: recursive and iterative versions
Binary search is an algorithm for searching a sorted array for a particular
target element, similar to playing Twenty Questions when the answer is a number
(hopefully in a range that includes at most 22 0 numbers). The algorithm starts
by picking an value in the middle of the array. If the target is less than this
value, we recurse on the bottom half of the array; else we recurse on the top half.
Here is an interface for binary search on an array of ints:
{.c include=examples/binarySearch/binarySearch.h}
Written recursively, we might implement the algorithm like this:
{.c include=examples/binarySearch/binarySearchRecursive.c}
This will work just fine, and indeed it finds the target element (or not) in O(log n)
time, because we can only recurse O(log n) times before running out of elements
and we only pay O(1) cost per recursive call to binarySearch. But we do have
to pay function call overhead for call, and there is a potential to run into stack
overflow if our stack is very constrained.
Fortunately, we don’t do anything with the return value from binarySearch
but pass it on up the stack: the function is tail-recursive. This means that we
can get rid of the recursion by reusing the stack from from the initial call. The
mechanical way to do this is wrap the body of the routine in a for(;;) loop
(so that we jump back to the top whenever we hit the bottom), and replace
each recursive call with one or more assignments to update any parameters that
change in the recursive call. The result looks like this:
{.c include=examples/binarySearch/binarySearchIterative.c}
Here’s some simple test code to demonstrate that these two implementations in
fact do the same thing: Makefile, testBinarySearch.c.
5.6.4
Mergesort: a recursive sorting algorithm
So far the examples we have given have not been very useful, or have involved
recursion that we can easily replace with iteration. Here is an example of a
recursive procedure that cannot be as easily turned into an iterative version.
263
We are going to implement the mergesort algorithm on arrays. This is a classic
divide and conquer sorting algorithm that splits an array into two pieces, sorts
each piece (recursively!), then merges the results back together. Here is the code,
together with a simple test program.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
/* merge sorted arrays a1 and a2, putting result in out */
void
merge(int n1, const int a1[], int n2, const int a2[], int out[])
{
int i1;
int i2;
int iout;
i1 = i2 = iout = 0;
}
while(i1 < n1 || i2 < n2) {
if(i2 >= n2 || ((i1 < n1) && (a1[i1] < a2[i2]))) {
/* a1[i1] exists and is smaller */
out[iout++] = a1[i1++];
} else {
/* a1[i1] doesn't exist, or is bigger than a2[i2] */
out[iout++] = a2[i2++];
}
}
/* sort a, putting result in out */
/* we call this mergeSort to avoid conflict with mergesort in libc */
void
mergeSort(int n, const int a[], int out[])
{
int *a1;
int *a2;
if(n < 2) {
/* 0 or 1 elements is already sorted */
memcpy(out, a, sizeof(int) * n);
} else {
/* sort into temp arrays */
a1 = malloc(sizeof(int) * (n/2));
a2 = malloc(sizeof(int) * (n - n/2));
264
mergeSort(n/2, a, a1);
mergeSort(n - n/2, a + n/2, a2);
/* merge results */
merge(n/2, a1, n - n/2, a2, out);
}
}
/* free the temp arrays */
free(a1);
free(a2);
#define N (20)
int
main(int argc, char **argv)
{
int a[N];
int b[N];
int i;
for(i = 0; i < N; i++) {
a[i] = N-i-1;
}
for(i = 0; i < N; i++) {
printf("%d ", a[i]);
}
putchar('\n');
mergeSort(N, a, b);
for(i = 0; i < N; i++) {
printf("%d ", b[i]);
}
putchar('\n');
}
return 0;
examples/sorting/mergesort.c
The cost of this is pretty cheap: O(n log n), since each element of a is processed
through merge once for each array it gets put in, and the recursion only goes
O(log n) layers deep before we hit 1-element arrays.
The reason that we can’t easily transform this into an iterative version is that
265
the mergeSort function is not tail-recursive: not only does it call itself twice,
but it also needs to free the temporary arrays at the end. Because the algorithm
has to do these tasks on the way back up the stack, we need to keep the stack
around to track them.
5.6.5
Asymptotic complexity of recursive functions
One issue with a recursive functions is that it becomes harder to estimate its
asymptotic complexity. Unlike loops, where we can estimate the cost by simply
multiplying the number of iterations by the cost of each iteration, the cost of a
recursive function depends on the cost of its recursive calls. This would make it
seem that we would need to be able to compute the cost of the function before
we could compute the cost of the function.
Fortunately, for most recursive functions, the size of the input drops whenever
we recurse. So the cost can be expressed in terms of a recurrence, a formula
for the cost T (n) on an input of size n in terms of the cost on smaller inputs.
Some examples:
T (n) = O(1) + T (n/2) This is the cost of binary search. To search an array of
n elements, look up the middle element (O(1) time) and, in the worst case,
recurse on an array of n/2 elements. T (n) = 2T (n/2) + O(n)
This is the cost of mergesort. Sort two half-size arrays recursively, then
merge them in O(n) time. T (n) = O(1) + T (n − 1)
This is the cost of most simple loops, if we think of them as a recursive
process. Do O(1) work on the first element, then do T (n − 1) work on the
rest.
There are standard tools for solving many of the recurrences that arise in common
algorithms, but these are overkill for our purposes, since there are only a handful
of recurrences that are likely to come up in practice and we already solved most
of them. Here is a table of some of the more common possibilities:
5.7
Recurrence
Solution
Example
T (n) = T (n − 1) + O(1)
T (n) = T (n − 1) + O(n)
T (n) = T (n/2) + O(1)
T (n) = 2T (n/2) + O(n)
T (n) = O(n)
T (n) = O(n2 )
T (n) = O(log n)
T (n) = O(n log n)
Finding a maximum
Selection sort
Binary search
Mergesort
Binary trees
Divide and conquer yields algorithms whose execution has a tree structure.
Sometimes we build data structures that are also trees. It is probably not
surprising that divide and conquer is the natural way to build algorithms that
266
use such trees as inputs.
5.7.1
Tree basics
Here is a typical binary tree. It is binary because every node has at most two
children. This particular tree is also complete because the nodes consist only
of internal nodes with exactly two children and leaves with no children. Not
all binary trees will be complete.
0
/ \
1
2
/ \
3
4
/ \
5
6
/ \
7
8
Structurally, a complete binary tree consists of either a single node (a leaf) or a
root node with a left and right subtree, each of which is itself either a leaf or a
root node with two subtrees. The set of all nodes underneath a particular node
x is called the subtree rooted at x.
The size of a tree is the number of nodes; a leaf by itself has size 1. The height
of a tree is the length of the longest path; 0 for a leaf, at least one in any larger
tree. The depth of a node is the length of the path from the root to that node.
The height of a node is the height of the subtree of which it is the root, i.e. the
length of the longest path from that node to some leaf below it. A node u is
an ancestor of a node v if v is contained in the subtree rooted at u; we may
write equivalently that v is a descendant of u. Note that every node is both
and ancestor and descendant of itself; if we wish to exclude the node itself, we
refer to a proper ancestor or proper descendant.
5.7.2
Binary tree implementations
In a low-level programming language like C, a binary tree typically looks a lot
like a linked list with an extra outgoing pointer from each element, e.g.
struct node {
int key;
struct node *left; /* left child */
struct node *right; /* right child */
};
Missing children (and the empty tree) are represented by null pointers. Typically,
individual tree nodes are allocated separately using malloc; however, for high267
performance use it is not unusual for tree libraries to do their own storage
allocation out of large blocks obtained from malloc.
Optionally, the struct may be extended to include additional information such
as a pointer to the node’s parent, hints for balancing, or aggregate information
about the subtree rooted at the node such as its size or the sum/max/average of
the keys of its nodes.
When it is not important to be able to move large subtrees around simply by
adjusting pointers, a tree may be represented implicitly by packing it into an
array. This is a standard approach for implementing heaps, which we will see
soon.
5.7.3
The canonical binary tree algorithm
Pretty much every divide and conquer algorithm for binary trees looks like this:
void
doSomethingToAllNodes(struct node *root)
{
if(root) {
doSomethingTo(root);
doSomethingToAllNodes(root->left);
doSomethingToAllNodes(root->right);
}
}
The function processes all nodes in what is called a preorder traversal, where
the “preorder” part means that the root of any tree is processed first. Moving
the call to doSomethingTo in between or after the two recursive calls yields an
inorder or postorder traversal, respectively.
In practice we usually want to extract some information from the tree. For
example, this function computes the size of a tree:
int
treeSize(struct node *root)
{
if(root == 0) {
return 0;
} else {
return 1 + treeSize(root->left) + treeSize(root->right);
}
}
and this function computes the height:
int
treeHeight(struct node *root)
268
{
}
int lh;
int rh;
/* height of left subtree */
/* height of right subtree */
if(root == 0) {
return -1;
} else {
lh = treeHeight(root->left);
rh = treeHeight(root->right);
return 1 + (lh > rh ? lh : rh);
}
Since both of these algorithms have the same structure, they both have the same
asymptotic running time. We can compute this running time by observing that
each recursive call to treeSize or treeHeight that does not get a null pointer
passed to it gets a different node (so there are n such calls), and each call that
does get a null pointer passed to it is called by a routine that doesn’t, and that
there are at most two such calls per node. Since the body of each call itself costs
O(1) (no loops), this gives a total cost of Θ(n).
So these are all Θ(n) algorithms.
5.7.4
Nodes vs leaves
For some binary trees we don’t store anything interesting in the internal nodes,
using them only to provide a route to the leaves. We might reasonably ask if an
algorithm that runs in O(n) time where n is the total number of nodes still runs
in O(m) time, where m counts only the leaves. For complete binary trees, we
can show that we get the same asymptotic performance whether we count leaves
only, internal nodes only, or both leaves and internal nodes.
Let T (n) be the number of internal nodes in a complete binary tree with n leaves.
It is easy to see that T (1) = 0 and T (2) = 1, but for larger trees there are multiple
structures and so it makes sense to write a recurrence: T (n) = 1+T (k)+T (n−k).
We can show by induction that the solution to this recurrence is exactly T (n) =
n − 1. We already have the base case T (1) = 0. For larger n, we have T (n) =
1 + T (k) + T (n − k) = 1 + (k − 1) + (n − k − 1) = n − 1.
So a complete binary tree with Θ(n) nodes has Θ(n) internal nodes and Θ(n)
leaves; if we don’t care about constant factors, we won’t care which number we
use.
269
5.7.5
Special classes of binary trees
So far we haven’t specified where particular nodes are placed in the binary tree.
Most applications of binary trees put some constraints on how nodes relate to
one another. Some possibilities:
• Heaps: Each node has a key that is less than the keys of both of its children.
These allow for a very simple implementation using arrays, so we will look
at these first.
• BinarySearchTrees: Each node has a key, and a node’s key must be greater
than all keys in the subtree of its left-hand child and less than all keys in
the subtree of its right-hand child.
5.8
Heaps
A heap is a binary tree in which each element has a key (or sometimes priority)
that is less than the keys of its children. Heaps are used to implement the
priority queue abstract data type, which we’ll talk about first.
5.8.1
Priority queues
In a standard queue, elements leave the queue in the same order as they arrive.
In a priority queue, elements leave the queue in order of decreasing priority:
the DEQUEUE operation becomes a DELETE-MIN operation (or DELETEMAX, if higher numbers mean higher priority), which removes and returns the
highest-priority element of the priority queue, regardless of when it was inserted.
Priority queues are often used in operating system schedulers to determine which
job to run next: a high-priority job (e.g., turn on the fire suppression system)
runs before a low-priority job (floss the cat) even if the low-priority job has been
waiting longer.
5.8.2
Expensive implementations of priority queues
Implementing a priority queue using an array or linked list is likely to be
expensive. If the array or list is unsorted, it takes O(n) time to find the
minimum element; if it is sorted, it takes O(n) time (in the worst case) to add
a new element. So such implementations are only useful when the numbers
of INSERT and DELETE-MIN operations are very different. For example, if
DELETE-MIN is called only rarely but INSERT is called often, it may actually
be cheapest to implement a priority queue as an unsorted linked list with O(1)
INSERTs and O(n) DELETE-MINs. But if we expect that every element that
is inserted is eventually removed, we want something for which both INSERT
and DELETE-MIN are cheap operations.
270
5.8.3
Structure of a heap
A heap is a binary tree in which each node has a smaller key than its children;
this property is called the heap property or heap invariant.
To insert a node in the heap, we add it as a new leaf, which may violate the heap
property if the new node has a lower key than its parent. But we can restore the
heap property (at least between this node and its parent) by swapping either
the new node or its sibling with the parent, where in either case we move up
the node with the smaller key. This may still leave a violation of the heap
property one level up in the tree, but by continuing to swap small nodes with
their parents we eventually reach the top and have a heap again. The time to
complete this operation is proportional to the depth of the heap, which will
typically be O(log n) (we will see how to enforce this in a moment).
To implement DELETE-MIN, we can easily find the value to return at the top of
the heap. Unfortunately, removing it leaves a vacuum that must be filled in by
some other element. The easiest way to do this is to grab a leaf (which probably
has a very high key), and then float it down to where it belongs by swapping it
with its smaller child at each iteration. After time proportional to the depth
(again O(log n) if we are doing things right), the heap invariant is restored.
Similar local swapping can be used to restore the heap invariant if the priority
of some element in the middle changes; we will not discuss this in detail.
5.8.4
Packed heaps
It is possible to build a heap using structs and pointers, where each element
points to its parent and children. In practice, heaps are instead stored in arrays,
with an implicit pointer structure determined by array indices. For zero-based
arrays as in C, the rule is that a node at position i has children at positions
2*i+1 and 2*i+2; in the other direction, a node at position i has a parent at
position (i-1)/2 (which rounds down in C). This is equivalent to storing a heap
in an array by reading through the tree in breadth-first search order:
0
/ \
1
2
/ \ / \
3 4 5 6
becomes
0 1 2 3 4 5 6
This approach works best if there are no gaps in the array. So to maximize
efficiency we make this “no gaps” policy part of the invariant. We can do so
because we don’t care which leaf gets added when we do an INSERT, so we
271
can make it be the position at the end of the array. Similarly, in a DELETEMIN operation we can promote the last element to the root before floating it
down. Both these operations change the number of elements in the array, and
INSERTs in particular might force us to reallocate eventually. So in the worst
case INSERT can be an expensive operation, although as with growing hash
tables, the amortized cost may still be small.
5.8.5
Bottom-up heapification
If we are presented with an unsorted array, we can turn it into a heap more
quickly than the O(n log n) time required to do n INSERTs. The trick is to
build the heap from the bottom up (i.e. starting with position n − 1 and working
back to position 0, so that when it comes time to fix the heap invariant at
position i we have already fixed it at all later positions (this is a form of dynamic
programming). Unfortunately, it is not quite enough simply to swap a[i] with
its smaller child when we get there, because we could find that a[0] (say) was
the largest element in the heap. But the cost of floating a[i] down to its proper
place will be proportional to its own height rather than the height of the entire
heap. Since most of the elements of the heap are close to the bottom, the total
cost will turn out to be O(n).
5.8.6
Heapsort
Bottom-up heapification is used in the Heapsort algorithm, which first does
bottom-up heapification in O(n) time and then repeatedly calls DELETE-MAX
to extract the largest remaining element. This is no faster than the O(n log n)
cost of mergesort or quicksort in typical use, but requires very little auxiliary
storage since we can maintain the heap in the bottom part of the same array
whose top part stores the max elements extracted so far.
Here is a simple implementation of heapsort, that demonstrates how both bottomup heapification and the DELETE-MAX procedure work by floating elements
down to their proper places:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
/* max heap implementation */
/* compute child 0 or 1 */
#define Child(x, dir) (2*(x)+1+(dir))
/* float value at position pos down */
static void
272
floatDown(int n, int *a, int pos)
{
int x;
/* save original value once */
x = a[pos];
for(;;) {
if(Child(pos, 1) < n && a[Child(pos, 1)] > a[Child(pos, 0)]) {
/* maybe swap with Child(pos, 1) */
if(a[Child(pos, 1)] > x) {
a[pos] = a[Child(pos, 1)];
pos = Child(pos, 1);
} else {
/* x is bigger than both kids */
break;
}
} else if(Child(pos, 0) < n && a[Child(pos, 0)] > x) {
/* swap with Child(pos, 0) */
a[pos] = a[Child(pos, 0)];
pos = Child(pos, 0);
} else {
/* done */
break;
}
}
}
a[pos] = x;
/* construct a heap bottom-up */
static void
heapify(int n, int *a)
{
int i;
}
for(i = n-1; i >= 0; i--) {
floatDown(n, a, i);
}
/* sort an array */
void
heapSort(int n, int *a)
{
int i;
273
int tmp;
heapify(n, a);
for(i = n-1; i > 0; i--) {
/* swap max to a[i] */
tmp = a[0];
a[0] = a[i];
a[i] = tmp;
}
}
/* float new a[0] down */
floatDown(i, a, 0);
#define N (100)
#define MULTIPLIER (17)
int
main(int argc, char **argv)
{
int a[N];
int i;
if(argc != 1) {
fprintf(stderr, "Usage: %s\n", argv[0]);
return 1;
}
for(i = 0; i < N; i++) { a[i] = (i*MULTIPLIER) % N; }
for(i = 0; i < N; i++) { printf("%d ", a[i]); }
putchar('\n');
heapSort(N, a);
for(i = 0; i < N; i++) { printf("%d ", a[i]); }
putchar('\n');
}
return 0;
examples/sorting/heapsort.c
274
5.8.7
More information
• Priority_queue
• Binary_heap
• http://mathworld.wolfram.com/Heap.html
5.9
Binary search trees
A binary search tree is a binary tree in which each node has a key, and a
node’s key must be greater than all keys in the subtree of its left-hand child and
less than all keys in the subtree of its right-hand child. This allows a node to be
searched for using essentially the same binary search algorithm used on sorted
arrays.
5.9.1
Searching for a node
/* returns node with given target key */
/* or null if no such node exists */
struct node *
treeSearch(struct node *root, int target)
{
if(root->key == target) {
return root;
} else if(root->key > target) {
return treeSearch(root->left, target);
} else {
return treeSearch(root->right, target);
}
}
This procedure can be rewritten iteratively, which avoids stack overflow and is
likely to be faster:
struct node *
treeSearch(struct node *root, int target)
{
while(root != 0 && root->key != target) {
if(root->key > target) {
root = root->left;
} else {
root = root->right;
}
}
275
}
return root;
These procedures can be modified in the obvious way to deal with keys that
aren’t ints, as long as they can be compared (e.g., by using strcmp on strings).
5.9.2
Inserting a new node
As in a hash table, the insertion procedure mirrors the search procedure. We
must be a little careful to avoid actually walking all the way down to a null
pointer, since a null pointer now indicates a missing space for a leaf that we can
fill with our new node. So the code is a little more complex.
void
treeInsert(struct tree *root, int key)
{
struct tree *newNode;
newNode = malloc(sizeof(*newNode));
assert(newNode);
newNode->key = key;
newNode->left = 0;
newNode->right = 0;
for(;;) {
if(root->key > key) {
/* try left child */
if(root->left) {
root = root->left;
} else {
/* put it in */
root->left = newNode;
return;
}
} else {
/* right child case is symmetric */
if(root->right) {
root = root->right;
} else {
/* put it in */
root->right = newNode;
return;
}
}
}
276
}
Note that this code happily inserts duplicate keys. It also makes no attempt
to keep the tree balanced. This may lead to very long paths if new keys are
inserted in strictly increasing or strictly decreasing order.
5.9.3
Deleting a node
Deletion is more complicated. If a node has no children, we can just remove it,
and the rest of the tree stays the same. A node with one child can be spliced
out, connecting its parent directly to its child. But with two children, we can’t
do this.
The trick is to find the leftmost node in our target’s right subtree (or vice versa).
This node exists assuming the target has two children. As in a hash table, we
can then swap our target node with this more convenient node. Because it is
the leftmost node, it has no left child, so we can delete it using the no-children
or one-child case.
5.9.4
Costs
Searching for or inserting a node in a binary search tree with n nodes takes time
proportional to the depth of the node. In balanced trees, where the nodes in
each subtree are divided roughly evenly between the two child subtrees, this will
be O(log n), but for a badly unbalanced tree, this might be as much as O(n).
So making a binary search tree work efficiently requires keeping it balanced.
5.10
Augmented trees
An augmented data structure stores additional information in each of its nodes
that caches values that might otherwise be expensive to compute. For trees,
this might include information like the size of a subtree (which can be useful for
ranking values, where we want to determine how many elements of the tree are
smaller), the height of a subtree, or other summary information like the sum of
all the keys in a subtree.
Augmented data structures, in a sense, violate the no-separate-but-equal rule
that says we shouldn’t store the same information in different places. The reason
we try to avoid this is that it’s trouble if the two copies diverge, and by not
having two copies in the first place there is no possibility that they contradict
each other. But in this case the reduced cost justifies breaking this rule.
The idea is that when we insert a new element into an augmented tree, it only
changes the height/size/sum/etc. values for nodes on the path from the root to
the new value. Since each of these aggregate values can be computed for a node
277
in O(1) time from the values in its children, we can update all the aggregate
values on our way back up the stack after doing the insertion at a cost of O(1)
per node. This will give a total cost of O(log n) assuming our tree is reasonably
balanced.
5.10.1
Applications
Storing the height field can be useful for balancing, as in AVL trees.
Storing the size allows ranking (computing the number of elements less than
a given target value) and unraking (find an element with a particular rank).
Sample code for doing this is given in the AVL tree sample implementation
below.
Storing other aggregates like the sum of keys or values allows range queries,
where we ask, for example, for some aggregate statistic (like the sum or average)
of all the elements between some goven minimum and maximum.
Assuming we keep the tree balanced and correctly maintain the aggregate data
or each subtree, all of these operations can be done in O(log n) time.
5.11
Balanced trees
Binary search trees are a fine idea, but they only work if they are balanced—if
moving from a tree to its left or right subtree reduces the size by a constant
fraction. Balanced binary trees add some extra mechanism to the basic binary
search tree to ensure balance. Finding efficient ways to balance a tree has been
studied for decades, and several good mechanisms are known. We’ll try to hit
the high points of all of them.
5.11.1
Tree rotations
The problem is that as we insert new nodes, some paths through the tree may
become very long. So we need to be able to shrink the long paths by moving
nodes elsewhere in the tree.
But how do we do this? The idea is to notice that there may be many binary
search trees that contain the same data, and that we can transform one into
another by a local modification called a rotation:
y
/ \
x
C
/ \
A
B
<==>
x
/ \
A
y
/ \
B
C
278
Single rotation on x-y edge
If A < x < B < y < C, then both versions of this tree have the binary search
tree property. By doing the rotation in one direction, we move A up and C
down; in the other direction, we move A down and C up. So rotations can be
used to transfer depth from the leftmost grandchild of a node to the rightmost
and vice versa.
But what if it’s the middle grandchild B that’s the problem? A single rotation
as above doesn’t move B up or down. To move B, we have to reposition it so
that it’s on the end of something. We do this by splitting B into two subtrees
B1 and B2 , and doing two rotations that split the two subtrees while moving
both up. For this we need to do two rotations:
z
/ \
x
C
/ \
A
y
/ \
B1 B2
===>
z
/ \
y
C
/ \
x B2
/ \
A B1
===>
y
/ \
x
z
/|
|\
A B1 B2 C
Double rotation: rotate xy then zy
5.11.2
AVL trees
Rotations in principle let us rebalance a tree, but we still need to decide when
to do them. If we try to keep the tree in perfect balance (all paths nearly the
same length), we’ll spend so much time rotating that we won’t be able to do
anything else.
AVL trees solve this problem by enforcing the invariant that the heights of
the two subtrees sitting under each node differ by at most one. This does not
guarantee perfect balance, but it does get close. Let S(k) be the size of the
smallest AVL tree with height k. This tree will have at least one subtree of height
k − 1, but its other subtree can be of height k − 2 (and should be, to keep it as
small as possible). We thus have the recurrence S(k) = 1 + S(k − 1) + S(k − 2),
which is very close to the Fibonacci recurrence.
It is possible to solve this exactly using generating functions. But we can get
close by guessing that S(k) ≥ ak for some constant a. This clearly works for
S(0) = a0 = 1. For larger k, compute
• S(k) = 1 + ak−1 + ak−2 = 1 + ak (1/a + 1/a2 ) > ak (1/a + 1/a2 ).
This last quantity is at least ak provided (1/a + 1/a2 ) is at least 1. We can
solve exactly for the largest a that makes this work, but a very quick calculation
shows that a = 3/2 works: 2/3 + 4/9 = 10/9 > 1. It follows that any AVL tree
279
with height k has at least (3/2)k nodes, or conversely that any AVL tree with
(3/2)k nodes has height at most k. So the height of an arbitrary AVL tree with
n nodes is no greater than log3/2 n = O(log n).
How do we maintain this invariant? The first thing to do is add extra information
to the tree, so that we can tell when the invariant has been violated. AVL
trees store in each node the difference between the heights of its left and right
subtrees, which will be one of −1, 0, or 1. In an ideal world this would require
log2 3 ≈ 1.58 bits per node, but since fractional bits are difficult to represent on
modern computers a typical implementation uses two bits. Inserting a new node
into an AVL tree involves
1. Doing a standard binary search tree insertion.
2. Updating the balance fields for every node on the insertion path.
3. Performing a single or double rotation to restore balance if needed.
Implementing this correctly is tricky. Intuitively, we can imagine a version of an
AVL tree in which we stored the height of each node (using O(log log n) bits).
When we insert a new node, only the heights of its ancestors change—so step
2 requires updating O(log n) height fields. Similarly, it is only these ancestors
that can be too tall. It turns out that fixing the closest ancestor fixes all the
ones above it (because it shortens their longest paths by one as well). So just
one single or double rotation restores balance.
Deletions are also possible, but are uglier: a deletion in an AVL tree may require
as many as O(log n) rotations. The basic idea is to use the standard binary
search tree deletion trick of either splicing out a node if it has no right child, or
replacing it with the minimum value in its right subtree (the node for which is
spliced out); we then have to check to see if we need to rebalance at every node
above whatever node we removed.
Which rotations we need to do to rebalance depends on how some pair of siblings
are unbalanced. Below, we show the possible cases.
Zig-zig case: This can occur after inserting in A or deleting in C. Here we rotate
A up:
y
/ \
x
C
/ \
A
B
|
#
===>
x
/ \
A
y
| / \
# B
C
Zig-zag case: This can occur after inserting in B or deleting in C. This requires
a double rotation.
z
/ \
===>
z
/ \
y
/ \
===>
280
x
C
/ \
A
y
/ \
B1 B2
y
C
/ \
x B2
/ \
A B1
x
z
/|
|\
A B1 B2 C
Zig-zag case, again: This last case comes up after deletion if both nephews of
the short node are too tall. The same double rotation we used in the previous
case works here, too. Note that one of the subtrees is still one taller than the
others, but that’s OK.
z
/ \
x
C
/ \
A
y
| / \
# B1 B2
5.11.2.1
===>
z
/ \
y
C
/ \
x B2
/ \
A B1
|
#
===>
y
/ \
x
z
/|
|\
A B1 B2 C
|
#
Sample implementation
If we are not fanatical about space optimization, we can just keep track of the
heights of all nodes explicitly, instead of managing the −1, 0, 1 balance values.
Below, we give a not-very-optimized example implementation that uses this
approach to store a set of ints. This is pretty much our standard unbalanced
BST (although we have to make sure that the insert and delete routines are
recursive, so that we can fix things up on the way back out), with a layer on
top, implemented in the treeFix function, that tracks the height and size of
each subtree (although we don’t use size), and another layer on top of that,
implemented in the treeBalance function, that fixes any violations of the AVL
balance rule.
/*
* Basic binary search tree data structure without balancing info.
*
* Convention:
*
* Operations that update a tree are passed a struct tree **,
* so they can replace the argument with the return value.
*
* Operations that do not update the tree get a const struct tree *.
*/
#define LEFT (0)
#define RIGHT (1)
281
#define TREE_NUM_CHILDREN (2)
struct tree {
/* we'll make this an array so that we can make some operations symmetric */
struct tree *child[TREE_NUM_CHILDREN];
int key;
int height;
/* height of this node */
size_t size;
/* size of subtree rooted at this node */
};
#define TREE_EMPTY (0)
#define TREE_EMPTY_HEIGHT (-1)
/* free all elements of a tree, replacing it with TREE_EMPTY */
void treeDestroy(struct tree **root);
/* insert an element into a tree pointed to by root */
void treeInsert(struct tree **root, int newElement);
/* return 1 if target is in tree, 0 otherwise */
/* we allow root to be modified to allow for self-balancing trees */
int treeContains(const struct tree *root, int target);
/* delete minimum element from the tree and return its key */
/* do not call this on an empty tree */
int treeDeleteMin(struct tree **root);
/* delete target from the tree */
/* has no effect if target is not in tree */
void treeDelete(struct tree **root, int target);
/* return height of tree */
int treeHeight(const struct tree *root);
/* return size of tree */
size_t treeSize(const struct tree *root);
/* pretty-print the contents of a tree */
void treePrint(const struct tree *root);
/* return the number of elements in tree less than target */
size_t treeRank(const struct tree *root, int target);
/* return an element with the given rank */
/* rank must be less than treeSize(root) */
int treeUnrank(const struct tree *root, size_t rank);
282
/* check that aggregate data is correct throughout the tree */
void treeSanityCheck(const struct tree *root);
examples/trees/AVL/tree.h
#include
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<assert.h>
<stdint.h>
<stdlib.h>
#include "tree.h"
int
treeHeight(const struct tree *root)
{
if(root == 0) {
return TREE_EMPTY_HEIGHT;
} else {
return root->height;
}
}
/* recompute height from height of kids */
static int
treeComputeHeight(const struct tree *root)
{
int childHeight;
int maxChildHeight;
int i;
if(root == 0) {
return TREE_EMPTY_HEIGHT;
} else {
maxChildHeight = TREE_EMPTY_HEIGHT;
for(i = 0; i < TREE_NUM_CHILDREN; i++) {
childHeight = treeHeight(root->child[i]);
if(childHeight > maxChildHeight) {
maxChildHeight = childHeight;
}
}
}
}
return maxChildHeight + 1;
283
size_t
treeSize(const
{
if(root ==
return
} else {
return
}
}
struct tree *root)
0) {
0;
root->size;
/* recompute size from size of kids */
static int
treeComputeSize(const struct tree *root)
{
int size;
int i;
if(root == 0) {
return 0;
} else {
size = 1;
for(i = 0; i < TREE_NUM_CHILDREN; i++) {
size += treeSize(root->child[i]);
}
}
}
return size;
/* fix aggregate data in root */
/* assumes children are correct */
static void
treeAggregateFix(struct tree *root)
{
if(root) {
root->height = treeComputeHeight(root);
root->size = treeComputeSize(root);
}
}
/* rotate child in given direction to root */
static void
treeRotate(struct tree **root, int direction)
{
284
struct tree *x;
struct tree *y;
struct tree *b;
/*
*
y
*
/ \
*
x
C
*
/ \
* A
B
*/
<=>
x
/ \
A
y
/ \
B
C
y = *root;
x = y->child[direction];
b = x->child[!direction];
assert(y);
assert(x);
/* do the rotation */
*root = x;
x->child[!direction] = y;
y->child[direction] = b;
}
/* fix aggregate data for y then x */
treeAggregateFix(y);
treeAggregateFix(x);
/* restore AVL property at *root after an insertion or deletion */
/* assumes subtrees already have AVL property */
static void
treeRebalance(struct tree **root)
{
int tallerChild;
if(*root) {
for(tallerChild = 0; tallerChild < TREE_NUM_CHILDREN; tallerChild++) {
if(treeHeight((*root)->child[tallerChild]) >= treeHeight((*root)->child[!tallerC
/* which grandchild is the problem? */
if(treeHeight((*root)->child[tallerChild]->child[!tallerChild])
> treeHeight((*root)->child[tallerChild]->child[tallerChild])) {
/* opposite-direction grandchild is too tall */
/* rotation at root will just change its parent but not change height */
/* so we rotate it up first */
treeRotate(&(*root)->child[tallerChild], !tallerChild);
}
285
/* now rotate up the taller child */
treeRotate(root, tallerChild);
/* don't bother with other child */
break;
}
}
/* test that we actually fixed it */
assert(abs(treeHeight((*root)->child[LEFT]) - treeHeight((*root)->child[RIGHT])) <=
#ifdef PARANOID_REBALANCE
treeSanityCheck(*root);
#endif
}
}
/* free all elements of a tree, replacing it with TREE_EMPTY */
void
treeDestroy(struct tree **root)
{
int i;
}
if(*root) {
for(i = 0; i < TREE_NUM_CHILDREN; i++) {
treeDestroy(&(*root)->child[i]);
}
free(*root);
*root = TREE_EMPTY;
}
/* insert an element into a tree pointed to by root */
void
treeInsert(struct tree **root, int newElement)
{
struct tree *e;
if(*root == 0) {
/* not already there, put it in */
e = malloc(sizeof(*e));
assert(e);
286
e->key = newElement;
e->child[LEFT] = e->child[RIGHT] = 0;
*root = e;
} else if((*root)->key == newElement) {
/* already there, do nothing */
return;
} else {
/* do this recursively so we can fix data on the way back out */
treeInsert(&(*root)->child[(*root)->key < newElement], newElement);
}
}
/* fix the aggregate data */
treeAggregateFix(*root);
treeRebalance(root);
/* return 1 if target is in tree, 0 otherwise */
int
treeContains(const struct tree *t, int target)
{
while(t && t->key != target) {
t = t->child[t->key < target];
}
}
return t != 0;
/* delete minimum element from the tree and return its key */
/* do not call this on an empty tree */
int
treeDeleteMin(struct tree **root)
{
struct tree *toFree;
int retval;
assert(*root);
/* can't delete min from empty tree */
if((*root)->child[LEFT]) {
/* recurse on left subtree */
retval = treeDeleteMin(&(*root)->child[LEFT]);
} else {
/* delete the root */
toFree = *root;
retval = toFree->key;
287
}
*root = toFree->child[RIGHT];
free(toFree);
/* fix the aggregate data */
treeAggregateFix(*root);
treeRebalance(root);
}
return retval;
/* delete target from the tree */
/* has no effect if target is not in tree */
void
treeDelete(struct tree **root, int target)
{
struct tree *toFree;
/* do nothing if target not in tree */
if(*root) {
if((*root)->key == target) {
if((*root)->child[RIGHT]) {
/* replace root with min value in right subtree */
(*root)->key = treeDeleteMin(&(*root)->child[RIGHT]);
} else {
/* patch out root */
toFree = *root;
*root = toFree->child[LEFT];
free(toFree);
}
} else {
treeDelete(&(*root)->child[(*root)->key < target], target);
}
}
}
/* fix the aggregate data */
treeAggregateFix(*root);
treeRebalance(root);
/* how far to indent each level of the tree */
#define INDENTATION_LEVEL (2)
/* print contents of a tree, indented by depth */
static void
288
treePrintIndented(const struct tree *root, int depth)
{
int i;
if(root != 0) {
treePrintIndented(root->child[LEFT], depth+1);
for(i = 0; i < INDENTATION_LEVEL*depth; i++) {
putchar(' ');
}
printf("%d Height: %d Size: %zu (%p)\n", root->key, root->height, root->size, (void
}
}
treePrintIndented(root->child[RIGHT], depth+1);
/* print the contents of a tree */
void
treePrint(const struct tree *root)
{
treePrintIndented(root, 0);
}
size_t
treeRank(const struct tree *t, int target)
{
size_t rank = 0;
while(t && t->key != target) {
if(t->key < target) {
/* go right */
/* root and left subtree are all less than target */
rank += (1 + treeSize(t->child[LEFT]));
t = t->child[RIGHT];
} else {
/* go left */
t = t->child[LEFT];
}
}
}
/* we must also count left subtree */
return rank + treeSize(t->child[LEFT]);
int
treeUnrank(const struct tree *t, size_t rank)
289
{
size_t leftSize;
/* basic idea: if rank < treeSize(child[LEFT]), recurse in left child */
/* if it's equal, return the root */
/* else recurse in right child with rank = rank - treeSize(child[LEFT]) - 1 */
while(rank != (leftSize = treeSize(t->child[LEFT]))) {
if(rank < leftSize) {
t = t->child[LEFT];
} else {
t = t->child[RIGHT];
rank -= (leftSize + 1);
}
}
}
return t->key;
/* check that aggregate data is correct throughout the tree */
void
treeSanityCheck(const struct tree *root)
{
int i;
if(root) {
assert(root->height == treeComputeHeight(root));
assert(root->size == treeComputeSize(root));
assert(abs(treeHeight(root->child[LEFT]) - treeHeight(root->child[RIGHT])) < 2);
}
}
for(i = 0; i < TREE_NUM_CHILDREN; i++) {
treeSanityCheck(root->child[i]);
}
#ifdef TEST_MAIN
int
main(int argc, char **argv)
{
int key;
int i;
const int n = 10;
const int randRange = 1000;
const int randTrials = 1000000;
struct tree *root = TREE_EMPTY;
290
if(argc != 1) {
fprintf(stderr, "Usage: %s\n", argv[0]);
return 1;
}
/* original test */
for(i = 0; i < n; i++) {
assert(!treeContains(root, i));
treeInsert(&root, i);
assert(treeContains(root, i));
treeSanityCheck(root);
#ifdef PRINT_AFTER_OPERATIONS
treePrint(root);
puts("---");
#endif
}
/* check ranks */
for(i = 0; i < n; i++) {
assert(treeRank(root, i) == i);
assert(treeUnrank(root, i) == i);
}
treeSanityCheck(root);
/* now delete everything */
for(i = 0; i < n; i++) {
assert(treeContains(root, i));
treeDelete(&root, i);
assert(!treeContains(root, i));
treeSanityCheck(root);
#ifdef PRINT_AFTER_OPERATIONS
treePrint(root);
puts("---");
#endif
}
treeSanityCheck(root);
treeDestroy(&root);
/* random test */
srand(1);
for(i = 0; i < randTrials; i++) {
treeInsert(&root, rand() % randRange);
291
}
treeSanityCheck(root);
treeDelete(&root, rand() % randRange);
treeSanityCheck(root);
treeDestroy(&root);
#ifdef TEST_USE_STDIN
while(scanf("%d", &key) == 1) {
/* insert if positive, delete if negative */
if(key > 0) {
treeInsert(&root, key);
assert(treeContains(root, key));
} else if(key < 0) {
treeDelete(&root, -key);
assert(!treeContains(root, key));
}
/* else ignore 0 */
#ifdef PRINT_AFTER_OPERATIONS
treePrint(root);
puts("---");
#endif
}
treeSanityCheck(root);
treeDestroy(&root);
#endif /* TEST_USE_STDIN */
return 0;
}
#endif /* TEST_MAIN */
examples/trees/AVL/tree.c
This Makefile will compile and run some demo code in tree.c if run with make
test.
(An older implementation can be found in the directory examples/trees/
oldAvlTree.
5.11.3
2–3 trees
An early branch in the evolution of balanced trees was the 2–3 tree. Here all
paths have the same length, but internal nodes have either 2 or 3 children. So a
2–3 tree with height k has between 2k and 3k leaves and a comparable number
292
of internal nodes. The maximum path length in a tree with n nodes is at most
dlg ne, as in a perfectly balanced binary tree.
An internal node in a 2–3 tree holds one key if it has two children (including two
nil pointers) and two if it has three children. A search that reaches a three-child
node must compare the target with both keys to decide which of the three
subtrees to recurse into. As in binary trees, these comparisons take constant
time, so we can search a 2–3 tree in O(log n) time.
Insertion is done by expanding leaf nodes. This may cause a leaf to split when
it acquires a third key. When a leaf splits, it becomes two one-key nodes and
the middle key moves up into its parent. This may cause further splits up the
ancestor chain; the tree grows in height by adding a new root when the old root
splits. In practice only a small number of splits are needed for most insertions,
but even in the worst case this entire process takes O(log n) time.
It follows that 2–3 trees have the same performance as AVL trees. Conceptually,
they are simpler, but having to write separate cases for 2-child and 3-child nodes
doubles the size of most code that works on 2–3 trees. The real significance of
2–3 trees is as a precursor to two other kinds of trees, the red-black tree and the
B-tree.
5.11.4
Red-black trees
A red-black tree is a 2–3–4 tree (i.e. all nodes have 2, 3, or 4 children and 1, 2,
or 3 internal keys) where each node is represented by a little binary tree with a
black root and zero, one, or two red extender nodes as follows:
The invariant for a red-black tree is that
1. No two red nodes are adjacent.
2. Every path contains the same number of black nodes.
For technical reasons, we include the null pointers at the bottom of the tree as
black nodes; this has no effect on the invariant, but simplifies the description of
the rebalancing procedure.
From the invariant it follows that every path has between k and 2k nodes, where
k is the black-height, the common number of black nodes on each path. From
this we can prove that the height of the tree is O(log n).
Searching in a red-black tree is identical to searching in any other binary search
tree; we simply ignore the color bit on each node. So search takes O(log n) time.
For insertions, we use the standard binary search tree insertion algorithm, and
insert the new node as a red node. This may violate the first part of the invariant
(it doesn’t violate the second because it doesn’t change the number of black
nodes on any path). In this case we need to fix up the constraint by recoloring
nodes and possibly performing a single or double rotation.
293
Figure 1: redblacknodes.png
294
Figure 2: redblackrebalance.png
295
Which operations we need to do depend on the color of the new node’s uncle.
If the uncle is red, we can recolor the node’s parent, uncle, and grandparent
and get rid of the double-red edge between the new node and its parent without
changing the number of black nodes on any path. In this case, the grandparent
becomes red, which may create a new double-red edge which must be fixed
recursively. Thus up to O(log n) such recolorings may occur at a total cost of
O(log n).
If the uncle is black (which includes the case where the uncle is a null pointer),
a rotation (possibly a double rotation) and recoloring is necessary. In this case
(depicted at the bottom of the picture above), the new grandparent is always
black, so there are no more double-red edges. So at most two rotations occur
after any insertion.
Deletion is more complicated but can also be done in O(log n) recolorings and
O(1) (in this case up to 3) rotations. Because deletion is simpler in red-black
trees than in AVL trees, and because operations on red-black trees tend to have
slightly smaller constants than corresponding operation on AVL trees, red-black
trees are more often used that AVL trees in practice.
5.11.5
B-trees
Neither is used as much as a B-tree, a specialized data structure optimized for
storage systems where the cost of reading or writing a large block (of typically
4096 or 8192 bytes) is no greater than the cost of reading or writing a single bit.
Such systems include typical disk drives, where the disk drive has to spend so
long finding data on disk that it tries to amortize the huge (tens of millions of
CPU clock cycles) seek cost over many returned bytes.
A B-tree is a generalization of a 2–3 tree where each node has between M/2 and
M − 1 children, where M is some large constant chosen so that a node (including
up to M − 1 pointers and up to M − 2 keys) will just fit inside a single block.
When a node would otherwise end up with M children, it splits into two nodes
with M/2 children each, and moves its middle key up into its parent. As in 2–3
trees this may eventually require the root to split and a new root to be created;
in practice, M is often large enough that a small fixed height is enough to span
as much data as the storage system is capable of holding.
Searches in B-trees require looking through logM n nodes, at a cost of O(M )
time per node. If M is a constant the total time is asymptotically O(log n). But
the reason for using B-trees is that the O(M ) cost of reading a block is trivial
compare to the much larger constant time to find the block on the disk; and so
it is better to minimize the number of disk accesses (by making M large) than
reduce the CPU time.
296
5.11.6
Splay trees
Yet another approach to balancing is to do it dynamically. Splay trees, described
by Sleator and Tarjan in the paper “Self-adjusting binary search trees” (JACM
32(3):652–686, July 1985) are binary search trees in which every search operation
rotates the target to the root. If this is done correctly, the amortized cost of
each tree operation is O(log n), although particular rare operations might take
as much as O(n) time. Splay trees require no extra space because they store no
balancing information; however, the constant factors on searches can be larger
because every search requires restructuring the tree. For some applications this
additional cost is balanced by the splay tree’s ability to adapt to data access
patterns; if some elements of the tree are hit more often than others, these
elements will tend to migrate to the top, and the cost of a typical search will
drop to O(log m), where m is the size of the “working set” of frequently-accessed
elements.
5.11.6.1
How splaying works
The basic idea of a splay operation is that we move some particular node to the
root of the tree, using a sequence of rotations that tends to fix the balance of the
tree if the node starts out very deep. So while we might occasionally drive the
tree into a state that is highly unbalanced, as soon as we try to exploit this by
searching for a deep node, we’ll start balancing the tree so that we can’t collect
too much additional cost. In fact, in order to set up the bad state in the first
place we will have to do a lot of cheap splaying operations: the missing cost of
these cheap splays ends up paying for the cost of the later expensive search.
Splaying a node to the root involves performing rotations two layers at a time.
There are two main cases, depending on whether the node’s parent and grandparent are in the same direction (zig-zig) or in opposite directions (zig-zag), plus
a third case when the node is only one step away from the root. At each step,
we pick one of these cases and apply it, until the target node reaches the root of
the tree.
This is probably best understood by looking at a figure from the original paper:
297
The bottom two cases are the ones we will do most of the time.
Just looking at the picture, it doesn’t seem like zig-zig will improve balance
much. But if we have a long path made up of zig-zig cases, each operation will
push at least one node off of this path, cutting the length of the path in half. So
the rebalancing happens as much because we are pushing nodes off of the long
path as because the specific rotation operations improve things locally.
5.11.6.2
Analysis
Sleator and Tarjan show that any sequence of m splay operations on an n-node
splay tree has total cost at most O((m+n) log n+m). For large m (at least linear
in n), the O(m log n) term dominates, giving an amortized cost per operation of
O(log n), the same as we get from any balanced binary tree. This immediately
gives a bound on search costs, because the cost of plunging down the tree to find
the node we are looking for is proportional to the cost of splaying it up to the
root.
Splay trees have a useful “caching” property in that they pull frequently-accessed
298
nodes to the to the top and push less-frequently-accessed nodes down. The
authors show that if only k of the n nodes are accessed, the long-run amortized
cost per search drops to O(log k). For more general access sequences, it is
conjectured that the cost to perform a sufficiently long sequence of searches using
a splay tree is in fact optimal up to a constant factor (the “dynamic optimality
conjecture”), but no one has yet been able to prove this conjecture (or provide a
counterexample).21
5.11.6.3
Other operations
A search operation consists of a standard binary tree search followed by splaying
the target node to the root (if present) or the last non-null node we reached to
the root instead (if not).
Insertion and deletion are built on top of procedures to split and join trees.
A split divides a single splay tree into two splay trees, consisting of all elements
less than or equal to some value x and all elements greater than x. This is done
by searching for x, which brings either x or the first element less than or greater
than x to the root, then breaking the link between the root and its left or right
child depending on whether the root should go in the right or left tree.
A join merges two splay trees L and R, where every element in L is less than
every element in R. This involves splaying the largest element in L to the root,
and then making the root of R the right child of this element.
To do an insert of x, we do a split around x, then make the roots of the two
trees the children of a new element holding x (unless x is already present in the
tree, in which case we stop before breaking the trees apart).
To do a delete of an element x, we splay x to the root, remove it, then join the
two orphaned subtrees.
For each operation, we are doing a constant number of splays (amortized cost
O(log n) each), plus O(1) additional work. A bit of work is needed to ensure
that the joins and splits don’t break the amortized cost analysis, but this is done
in the paper, so we will sweep it under the carpet with the rest of the analysis.
5.11.6.4
Top-down splaying
There are a few remaining details that we need to deal with before trying to
implement a splay trees. Because the splay tree could become very deep, we
probably don’t want to implement a splay recursively in a language like C,
because we’ll blow out our stack. We also have a problem if we are trying to
rotate our target up from the bottom of figuring out what its ancestors are. We
could solve both of these problems by including parent pointers in our tree, but
21 A summary of the state of this problem as of 2013 can be found in http://arxiv.org/pdf/
1306.0207v1.pdf.
299
this would add a lot of complexity and negate the space improvement over AVL
trees of not having to store heights.
The solution given in the Sleator-Tarjan paper is to replace the bottom-up splay
procedure with a top-down splay procedure that accomplishes the same task.
The idea is that rotating a node up from the bottom effectively splits the tree
above it into two new left and right subtrees by pushing ancestors sideways
according to the zig-zig and zig-zag patters. But we can recognize these zig-zig
and zig-zag patterns from the top as well, and so we can construct these same
left and right subtrees from the top down instead of the bottom up. When we
do this, instead of adding new nodes to the tops of the trees, we will be adding
new nodes to the bottoms, as the right child of the rightmost node in the left
tree or the left child of the rightmost node in the left tree.
Here’s the picture, from the original paper:
300
To implement this, we need to keep track of the roots of the three trees, as well
as the locations in the left and right trees where we will be adding new vertices.
The roots we can just keep pointers to. For the lower corners of the trees, it
makes sense to store instead a pointer to the pointer location, so that we can
modify the pointer in the tree (and then move the pointer to point to the pointer
in the new corner). Initially, these corner pointers will just point to the left and
right tree roots, which will start out empty.
The last step (shown as Figure 12 from the paper) pastes the tree back together
301
by inserting the left and right trees between the new root and its children.
5.11.6.5
An implementation
Here is an implementation of a splay tree, with an interface similar to the
previous AVL tree implementation.
/*
* Basic binary search tree data structure without balancing info.
*
* Convention:
*
* Operations that update a tree are passed a struct tree **,
* so they can replace the argument with the return value.
*
* Operations that do not update the tree get a const struct tree *.
*/
#define LEFT (0)
#define RIGHT (1)
#define TREE_NUM_CHILDREN (2)
struct tree {
/* we'll make this an array so that we can make some operations symmetric */
struct tree *child[TREE_NUM_CHILDREN];
int key;
};
#define TREE_EMPTY (0)
/* free all elements of a tree, replacing it with TREE_EMPTY */
void treeDestroy(struct tree **root);
/* insert an element into a tree pointed to by root */
void treeInsert(struct tree **root, int newElement);
/* return 1 if target is in tree, 0 otherwise */
/* we allow root to be modified to allow for self-balancing trees */
int treeContains(struct tree **root, int target);
/* delete target from the tree */
/* has no effect if target is not in tree */
void treeDelete(struct tree **root, int target);
/* pretty-print the contents of a tree */
void treePrint(const struct tree *root);
302
examples/trees/splay/tree.h
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<assert.h>
<limits.h>
#include "tree.h"
/* free all elements of a tree, replacing it with TREE_EMPTY */
void
treeDestroy(struct tree **root)
{
/* we want to avoid doing this recursively, because the tree might be deep */
/* so we will repeatedly delete the root until the tree is empty */
while(*root) {
treeDelete(root, (*root)->key);
}
}
/* rotate child in given direction to root */
void
treeRotate(struct tree **root, int direction)
{
struct tree *x;
struct tree *y;
struct tree *b;
/*
*
y
*
/ \
*
x
C
*
/ \
* A
B
*/
<=>
A
x
/ \
y
/ \
B
C
y = *root;
x = y->child[direction];
b = x->child[!direction];
}
assert(y);
assert(x);
/* do the rotation */
*root = x;
x->child[!direction] = y;
y->child[direction] = b;
303
/* link operations for top-down splay */
/* this pastes a node in as !d-most node in subtree on side d */
static inline void
treeLink(struct tree ***hook, int d, struct tree *node)
{
*hook[d] = node;
/* strictly speaking we don't need to do this, but it allows printing the partial trees
node->child[!d] = 0;
hook[d] = &node->child[!d];
}
/* splay last element on path to target to root */
static void
treeSplay(struct tree **root, int target)
{
struct tree *t;
struct tree *child;
struct tree *grandchild;
struct tree *top[TREE_NUM_CHILDREN];
/* accumulator trees that will become subtrees of
struct tree **hook[TREE_NUM_CHILDREN]; /* where to link new elements into accumulator tr
int d;
int dChild;
/* direction of child */
int dGrandchild;
/* direction of grandchild */
/* we don't need to keep following this pointer, we'll just fix it at the end */
t = *root;
/* don't do anything to an empty tree */
if(t == 0) { return; }
/* ok, tree is not empty, start chopping it up */
for(d = 0; d < TREE_NUM_CHILDREN; d++) {
top[d] = 0;
hook[d] = &top[d];
}
/* keep going until we hit the key or we would hit a null pointer in the child */
while(t->key != target && (child = t->child[dChild = t->key < target]) != 0) {
/* child is not null */
grandchild = child->child[dGrandchild = child->key < target];
#ifdef DEBUG_SPLAY
treePrint(top[0]);
puts("---");
treePrint(t);
puts("---");
304
#endif
}
treePrint(top[1]);
puts("===");
if(grandchild == 0 || child->key == target) {
/* zig case; paste root into opposite-side hook */
treeLink(hook, !dChild, t);
t = child;
/* we can break because we know we will hit child == 0 next */
break;
} else if(dChild == dGrandchild) {
/* zig-zig case */
/* rotate and then hook up child */
/* grandChild becomes new root */
treeRotate(&t, dChild);
treeLink(hook, !dChild, child);
t = grandchild;
} else {
/* zig-zag case */
/* root goes to !dChild, child goes to dChild, grandchild goes to root */
treeLink(hook, !dChild, t);
treeLink(hook, dChild, child);
t = grandchild;
}
/* now reassemble the tree */
/* t's children go in hooks, top nodes become t's new children */
for(d = 0; d < TREE_NUM_CHILDREN; d++) {
*hook[d] = t->child[d];
t->child[d] = top[d];
}
}
/* and put t back in *root */
*root = t;
/* return 1 if target is in tree, 0 otherwise */
int
treeContains(struct tree **root, int target)
{
treeSplay(root, target);
return *root != 0 && (*root)->key == target;
}
305
/* insert an element into a tree pointed to by root */
void
treeInsert(struct tree **root, int newElement)
{
struct tree *e;
struct tree *t;
int d;
/* which side of e to put old root on */
treeSplay(root, newElement);
t = *root;
/* skip if already present */
if(t && t->key == newElement) { return; }
/* otherwise split the tree */
e = malloc(sizeof(*e));
assert(e);
e->key = newElement;
if(t == 0) {
e->child[LEFT] = e->child[RIGHT] = 0;
} else {
/* split tree and put e on top */
/* we know t is closest to e, so we don't have to move anything else */
d = (*root)->key > newElement;
e->child[d] = t;
e->child[!d] = t->child[!d];
t->child[!d] = 0;
}
}
/* either way we stuff e in *root */
*root = e;
/* delete target from the tree */
/* has no effect if target is not in tree */
void
treeDelete(struct tree **root, int target)
{
struct tree *left;
struct tree *right;
treeSplay(root, target);
306
if(*root && (*root)->key == target) {
/* save pointers to kids */
left = (*root)->child[LEFT];
right = (*root)->child[RIGHT];
/* free the old root */
free(*root);
/* if left is empty, just return right */
if(left == 0) {
*root = right;
} else {
/* first splay max element in left to top */
treeSplay(&left, INT_MAX);
/* now paste in right subtree */
left->child[RIGHT] = right;
}
}
}
/* return left */
*root = left;
/* how far to indent each level of the tree */
#define INDENTATION_LEVEL (2)
/* print contents of a tree, indented by depth */
static void
treePrintIndented(const struct tree *root, int depth)
{
int i;
if(root != 0) {
treePrintIndented(root->child[LEFT], depth+1);
for(i = 0; i < INDENTATION_LEVEL*depth; i++) {
putchar(' ');
}
printf("%d (%p)\n", root->key, (void *) root);
}
}
treePrintIndented(root->child[RIGHT], depth+1);
/* print the contents of a tree */
307
void
treePrint(const struct tree *root)
{
treePrintIndented(root, 0);
}
#ifdef TEST_MAIN
int
main(int argc, char **argv)
{
int i;
const int n = 10;
struct tree *root = TREE_EMPTY;
if(argc != 1) {
fprintf(stderr, "Usage: %s\n", argv[0]);
return 1;
}
for(i = 0; i < n; i++) {
assert(!treeContains(&root, i));
treeInsert(&root, i);
assert(treeContains(&root, i));
treePrint(root);
puts("===");
}
/* now delete everything */
for(i = 0; i < n; i++) {
assert(treeContains(&root, i));
treeDelete(&root, i);
assert(!treeContains(&root, i));
treePrint(root);
puts("===");
}
treeDestroy(&root);
return 0;
}
#endif
examples/trees/splay/tree.c
Makefile. The file speedTest.c can be used to do a simple test of the efficiency
of inserting many random elements. On my machine, the splay tree version is
308
about 10% slower than the AVL tree for this test on a million elements. This
probably indicates a bigger slowdown for treeInsert itself, because some of the
time will be spent in rand and treeDestroy, but I was too lazy to actually test
this further.
5.11.6.6
More information
For more details on splay trees, see the original paper, or any number of demos,
animations, and other descriptions that can be found via Google.
5.11.7
Scapegoat trees
Scapegoat trees are another amortized balanced tree data structure. The idea
of a scapegoat tree is that if we ever find ourselves doing an insert at the end of
a path that is too long, we can find some subtree rooted at a node along this
path that is particularly imbalanced and rebalance it all at once at a cost of O(k)
where k is the size of the subtree. These were shown by Galperin and Rivest
(SODA 1993) to give O(log n) amortized cost for inserts, while guaranteeing
O(log n) depth, so that inserts also run in O(log n) worst-case time; they also
came up with the name “scapegoat tree”, although it turns out the same data
structure had previously been published by Andersson in 1989. Unlike splay
trees, scapegoat trees do not require modifying the tree during a search, and
unlike AVL trees, scapegoat trees do not require tracking any information in
nodes (although they do require tracking the total size of the tree and, to allow
for rebalancing after many deletes, the maximum size of the tree since the last
time the entire tree was rebalanced).
Unfortunately, scapegoat trees are not very fast, so one is probably better off
with an AVL tree.
5.11.8
Skip lists
Skip lists are yet another balanced tree data structure, where the tree is disguised
as a tower of linked lists. Since they use randomization for balance, we describe
them with other randomized data structures.
5.11.9
Implementations
AVL trees and red-black trees have been implemented for every reasonable
programming language you’ve ever heard of. For C implementations, a good
place to start is at http://adtinfo.org/.
309
5.12
Graphs
These are notes on implementing graphs and graph algorithms in C.
5.12.1
Basic definitions
A graph consists of a set of nodes or vertices together with a set of edges
or arcs where each edge joins two vertices. Unless otherwise specified, a graph
is undirected: each edge is an unordered pair {u, v} of vertices, and we don’t
regard either of the two vertices as having a distinct role from the other. However,
it is more common in computing to consider directed graphs or digraphs in
which edges are ordered pairs (u, v); here the vertex u is the source of the edge
and vertex v is the sink or target of the edge. Directed edges are usually drawn
as arrows and undirected edges as curves or line segments. It is always possible
to represent an undirected graph as a directed graph where each undirected edge
{u, v} becomes two oppositely directed edges (u, v) and (v, u).
Here is an example of a small graph, drawn using this file using the circo
program from the GraphViz library:
Here is a similar directed graph, drawn using this file:
Given an edge (u, v), the vertices u and v are said to be incident to the edge
and adjacent to each other. The number of vertices adjacent to a given vertex u
is the degree of u; this can be divided into the out-degree (number of vertices
v such that (u, v) is an edge) and the in-degree (number of vertices v such that
(v, u) is an edge). A vertex v adjacent to u is called a neighbor of u, and (in a
directed graph) is a predecessor of u if (v, u) is an edge and a successor of u
if (u, v) is an edge. We will allow a node to be its own predecessor and successor.
5.12.2
Why graphs are useful
Graphs can be used to model any situation where we have things that are related
to each other in pairs; for example, all of the following can be represented by
graphs:
Family trees Nodes are members, with an edge from each parent to each of
their children. Transportation networks
Nodes are airports, intersections, ports, etc. Edges are airline flights,
one-way roads, shipping routes, etc. Assignments
Suppose we are assigning classes to classrooms. Let each node be either a
class or a classroom, and put an edge from a class to a classroom if the
class is assigned to that room. This is an example of a bipartite graph,
where the nodes can be divided into two sets S and T and all edges go
from S to T .
310
Figure 3: A graph
311
Figure 4: A directed graph
312
5.12.3
Operations on graphs
What would we like to do to graphs? Generally, we first have to build a graph by
starting with a set of nodes and adding in any edges we need, and then we want
to extract information from it, such as “Is this graph connected?”, “What is the
shortest path in this graph from s to t?”, or “How many edges can I remove from
this graph before some nodes become unreachable from other nodes?” There
are standard algorithms for answering all of these questions; the information
these algorithms need is typically (a) given a vertex u, what successors does it
have; and sometimes (b) given vertices u and v, does the edge (u, v) exist in the
graph?
5.12.4
Representations of graphs
A good graph representation will allow us to answer one or both of these questions
quickly. There are generally two standard representations of graphs that are
used in graph algorithms, depending on which question is more important.
For both representations, we simplify the representation task by insisting that
vertices be labeled 0, 1, 2, . . . , n − 1, where n is the number of vertices in the
graph. If we have a graph with different vertex labels (say, airport codes), we
can enforce an integer labeling by a preprocessing step where we assign integer
labels, and then translate the integer labels back into more useful user labels
afterwards. The preprocessing step can usually be done using a hash table in
O(n) time, which is likely to be smaller than the cost of whatever algorithm we
are running on our graph, and the savings in code complexity and running time
from working with just integer labels will pay this cost back many times over.
5.12.4.1
Adjacency matrices
An adjacency matrix is just a matrix a where a[i][j] is 1 if (i,j) is an edge in
the graph and 0 otherwise. It’s easy to build an adjacency matrix, and adding or
testing for the existence of an edges takes O(1) time. The downsides of adjacency
matrices are that finding all the outgoing edges from a vertex takes O(n) time
even if there aren’t very many, and the O(n2 ) space cost is high for “sparse
graphs,” those with much fewer than n2 edges.
5.12.4.2
Adjacency lists
An adjacency list representation of a graph creates a list of successors for each
node u. These lists may be represented as linked lists (the typical assumption in
algorithms textbooks), or in languages like C may be represented by variablelength arrays. The cost for adding an edge is still O(1), but testing for the
existence of an edge (u, v) rises to O(d+ (u)), where d+ (u) is the out-degree of
u (i.e., the length of the list of u’s successors). The cost of enumerating the
313
successors of u is also O(d+ (u)), which is clearly the best possible since it takes
that long just to write them all down. Finding predecessors of a node u is
extremely expensive, requiring looking through every list of every node in time
O(n + m), where m is the total number of edges, although if this is something
we actually need to do often we can store a second copy of the graph with the
edges reversed.
Adjacency lists are thus most useful when we mostly want to enumerate outgoing
edges of each node. This is common in search tasks, where we want to find a
path from one node to another or compute the distances between pairs of nodes.
If other operations are important, we can optimize them by augmenting the
adjacency list representation; for example, using sorted arrays for the adjacency
lists reduces the cost of edge existence testing to O(log(d+ (u))), and adding a
second copy of the graph with reversed edges lets us find all predecessors of u in
O(d− (u)) time, where d− (u) is u’s in-degree.
Adjacency lists also require much less space than adjacency matrices for sparse
graphs: O(n + m) vs O(n2 ) for adjacency matrices. For this reason adjacency
lists are more commonly used than adjacency matrices.
5.12.4.2.1
An implementation
Here is an implementation of a basic graph type using adjacency lists.
/* basic directed graph type */
typedef struct graph *Graph;
/* create a new graph with n vertices labeled 0..n-1 and no edges */
Graph graphCreate(int n);
/* free all space used by graph */
void graphDestroy(Graph);
/* add an edge to an existing graph */
/* doing this more than once may have unpredictable results */
void graphAddEdge(Graph, int source, int sink);
/* return the number of vertices/edges in the graph */
int graphVertexCount(Graph);
int graphEdgeCount(Graph);
/* return the out-degree of a vertex */
int graphOutDegree(Graph, int source);
/* return 1 if edge (source, sink) exists), 0 otherwise */
int graphHasEdge(Graph, int source, int sink);
314
/* invoke f on all edges (u,v) with source u */
/* supplying data as final parameter to f */
/* no particular order is guaranteed */
void graphForeach(Graph g, int source,
void (*f)(Graph g, int source, int sink, void *data),
void *data);
examples/graphs/graph.h
#include <stdlib.h>
#include <assert.h>
#include "graph.h"
/* basic directed graph type */
/* the implementation uses adjacency lists
* represented as variable-length arrays */
/* these arrays may or may not be sorted: if one gets long enough
* and you call graphHasEdge on its source, it will be */
struct graph {
int n;
/* number of vertices */
int m;
/* number of edges */
struct successors {
int d;
/* number of successors */
int len;
/* number of slots in array */
int isSorted; /* true if list is already sorted */
int list[];
/* actual list of successors starts here */
} *alist[];
};
/* create a new graph with n vertices labeled 0..n-1 and no edges */
Graph
graphCreate(int n)
{
Graph g;
int i;
g = malloc(sizeof(struct graph) + sizeof(struct successors *) * n);
assert(g);
g->n = n;
g->m = 0;
for(i = 0; i < n; i++) {
315
g->alist[i] = malloc(sizeof(struct successors));
assert(g->alist[i]);
}
}
g->alist[i]->d = 0;
g->alist[i]->len = 0;
g->alist[i]->isSorted= 1;
return g;
/* free all space used by graph */
void
graphDestroy(Graph g)
{
int i;
}
for(i = 0; i < g->n; i++) free(g->alist[i]);
free(g);
/* add an edge to an existing graph */
void
graphAddEdge(Graph g, int u, int v)
{
assert(u >= 0);
assert(u < g->n);
assert(v >= 0);
assert(v < g->n);
/* do we need to grow the list? */
while(g->alist[u]->d >= g->alist[u]->len) {
g->alist[u]->len = g->alist[u]->len * 2 + 1;
/* +1 because it might have been 0 */
g->alist[u] =
realloc(g->alist[u],
sizeof(struct successors) + sizeof(int) * g->alist[u]->len);
}
/* now add the new sink */
g->alist[u]->list[g->alist[u]->d++] = v;
g->alist[u]->isSorted = 0;
}
/* bump edge count */
g->m++;
316
/* return the number of vertices in the graph */
int
graphVertexCount(Graph g)
{
return g->n;
}
/* return the number of vertices in the graph */
int
graphEdgeCount(Graph g)
{
return g->m;
}
/* return the out-degree of a vertex */
int
graphOutDegree(Graph g, int source)
{
assert(source >= 0);
assert(source < g->n);
}
return g->alist[source]->d;
/* when we are willing to call bsearch */
#define BSEARCH_THRESHOLD (10)
static int
intcmp(const void *a, const void *b)
{
return *((const int *) a) - *((const int *) b);
}
/* return 1 if edge (source, sink) exists), 0 otherwise */
int
graphHasEdge(Graph g, int source, int sink)
{
int i;
assert(source >= 0);
assert(source < g->n);
assert(sink >= 0);
assert(sink < g->n);
if(graphOutDegree(g, source) >= BSEARCH_THRESHOLD) {
/* make sure it is sorted */
317
if(! g->alist[source]->isSorted) {
qsort(g->alist[source]->list,
g->alist[source]->d,
sizeof(int),
intcmp);
}
}
/* call bsearch to do binary search for us */
return
bsearch(&sink,
g->alist[source]->list,
g->alist[source]->d,
sizeof(int),
intcmp)
!= 0;
} else {
/* just do a simple linear search */
/* we could call lfind for this, but why bother? */
for(i = 0; i < g->alist[source]->d; i++) {
if(g->alist[source]->list[i] == sink) return 1;
}
/* else */
return 0;
}
/* invoke f on all edges (u,v) with source u */
/* supplying data as final parameter to f */
void
graphForeach(Graph g, int source,
void (*f)(Graph g, int source, int sink, void *data),
void *data)
{
int i;
assert(source >= 0);
assert(source < g->n);
}
for(i = 0; i < g->alist[source]->d; i++) {
f(g, source, g->alist[source]->list[i], data);
}
examples/graphs/graph.c
And here is some test code: graphTest.c.
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5.12.4.3
Implicit representations
For some graphs, it may not make sense to represent them explicitly. An example
might be the word-search graph from CS223/2005/Assignments/HW10, which
consists of all words in a dictionary with an edge between any two words that
differ only by one letter. In such a case, rather than building an explicit data
structure containing all the edges, we might generate edges as needed when
computing the neighbors of a particular vertex. This gives us an implicit or
procedural representation of a graph.
Implicit representations require the ability to return a vector or list of values
from the neighborhood-computing function. There are various way to do this, of
which the most sophisticated might be to use an iterator.
5.12.5
Searching for paths in a graph
A path is a sequence of vertices v1 , v2 , . . . vk where each pair (vi , vi+1 ) is an
edge. Often we want to find a path from a source vertex s to a target vertex t, or
more generally to detect which vertices are reachable from a given source vertex
s. We can solve these problems by using any of several standard graph search
algorithms, of which the simplest and most commonly used are depth-first
search and breadth-first search.
Both of these search algorithms are a special case of a more general algorithm
for growing a directed tree in a graph rooted at a given node s. Here we are
using tree as a graph theorist would, to mean any set of k nodes joined by k − 1
edges. This is similar to trees used in data structures except that there are no
limits on the number of children a node can have and no ordering constraints
within the tree.
The general tree-growing algorithm might be described as follows:
1. Start with a tree consisting of just s.
2. If there is at least one edge that leaves the tree (i.e. goes from a node in
the current tree to a node outside the current tree), pick the “best” such
edge and add it and its sink to the tree.
3. Repeat step 2 until no edges leave the tree.
Practically, steps 2 and 3 are implemented by having some sort of data structure
that acts as a bucket for unprocessed edges. When a new node is added to the
tree, all of its outgoing edges are thrown into the bucket. The “best” outgoing
edge is obtained by applying some sort of pop, dequeue, or delete-min operation
to the bucket, depending on which it provides; if this edge turns out to be an
internal edge of the tree (maybe we added its sink after putting it in the bucket),
we throw it away. Otherwise we mark the edge and its sink as belonging to the
tree and repeat.
The output of the generic tree-growing algorithm typically consists of (a) marks
319
on all the nodes that are reachable from s, and (b) for each such node v, a
parent pointer back to the source of the edge that brought v into the tree. Often
these two values can be combined by using a null parent pointer to represent the
absence of a mark (this usually requires making the root point to itself so that
we know it’s in the tree). Other values that may be useful are a table showing
the order in which nodes were added to the tree.
What kind of tree we get depends on what we use for the bucket—specifically,
on what edge is returned when we ask for the “best” edge. Two easy cases are:
1. The bucket is a stack. When we ask for an outgoing edge, we get the
last edge inserted. This has the effect of running along as far as possible
through the graph before backtracking, since we always keep going from
the last node if possible. The resulting algorithm is called depth-first
search and yields a depth-first search tree. If we don’t care about the
lengths of the paths we consider, depth-first search is a perfectly good
algorithm for testing connectivity. It can also be implemented without any
auxiliary data structures as a recursive procedure, as long as we don’t go
so deep as to blow out the system stack.
2. The bucket is a queue. Now when we ask for an outgoing edge, we get the
first edge inserted. This favors edges that are close to the root: we don’t
start consider edges from nodes adjacent to the root until we have already
added all the root’s successors to the tree, and similarly we don’t start
considering edges at distance k until we have already added all the closer
nodes to the tree. This gives breadth-first search, which constructs a
shortest-path tree in which every path from the root to a node in the
tree has the minimum length.
Structurally, these algorithms are almost completely identical; indeed, if we
organize the stack/queue so that it can pop from both ends, we can switch
between depth-first search and breadth-first search just by choosing which end
to pop from.
Below, we give a [combined implementation](#combinedDFSBFS} of both depthfirst search and breadth-first search that does precisely this, although this is
mostly for show. Typical implementations of breadth-first search include a
further optimization, where we test an edge to see if we should add it to the
tree (and possibly add it) before inserting into the queue. This gives the same
result as the DFS-like implementation but only requires O(n) space for the
queue instead of O(m), with a smaller constant as well since don’t need to
bother storing source edges in the queue. An example of this approach is given
[below]{#graphSearchImplementation}.
The running time of any of these algorithms is very fast: we pay O(1) per
vertex in setup costs and O(1) per edge during the search (assuming the input
is in adjacency-list form), giving a linear O(n + m) total cost. Often it is more
expensive to set up the graph in the first place than to run a search on it.
320
5.12.5.1
Implementation of depth-first and breadth-first search
Here is a simple implementation of depth-first search, using a recursive algorithm,
and breadth-first search, using an iterative algorithm that maintains a queue of
vertices. In both cases the algorithm is applied to a sample graph whose vertices
are the integers 0 through n − 1 for some n, and in which vertex x has edges to
vertices x/2, 3 · x, and x + 1, whenever these values are also integers in the range
0 through n − 1. For large graphs it may be safer to run an iterative version of
DFS that uses an explicit stack instead of a possibly very deep recursion.
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<assert.h>
<stdint.h>
typedef int Vertex;
#define VERTEX_NULL (-1)
struct node {
Vertex *neighbors;
Vertex parent;
};
/* array of outgoing edges, terminated by VERTEX_NULL */
/* for search */
struct graph {
size_t n;
struct node *v;
};
/* number of vertices */
/* list of vertices */
void
graphDestroy(struct graph *g)
{
Vertex v;
for(v = 0; v < g->n; v++) {
free(g->v[v].neighbors);
}
}
free(g);
/* this graph has edges from x to x+1, x to 3*x, and x to x/2 (when x is even) */
struct graph *
makeSampleGraph(size_t n)
{
struct graph *g;
Vertex v;
321
const int allocNeighbors = 4;
int i;
g = malloc(sizeof(*g));
assert(g);
g->n = n;
g->v = malloc(sizeof(struct node) * n);
assert(g->v);
for(v = 0; v < n; v++) {
g->v[v].parent = VERTEX_NULL;
}
}
/* fill in neighbors */
g->v[v].neighbors = malloc(sizeof(Vertex) * allocNeighbors);
i = 0;
if(v % 2 == 0) { g->v[v].neighbors[i++] = v/2; }
if(3*v < n) { g->v[v].neighbors[i++] = 3*v; }
if(v+1 < n) { g->v[v].neighbors[i++] = v+1; }
g->v[v].neighbors[i++] = VERTEX_NULL;
return g;
/* output graph in dot format */
void
printGraph(const struct graph *g)
{
Vertex u;
size_t i;
puts("digraph G {");
for(u = 0; u < g->n; u++) {
for(i = 0; g->v[u].neighbors[i] != VERTEX_NULL; i++) {
printf("%d -> %d;\n", u, g->v[u].neighbors[i]);
}
}
}
puts("}");
/* reconstruct path back to root from u */
void
printPath(const struct graph *g, Vertex u)
322
{
}
do {
printf(" %d", u);
u = g->v[u].parent;
} while(g->v[u].parent != u);
/* print the tree in dot format */
void
printTree(const struct graph *g)
{
Vertex u;
puts("digraph G {");
for(u = 0; u < g->n; u++) {
if(g->v[u].parent != VERTEX_NULL) {
printf("%d -> %d;\n", u, g->v[u].parent);
}
}
}
puts("}");
/* compute DFS tree starting at root */
/* this uses a recursive algorithm and will not work on large graphs! */
static void
dfsHelper(struct graph *g, Vertex parent, Vertex child)
{
int i;
Vertex neighbor;
}
if(g->v[child].parent == VERTEX_NULL) {
g->v[child].parent = parent;
for(i = 0; (neighbor = g->v[child].neighbors[i]) != VERTEX_NULL; i++) {
dfsHelper(g, child, neighbor);
}
}
void
dfs(struct graph *g, Vertex root)
{
dfsHelper(g, root, root);
}
323
/* compute BFS tree starting at root */
void
bfs(struct graph *g, Vertex root)
{
Vertex *q;
int head;
/* deq from here */
int tail;
/* enq from here */
Vertex current;
Vertex nbr;
int i;
q = malloc(sizeof(Vertex) * g->n);
assert(q);
head = tail = 0;
/* push root onto q */
g->v[root].parent = root;
q[tail++] = root;
while(head < tail) {
current = q[head++];
}
}
for(i = 0; (nbr = g->v[current].neighbors[i]) != VERTEX_NULL; i++) {
if(g->v[nbr].parent == VERTEX_NULL) {
/* haven't seen this guy */
/* push it */
g->v[nbr].parent = current;
q[tail++] = nbr;
}
}
free(q);
int
main(int argc, char **argv)
{
int n;
struct graph *g;
if(argc != 3) {
fprintf(stderr, "Usage: %s action n\nwhere action =\n
324
g - print graph\n
d - print
}
return 1;
n = atoi(argv[2]);
g = makeSampleGraph(n);
switch(argv[1][0]) {
case 'g':
printGraph(g);
break;
case 'd':
dfs(g, 0);
printTree(g);
break;
case 'b':
bfs(g, 0);
printTree(g);
break;
default:
fprintf(stderr, "%s: unknown action '%c'\n", argv[0], argv[1][0]);
return 1;
}
graphDestroy(g);
}
return 0;
examples/graphSearch/search.c
The output of the program is either the graph, a DFS tree of the graph rooted
at 0, or a BFS tree of the graph rooted at 0, in a format suitable for feeding to
the GraphViz program dot, which draws pictures of graphs.
Here are the pictures for n = 20.
5.12.5.2
search
Combined implementation of depth-first and breadth-first
These are some older implementations of BFS and DFS that demonstrate how
both can be written using the same code just by changing the behavior of the
core data structure. This also demonstrates how to construct DFS iteratively;
for BFS, the preceding implementation is better in every respect.
/* Typical usage:
*
325
Figure 5: The full graph
326
Figure 6: DFS tree
327
Figure 7: BFS tree
328
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*/
struct searchInfo *s;
int n;
s = searchInfoCreate(g);
n = graph_vertices(g);
for(i = 0; i < n; i++) {
dfs(s, i);
}
... use results in s ...
searchInfoDestroy(s);
/* summary information per node for dfs and bfs */
/* this is not intended to be opaque---user can read it */
/* (but should not write it!) */
#define SEARCH_INFO_NULL (-1) /* for empty slots */
struct searchInfo {
Graph graph;
int reached;
int *preorder;
int *time;
int *parent;
int *depth;
};
/*
/*
/*
/*
/*
count of reached nodes */
list of nodes in order first reached */
time[i] == position of node i in preorder list */
parent in DFS or BFS forest */
distance from root */
/* allocate and initialize search results structure */
/* you need to do this before passing it to dfs or bfs */
struct searchInfo *searchInfoCreate(Graph g);
/* free searchInfo data---does NOT free graph pointer */
void searchInfoDestroy(struct searchInfo *);
/* perform depth-first search starting at root, updating results */
void dfs(struct searchInfo *results, int root);
/* perform breadth-first search starting at root, updating results */
void bfs(struct searchInfo *results, int root);
examples/graphs/genericSearch.h
#include <stdlib.h>
329
#include <assert.h>
#include "graph.h"
#include "genericSearch.h"
/* create an array of n ints initialized to SEARCH_INFO_NULL */
static int *
createEmptyArray(int n)
{
int *a;
int i;
a = malloc(sizeof(*a) * n);
assert(a);
for(i = 0; i < n; i++) {
a[i] = SEARCH_INFO_NULL;
}
}
return a;
/* allocate and initialize search results structure */
/* you need to do this before passing it to dfs or bfs */
struct searchInfo *
searchInfoCreate(Graph g)
{
struct searchInfo *s;
int n;
s = malloc(sizeof(*s));
assert(s);
s->graph = g;
s->reached = 0;
n = graphVertexCount(g);
s->preorder = createEmptyArray(n);
s->time = createEmptyArray(n);
s->parent = createEmptyArray(n);
s->depth = createEmptyArray(n);
}
return s;
330
/* free searchInfo data---does NOT free graph pointer */
void
searchInfoDestroy(struct searchInfo *s)
{
free(s->depth);
free(s->parent);
free(s->time);
free(s->preorder);
free(s);
}
/* used inside search routines */
struct edge {
int u;
/* source */
int v;
/* sink */
};
/* stack/queue */
struct queue {
struct edge *e;
int bottom;
int top;
};
static void
pushEdge(Graph g, int u, int v, void *data)
{
struct queue *q;
q = data;
assert(q->top < graphEdgeCount(g) + 1);
}
q->e[q->top].u = u;
q->e[q->top].v = v;
q->top++;
/* this rather horrible function implements dfs if useQueue == 0 */
/* and bfs if useQueue == 1 */
static void
genericSearch(struct searchInfo *r, int root, int useQueue)
{
/* queue/stack */
struct queue q;
331
/* edge we are working on */
struct edge cur;
/* start with empty q */
/* we need one space per edge */
/* plus one for the fake (root, root) edge */
q.e = malloc(sizeof(*q.e) * (graphEdgeCount(r->graph) + 1));
assert(q.e);
q.bottom = q.top = 0;
/* push the root */
pushEdge(r->graph, root, root, &q);
/* while q.e not empty */
while(q.bottom < q.top) {
if(useQueue) {
cur = q.e[q.bottom++];
} else {
cur = q.e[--q.top];
}
/* did we visit sink already? */
if(r->parent[cur.v] != SEARCH_INFO_NULL) continue;
/* no */
assert(r->reached < graphVertexCount(r->graph));
r->parent[cur.v] = cur.u;
r->time[cur.v] = r->reached;
r->preorder[r->reached++] = cur.v;
if(cur.u == cur.v) {
/* we could avoid this if we were certain SEARCH_INFO_NULL */
/* would never be anything but -1 */
r->depth[cur.v] = 0;
} else {
r->depth[cur.v] = r->depth[cur.u] + 1;
}
}
}
/* push all outgoing edges */
graphForeach(r->graph, cur.v, pushEdge, &q);
free(q.e);
void
332
dfs(struct searchInfo *results, int root)
{
genericSearch(results, root, 0);
}
void
bfs(struct searchInfo *results, int root)
{
genericSearch(results, root, 1);
}
examples/graphs/genericSearch.c
And here is some test code: genericSearchTest.c. You will need to compile
genericSearchTest.c together with both genericSearch.c and graph.c to
get it to work. This Makefile will do this for you.
5.12.5.3
Other variations on the basic algorithm
Stacks and queues are not the only options for the bucket in the generic search
algorithm. Some other choices are:
• A priority queue keyed by edge weights. If the edges have weights, the
generic tree-builder can be used to find a tree containing s with minimum
total edge weight.22 The basic idea is to always pull out the lightest edge.
The resulting algorithm runs in O(n + m log m) time (since each heap
operation takes O(log m) time), and is known as Prim’s algorithm. See
Prim’s algorithm for more details.
• A priority queue keyed by path lengths. Here we assume that edges have
lengths, and we want to build a shortest-path tree where the length of
the path is no longer just the number of edges it contains but the sum of
their weights. The basic idea is to keep track of the distance from the root
to each node in the tree, and assign each edge a key equal to the sum of
the distance to its source and its length. The resulting search algorithm,
known as Dijkstra’s algorithm, will give a shortest-path tree if all the
edge weights are non-negative. See Dijkstra’s algorithm.
5.13
Dynamic programming
Dynamic programming is a general-purpose algorithm design technique that
is most often used to solve combinatorial optimization problems, where
we are looking for the best possible input to some function chosen from an
exponentially large search space.
22 This only works if the graph is undirected, which means that for every edge uv there is a
matching edge vu with the same weight.
333
There are two parts to dynamic programming. The first part is a programming
technique: dynamic programming is essentially divide and conquer run in reverse:
we solve a big instance of a problem by breaking it up recursively into smaller
instances; but instead of carrying out the computation recursively from the top
down, we start from the bottom with the smallest instances of the problem,
solving each increasingly large instance in turn and storing the result in a
table. The second part is a design principle: in building up our table, we are
careful always to preserve alternative solutions we may need later, by delaying
commitment to particular choices to the extent that we can.
The bottom-up aspect of dynamic programming is most useful when a straightforward recursion would produce many duplicate subproblems. It is most efficient
when we can enumerate a class of subproblems that doesn’t include too many
extraneous cases that we don’t need for our original problem.
To take a simple example, suppose that we want to compute the n-th Fibonacci
number using the defining recurrence
• F (n) = F (n − 1) + F (n − 2)
• F (1) = F (0) = 1.
A naive approach would simply code the recurrence up directly:
int
fib(int n)
{
if(n < 2) {
return 1
} else {
return fib(n-1) + fib(n-2);
}
}
The running time of this procedure is easy to compute. The recurrence is
• T (n) = T (n − 1) + T (n − 2) + Θ(1),
whose solution is Θ(an ) where a is the golden ratio 1.6180339887498948482 . . ..
This is badly exponential.23
5.13.1
Memoization
The problem is that we keep recomputing values of fib that we’ve already
computed. We can avoid this by memoization, where we wrap our recursive
solution in a memoizer that stores previously-computed solutions in a hash
23 But it’s linear in the numerical value of the output, which means that fib(n) will actually
terminate in a reasonable amount of time on a typical modern computer when run on any n
small enough that F (n) fits in 32 bits. Running it using 64-bit (or larger) integer representations
will be slower.
334
table. Sensible programming languages will let you write a memoizer once and
apply it to arbitrary recursive functions. In less sensible programming languages
it is usually easier just to embed the memoization in the function definition itself,
like this:
int
memoFib(int n)
{
int ret;
}
if(hashContains(FibHash, n)) {
return hashGet(FibHash, n);
} else {
ret = memoFib(n-1) + memoFib(n-2);
hashPut(FibHash, n, ret);
return ret;
}
The assumption here is that FibHash is a global hash table that we have initialized
to map 0 and 1 to 1. The total cost of running this procedure is O(n), because
fib is called at most twice for each value k in 0 . . . n.
Memoization is a very useful technique in practice, but it is not popular with
algorithm designers because computing the running time of a complex memoized
procedure is often much more difficult than computing the time to fill a nice
clean table. The use of a hash table instead of an array may also add overhead
(and code complexity) that comes out in the constant factors. But it is always
the case that a memoized recursive procedure considers no more subproblems
than a table-based solution, and it may consider many fewer if we are sloppy
about what we put in our table (perhaps because we can’t easily predict what
subproblems will be useful).
5.13.2
Dynamic programming
Dynamic programming comes to the rescue. Because we know what smaller cases
we have to reduce F(n) to, instead of computing F(n) top-down, we compute it
bottom-up, hitting all possible smaller cases and storing the results in an array:
int
fib2(int n)
{
int *a;
int i;
int ret;
if(n < 2) {
335
return 1;
} else {
a = malloc(sizeof(*a) * (n+1));
assert(a);
a[1] = a[2] = 1;
}
}
for(i = 3; i <= n; i++) {
a[i] = a[i-1] + a[i-2];
}
ret = a[n];
free(a);
return ret;
Notice the recurrence is exactly the same in this version as in our original
recursive version, except that instead of computing F(n-1) and F(n-2) recursively,
we just pull them out of the array. This is typical of dynamic-programming
solutions: often the most tedious editing step in converting a recursive algorithm
to dynamic programming is changing parentheses to square brackets. As with
memoization, the effect of this conversion is dramatic; what used to be an
exponential-time algorithm is now linear-time.
5.13.2.1
5.13.2.1.1
More examples
Longest increasing subsequence
Suppose that we want to compute the longest increasing subsequence of
an array. This is a sequence, not necessarily contiguous, of elements from the
array such that each is strictly larger than the one before it. Since there are 2n
different subsequences of an n-element array, the brute-force approach of trying
all of them might take a while.
What makes this problem suitable for dynamic programming is that any prefix
of a longest increasing subsequence is a longest increasing subsequence of the
part of the array that ends where the prefix ends; if it weren’t, we could
make the big sequence longer by choosing a longer prefix. So to find the
longest increasing subsequence of the whole array, we build up a table of longest
increasing subsequences for each initial prefix of the array. At each step, when
finding the longest increasing subsequence of elements 0 . . . i, we can just scan
through all the possible values for the second-to-last element and read the length
of the best possible subsequence ending there out of the table. When the table
is complete, we can scan for the best last element and then work backwards to
reconstruct the actual subsequence.
336
This last step requires some explanation. We don’t really want to store in
table[i] the full longest increasing subsequence ending at position i, because it
may be very big. Instead, we store the index of the second-to-last element of this
sequence. Since that second-to-last element also has a table entry that stores the
index of its predecessor, by following the indices we can generate a subsequence
of length O(n), even though we only stored O(1) pieces of information in each
table entry. This is similar to the parent pointer technique used in graph search
algorithms.
Here’s what the code looks like:
/* compute a longest strictly increasing subsequence of an array of ints */
/* input is array a with given length n */
/* returns length of LIS */
/* If the output pointer is non-null, writes LIS to output pointer. */
/* Caller should provide at least sizeof(int)*n space for output */
/* If there are multiple LIS's, which one is returned is arbitrary. */
unsigned long
longest_increasing_subsequence(const int a[], unsigned long n, int *output);
examples/dynamicProgramming/lis/lis.h
#include <stdlib.h>
#include <assert.h>
#include "lis.h"
unsigned long
longest_increasing_subsequence(const int a[], unsigned long n, int *output)
{
struct lis_data {
unsigned long length;
/* length of LIS ending at this point */
unsigned long prev;
/* previous entry in the LIS ending at this point
} *table;
unsigned long best;
unsigned long scan;
/* best entry in table */
/* used to generate output */
unsigned long i;
unsigned long j;
unsigned long best_length;
/* special case for empty table */
if(n == 0) return 0;
table = malloc(sizeof(*table) * n);
for(i = 0; i < n; i++) {
337
/* default best is just this element by itself */
table[i].length = 1;
table[i].prev = n;
/* default end-of-list value */
}
/* but try all other possibilities */
for(j = 0; j < i; j++) {
if(a[j] < a[i] && table[j].length + 1 > table[i].length) {
/* we have a winner */
table[i].length = table[j].length + 1;
table[i].prev = j;
}
}
/* now find the best of the lot */
best = 0;
for(i = 1; i < n; i++) {
if(table[i].length > table[best].length) {
best = i;
}
}
/* table[best].length is now our return value */
/* save it so that we don't lose it when we free table */
best_length = table[best].length;
/* do we really have to compute the output? */
if(output) {
/* yes :-( */
scan = best;
for(i = 0; i < best_length; i++) {
assert(scan >= 0);
assert(scan < n);
output[best_length - i - 1] = a[scan];
}
}
scan = table[scan].prev;
free(table);
}
return best_length;
338
examples/dynamicProgramming/lis/lis.c
A sample program that runs longest_increasing_subsequence on a list of
numbers passed in by stdin is given in test_lis.c. Here is a Makefile.
Implemented like this, the cost of finding an LIS is O(n2 ), because to compute
each entry in the array, we have to search through all the previous entries to
find the longest path that ends at a value less than the current one. This can be
improved by using a more clever data structure. If we use a binary search tree
that stores path keyed by the last value, and augment each node with a field
that represents the maximum length of any path in the subtree under that node,
then we can find the longest feasible path that we can append the current node
to in O(log n) time instead of O(n) time. This brings the total cost down to
only O(n log n).
5.13.2.1.2
All-pairs shortest paths
Suppose we want to compute the distance between any two points in a graph,
where each edge uv has a length `u v (+∞ for edges not in the graph) and
the distance between two vertices s and t$ is the minimum over all s–t paths
of the total length of the edges. There are various algorithms for doing this
for a particular s and t, but there is also a very simple dynamic programming
algorithm known as Floyd-Warshall that computes the distance between all
n2 pairs of vertices in Θ(n3 ) time.
The assumption is that the graph does not contain a negative cycle (a cycle
with total edge weight less than zero), so that for two connected nodes there
is always a shortest path that uses each intermediate vertex at most once. If a
graph does contain a negative cycle, the algorithm will detect it by reporting
the distance from i to i less than zero for some i.
Negative cycles don’t generally exist in distance graphs (unless you have the
ability to move faster than the speed of light), but they can come up in other
contexts. One example would be in currency arbitrage, where each node is some
currency, the weight of an edge uv is the logarithm of the exchange rate from
u to v, and the total weight of a path from s to t gives the logarithm of the
number of units of t you can get for one unit of s, since adding the logs along
the path corresponds to multiplying all the exchange rates. In this context a
negative cycle gives you a way to turn a dollar into less than a dollar by running
it through various other currencies, which is not useful, but a positive cycle lets
you pay for the supercomputer you bought to find it before anybody else did. If
we negate all the edge weights, we turn a positive cycle into a negative cycle,
making a fast algorithm for finding this negative cycle potentially valuable.
However, if we don’t have any negative cycles, the idea is that we can create
restricted instances of the shortest-path problem by limiting the maximum index
of any node used on the path. Let L(i, j, k) be the length of a shortest path from
i to j that uses only the vertices 0, . . . , k − 1 along the path (not counting the
339
endpoints i and j, which can be anything). When k = 0, this is just the length
of the i–j edge, or +∞ if there is no such edge. So we can start by computing
L(i, j, 0) for all i. Now given L(i, j, k) for all i and some k, we can compute
L(i, j, k + 1) by observing that any shortest i–j path that has intermediate
vertices in 0 . . . k either consists of a path with intermediate vertices in 0 . . . k − 1,
or consists of a path from i to k followed by a path from k to j, where both of
these paths have intermediate vertices in 0 . . . k − 1. So we get
• L(i, j, k + 1) = min(L(i, j, k), L(i, k, k) + L(k, j, k).
Since this takes O(1) time to compute if we have previously computed L(i, j, k)
for all i and j, we can fill in the entire table in O(n3 ) time.
Implementation details:
• If we want to reconstruct the shortest path in addition to computing its
length, we can store the first vertex for each i–j path. This will either be
(a) the first vertex in the i–j path for the previous k, or (b) the first vertex
in the i–k path.
• We don’t actually need to use a full three-dimensional array. It’s enough
to store one value for each pair i, j and let k be implicit. At each step we
let L[i][j] be min(L[i][j], L[i][k] + L[k][j]). The trick is that we don’t care
if L[i][k] or L[k][j] has already been updated, because that will only give
us paths with a few extra k vertices, which won’t be the shortest paths
anyway assuming no negative cycles.
5.13.2.1.3
Longest common subsequence
Given sequences of characters v and w, v is a subsequence of w if every character
in v appears in w in the same order. For example, aaaaa, brac, and badar
are all subsequences of abracadabra, but badcar is not. A longest common
subsequence (LCS for short) of two sequences x and y is the longest sequence that
is a subsequence of both: two longest common subsequences of abracadabra
and badcar are badar and bacar.
As with longest increasing subsequence, one can find the LCS of two sequence by
brute force, but it will take even longer. Not only are there are 2n subsequences
of a sequence of length n, but checking each subsequence of the first to see
if it is also a subsequence of the second may take some time. It is better to
solve the problem using dynamic programming. Having sequences gives an
obvious linear structure to exploit: the basic strategy will be to compute LCSs
for increasingly long prefixes of the inputs. But with two sequences we will have
to consider prefixes of both, which will give us a two-dimensional table where
rows correspond to prefixes of sequence x and columns correspond to prefixes of
sequence y.
The recursive decomposition that makes this technique work looks like this. Let
L(x, y) be the length of the longest common subsequence of x and y, where
340
x and y are strings. Let a and b be single characters. Then L(xa, yb) is the
maximum of:
• L(x, y) + 1, if a = b,
• L(xa, y), or
• L(x, yb).
The idea is that we either have a new matching character we couldn’t use before
(the first case), or we have an LCS that doesn’t use one of a or b (the remaining
cases). In each case the recursive call to LCS involves a shorter prefix of xa or
yb, with an ultimate base case L(x, y) = 0 if at least one of x or y is the empty
string. So we can fill in these values in a table, as long as we are careful to
make sure that the shorter prefixes are always filled first. If we are smart about
remembering which case applies at each step, we can even go back and extract
an actual LCS, by stitching together to places where a = b. Here’s a short C
program that does this:
#include
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<assert.h>
<string.h>
<limits.h>
/* compute longest common subsequence of argv[1] and argv[2] */
/* computes longest common subsequence of x and y, writes result to lcs */
/* lcs should be pre-allocated by caller to 1 + minimum length of x or y */
void
longestCommonSubsequence(const char *x, const char *y, char *lcs)
{
int xLen;
int yLen;
int i;
/* position in x */
int j;
/* position in y */
xLen = strlen(x);
yLen = strlen(y);
/* best choice at each position */
/* length gives length of LCS for these prefixes */
/* prev points to previous substring */
/* newChar if non-null is new character */
struct choice {
int length;
struct choice *prev;
char newChar;
} best[xLen][yLen];
341
for(i = 0; i < xLen; i++) {
for(j = 0; j < yLen; j++) {
/* we can always do no common substring */
best[i][j].length = 0;
best[i][j].prev = 0;
best[i][j].newChar = 0;
/* if we have a match, try adding new character */
/* this is always better than the nothing we started with */
if(x[i] == y[j]) {
best[i][j].newChar = x[i];
if(i > 0 && j > 0) {
best[i][j].length = best[i-1][j-1].length + 1;
best[i][j].prev = &best[i-1][j-1];
} else {
best[i][j].length = 1;
}
}
/* maybe we can do even better by ignoring a new character */
if(i > 0 && best[i-1][j].length > best[i][j].length) {
/* throw away a character from x */
best[i][j].length = best[i-1][j].length;
best[i][j].prev = &best[i-1][j];
best[i][j].newChar = 0;
}
if(j > 0 && best[i][j-1].length > best[i][j].length) {
/* throw away a character from x */
best[i][j].length = best[i][j-1].length;
best[i][j].prev = &best[i][j-1];
best[i][j].newChar = 0;
}
}
}
/* reconstruct string working backwards from best[xLen-1][yLen-1] */
int outPos;
/* position in output string */
struct choice *p; /* for chasing linked list */
outPos = best[xLen-1][yLen-1].length;
lcs[outPos--] = '\0';
for(p = &best[xLen-1][yLen-1]; p; p = p->prev) {
342
}
if(p->newChar) {
assert(outPos >= 0);
lcs[outPos--] = p->newChar;
}
}
int
main(int argc, char **argv)
{
if(argc != 3) {
fprintf(stderr, "Usage: %s string1 string2\n", argv[0]);
return 1;
}
char output[strlen(argv[1]) + 1];
longestCommonSubsequence(argv[1], argv[2], output);
puts(output);
}
return 0;
examples/dynamicProgramming/lcs/lcs.c
The whole thing takes O(nm) time where n and m are the lengths of A and B.
5.14
Randomization
Randomization is a fundamental technique in algorithm design, that allows
programs to run quickly when the average-case behavior of an algorithm is
better than the worst-case behavior. It is also heavily used in games, both in
entertainment and gambling. The latter application gives the only example I
know of a programmer killed for writing bad code, which shows how serious
good random-number generation is.
5.14.1
Generating random values in C
If you want random values in a C program, there are three typical ways of getting
them, depending on how good (i.e. uniform, uncorrelated, and unpredictable)
you want them to be.
5.14.1.1
The rand function from the standard library
343
E.g.
#include <stdio.h>
#include <stdlib.h>
int
main(int argc, char **argv)
{
printf("%d\n", rand());
return 0;
}
examples/randomization/randOnce.c
The rand function, declared in stdlib.h, returns a random-looking integer in
the range 0 to RAND_MAX (inclusive) every time you call it. On machines using
the GNU C library RAND_MAX is equal to INT_MAX which is typically 23 1 − 1, but
RAND_MAX may be as small as 32767. There are no particularly strong guarantees
about the quality of random numbers that rand returns, but it should be good
enough for casual use, and it has the advantage that as part of the C standard
you can assume it is present almost everywhere.
Note that rand is a pseudorandom number generator: the sequence of values
it returns is predictable if you know its starting state (and is still predictable
from past values in the sequence even if you don’t know the starting state, if
you are clever enough). It is also the case that the initial seed is fixed, so that
the program above will print the same value every time you run it.
This is a feature: it permits debugging randomized programs. As John von
Neumann, who proposed pseudorandom number generators in his 1946 talk
“Various Techniques Used in Connection With Random Digits,” explained:
We see then that we could build a physical instrument to feed random
digits directly into a high-speed computing machine and could have
the control call for these numbers as needed. The real objection
to this procedure is the practical need for checking computations.
If we suspect that a calculation is wrong, almost any reasonable
check involves repeating something done before. At that point the
introduction of new random numbers would be intolerable.
5.14.1.1.1
Supplying a seed with srand
If you want to get different sequences, you need to seed the random number
generator using srand. A typical use might be:
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
344
int
main(int argc, char **argv)
{
srand(time(0));
printf("%d\n", rand());
return 0;
}
examples/randomization/srandFromTime.c
Here time(0) returns the number of seconds since the epoch (00:00:00 UTC,
January 1, 1970, for POSIX systems, not counting leap seconds). Note that this
still might give repeated values if you run it twice in the same second, and it’s
extremely dangerous if you expect to distribute your code to a lot of people who
want different results, since two of your users are likely to run it twice in the
same second. See the discussion of /dev/urandom below for a better method.
5.14.1.2
Better pseudorandom number generators
There has been quite a bit of research on pseudorandom number generators over
the years, and much better pseudorandom number generators than rand are
available. The current champion for simulation work is Mersenne Twister,
which runs about 4 times faster than rand in its standard C implementation
and passes a much wider battery of statistical tests. Its English-language home
page is at http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html. As
with rand, you still need to provide an initial seed value.
There are also cryptographically secure pseudorandom number generators, of which the most famous is Blum Blum Shub. These cannot be predicted
based on their output if seeded with a true random value (under certain cryptographic assumptions: hardness of factoring for Blum Blum Shub). Unfortunately,
cryptographic PRNGs are usually too slow for day-to-day use.
5.14.1.3
Random numbers without the pseudo
If you really need actual random numbers and are on a Linux or BSD-like operating system, you can use the special device files /dev/random and /dev/urandom.
These can be opened for reading like ordinary files, but the values read from
them are a random sequence of bytes (including null characters). A typical use
might be:
#include <stdio.h>
int
main(int argc, char **argv)
{
unsigned int randval;
345
FILE *f;
f = fopen("/dev/random", "r");
fread(&randval, sizeof(randval), 1, f);
fclose(f);
printf("%u\n", randval);
}
return 0;
examples/randomization/devRandom.c
(A similar construction can also be used to obtain a better initial seed for srand
than time(0).)
Both /dev/random and /dev/urandom derive their random bits from physically
random properties of the computer, like time between keystrokes or small
variations in hard disk rotation speeds. The difference between the two is that
/dev/urandom will always give you some random-looking bits, even if it has to
generate extra ones using a cryptographic pseudo-random number generator,
while /dev/random will only give you bits that it is confident are in fact random.
Since your computer only generates a small number of genuinely random bits per
second, this may mean that /dev/random will exhaust its pool if read too often.
In this case, a read on /dev/random will block (just like reading a terminal with
no input on it) until the pool has filled up again.
Neither /dev/random nor /dev/urandom is known to be secure against a determined attacker, but they are about the best you can do without resorting to
specialized hardware.
5.14.1.4
Range issues
The problem with rand is that getting a uniform value between 0 and 231 -1 may
not be what you want. It could be that RAND_MAX is be too small; in this case,
you may have to call rand more than once and paste together the results. But
there can be problems with RAND_MAX even if it is bigger than the values you
want.
For example, suppose you want to simulate a die roll for your video craps machine,
but you don’t want to get whacked by Johnny “The Debugger” when the Nevada
State Gaming Commission notices that 6-6 is coming up slightly less often than
it’s supposed to. A natural thing to try would be to take the output of rand
mod 6:
int d6(void) {
return rand() % 6 + 1;
}
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The problem here is that there are 23 1 outputs from rand, and 6 doesn’t divide
23 1. So 1 and 2 are slightly more likely to come up than 3, 4, 5, or 6. This can
be particularly noticeable if we want a uniform variable from a larger range, e.g.
[0 . . . b(2/3) · 23 1c].
We can avoid this with a technique called rejection sampling, where we reject
excess parts of the output range of rand. For rolling a die, the trick is to reject
anything in the last extra bit of the range that is left over after the largest
multiple of the die size. Here’s a routine that does this, returning a uniform
value in the range 0 to n-1 for any positive n, together with a program that
demonstrates its use for rolling dice:
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<assert.h>
<time.h>
/* return a uniform random value in the range 0..n-1 inclusive */
int
randRange(int n)
{
int limit;
int r;
limit = RAND_MAX - (RAND_MAX % n);
while((r = rand()) >= limit);
}
return r % n;
int
main(int argc, char **argv)
{
int i;
srand(time(0));
for(i = 0; i < 40; i++) {
printf("%d ", randRange(6)+1);
}
putchar('\n');
}
return 0;
347
examples/randomization/randRange.c
More generally, rejection sampling can be used to get random values with
particular properties, where it’s hard to generate a value with that property
directly. Here’s a program that generates random primes:
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<assert.h>
<time.h>
/* return 1 if n is prime */
int
isprime(int n)
{
int i;
if(n % 2 == 0 || n == 1) { return 0; }
for(i = 3; i*i <= n; i += 2) {
if(n % i == 0) { return 0; }
}
}
return 1;
/* return a uniform random value in the range 0..n-1 inclusive */
int
randPrime(void)
{
int r;
/* extra parens avoid warnings */
while(!isprime((r = rand())));
}
return r;
int
main(int argc, char **argv)
{
int i;
srand(time(0));
for(i = 0; i < 10; i++) {
printf("%d\n", randPrime());
348
}
}
return 0;
examples/randomization/randPrime.c
One temptation to avoid is to re-use your random values. If, for example, you
try to find a random prime by picking a random x and trying x, x+1, x+2, etc.,
until you hit a prime, some primes are more likely to come up than others.
5.14.2
Randomized algorithms
Randomized algorithms typically make random choices to get good average
worst-case performance in situations where a similar deterministic algorithm
would fail badly for some inputs but perform well on most inputs. The idea is
that the randomization scrambles the input space so that the adversary can’t
predict which possible input values will be bad for us. This still allows him to
make trouble if he gets lucky, but most of the time our algorithm should run
quickly.
5.14.2.1
Randomized search
This is essentially rejection sampling in disguise. Suppose that you want to find
one of many needles in a large haystack. One approach is to methodically go
through the straws/needles one at a time until you find a needle. But you may
find that your good friend the adversary has put all the needles at the end of
your list. Picking candidate at random is likely to hit a needle faster if there are
many of them.
Here is a (silly) routine that quickly finds a number whose high-order bits match
a particular pattern:
int
matchBits(int pattern)
{
int r;
while(((r = rand()) & 0x70000000) != (pattern & 0x70000000));
}
return r;
This will find a winning value in 8 tries on average. In contrast, this deterministic
version will take a lot longer for nonzero patterns:
int
matchBitsDeterministic(int pattern)
349
{
int i;
for(i = 0; (i & 0x70000000) != (pattern & 0x70000000); i++);
}
return i;
The downside of the randomized approach is that it’s hard to tell when to quit
if there are no matches; if we stop after some fixed number of trials, we get a
Monte Carlo algorithm that may give the wrong answer with small probability.
The usual solution is to either accept a small probability of failure, or interleave
a deterministic backup algorithm that always works. The latter approach gives
a Las Vegas algorithm whose running time is variable but whose correctness is
not.
5.14.2.2
Quickselect and quicksort
Quickselect, or Hoare’s FIND (Hoare, C. A. R. Algorithm 65: FIND, CACM
4(7):321–322, July 1961), is an algorithm for quickly finding the k-th largest
element in an unsorted array of n elements. It runs in O(n) time on average,
which is the best one can hope for (we have to look at every element of the array
to be sure we didn’t miss a small one that changes our answer) and better than
the O(n log n) time we get if we sort the array first using a comparison-based
sorting algorithm.
The idea is to pick a random pivot and divide the input into two piles, each
of which is likely to be roughly a constant fraction of the size of the original
input.24 It takes O(n) time to split the input up (we have to compare each
element to the pivot once), and in the recursive calls this gives a geometric series.
We can even do the splitting up in place if we are willing to reorder the elements
of our original array.
If we recurse into both piles instead of just one, we get quicksort (Hoare, C.
A. R. Algorithm 64: Quicksort. CACM 4(7):321, July 1961), a very fast and
simple comparison-based sorting algorithm. Here is an implementation of both
algorithms:
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
/* reorder an array to put elements <= pivot
* before elements > pivot.
24 The actual analysis is pretty complicated, since we are more likely to land in a bigger pile,
but it’s not hard to show that on average even the bigger pile has no more than 3/4 of the
elements.
350
* Returns number of elements <= pivot */
static int
splitByPivot(int n, int *a, int pivot)
{
int lo;
int hi;
int temp; /* for swapping */
assert(n >= 0);
/*
/*
/*
/*
lo
hi
Dutch Flag algorithm */
swap everything <= pivot to bottom of array */
invariant is i < lo implies a[i] <= pivot */
and i > hi implies a[i] > pivot */
= 0;
= n-1;
while(lo <= hi) {
if(a[lo] <= pivot) {
lo++;
} else {
temp = a[hi];
a[hi--] = a[lo];
a[lo] = temp;
}
}
}
return lo;
/* find the k-th smallest element of an n-element array */
/* may reorder elements of the original array */
int
quickselectDestructive(int k, int n, int *a)
{
int pivot;
int lo;
assert(0 <= k);
assert(k < n);
if(n == 1) {
return a[0];
}
/* else */
351
pivot = a[rand() % n];
/* we will tolerate non-uniformity */
lo = splitByPivot(n, a, pivot);
}
/* lo is now number of values <= pivot */
if(k < lo) {
return quickselectDestructive(k, lo, a);
} else {
return quickselectDestructive(k - lo, n - lo, a + lo);
}
/* sort an array in place */
void
quickSort(int n, int *a)
{
int pivot;
int lo;
if(n <= 1) {
return;
}
/* else */
pivot = a[rand() % n];
/* we will tolerate non-uniformity */
lo = splitByPivot(n, a, pivot);
}
quickSort(lo, a);
quickSort(n - lo, a + lo);
/* shuffle an array */
void
shuffle(int n, int *a)
{
int i;
int r;
int temp;
for(i = n - 1; i > 0; i--) {
r = rand() % i;
temp = a[r];
a[r] = a[i];
a[i] = temp;
352
}
}
#define N (1024)
int
main(int argc, char **argv)
{
int a[N];
int i;
srand(0);
/* use fixed value for debugging */
for(i = 0; i < N; i++) {
a[i] = i;
}
shuffle(N, a);
for(i = 0; i < N; i++) {
assert(quickselectDestructive(i, N, a) == i);
}
shuffle(N, a);
quickSort(N, a);
for(i = 0; i < N; i++) {
assert(a[i] == i);
}
}
return 0;
examples/randomization/quick.c
5.14.3
Randomized data structures
Suppose we insert n elements into an initially-empty binary search tree in random
order with no rebalancing. Then each element is equally likely to be the root,
and all the elements less than the root end up in the left subtree, while all
the elements greater than the root end up in the right subtree, where they are
further partitioned recursively. This is exactly what happens in quicksort, so
the structure of the tree will exactly mirror the structure of an execution of
quicksort. In particular, the average depth of a node will be O(log n), giving us
353
the same expected search cost as in a balanced binary tree.
The problem with this approach is that we don’t have any guarantees that the
input will be supplied in random order, and in the worst case we end up with a
linked list. The solution is to put the randomization into the algorithm itself,
making the structure of the tree depend on random choices made by the program
itself.
5.14.3.1
Skip lists
A skip list (Pugh, 1990) is a randomized tree-like data structure based on linked
lists. It consists of a level 0 list that is an ordinary sorted linked list, together
with higher-level lists that contain a random sampling of the elements at lower
levels. When inserted into the level i list, an element flips a coin that tells it
with probability p to insert itself in the level i+1 list as well.
Searches in a skip list are done by starting in the highest-level list and searching
forward for the last element whose key is smaller than the target; the search then
continues in the same way on the next level down. The idea is that the higherlevel lists act as express lanes to get us to our target value faster. To bound the
expected running time of a search, it helps to look at this process backwards;
the reversed search path starts at level 0 and continues going backwards until it
reaches the first element that is also in a higher level; it then jumps to the next
level up and repeats the process. On average, we hit 1 + 1/p nodes at each level
before jumping back up; for constant p (e.g. 1/2), this gives us O(log n) steps
for the search.
The space per element of a skip list also depends on p. Every element has at
least one outgoing pointer (on level 0), and on average has exactly 1/(1 − p)
expected pointers. So the space cost can also be adjusted by adjusting p. For
example, if space is at a premium, setting p = 1/10 produces 10/9 pointers per
node on average—not much more than in a linked list—but still gives O(log n)
search time.
Below is an implementation of a skip list. To avoid having to allocate a separate
array of pointers for each element, we put a length-1 array at the end of
struct skiplist and rely on C’s lack of bounds checking to make the array
longer if necessary. A dummy head element stores pointers to all the initial
elements in each level of the skip list; it is given the dummy key INT_MIN so that
searches for values less than any in the list will report this value. Aside from
these nasty tricks, the code for search and insertion is pretty straightforward.
Code for deletion is a little more involved, because we have to make sure that
we delete the leftmost copy of a key if there are duplicates (an alternative would
be to modify skiplistInsert to ignore duplicates).
#include <stdlib.h>
#include <assert.h>
#include <limits.h>
354
#include "skiplist.h"
#define MAX_HEIGHT (32)
struct skiplist {
int key;
int height;
struct skiplist *next[1];
};
/* number of next pointers */
/* first of many */
/* choose a height according to a geometric distribution */
static int
chooseHeight(void)
{
int i;
for(i = 1; i < MAX_HEIGHT && rand() % 2 == 0; i++);
}
return i;
/* create a skiplist node with the given key and height */
/* does not fill in next pointers */
static Skiplist
skiplistCreateNode(int key, int height)
{
Skiplist s;
assert(height > 0);
assert(height <= MAX_HEIGHT);
s = malloc(sizeof(struct skiplist) + sizeof(struct skiplist *) * (height - 1));
assert(s);
s->key = key;
s->height = height;
}
return s;
/* create an empty skiplist */
Skiplist
skiplistCreate(void)
{
355
Skiplist s;
int i;
/* s is a dummy head element */
s = skiplistCreateNode(INT_MIN, MAX_HEIGHT);
/* this tracks the maximum height of any node */
s->height = 1;
for(i = 0; i < MAX_HEIGHT; i++) {
s->next[i] = 0;
}
}
return s;
/* free a skiplist */
void
skiplistDestroy(Skiplist s)
{
Skiplist next;
}
while(s) {
next = s->next[0];
free(s);
s = next;
}
/* return maximum key less than or equal to key */
/* or INT_MIN if there is none */
int
skiplistSearch(Skiplist s, int key)
{
int level;
for(level = s->height - 1; level >= 0; level--) {
while(s->next[level] && s->next[level]->key <= key) {
s = s->next[level];
}
}
}
return s->key;
/* insert a new key into s */
356
void
skiplistInsert(Skiplist s, int key)
{
int level;
Skiplist elt;
elt = skiplistCreateNode(key, chooseHeight());
assert(elt);
if(elt->height > s->height) {
s->height = elt->height;
}
/* search through levels taller than elt */
for(level = s->height - 1; level >= elt->height; level--) {
while(s->next[level] && s->next[level]->key < key) {
s = s->next[level];
}
}
/* now level is elt->height - 1, we can start inserting */
for(; level >= 0; level--) {
while(s->next[level] && s->next[level]->key < key) {
s = s->next[level];
}
}
}
/* s is last entry on this level < new element */
/* do list insert */
elt->next[level] = s->next[level];
s->next[level] = elt;
/* delete a key from s */
void
skiplistDelete(Skiplist s, int key)
{
int level;
Skiplist target;
/* first we have to find leftmost instance of key */
target = s;
for(level = s->height - 1; level >= 0; level--) {
while(target->next[level] && target->next[level]->key < key) {
357
}
}
target = target->next[level];
/* take one extra step at bottom */
target = target->next[0];
if(target == 0 || target->key != key) {
return;
}
/* now we found target, splice it out */
for(level = s->height - 1; level >= 0; level--) {
while(s->next[level] && s->next[level]->key < key) {
s = s->next[level];
}
}
}
if(s->next[level] == target) {
s->next[level] = target->next[level];
}
free(target);
examples/trees/skiplist/skiplist.c
Here is the header file, Makefile, and test code:
test_skiplist.c.
5.14.3.2
skiplist.h, Makefile,
Universal hash families
Randomization can also be useful in hash tables. Recall that in building a hash
table, we are relying on the hash function to spread out bad input distributions
over the indices of our array. But for any fixed hash function, in the worst case
we may get inputs where every key hashes to the same location. Universal
hashing (Carter and Wegman, 1979) solves this problem by choosing a hash
function at random. We may still get unlucky and have the hash function hash
all our values to the same location, but now we are relying on the random number
generator to be nice to us instead of the adversary. We can also rehash with a
new random hash function if we find out that the one we are using is bad.
The problem here is that we can’t just choose a function uniformly at random
out of the set of all possible hash functions, because there are too many of them,
meaning that we would spend more space representing our hash function than
we would on the table. The solution is to observe that we don’t need our hash
function h to be truly random; it’s enough if the probability of collision Pr[h(x)
358
= h(y)] for any fixed keys x =
6 y is 1/m, where m is the size of the hash table.
The reason is that the cost of searching for x (with chaining) is linear in the
number of keys already in the table that collide with it. The expected number
of such collisions is the sum of Pr[h(x) = h(y)] over all keys y in the table, or
n/m if we have n keys. So this pairwise collision probability bound is enough to
get the desired n/m behavior out of our table. A family of hash function with
this property is called universal.
How do we get a universal hash family? For strings, we can use a table of random
values, one for each position and possible character in the string. The hash
of a string is then the exclusive or of the random values hashArray[i][s[i]]
corresponding to the actual characters in the string. If our table has size a power
of two, this has the universal property, because if two strings x and y differ in
some position i, then there is only one possible value of hashArray[i][y[i]]
(mod m) that will make the hash functions equal.
Typically to avoid having to build an arbitrarily huge table of random values,
we only has an initial prefix of the string. Here is a hash function based on this
idea, which assumes that the d data structure includes a hashArray field that
contains the random values for this particular hash table:
static unsigned long
hash_function(Dict d, const char *s)
{
unsigned const char *us;
unsigned long h;
int i;
h = 0;
us = (unsigned const char *) s;
for(i = 0; i < HASH_PREFIX_LENGTH && us[i] != '\0'; i++) {
h ^= d->hashArray[i][us[i]];
}
}
return h;
A modified version of the Dict hash table from the chapter on hash tables that
uses this hash function is given here: dict.c, dict.h, test_dict.c, Makefile.
5.15
String processing
Most of the time, when we’ve talked about the asymptotic performance of data
structures, we have implicitly assumed that the keys we are looking up are of
359
constant size. This means that computing a hash function or comparing two
keys (as in a binary search tree) takes O(1) time. What if this is not the case?
If we consider an m-character string, any reasonable hash function is going to
take O(m) time since it has to look at all of the characters. Similarly, comparing
two m-character strings may also take O(m) time. If we charge for this (as we
should!) then the cost of hash table operations goes from O(1) to O(m) and the
cost of binary search tree operations, even in a balanced tree, goes from O(log n)
to O(m log n). Even sorting becomes more expensive: a sorting algorithm that
does O(n log n) comparisons may now take O(mn log n) time. But maybe we
can exploit the structure of strings to get better performance.
5.15.1
Radix search
Radix search refers to a variety of data structures that support searching for
strings considered as sequences of digits in some large base (or radix). These
are generally faster than simple binary search trees because they usually only
require examining one byte or less of the target string at each level of the tree,
as compared to every byte in the target in a full string comparison. In many
cases, the best radix search trees are even faster than hash tables, because they
only need to look at a small part of the target string to identify it.
We’ll describe several radix search trees, starting with the simplest and working
up.
5.15.1.1
Tries
A trie is a binary tree (or more generally, a k-ary tree where k is the radix)
where the root represents the empty bit sequence and the two children of a
node representing sequence x represent the extended sequences x0 and x1 (or
generally x0, x1, . . . , x(k − 1)). So a key is not stored at a particular node but
is instead represented bit-by-bit (or digit-by-digit) along some path. Typically
a trie assumes that the set of keys is prefix-free, i.e. that no key is a prefix of
another; in this case there is a one-to-one correspondence between keys and
leaves of the trie. If this is not the case, we can mark internal nodes that
also correspond to the ends of keys, getting a slightly different data structure
known as a digital search tree. For null-terminated strings as in C, the null
terminator ensures that any set of strings is prefix-free.
Given this simple description, a trie storing a single long key would have a very
large number of nodes. A standard optimization is to chop off any path with no
branches in it, so that each leaf corresponds to the shortest unique prefix of a
key. This requires storing the key in the leaf so that we can distinguish different
keys with the same prefix.
The name trie comes from the phrase “information retrieval.” Despite the
etymology, trie is now almost always pronounced like try instead of tree to avoid
360
confusion with other tree data structures.
5.15.1.1.1
Searching a trie
Searching a trie is similar to searching a binary search tree, except that instead
of doing a comparison at each step we just look at the next bit in the target.
The time to perform a search is proportional to the number of bits in the longest
path in the tree matching a prefix of the target. This can be very fast for search
misses if the target is wildly different from all the keys in the tree.
5.15.1.1.2
Inserting a new element into a trie
Insertion is more complicated for tries than for binary search trees. The reason
is that a new element may add more than one new node. There are essentially
two cases:
1. (The simple case.) In searching for the new key, we reach a null pointer
leaving a non-leaf node. In this case we can simply add a new leaf. The
cost of this case is essentially the same as for search plus O(1) for building
the new leaf.
2. (The other case.) In searching for the new key, we reach a leaf, but the
key stored there isn’t the same as the new key. Now we have to generate a
new path for as long as the old key and the new key have the same bits,
branching out to two different leaves at the end. The cost of this operation
is within a constant factor of the cost for searching for the new leaf after
it is inserted, since that’s how long the newly-built search path will be.
In either case, the cost is bounded by the length of the new key, which is about
the best we can hope for in the worst case for any data structure.
5.15.1.1.3
Implementation
A typical trie implementation in C might look like this. It uses a GET_BIT macro
similar to the one from the chapter on bit manipulation, except that we reverse
the bits within each byte to get the right sorting order for keys.
typedef struct trie_node *Trie;
#define EMPTY_TRIE (0)
/* returns 1 if trie contains target */
int trie_contains(Trie trie, const char *target);
/* add a new key to a trie */
/* and return the new trie */
Trie trie_insert(Trie trie, const char *key);
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/* free a trie */
void trie_destroy(Trie);
/* debugging utility: print all keys in trie */
void trie_print(Trie);
examples/trees/trie/trie.h
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<string.h>
<assert.h>
#include "trie.h"
#define BITS_PER_BYTE (8)
/* extract the n-th bit of x */
/* here we process bits within bytes in MSB-first order */
/* this sorts like strcmp */
#define GET_BIT(x, n) ((((x)[(n) / BITS_PER_BYTE]) & (0x1 << (BITS_PER_BYTE - 1 - (n) % BITS
#define TRIE_BASE (2)
struct trie_node {
char *key;
struct trie_node *kids[TRIE_BASE];
};
#define IsLeaf(t) ((t)->kids[0] == 0 && (t)->kids[1] == 0)
/* returns 1 if trie contains target */
int
trie_contains(Trie trie, const char *target)
{
int bit;
for(bit = 0; trie && !IsLeaf(trie); bit++) {
/* keep going */
trie = trie->kids[GET_BIT(target, bit)];
}
if(trie == 0) {
/* we lost */
return 0;
} else {
/* check that leaf really contains the target */
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}
}
return !strcmp(trie->key, target);
/* gcc -pedantic kills strdup! */
static char *
my_strdup(const char *s)
{
char *s2;
s2 = malloc(strlen(s) + 1);
assert(s2);
}
strcpy(s2, s);
return s2;
/* helper functions for insert */
static Trie
make_trie_node(const char *key)
{
Trie t;
int i;
t = malloc(sizeof(*t));
assert(t);
if(key) {
t->key = my_strdup(key);
assert(t->key);
} else {
t->key = 0;
}
for(i = 0; i < TRIE_BASE; i++) t->kids[i] = 0;
}
return t;
/* add a new key to a trie */
/* and return the new trie */
Trie
trie_insert(Trie trie, const char *key)
{
int bit;
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int bitvalue;
Trie t;
Trie kid;
const char *oldkey;
if(trie == 0) {
return make_trie_node(key);
}
/* else */
/* first we'll search for key */
for(t = trie, bit = 0; !IsLeaf(t); bit++, t = kid) {
kid = t->kids[bitvalue = GET_BIT(key, bit)];
if(kid == 0) {
/* woohoo! easy case */
t->kids[bitvalue] = make_trie_node(key);
return trie;
}
}
/* did we get lucky? */
if(!strcmp(t->key, key)) {
/* nothing to do */
return trie;
}
/* else */
/* hard case---have to extend the trie */
oldkey = t->key;
#ifdef EXCESSIVE_TIDINESS
t->key = 0;
/* not required but makes data structure look tidier */
#endif
/* walk the common prefix */
while(GET_BIT(oldkey, bit) == (bitvalue = GET_BIT(key, bit))) {
kid = make_trie_node(0);
t->kids[bitvalue] = kid;
bit++;
t = kid;
}
/* then split */
t->kids[bitvalue] = make_trie_node(key);
t->kids[!bitvalue] = make_trie_node(oldkey);
}
return trie;
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/* kill it */
void
trie_destroy(Trie trie)
{
int i;
if(trie) {
for(i = 0; i < TRIE_BASE; i++) {
trie_destroy(trie->kids[i]);
}
if(IsLeaf(trie)) {
free(trie->key);
}
}
}
free(trie);
static void
trie_print_internal(Trie t, int bit)
{
int i;
int kid;
}
if(t != 0) {
if(IsLeaf(t)) {
for(i = 0; i < bit; i++) putchar(' ');
puts(t->key);
} else {
for(kid = 0; kid < TRIE_BASE; kid++) {
trie_print_internal(t->kids[kid], bit+1);
}
}
}
void
trie_print(Trie t)
{
trie_print_internal(t, 0);
}
examples/trees/trie/trie.c
Here is a short test program that demonstrates how to use it:
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#include <stdio.h>
#include <stdlib.h>
#include "trie.h"
/* test for trie.c */
/* reads lines from stdin and echoes lines that haven't appeared before */
/* read a line of text from stdin
* and return it (without terminating newline) as a freshly-malloc'd block.
* Caller is responsible for freeing this block.
* Returns 0 on error or EOF.
*/
char *
getline(void)
{
char *line;
/* line buffer */
int n;
/* characters read */
int size;
/* size of line buffer */
int c;
size = 1;
line = malloc(size);
if(line == 0) return 0;
n = 0;
while((c = getchar()) != '\n' && c != EOF) {
while(n >= size - 1) {
size *= 2;
line = realloc(line, size);
if(line == 0) return 0;
}
line[n++] = c;
}
}
if(c == EOF && n == 0) {
/* got nothing */
free(line);
return 0;
} else {
line[n++] = '\0';
return line;
}
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int
main(int argc, char **argv)
{
Trie t;
char *line;
t = EMPTY_TRIE;
while((line = getline()) != 0) {
if(!trie_contains(t, line)) {
puts(line);
}
/* try to insert it either way */
/* this tests that insert doesn't blow up on duplicates */
t = trie_insert(t, line);
}
free(line);
puts("===");
trie_print(t);
trie_destroy(t);
}
return 0;
examples/trees/trie/test_trie.c
5.15.1.2
Patricia trees
Tries perform well when all keys are short (or are distinguished by short prefixes),
but can grow very large if one inserts two keys that have a long common prefix.
The reason is that a trie has to store an internal node for every bit of the
common prefix until the two keys become distinguishable, leading to long chains
of internal nodes each of which has only one child. An optimization (described
in this paper) known as a Patricia tree eliminates these long chains by having
each node store the number of the bit to branch on, like this:
struct patricia_node {
char *key;
int bit;
struct patricia_node *kids[2];
};
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typedef struct patricia_node *Patricia;
Now when searching for a key, instead of using the number of nodes visited so
far to figure out which bit to look at, we just read the bit out of the table. This
means in particular that we can skip over any bits that we don’t actually branch
on. We do however have to be more careful to make sure we don’t run off the
end of our target key, since it is possible that when skipping over intermediate
bits we might skip over some that distinguish our target from all keys in the
table, including longer keys. For example, a Patricia tree storing the strings
abc and abd will first test bit position 22, since that’s where abc and abd differ.
This can be trouble if we are looking for x.
Here’s the search code:
int
patricia_contains(Patricia t, const char *key)
{
int key_bits;
key_bits = BITS_PER_BYTE * (strlen(key)+1);
/* +1 for the NUL */
while(t && !IsLeaf(t)) {
if(t->bit >= key_bits) {
/* can't be there */
return 0;
} else {
t = t->kids[GET_BIT(key, t->bit)];
}
}
}
return t && !strcmp(t->key, key);
The insertion code is similar in many respects to the insertion code for a trie.
The differences are that we never construct a long chain of internal nodes when
splitting a leaf (although we do have to scan through both the old and new keys
to find the first bit position where they differ), but we may sometimes have to
add a new internal node between two previously existing nodes if a new key
branches off at a bit position that was previously skipped over.
In the worst case Patricia trees are much more efficient than tries, in both space
(linear in the number of keys instead of linear in the total size of the keys) and
time complexity, often needing to examine only a very small number of bits for
misses (hits still require a full scan in strcmp to verify the correct key). The
only downside of Patricia trees is that since they work on bits, they are not quite
perfectly tuned to the byte or word-oriented structure of modern CPUs.
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5.15.1.3
Ternary search trees
Ternary search trees were described by Jon Bentley and Bob Sedgewick in
an article in the April 1988 issue of Dr. Dobb’s Journal, available here.
The basic idea is that each node in the tree stores one character from the key
and three child pointers lt, eq, and gt. If the corresponding character in the
target is equal to the character in the node, we move to the next character in
the target and follow the eq pointer out of the node. If the target is less, follow
the lt pointer but stay at the same character. If the target is greater, follow the
gt pointer and again stay at the same character. When searching for a string,
we walk down the tree until we either reach a node that matches the terminating
NUL (a hit), or follow a null pointer (a miss).
A TST acts a bit like a 256-way trie, except that instead of storing an array
of 256 outgoing pointers, we build something similar to a small binary search
tree for the next character. Note that no explicit balancing is done within these
binary search trees. From a theoretical perspective, the worst case is that we
get a 256-node deep linked-list equivalent at each step, multiplying our search
time by 256 = O(1). In practice, only those characters that actual appear in
some key at this stage will show up, so the O(1) is likely to be a small O(1),
especially if keys are presented in random order.
TSTs are one of the fastest known data structures for implementing dictionaries
using strings as keys, beating both hash tables and tries in most cases. They
can be slower than Patricia trees if the keys have many keys with long matching
prefixes; however, a Patricia-like optimization can be applied to give a compressed ternary search tree that works well even in this case. In practice,
regular TSTs are usually good enough.
Here is a simple implementation of an insert-only TST. The C code includes two
versions of the insert helper routine; the first is the original recursive version
and the second is an iterative version generated by eliminating the tail recursion
from the first.
typedef struct tst_node *TST;
#define EMPTY_TST (0)
/* returns 1 if t contains target */
int tst_contains(TST t, const char *target);
/* add a new key to a TST */
/* and return the new TST */
TST tst_insert(TST t, const char *key);
/* free a TST */
void tst_destroy(TST);
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examples/trees/tst/tst.h
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
#include "tst.h"
struct tst_node {
char key;
struct tst_node *lt;
struct tst_node *eq;
struct tst_node *gt;
};
/*
/*
/*
/*
value to split on */
go here if target[index] < value */
go here if target[index] == value */
go here if target[index] > value */
/* returns 1 if t contains key */
int
tst_contains(TST t, const char *key)
{
assert(key);
while(t) {
if(*key < t->key) {
t = t->lt;
} else if(*key > t->key) {
t = t->gt;
} else if(*key == '\0') {
return 1;
} else {
t = t->eq;
key++;
}
}
}
return 0;
/* original recursive insert */
static void
tst_insert_recursive(TST *t, const char *key)
{
if(*t == 0) {
*t = malloc(sizeof(**t));
assert(*t);
(*t)->key = *key;
(*t)->lt = (*t)->eq = (*t)->gt = 0;
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}
}
/* now follow search */
if(*key < (*t)->key) {
tst_insert_recursive(&(*t)->lt, key);
} else if(*key > (*t)->key) {
tst_insert_recursive(&(*t)->gt, key);
} else if(*key == '\0') {
/* do nothing, we are done */
;
} else {
tst_insert_recursive(&(*t)->eq, key+1);
}
/* iterative version of above, since somebody asked */
/* This is pretty much standard tail-recursion elimination: */
/* The whole function gets wrapped in a loop, and recursive
* calls get replaced by assignment */
static void
tst_insert_iterative(TST *t, const char *key)
{
for(;;) {
if(*t == 0) {
*t = malloc(sizeof(**t));
assert(*t);
(*t)->key = *key;
(*t)->lt = (*t)->eq = (*t)->gt = 0;
}
}
}
/* now follow search */
if(*key < (*t)->key) {
t = &(*t)->lt;
} else if(*key > (*t)->key) {
t = &(*t)->gt;
} else if(*key == '\0') {
/* do nothing, we are done */
return;
} else {
t = &(*t)->eq;
key++;
}
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/* add a new key to a TST */
/* and return the new TST */
TST
tst_insert(TST t, const char *key)
{
assert(key);
#ifdef USE_RECURSIVE_INSERT
tst_insert_recursive(&t, key);
#else
tst_insert_iterative(&t, key);
#endif
return t;
}
/* free a TST */
void
tst_destroy(TST t)
{
if(t) {
tst_destroy(t->lt);
tst_destroy(t->eq);
tst_destroy(t->gt);
free(t);
}
}
examples/trees/tst/tst.c
And here is some test code, almost identical to the test code for tries: test_tst.c.
The Dr. Dobb’s article contains additional code for doing deletions and partial
matches, plus some optimizations for inserts.
5.15.1.4
More information
• http://imej.wfu.edu/articles/2002/2/02/index.asp has some good Javabased animations of radix tries, Patricia tries, and other tree-like data
structures.
5.15.2
Radix sort
The standard quicksort routine is an example of a comparison-based sorting
algorithm. This means that the only information the algorithm uses about the
items it is sorting is the return value of the compare routine. This has a rather
nice advantage of making the algorithm very general, but has the disadvantage
372
that the algorithm can extract only one bit of information from every call to
compare. Since there are n! possible ways to reorder the input sequence, this
means we need at least log(n!) = O(n log n) calls to compare to finish the sort. If
we are sorting something like strings, this can get particularly expensive, because
calls to strcmp may take time linear in the length of the strings being compared.
In the worst case, sorting n strings of length m each could take O(nm log n)
time.
5.15.2.1
Bucket sort
The core idea of radix sort is that if we want to sort values from a small range,
we can do it by making one bucket for each possible value and throw any object
with that value into the corresponding bucket. In the old days, when Solitaire
was a game played with physical pieces of cardboard, a player who suspected
that that one of these “cards” had fallen under the couch might sort the deck by
dividing it up into Spades, Hearts, Diamonds, and Club piles and then sorting
each pile recursively. The same trick works in a computer, but there the buckets
are typically implemented as an array of some convenient data structure.
If the number of possible values is too big, we may still be able to use bucket
sort digit-by-digit. The resulting algorithms are known generally as radix sort.
These are a class of algorithms designed for sorting strings in lexicographic
order—the order used by dictionaries where one string is greater than another if
the first character on which they differ is greater. One particular variant, mostsignificant-byte radix sort or MSB radix sort, has the beautiful property
that its running time is not only linear in the size of the input in bytes, but
is also linear in the smallest number of characters in the input that need to
be examined to determine the correct order. This algorithm is so fast that
it takes not much more time to sort data than it does to read the data from
memory and write it back. But it’s a little trickier to explain that the original
least-significant-byte radix sort or LSB radix sort.
5.15.2.2
Classic LSB radix sort
This is the variant used for punch cards, and works well for fixed-length strings.
The idea is to sort on the least significant position first, then work backwards to
the most significant position. This works as long as each sort is stable, meaning
that it doesn’t reorder values with equal keys. For example, suppose we are
sorting the strings:
sat
bat
bad
The first pass sorts on the third column, giving:
bad
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sat
bat
The second pass sorts on the second column, producing no change in the order
(all the characters are the same). The last pass sorts on the first column. This
moves the s after the two bs, but preserves the order of the two words starting
with b. The result is:
bad
bat
sat
There are three downsides to LSB radix sort:
1. All the strings have to be the same length (this is not necessarily a problem
if they are really fixed-width data types like ints).
2. The algorithm used to sort each position must be stable, which may require
more effort than most programmers would like to take.
3. It may be that the late positions in the strings don’t affect the
order, but we have to sort on them anyway. If we are sorting
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa and baaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa,
we spend a lot of time matching up as against each other.
5.15.2.3
MSB radix sort
For these reasons, MSB radix sort is used more often. This is basically the radix
sort version of quicksort, where instead of partitioning our input data into two
piles based on whether each element is less than or greater than a random pivot,
we partition the input into 256 piles, one for each initial byte. We can then
sort each pile recursively using the same algorithm, taking advantage of the fact
that we know that the first byte (or later, the first k bytes) are equal and so we
only need to look at the next one. The recursion stops when we get down to 1
value, or in practice where we get down to a small enough number of values that
the cost of doing a different sorting algorithm becomes lower than the cost of
creating and tearing down the data structures for managing the piles.
5.15.2.3.1
Issues with recursion depth
The depth of recursion for MSB radix sort is equal to the length of the secondlongest string in the worst case. Since strings can be pretty long, this creates
a danger of blowing out the stack. The solution (as in quicksort) is to use
tail recursion for the largest pile. Now any pile we recurse into with an actual
procedure call is at most half the size of the original pile, so we get stack depth
at most O(log n).
5.15.2.3.2
Implementing the buckets
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There is a trick we can do analagous to the Dutch flag algorithm where we sort
the array in place. The idea is that we first count the number of elements that
land in each bucket and assign a block of the array for each bucket, keeping
track in each block of an initial prefix of values that belong in the bucket with
the rest not yet processed. We then walk through the buckets swapping out any
elements at the top of the good prefix to the bucket they are supposed to be in.
This procedure puts at least one element in the right bucket for each swap, so
we reorder everything correctly in at most n swaps and O(n) additional work.
To keep track of each bucket, we use two pointers bucket[i] for the first element
of the bucket and top[i] for the first unused element. We could make these be
integer array indices, but this slows the code down by about 10%. This seems to
be a situation where our use of pointers is complicated enough that the compiler
can’t optimize out the array lookups.
5.15.2.3.3
Further optimization
Since we are detecting the heaviest bucket anyway, there is a straightforward
optimization that speeds the sort up noticeably on inputs with a lot of duplicates:
if everything would land in the same bucket, we can skip the bucket-sort and
just go directly to the next character.
5.15.2.3.4
Sample implementation
Here is an implementation of MSB radix sort using the ideas above:
#include <assert.h>
#include <limits.h>
#include <string.h>
#include "radixSort.h"
/* in-place MSB radix sort for null-terminated strings */
/* helper routine for swapping */
static void
swapStrings(const char **a, const char **b)
{
const char *temp;
}
temp = *a;
*a = *b;
*b = temp;
/* this is the internal routine that assumes all strings are equal for the
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* first k characters */
static void
radixSortInternal(int n, const char **a, int k)
{
int i;
int count[UCHAR_MAX+1]; /* number of strings with given character in position k */
int mode;
/* most common position-k character */
const char **bucket[UCHAR_MAX+1]; /* position of character block in output */
const char **top[UCHAR_MAX+1];
/* first unused index in this character block */
/* loop implements tail recursion on most common character */
while(n > 1) {
/* count occurrences of each character */
memset(count, 0, sizeof(int)*(UCHAR_MAX+1));
for(i = 0; i < n; i++) {
count[(unsigned char) a[i][k]]++;
}
/* find the most common nonzero character */
/* we will handle this specially */
mode = 1;
for(i = 2; i < UCHAR_MAX+1; i++) {
if(count[i] > count[mode]) {
mode = i;
}
}
if(count[mode] < n) {
/* generate bucket and top fields */
bucket[0] = top[0] = a;
for(i = 1; i < UCHAR_MAX+1; i++) {
top[i] = bucket[i] = bucket[i-1] + count[i-1];
}
/* reorder elements by k-th character */
/* this is similar to dutch flag algorithm */
/* we start at bottom character and swap values out until everything is in place
/* invariant is that for all i, bucket[i] <= j < top[i] implies a[j][k] == i */
for(i = 0; i < UCHAR_MAX+1; i++) {
while(top[i] < bucket[i] + count[i]) {
if((unsigned char) top[i][0][k] == i) {
/* leave it in place, advance bucket */
top[i]++;
376
}
}
} else {
/* swap with top of appropriate block */
swapStrings(top[i], top[(unsigned char) top[i][0][k]]++);
}
/* we have now reordered everything */
/* recurse on all but 0 and mode */
for(i = 1; i < UCHAR_MAX+1; i++) {
if(i != mode) {
radixSortInternal(count[i], bucket[i], k+1);
}
}
/* tail recurse on mode */
n = count[mode];
a = bucket[mode];
k = k+1;
} else {
}
}
}
/* tail recurse on whole pile */
k = k+1;
void
radixSort(int n, const char **a)
{
radixSortInternal(n, a, 0);
}
examples/sorting/radixSort/radixSort.c
Some additional files: radixSort.h, test_radixSort.c, Makefile, sortInput.c.
The last is a program that sorts lines on stdin and writes the result to
stdout, similar to the GNU sort utility. When compiled with -O3 and
run on my machine, this runs in about the same time as the standard sort
program when run on a 4.7 million line input file consisting of a random
shuffle of 20 copies of /usr/share/dict/words, provided sort is run with
LANG=C sort < /usr/share/dict/words to keep it from having to deal with
locale-specific collating issues. On other inputs, sort is faster. This is not
bad given how thoroughly sort has been optimized, but is a sign that further
optimization is possible.
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6
Other topics not covered in detail in 2015
These are mostly leftovers from previous versions of the class where different
topics were emphasized.
6.1
More applications of function pointers
Here we show how to implement various mechanisms often found in more
sophisticated programming languages in C using function pointers.
6.1.1
Iterators
Suppose we have an abstract data type that represents some sort of container,
such as a list or dictionary. We’d like to be able to do something to every
element of the container; say, count them up. How can we write operations on
the abstract data type to allow this, without exposing the implementation?
To make the problem more concrete, let’s suppose we have an abstract data type
that represents the set of all non-negative numbers less than some fixed bound.
The core of its interface might look like this:
/*
* Abstract data type representing the set of numbers from 0 to
* bound-1 inclusive, where bound is passed in as an argument at creation.
*/
typedef struct nums *Nums;
/* Create a Nums object with given bound. */
Nums nums_create(int bound);
/* Destructor */
void nums_destroy(Nums);
/* Returns 1 if nums contains element, 0 otherwise */
int nums_contains(Nums nums, int element);
#include <stdlib.h>
#include "nums.h"
struct nums {
int bound;
};
Nums nums_create(int bound)
{
378
}
struct nums *n;
n = malloc(sizeof(*n));
n->bound = bound;
return n;
void nums_destroy(Nums n) { free(n); }
int nums_contains(Nums n, int element)
{
return element >= 0 && element < n->bound;
}
From the outside, a Nums acts like the set of numbers from 0 to bound - 1;
nums_contains will insist that it contains any int that is in this set and contains
no int that is not in this set.
Let’s suppose now that we want to loop over all elements of some Nums, say to add
them together. In particular, we’d like to implement the following pseudocode,
where nums is some Nums instance:
sum = 0;
for(each i in nums) {
sum += i;
}
One way to do this would be to build the loop into some operation in nums.c,
including its body. But we’d like to be able to substitute any body for the
sum += i line. Since we can’t see the inside of a Nums, we need to have some
additional operation or operations on a Nums that lets us write the loop. How
can we do this?
6.1.1.1
Option 1: Function that returns a sequence
A data-driven approach might be to add a nums_contents function that returns
a sequence of all elements of some instance, perhaps in the form of an array or
linked list. The advantage of this approach is that once you have the sequence,
you don’t need to worry about changes to (or destruction of) the original object.
The disadvantage is that you have to deal with storage management issues, and
have to pay the costs in time and space of allocating and filling in the sequence.
This can be particularly onerous for a “virtual” container like Nums, since we
could conceivably have a Nums instance with billions of elements.
Bearing these facts in mind, let’s see what this approach might look like. We’ll
define a new function nums_contents that returns an array of ints, terminated
by a -1 sentinel:
int *
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nums_contents(Nums n)
{
int *a;
int i;
a = malloc(sizeof(*a) * (n->bound + 1));
for(i = 0; i < n->bound; i++) a[i] = i;
a[n->bound] = -1;
return a;
}
We might use it like this:
sum = 0;
contents = nums_contents(nums);
for(p = contents; *p != -1; p++) {
sum += *p;
}
free(contents);
Despite the naturalness of the approach, returning a sequence in this case leads
to the most code complexity of the options we will examine.
6.1.1.2
Option 2: Iterator with first/done/next operations
If we don’t want to look at all the elements at once, but just want to process them
one at a time, we can build an iterator. An iterator is an object that allows you to
step through the contents of another object, by providing convenient operations
for getting the first element, testing when you are done, and getting the next
element if you are not. In C, we try to design iterators to have operations that
fit well in the top of a for loop.
For the Nums type, we’ll make each Nums its own iterator. The new operations
are given here:
int nums_first(Nums n) { return 0; }
int nums_done(Nums n, int val) { return val >= n->bound; }
int nums_next(Nums n, int val) { return val+1; }
And we use them like this:
sum = 0;
for(i = nums_first(nums); !nums_done(nums, i); i = nums_next(nums, i)) {
sum += i;
}
Not only do we completely avoid the overhead of building a sequence, we also
get much cleaner code. It helps in this case that all we need to find the next
value is the previous one; for a more complicated problem we might have to
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create and destroy a separate iterator object that holds the state of the loop.
But for many tasks in C, the first/done/next idiom is a pretty good one.
6.1.1.3
Option 3: Iterator with function argument
Suppose we have a very complicated iteration, say one that might require several
nested loops or even a recursion to span all the elements. In this case it might
be very difficult to provide first/done/next operations, because it would be hard
to encode the state of the iteration so that we could easily pick up in the next
operation where we previously left off. What we’d really like to do is to be
able to plug arbitrary code into the innermost loop of our horrible iteration
procedure, and do it in a way that is reasonably typesafe and doesn’t violate
our abstraction barrier. This is a job for function pointers, and an example of
the functional programming style in action.
We’ll define a nums_foreach function that takes a function as an argument:
void nums_foreach(Nums n, void (*f)(int, void *), void *f_data)
{
int i;
for(i = 0; i < n->bound; i++) f(i, f_data);
}
The f_data argument is used to pass extra state into the passed-in function f;
it’s a void * because we want to let f work on any sort of extra state it likes.
Now to do our summation, we first define an extra function sum_helper, which
adds each element to an accumulator pointed to by f_data:
static void sum_helper(int i, void *f_data)
{
*((int *) f_data) += i;
}
We then feed sum_helper to the nums_foreach function:
sum = 0;
nums_foreach(nums, sum_helper, (void *) &sum);
There is a bit of a nuisance in having to define the auxiliary sum_helper function
and in all the casts to and from void, but on the whole the complexity of this
solution is not substantially greater than the first/done/next approach. Which
you should do depends on whether it’s harder to encapsulate the state of the
iterator (in which case the functional approach is preferable) or of the loop body
(in which case the first/done/next approach is preferable), and whether you
need to bail out of the loop early (which would require special support from
the foreach procedure, perhaps checking a return value from the function).
However, it’s almost always straightforward to encapsulate the state of a loop
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body; just build a struct containing all the variables that it uses, and pass a
pointer to this struct as f_data.
6.1.2
Closures
A closure is a function plus some associated state. A simple way to implement
closures in C is to use a static local variable, but then you only get one. Better
is to allocate the state somewhere and pass it around with the function. For
example, here’s a simple functional implementation of infinite sequences:
/* a sequence is an object that returns a new value each time it is called */
struct sequence {
int (*next)(void *data);
void *data;
};
typedef struct sequence *Sequence;
Sequence
create_sequence(int (*next)(void *data), void *data)
{
Sequence s;
s = malloc(sizeof(*s));
assert(s);
s->next = next;
s->data = data;
}
return s;
int
sequence_next(Sequence s)
{
return s->next(s->data);
}
And here are some examples of sequences:
/* make a constant sequence that always returns x */
static int
constant_sequence_next(void *data)
{
return *((int *) data);
}
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Sequence
constant_sequence(int x)
{
int *data;
data = malloc(sizeof(*data));
if(data == 0) return 0;
*data = x;
}
return create_sequence(constant_sequence_next, data);
/* make a sequence x, x+a, x+2*a, x+3*a, ... */
struct arithmetic_sequence_data {
int cur;
int step;
};
static int
arithmetic_sequence_next(void *data)
{
struct arithmetic_sequence_data *d;
d = data;
d->cur += d->step;
}
return d->cur;
Sequence
arithmetic_sequence(int x, int a)
{
struct arithmetic_sequence_data *d;
d = malloc(sizeof(*d));
if(d == 0) return 0;
d->cur = x - a;
d->step = a;
}
/* back up so first value returned is x */
return create_sequence(arithmetic_sequence_next, d);
/* Return the sum of two sequences */
383
static int
add_sequences_next(void *data)
{
Sequence *s;
}
s = data;
return sequence_next(s[0]) + sequence_next(s[1]);
Sequence
add_sequences(Sequence s0, Sequence s1)
{
Sequence *s;
s = malloc(2*sizeof(*s));
if(s == 0) return 0;
s[0] = s0;
s[1] = s1;
}
return create_sequence(add_sequences_next, s);
/* Return the sequence x, f(x), f(f(x)), ... */
struct iterated_function_sequence_data {
int x;
int (*f)(int);
}
static int
iterated_function_sequence_next(void *data)
{
struct iterated_function_sequence_data *d;
int retval;
d = data;
retval = d->x;
d->x = d->f(d->x);
}
return retval;
Sequence
iterated_function_sequence(int (*f)(int), int x0)
{
384
struct iterated_function_sequence_data *d;
d = malloc(sizeof(*d));
if(d == 0) return 0;
d->x = x0;
d->f = f;
}
return create_sequence(iterated_function_sequence_next, d);
Note that we haven’t worried about how to free the data field inside a Sequence,
and indeed it’s not obvious that we can write a generic data-freeing routine
since we don’t know what structure it has. The solution is to add more function
pointers to a Sequence, so that we can get the next value, get the sequence to
destroy itself, etc. When we do so, we have gone beyond building a closure to
building an object.
6.1.3
Objects
Here’s an example of a hierarchy of counter objects. Each counter object has (at
least) three operations: reset, next, and destroy. To call the next operation
on counter c we include c and the first argument, e.g. c->next(c) (one could
write a wrapper to enforce this).
The main trick is that we define a basic counter structure and then extend it
to include additional data, using lots of pointer conversions to make everything
work.
/* use preprocessor to avoid rewriting these */
#define COUNTER_FIELDS \
void (*reset)(struct counter *);
\
int (*next)(struct counter *);
\
void (*destroy)(struct counter *);
struct counter {
COUNTER_FIELDS
};
typedef struct counter *Counter;
/* minimal counter--always returns zero */
/* we don't even allocate this, just have one global one */
static void noop(Counter c) { ; }
static int return_zero(Counter c) { return 0; }
static struct counter Zero_counter = { noop, return_zero, noop };
385
Counter
make_zero_counter(void)
{
return &Zero_counter;
}
/* a fancier counter that iterates a function sequence */
/* this struct is not exported anywhere */
struct ifs_counter {
/* copied from struct counter declaration */
COUNTER_FIELDS
};
/* new fields */
int init;
int cur;
int (*f)(int);
/* update rule */
static void
ifs_reset(Counter c)
{
struct ifs_counter *ic;
ic = (struct ifs_counter *) c;
}
ic->cur = ic->init;
static void
ifs_next(Counter c)
{
struct ifs_counter *ic;
int ret;
ic = (struct ifs_counter *) c;
ret = ic->cur;
ic->cur = ic->f(ic->cur);
}
return ret;
Counter
make_ifs_counter(int init, int (*f)(int))
386
{
struct ifs_counter *ic;
ic = malloc(sizeof(*ic));
ic->reset = ifs_reset;
ic->next = ifs_next;
ic->destroy = (void (*)(struct counter *)) free;
ic->init = init;
ic->cur = init;
ic->f = f;
}
/* it's always a Counter on the outside */
return (Counter) ic;
A typical use might be
static int
times2(int x)
{
return x*2;
}
void
print_powers_of_2(void)
{
int i;
Counter c;
c = make_ifs_counter(1, times2);
for(i = 0; i < 10; i++) {
printf("%d\n", c->next(c));
}
c->reset(c);
for(i = 0; i < 20; i++) {
printf("%d\n", c->next(c));
}
}
c->destroy(c);
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6.2
Suffix arrays
These are notes on practical implementations of suffix arrays, which are a data
structure for searching quickly for substrings of a given large string.
6.2.1
Why do we want to do this?
• Answer from the old days: Fast string searching is the key to dealing with
mountains of information. Why, a modern (c. 1960) electronic computer
can search the equivalent of hundreds of pages of text in just a few hours. . .
• More recent answer:
– We still need to search enormous corpuses of text (see http://www.
google.com).
– Algorithms for finding long repeated substrings or patterns can be
useful for data compression) or detecting plagiarism.
– Finding all occurrence of a particular substring in some huge corpus
is the basis of statistical machine translation.
– We are made out of strings over a particular finite alphabet GATC.
String search is a central tool in computational biology.
6.2.2
String search algorithms
Without preprocessing, searching an n-character string for an m-character substring can be done using algorithms of varying degrees of sophistication, the
worst of which run in time O(nm) (run strncmp on each position in the big
string), and best of which run in time O(n + m) (run the Boyer-Moore string
search algorithm). But we are interested in the case where we can preprocess
our big string into a data structure that will let us do lots of searches for cheap.
6.2.3
Suffix trees and suffix arrays
Suffix trees and suffix arrays are data structures for representing texts that
allow substring queries like “where does this pattern appear in the text” or “how
many times does this pattern occur in the text” to be answered quickly. Both
work by storing all suffixes of a text, where a suffix is a substring that runs to the
end of the text. Of course, storing actual copies of all suffixes of an n-character
text would take O(n2 ) space, so instead each suffix is represented by a pointer
to its first character.
A suffix array stores all the suffixes sorted in dictionary order. For example,
the suffix array of the string abracadabra is shown below. The actual contents
of the array are the indices in the left-hand column; the right-hand shows the
corresponding suffixes.
388
11
10
7
0
3
5
8
1
4
6
9
2
\0
a\0
abra\0
abracadabra\0
acadabra\0
adabra\0
bra\0
bracadabra\0
cadabra\0
dabra\0
ra\0
racadabra\0
A suffix tree is similar, but instead using an array, we use some sort of tree
data structure to hold the sorted list. A common choice given an alphabet of
some fixed size k is a trie, in which each node at depth d represents a string of
length d, and its up to k children represent all (d + 1)-character extensions of the
string. The advantage of using a suffix trie is that searching for a string of length
m takes O(m) time, since we can just walk down the trie at the rate of one
node per character in m. A further optimization is to replace any long chain of
single-child nodes with a compressed edge labeled with the concatenation all the
characters in the chain. Such compressed suffix tries can not only be searched in
linear time but can also be constructed in linear time with a sufficiently clever
algorithm (we won’t actually do this here). Of course, we could also use a simple
balanced binary tree, which might be preferable if the alphabet is large.
The disadvantage of suffix trees over suffix arrays is that they generally require
more space to store all the internal pointers in the tree. If we are indexing a
huge text (or collection of texts), this extra space may be too expensive.
6.2.3.1
Building a suffix array
A straightforward approach to building a suffix array is to run any decent
comparison-based sorting algorithm on the set of suffixes (represented by pointers
into the text). This will take O(n log n) comparisons, but in the worst case each
comparison will take O(n) time, for a total of O(n2 log n) time. This is the
approach used in the sample code below.
The original suffix array paper by Manber and Myers (“Suffix arrays: a new
method for on-line string searches,” SIAM Journal on Computing 22(5):935-948,
1993) gives an O(n log n) algorithm, somewhat resembling radix sort, for building
suffix arrays in place with only a small amount of additional space. They also
note that for random text, simple radix sorting works well, since most suffixes
become distinguishable after about logk n characters (where k is the size of the
alphabet); this gives a cost of O(n log n) to do the sort, since radix sort only
looks at the bytes it needs to once. For a comparison-based sort, the same
389
assumption gives an O(n log2 n) running time; this is a factor of log n slower,
but this may be acceptable if programmer time is more important.
The fastest approach is to build a suffix tree in O(n) time and extract the suffix
array by traversing the tree. The only complication is that we need the extra
space to build the tree, although we get it back when we throw the tree away.
6.2.3.2
Searching a suffix array
Suppose we have a suffix array corresponding to an n-character text and we
want to find all occurrences in the text of an m-character pattern. Since the
suffixes are ordered, the easiest solution is to do binary search for the first and
last occurrences of the pattern (if any) using O(log n) comparisons. (The code
below does something even lazier than this, searching for some match and then
scanning linearly for the first and last matches.) Unfortunately, each comparison
may take as much as O(m) time, since we may have to check all m characters of
the pattern. So the total cost will be O(m log n) in the worst case.
By storing additional information about the longest common prefix of consecutive
suffixes, it is possible to avoid having to re-examine every character in the
pattern for every comparison, reducing the search cost to O(m + log n). With a
sufficiently clever algorithm, this information can be computed in linear time,
and can also be used to solve quickly such problems as finding the longest
duplicate substrings, or most frequently occurring strings. For more details, see
(Gusfield, Dan. Algorithms on Strings, Trees, and Sequences: Computer Science
and Computational Biology. Cambridge University Press, 1997, §7.14.4]).
Using binary search on the suffix array, most searching tasks are now easy:
• Finding if a substring appears in the array uses binary search directly.
• Finding all occurrences requires two binary searches, one for the first
occurrence and one for the last. If we only want to count the occurrences
and not return their positions, this takes O(m + log n) time. If we want
to return their positions, it takes O(m + log n + k) time, where k is the
number of times the pattern occurs.
• Finding duplicate substrings of length m or more can be done by looking
for adjacent entries in the array with long common prefixes, which takes
O(mn) time in the worst case if done naively (and O(n) time if we have
already computed longest common prefix information.
6.3
Burrows-Wheeler transform
Closely related to suffix arrays is the Burrows-Wheeler transform (Burrows
and Wheeler, A Block-Sorting Lossless Data Compression Algorithm, DEC
Systems Research Center Technical Report number 124, 1994), which is the basis
for some of the best currently known algorithms for text compression (it’s the
technique that is used, for example, in bzip2).
390
The idea of the Burrows-Wheeler Transform is to construct an array whose rows
are all cyclic shifts of the input string in dictionary order, and return the last
column of the array. The last column will tend to have long runs of identical
characters, since whenever some substring (like the appears repeatedly in the
input, shifts that put the first character t in the last column will put the rest
of the substring he in the first columns, and the resulting rows will tend to be
sorted together. The relative regularity of the last column means that it will
compress well with even very simple compression algorithms like run-length
encoding.
Below is an example of the Burrows-Wheeler transform in action, with $ marking
end-of-text. The transformed value of abracadabra$ is $drcraaaabba, the last
column of the sorted array; note the long run of a’s (and the shorter run of b’s).
abracadabra$
abracadabra$
bracadabra$a
abra$abracad
racadabra$ab
acadabra$abr
acadabra$abr
adabra$abrac
cadabra$abra
a$abracadabr
adabra$abrac
bracadabra$a
dabra$abraca --> bra$abracada
abra$abracad
cadabra$abra
bra$abracada
dabra$abraca
ra$abracadab
racadabra$ab
a$abracadabr
ra$abracadab
$abracadabra
$abracadabra
The most useful property of the Burrows-Wheeler transform is that it can be
inverted; this distinguishes it from other transforms that produce long runs like
simply sorting the characters. We’ll describe two ways to do this; the first is
less efficient, but more easily grasped, and involves rebuilding the array one
column at a time, starting at the left. Observe that the leftmost column is just
all the characters in the string in sorted order; we can recover it by sorting the
rightmost column, which we have to start off with. If we paste the rightmost and
leftmost columns together, we have the list of all 2-character substrings of the
original text; sorting this list gives the first two columns of the array. (Remember
that each copy of the string wraps around from the right to the left.) We can
then paste the rightmost column at the beginning of these two columns, sort
the result, and get the first three columns. Repeating this process eventually
reconstructs the entire array, from which we can read off the original string from
any row. The initial stages of this process for abracadabra$ are shown below:
$
d
r
c
r
a
a
a
a
a
a
b
$a
da
ra
ca
ra
ab
ab
ab
ac
ad
a$
br
$ab
dab
rac
cad
ra$
abr
abr
abr
aca
ada
a$a
bra
391
a -> b
a
c
a
d
b
r
b
r
a
$
ab -> br
ac
ca
ad
da
br
ra
br
ra
a$
$a
abr -> bra
aca
cad
ada
dab
bra
rac
bra
ra$
a$a
$ab
Rebuilding the entire array in this fashion takes O(n2 ) time and O(n2 ) space.
In their paper, Burrows and Wheeler showed that one can in fact reconstruct
the original string from just the first and last columns in the array in O(n) time.
Here’s the idea: Suppose that all the characters were distinct. Then after
reconstructing the first column we would know all pairs of adjacent characters.
So we could just start with the last character $ and regenerate the string by
appending at each step the unique successor to the last character so far. If all
characters were distinct, we would never get confused about which character
comes next.
The problem is what to do with pairs with duplicate first characters, like ab
and ac in the example above. We can imagine that each a in the last column is
labeled in some unique way, so that we can talk about the first a or the third a,
but how do we know which a is the one that comes before b or d?
The trick is to look closely at how the original sort works. Look at the rows in
the original transformation. If we look at all rows that start with a, the order
they are sorted in is determined by the suffix after a. These suffixes also appear
as the prefixes of the rows that end with a, since the rows that end with a are
just the rows that start with a rotated one position. It follows that all instances
of the same letter occur in the same order in the first and last columns. So if
we use a stable sort to construct the first column, we will correctly match up
instances of letters.
This method is shown in action below. Each letter is annotated uniquely with
a count of how many identical letters equal or precede it. Sorting recovers the
first column, and combining the last and first columns gives a list of unique
pairs of adjacent annotated characters. Now start with $1 and construct the full
sequence $1 a1 b1 r1 a3 c1 a4 d1 a2 b2 r2 a5 $1. The original string is
obtained by removing the end-of-string markers and annotations: abracadabra.
$1
a1
d1
a2
r1
a3
c1
a4
r2
a5
a1
b1
a2 --> b2
a3
c1
a4
d1
b1
r1
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b2
a5
r2
$1
Because we are only sorting single characters, we can perform the sort in linear
time using counting sort. Extracting the original string also takes linear time if
implemented reasonably.
6.3.1
Suffix arrays and the Burrows-Wheeler transform
A useful property of the Burrows-Wheeler transform is that each row of the
sorted array is essentially the same as the corresponding row in the suffix array,
except for the rotated string prefix after the $ marker. This means, among
other things, that we can compute the Burrows-Wheeler transform in linear time
using suffix trees. Ferragina and Manzini (http://people.unipmn.it/~manzini/
papers/focs00draft.pdf) have further exploited this correspondence (and some
very clever additional ideas) to design compressed suffix arrays that compress
and index a text at the same time, so that pattern searches can be done directly
on the compressed text in time close to that needed for suffix array searches.
6.3.2
Sample implementation
As mentioned above, this is a pretty lazy implementation of suffix arrays, that
doesn’t include many of the optimizations that would be necessary to deal with
huge source texts.
/* we expose this so user can iterate through it */
struct suffixArray {
size_t n;
const char *string;
const char **suffix;
};
/* length of string INCLUDING final null */
/* original string */
/* suffix array of length n */
typedef struct suffixArray *SuffixArray;
/* construct a suffix array */
/* it is a bad idea to modify string before destroying this */
SuffixArray suffixArrayCreate(const char *string);
/* destructor */
void suffixArrayDestroy(SuffixArray);
/* return number of occurrences of substring */
/* if non-null, index of first occurrence is place in first */
size_t
393
suffixArraySearch(SuffixArray, const char *substring, size_t *first);
/* return the Burrows-Wheeler transform of the underlying string
* as malloc'd data of length sa->n */
/* note that this may have a null in the middle somewhere */
char *suffixArrayBWT(SuffixArray sa);
/* invert BWT of null-terminated string, returning a malloc'd copy of original */
char *inverseBWT(size_t len, const char *s);
examples/suffixArray/suffixArray.h
#include
#include
#include
#include
<stdlib.h>
<assert.h>
<string.h>
<limits.h>
#include "suffixArray.h"
static int
saCompare(const void *s1, const void *s2)
{
return strcmp(*((const char **) s1), *((const char **) s2));
}
SuffixArray
suffixArrayCreate(const char *s)
{
size_t i;
SuffixArray sa;
sa = malloc(sizeof(*sa));
assert(sa);
sa->n = strlen(s) + 1;
sa->string = s;
sa->suffix = malloc(sizeof(*sa->suffix) * sa->n);
assert(sa->suffix);
/* construct array of pointers to suffixes */
for(i = 0; i < sa->n; i++) {
sa->suffix[i] = s+i;
}
/* this could be a lot more efficient */
qsort(sa->suffix, sa->n, sizeof(*sa->suffix), saCompare);
394
}
return sa;
void
suffixArrayDestroy(SuffixArray sa)
{
free(sa->suffix);
free(sa);
}
size_t
suffixArraySearch(SuffixArray sa, const char *substring, size_t *first)
{
size_t lo;
size_t hi;
size_t mid;
size_t len;
int cmp;
len = strlen(substring);
/* invariant: suffix[lo] <= substring < suffix[hi] */
lo = 0;
hi = sa->n;
while(lo + 1 < hi) {
mid = (lo+hi)/2;
cmp = strncmp(sa->suffix[mid], substring, len);
if(cmp == 0) {
/* we have a winner */
/* search backwards and forwards for first and last */
for(lo = mid; lo > 0 && strncmp(sa->suffix[lo-1], substring, len) == 0; lo--);
for(hi = mid; hi < sa->n && strncmp(sa->suffix[hi+1], substring, len) == 0; hi++
if(first) {
*first = lo;
}
return hi - lo + 1;
} else if(cmp < 0) {
lo = mid;
} else {
hi = mid;
}
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}
}
return 0;
char *
suffixArrayBWT(SuffixArray sa)
{
char *bwt;
size_t i;
bwt = malloc(sa->n);
assert(bwt);
for(i = 0; i < sa->n; i++) {
if(sa->suffix[i] == sa->string) {
/* wraps around to nul */
bwt[i] = '\0';
} else {
bwt[i] = sa->suffix[i][-1];
}
}
}
return bwt;
char *
inverseBWT(size_t len, const char *s)
{
/* basic trick: stable sort of s gives successor indices */
/* then we just thread through starting from the nul */
size_t *successor;
int c;
size_t count[UCHAR_MAX+1];
size_t offset[UCHAR_MAX+1];
size_t i;
char *ret;
size_t thread;
successor = malloc(sizeof(*successor) * len);
assert(successor);
/* counting sort */
for(c = 0; c <= UCHAR_MAX; c++) {
count[c] = 0;
396
}
for(i = 0; i < len; i++) {
count[(unsigned char) s[i]]++;
}
offset[0] = 0;
for(c = 1; c <= UCHAR_MAX; c++) {
offset[c] = offset[c-1] + count[c-1];
}
for(i = 0; i < len; i++) {
successor[offset[(unsigned char) s[i]]++] = i;
}
/* find the nul */
for(thread = 0; s[thread]; thread++);
/* thread the result */
ret = malloc(len);
assert(ret);
for(i = 0, thread = successor[thread]; i < len; i++, thread = successor[thread]) {
ret[i] = s[thread];
}
}
return ret;
examples/suffixArray/suffixArray.c
Here is a Makefile and test code: Makefile, testSuffixArray.c.
The output of make test shows all occurrences of a target string, the BurrowsWheeler transform of the source string (second-to-last line), and its inversion
(last line, which is just the original string):
$ make test
/bin/echo -n abracadabra-abracadabra-shmabracadabra | ./testSuffixArray abra
Count: 6
abra
abra-abr
abra-shm
abracada
abracada
abracada
aaarrrdddm\x00-rrrcccaaaaaaaaaaaashbbbbbb397
abracadabra-abracadabra-shmabracadabra
6.4
C++
Here we will describe some basic features of C++ that are useful for implementing
abstract data types. Like all programming languages, C++ comes with an
ideology, which in this case emphasizes object-oriented features like inheritance.
We will be ignoring this ideology and treating C++ as an improved version of C.
The goal here is not to teach you all of C++, which would take a while, but
instead to give you some hints for why you might want to learn C++ on your own.
If you decide to learn C++ for real, Bjarne Stroustrup’s The C++ Programming
Language is the definitive source. A classic tutorial here aimed at C programmers
introduces C++ features one at a time (some of these features have since migrated
into C). The web site http://www.cplusplus.com has extensive tutorials and
documentation.
6.4.1
Hello world
The C++ version of “hello world” looks like this:
#include <iostream>
int
main(int argc, const char **argv)
{
std::cout << "hi\n";
}
return 0;
examples/c++/helloworld.cpp
Compile this using g++ instead of gcc. Make shows how it is done:
$ make helloworld
g++
helloworld.cpp
-o helloworld
Or we could use an explicit Makefile:
CPP=g++
CPPFLAGS=-g3 -Wall
helloworld: helloworld.o
$(CPP) $(CPPFLAGS) -o $@ $^
Now the compilation looks like this:
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$ make helloworld
g++ -g3 -Wall -c -o helloworld.o helloworld.cpp
g++ -g3 -Wall -o helloworld helloworld.o
The main difference from the C version:
1. #include <stdio.h> is replaced by #include <iostream>, which gets
the C++ version of the stdio library.
2. printf("hi\n") is replaced by std::cout << "hi\n". The stream
std::cout is the C++ wrapper for stdout; you should read this variable
name as cout in the std namespace. The << operator is overloaded for
streams so that it sends its right argument out on its left argument (see
the discussion of operator overloading below). You can also do things like
std::cout << 37, std::cout << 'q', std::cout << 4.7, etc. These
all do pretty much what you expect.
If you don’t like typing std:: before all the built-in functions and variables, you
can put using namespace std somewhere early in your program, like this:
#include <iostream>
using namespace std;
int
main(int argc, const char **argv)
{
cout << "hi\n";
}
return 0;
examples/c++/helloworld_using.cpp
6.4.2
References
Recall that in C we sometime pass objects into function by reference instead of
by value, by using a pointer:
void increment(int *x)
{
(*x)++;
}
This becomes even more useful in C++, since many of the objects we are dealing
with are quite large, and can defend themselves against dangerous modifications
by restricting access to their components. So C++ provides a special syntax
allowing function parameters to be declared as call-by-reference rather than
call-by-value. The function above could be rewritten in C++ as
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void increment(int &x)
{
x++;
}
The int &x declaration says that x is a reference to whatever variable is passed
as the argument to increment. A reference acts exactly like a pointer that has
already had * applied to it. You can even write &x to get a pointer to the original
variable if you want to for some reason.
As with pointers, it’s polite to mark a reference with const if you don’t intend
to modify the original object:
void reportWeight(const SumoWrestler &huge)
{
cout << huge.getWeight();
}
References are also used as a return type to chain operators together; in the
expression
cout << "hi" << '\n';
the return type of the first << operator is an ostream & reference (as is cout);
this means that the '\n' gets sent to the same object. We could make the
return value be just an ostream, but then cout would be copied, which could
be expensive and would mean that the copy was no longer working on the same
internal state as the original. This same trick is used when overloading the
assignment operator.
6.4.3
Function overloading
C++ lets you define multiple functions with the same name, where the choice of
which function to call depends on the type of its arguments. Here is a program
that demonstrates this feature:
#include <iostream>
using namespace std;
const char *
typeName(int x)
{
return "int";
}
const char *
typeName(double x)
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{
}
return "double";
const char *
typeName(char x)
{
return "char";
}
int
main(int
{
cout
cout
cout
}
argc, const char **argv)
<< "The type of " << 3 << " is " << typeName(3) << ".\n";
<< "The type of " << 3.1 << " is " << typeName(3.1) << ".\n";
<< "The type of " << 'c' << " is " << typeName('c') << ".\n";
return 0;
examples/c++/functionOverloading.cpp
And here is what it looks like when we compile and run it:
$ make functionOverloading
g++
functionOverloading.cpp
$ ./functionOverloading
The type of 3 is int.
The type of 3.1 is double.
The type of c is char.
-o functionOverloading
Internally, g++ compiles three separate functions with different (and ugly) names,
and when you use typeName on an object of a particular type, g++ picks the one
whose type matches. This is similar to what happens with built-in operators in
straight C, where + means different things depending on whether you apply it
to a pair of ints, a pair of doubles, or a pointer and an int, but C++ lets you
do it with your own functions.
6.4.4
Classes
C++ allows you to declare classes that look suspiciously like structs. The main
differences between a class and a C-style struct are that (a) classes provide
member functions or methods that operate on instances of the class and
that are called using a struct-like syntax; and (b) classes can distinguish between
private members (only accessible to methods of the class) and public members
(accessible to everybody).
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In C, we organize abstract data types by putting the representation in a struct
and putting the operations on the data type in functions that work on this struct,
often giving the functions a prefix that hints at the type of its target (mostly to
avoid namespace collisions). Classes in C++ make this connection between a
data structure and the operations on it much more explicit.
Here is a simple example of a C++ class in action:
#include <iostream>
using namespace std;
/* counters can be incremented or read */
class Counter {
int value;
/* private value */
public:
Counter();
/* constructor with default value */
Counter(int);
/* constructor with specified value */
~Counter();
/* useless destructor */
int read();
/* get the value of the counter */
void increment();
/* add one to the counter */
};
Counter::Counter() { value = 0; }
Counter::Counter(int initialValue) { value = initialValue; }
Counter::~Counter() { cerr << "counter de-allocated with value " << value << '\n'; }
int Counter::read() { return value; }
void Counter::increment() { value++; }
int
main(int argc, const char **argv)
{
Counter c;
Counter c10(10);
cout << "c starts at " << c.read() << '\n';
c.increment();
cout << "c after one increment is " << c.read() << '\n';
cout << "c10 starts at " << c10.read() << '\n';
c.increment();
c.increment();
cout <<"c10 after two increments is " << c10.read() << '\n';
}
return 0;
402
examples/c++/counter.cpp
Things to notice:
1. In the class Counter declaration, the public: label introduces the public
members of the class. The member value is only accessible to member
functions of Counter. This enforces much stronger information hiding
than the default in C, although one can still use void * trickery to hunt
down and extract supposedly private data in C++ objects.
2. In addition to the member function declarations in the class declaration, we also need to provide definitions. These look like ordinary function definitions, except that the class name is prepended using :: as in
Counter::read.
3. Member functions are called using struct access syntax, as in c.read().
Conceptually, each instance of a class has its own member functions, so
that c.read is the function for reading c while c10.read is the function
for reading c10. Inside a member function, names of class members refer to
members of the current instance; value inside c.read is c.value (which
otherwise is not accessible, since c.value is not public).
4. Two special member functions are Counter::Counter() and Counter::Counter(int).
These are constructors, and are identifiable as such because they are
named after the class. A constructor is called whenever a new instance
of the class is created. If you create an instance with no arguments
(as in the declaration Counter c;), you get the constructor with no
arguments. If you create an instance with arguments (as in the declaration
Counter c10(10);), you get the version with the appropriate arguments.
This is just another example of function overloading. If you don’t define
any constructors, C++ supplies a default constructor that takes no
arguments and does nothing. Note that constructors don’t have a return
type (you don’t need to preface them with void).
5. The special member function Counter::~Counter() is a destructor; it is
called when an object of type Counter is de-allocated (say, when returning
from a function with a local variable of this type). This particular destructor
is not very useful. Destructors are mostly important for objects that allocate
their own storage that needs to be de-allocated when the object is; see the
section on storage allocation below.
Compiling and running this program gives the following output. Note that the
last two lines are produced by the destructor.
c starts at 0
c after one increment is 1
c10 starts at 10
c10 after two increments is 10
counter de-allocated with value 10
counter de-allocated with value 3
One subtle difference between C and C++ is that C++ uses empty parentheses
403
() for functions with no arguments, where C would use (void). This is a bit
of a historical artifact, having to do with C allowing () for functions whose
arguments are not specified in the declaration (which was standard practice
before ANSI C).
Curiously, C++ also allows you to declare structs, with the interpretation that
a struct is exactly like a class except that all members are public by default.
So if you change class to struct in the program above, it will do exactly the
same thing. In practice, nobody who codes in C++ does this; the feature is
mostly useful to allow C code with structs to mix with C++ code.
6.4.5
Operator overloading
Sometimes when you define a new class, you also want to define new interpretations of operators on that class. Here is an example of a class that defines
elements of the max-plus algebra over ints. This gives us objects that act
like ints, except that the + operator now returns the larger of its arguments and
the * operator now returns the sum.25
The mechanism in C++ for doing this is to define member functions with names
operatorsomething where something is the name of the operator we want to
define. These member functions take one less argument that the operator they
define; in effect, x + y becomes syntactic sugar for x.operator+(y) (which,
amazingly, is actually legal C++). Because these are member functions, they
are allowed to access members of other instances of the same class that would
normally be hidden.
This same mechanism is also used to define automatic type conversions out
of a type: the MaxPlus::operator int() function allows C++ to convert a
MaxPlus object to an int whenever it needs to (for example, to feed it to cout).
(Automatic type conversions into a type happen if you provide an appropriate
constructor.)
#include <iostream>
#include <algorithm> // for max
using namespace std;
/* act like ints, except + does max and * does addition */
class MaxPlus {
int value;
25 This otherwise insane-looking modification is useful for modeling scheduling problems,
where a+b is the time to do a and b in parallel, and a*b is the time to do a and b sequentially.
The reason for making the first case + and the second case * is because this makes the
distributive law a*(b+c) = (a*b)+(a*c) work. It also allows tricks like matrix multiplication
using the standard definition. See http://maxplus.org for more than you probably want to
know about this.
404
public:
MaxPlus(int);
MaxPlus operator+(const MaxPlus &);
MaxPlus operator*(const MaxPlus &);
operator int();
};
MaxPlus::MaxPlus(int x) { value = x; }
MaxPlus
MaxPlus::operator*(const MaxPlus &other)
{
return MaxPlus(value + other.value);
}
MaxPlus
MaxPlus::operator+(const MaxPlus &other)
{
/* std::max does what you expect */
return MaxPlus(max(value, other.value));
}
MaxPlus::operator int() { return value; }
int
main(int argc, const char **argv)
{
cout << "2+3 == " << (MaxPlus(2) + MaxPlus(3)) << '\n';
cout << "2*3 == " << (MaxPlus(2) * MaxPlus(3)) << '\n';
}
return 0;
examples/c++/maxPlus.cpp
Avoid the temptation to overuse operator overloading, as it can be dangerous if
used to obfuscate what an operator normally does:
MaxPlus::operator--() { godzilla.eat(tokyo); }
The general rule of thumb is that you should probably only do operator overloading if you really are making things that act like numbers (yes, cout << violates
this).
Automatic type conversions can be particularly dangerous. The line
cout << (MaxPlus(2) + 3) << '\n';
is ambiguous: should the compiler convert MaxPlus(2) to an int using the
405
MaxPlus(int) constructor and use ordinary integer addition or convert 3 to
a MaxPlus using MaxPlus::operator int() and use funky MaxPlus addition?
Fortunately most C++ compilers will complain about the ambiguity and fail
rather than guessing wrong.
6.4.6
Templates
One of the things we kept running into in this class was that if we defined a
container type like a hash table, binary search tree, or priority queue, we had
to either bake in the type of the data it held or do horrible tricks with void *
pointers to work around the C type system. C++ includes a semi-principled
work-around for this problem known as templates. These are essentially macros
that take a type name as an argument, that are expanded as needed to produce
functions or classes with specific types (see Macros for an example of how to do
this if you only have C).
Typical use is to prefix a definition with template <class T> and then use T
as a type name throughout:
template <class T>
T add1(T x)
{
return x + ((T) 1);
}
Note the explicit cast to T of 1; this avoids ambiguities that might arise with
automatic type conversions.
If you put this definition in a program, you can then apply add1 to any type
that has a + operator and that you can convert 1 to. For example, the output of
this code fragment:
cout
cout
cout
cout
cout
<<
<<
<<
<<
<<
"add1(3) == " << add1(3) << '\n';
"add1(3.1) == " << add1(3.1) << '\n';
"add1('c') == " << add1('c') << '\n';
"add1(MaxPlus(0)) == " << add1(MaxPlus(0)) << '\n';
"add1(MaxPlus(2)) == " << add1(MaxPlus(2)) << '\n';
is
add1(3) == 4
add1(3.1) == 4.1
add1('c') == d
add1(MaxPlus(0)) == 1
add1(MaxPlus(2)) == 2
By default, C++ will instantiate a template to whatever type fits in its argument.
If you want to force a particular version, you can put the type in angle brackets
after the name of whatever you defined. For example,
406
cout << "add1<int>(3.1) == " << add1<int>(3.1) << '\n';
produces
add1<int>(3.1) == 4
because add1<int> forces its argument to be converted to an int (truncating
to 3) before adding one to it.
Because templates are really macros that get expanded as needed, it is common
to put templates in header (.h) files rather than in .cpp files. See the stack
implementation below for an example of this.
6.4.7
Exceptions
C provides no built-in mechanism for signaling that something bad happened.
So C programmers are left to come up with ad-hoc mechanisms like:
1. Calling abort to kill the program, either directly or via assert.
2. Calling exit with a nonzero exit code.
3. Returning a special error value from a function. This is often done in
library routines, because it’s rude for a library routine not to give the caller
a chance to figure out how to deal with the error. But it means coming up
with some special error value that won’t be returned normally, and these
can vary widely from one routine to another (null pointers, -1, etc.)
C++ provides a standard mechanism for signaling unusual events known as
exceptions. The actual mechanism is similar to return: the throw statement
throws an exception that may be caught by a try..catch statement anywhere
above it on the execution stack (not necessarily in the same function). Example:
#include <iostream>
using namespace std;
int fail()
{
throw "you lose";
}
return 5;
int
main(int argc, const char **argv)
{
try {
cout << fail() << '\n';
}
407
catch(const char *s) {
cerr << "Caught error: " << s << '\n';
}
}
return 0;
examples/c++/exception.cpp
In action:
$ make exception
g++ -g3 -Wall
exception.cpp
$ ./exception
Caught error: you lose
-o exception
Note the use of cerr instead of cout. This sends the error message to stderr.
A try..catch statement will catch an exception only if the type matches the
type of the argument to the catch part of the statement. This can be used to
pick and choose which exceptions you want to catch. See http://www.cplusplus.
com/doc/tutorial/exceptions/ for some examples and descriptions of some C++
standard library exceptions.
6.4.8
Storage allocation
C++ programs generally don’t use malloc and free, but instead use the built-in
C++ operators new and delete. The advantage of new and delete is that they
know about types: not only does this mean that you don’t have to play games
with sizeof to figure out how much space to allocate, but if you allocate a new
object from a class with a constructor, the constructor gets called to initialize
the object, and if you delete an object, its destructor (if it has one) is called.
There are two versions of new and delete, depending on whether you want
to allocate just one object or an array of objects, plus some special syntax for
passing constructor arguments:
• To allocate a single object, use new type.
• To allocate an array of objects, use new type[size]. As with malloc, both
operations return a pointer to type.
• If you want to pass arguments to a constructor for type, use new
type(args). This only works with the single-object version, so you can’t
do new SomeClass[12] unless SomeClass has a constructor that takes no
arguments.
• To de-allocate a single object, use delete pointer-to-object.
• To de-allocate an array, use delete [] pointer-to-base-of-array. Mixing
new with delete [] or vice versa is an error that may or may not be
408
detected by the compiler. Mixing either with malloc or free is a very bad
idea.
The program below gives examples of new and delete in action:
#include <iostream>
#include <cassert>
using namespace std;
class Noisy {
int id;
public:
Noisy(int);
~Noisy();
};
// create a noisy object with this id
Noisy::Noisy(int initId) {
id = initId;
cout << "Noisy object created with id " << id << '\n';
}
Noisy::~Noisy() {
cout << "Noisy object destroyed with id " << id << '\n';
}
int
main(int argc, const char **argv)
{
int *p;
int *a;
const int n = 100;
Noisy n1(1);
Noisy *n2;
p = new int;
a = new int[n];
n2 = new Noisy(2);
*p = 5;
assert(*p == 5);
for(int i = 0; i < n; i++) {
a[i] = i;
}
for(int i = 0; i < n; i++) {
409
}
assert(a[i] == i);
delete [] a;
delete p;
delete n2;
}
return 0;
examples/c++/allocation.cpp
6.4.8.1
Storage allocation inside objects
Inside objects, storage allocation gets complicated. The reason is that if the
object is copied, either by an assignment or by being passed as a call-by-value
parameter, the storage pointed to by the object will not be copied. This can
lead to two different objects that share the same internal data structures, which
is usually not something you want. Furthermore, when the object is deallocated,
it’s necessary to also deallocate any space it allocated, which can be done inside
the object’s destructor.
To avoid all these problems, any object of type T that uses new needs to have all
of:
1. A destructor T::~T().
2. A copy constructor T::T(const T &), which is a constructor that takes a
reference to another object of the same type as an argument and copies its
contents.
3. An overloaded assignment operator T::operator=(const T &) that does
the same thing, but also deallocates any internal storage of the current
object before copying new data in place of it (or possibly just copies the
contents of internal storage without doing any allocation and deallocation).
The overloaded assignment operator is particularly tricky, because you have
to make sure it doesn’t destroy the contents of the object if somebody writes
the useless self-assignment a = a, and you also need to return a reference
to *this so that you can chain assignments together as in a = b = c.
Here is an example of a Stack class that includes all of these members. Note
that it is defined using templates so we can make a stack of any type we like.
template <class T>
class Stack {
static const int initialSize = 32;
int top;
int size;
T* contents;
public:
410
/* static means this is shared across entire class
Stack();
/* create a new empty stack */
/* the unholy trinity of complex C++ objects */
~Stack();
/* destructor */
Stack(const Stack &);
/* copy constructor */
Stack& operator=(const Stack &); /* overloaded assignment */
};
void push(T);
int isEmpty();
T pop();
/* push an element onto the stack */
/* return 1 if empty */
/* pop top element from stack */
template <class T>
Stack<T>::Stack()
{
size = initialSize;
top = 0;
contents = new T[size];
}
template <class T>
Stack<T>::~Stack()
{
delete [] contents;
}
template <class T>
Stack<T>::Stack(const Stack<T> &other)
{
size = other.size;
top = other.top;
contents = new T[size];
}
for(int i = 0; i < top; i++) {
contents[i] = other.contents[i];
}
template <class T>
Stack<T> &
Stack<T>::operator=(const Stack<T> &other)
{
if(&other != this) {
/* this is a real assignment */
delete [] contents;
411
size = other.size;
top = other.top;
contents = new T[size];
}
}
for(int i = 0; i < top; i++) {
contents[i] = other.contents[i];
}
return *this;
template <class T>
void
Stack<T>::push(T elt)
{
if(top >= size) {
int newSize = 2*size;
T *newContents = new T[newSize];
for(int i = 0; i < top; i++) {
newContents[i] = contents[i];
}
delete [] contents;
}
}
contents = newContents;
size = newSize;
contents[top++] = elt;
template <class T>
T
Stack<T>::pop()
{
if(top > 0) {
return contents[--top];
} else {
throw "stack empty";
}
}
examples/c++/stack/stack.h
412
Here is some code demonstrating use of the stack:
#include <iostream>
#include "stack.h"
using namespace std;
int
main(int argc, const char **argv)
{
Stack<int> s;
Stack<int> s2;
try {
s.push(1);
s.push(2);
s.push(3);
s2 = s;
cout << s.pop() << '\n';
cout << s.pop() << '\n';
cout << s.pop() << '\n';
cout << s2.pop() << '\n';
cout << s2.pop() << '\n';
cout << s2.pop() << '\n';
try {
s2.pop();
} catch(const char *err) {
cout << "Caught expected exception " << err << '\n';
}
for(int i = 0; i < 1000; i++) {
s.push(i);
}
cout << s.pop() << '\n';
} catch(const char *err) {
cerr << "Caught error " << err << '\n';
}
}
return 0;
413
examples/c++/stack/testStack.cpp
6.4.9
Standard library
C++ has a large standard library that includes implementations of many of
the data structures we’ve seen in this class. In most situations, it is easier to
use the standard library implementations than roll your own, although you
have to be careful to make sure you understand just what the standard library
implementations do. For example, here is a reimplementation of the main routine
from testStack.cpp using the stack template from #include <stack>.
#include <iostream>
#include <stack>
using namespace std;
int
main(int argc, const char **argv)
{
stack<int> s;
stack<int> s2;
s.push(1);
s.push(2);
s.push(3);
s2 = s;
cout << s.top() << '\n'; s.pop();
cout << s.top() << '\n'; s.pop();
cout << s.top() << '\n'; s.pop();
cout << s2.top() << '\n'; s2.pop();
cout << s2.top() << '\n'; s2.pop();
cout << s2.top() << '\n'; s2.pop();
for(int i = 0; i < 1000; i++) {
s.push(i);
}
cout << s.top() << '\n';
}
return 0;
examples/c++/stack/stdStack.cpp
414
One difference between the standard stack and our stack is that std::stack’s
pop member function doesn’t return anything. So we have to use top to get the
top element before popping it.
There is a chart of all the standard library data structures at http://www.
cplusplus.com/reference/stl/.
6.4.10
Things we haven’t talked about
The main thing we’ve omitted here is any discussion of object-oriented features
of C++, particularly inheritance. These are not immediately useful for the
abstract-data-type style of programming we’ve used in CS223, but can be helpful
for building more complicated systems, where we might want to have various
specialized classes of objects that can all be approached using a common interface
represented by a class that they inherit from. If you are interested in exploring
these tools further, the CS department occasionally offers a class on objectoriented programming; Mike Fischer’s lecture notes from the last time this
course was offered can be found at http://zoo.cs.yale.edu/classes/cs427/2011a/
lectures.html.
6.5
Testing during development
It is a truth universally acknowledged that test code should be written early in
the development process. Unfortunately, most programmers (including me) tend
to assume that a program will work on the first attempt and there’s not much
point in testing it anyway, so writing and running test code often gets deferred
indefinitely. The solution is to write the test code first, and run it directly
from your Makefile every time you save and compile your program. Not only
will this guarantee that your program actually works when you are done (or at
least passes the tests you thought of), it allows you to see how the program is
improving with each positive change, and prevents you from accidentally making
new negative changes that break things that used to work.
Going one step further, we can often write our interface and test code first, build
a non-working stub implementation, and then slowly flesh out the missing pieces
until the implementation passes all the tests. This way there is always some
obvious step to do next, and we don’t find ourselves stuck staring at an empty
file.
6.5.1
Unit tests
A straightforward approach to testing is to include test code with every unit in
your program, where a unit is any part of the program that can be sensibly run
415
by itself. Typically, this will be a single function or a group of functions that
together implement some data structure.
In C, these will often make up the contents of a single source file. Though this is
probably not the best approach if you are building a production-quality testing
framework, a simple way to include unit tests in a program is to append to each
source file a test main function that can be enabled by defining a macro (I like
TEST_MAIN). You can then build this file by itself with the macro defined to get
a stand-alone test program for just this code.
6.5.1.1
What to put in the test code
Ideally, you want to use enough different inputs that every line of code in your
program is reached by some test, a goal called code coverage. For complex
programs, this may be hard to achieve, and there are programs, such as the
gcov program that comes with gcc, that will analyze how much code coverage
you get out of your tests. For simple programs, we can just try to come up with
a set of inputs that covers all our bases.
Testing can be done as black-box testing, where the test code assumes no
knowledge of the implementation, or white-box testing, where the test code
has direct access to the implementation and can observe the effects of its actions.
Black-box testing is handy if your implementation may change, and it is generally
a good idea to write black-box tests first. White-box testing can be useful if
some states of the data structure are hard to reach otherwise, or if black-box
testing is not very informative about why a particular operation is failing. The
example given below uses both.
6.5.1.2
Example
Here is an example of a simple data structure with some built-in test code
conditionally compiled by defining TEST_MAIN. The data structure implements a
counter with built-in overflow protection. The counter interface does not provide
the ability to read the counter value; instead, the user can only tell if it is zero
or not.
Because the counter is implemented internally as a uint64_t, black-box testing
of what happens with too many increments would take centuries. So we include
some white-box tests that directly access the counter value to set up this (arguably
unnecessary) test case.
The code is given below. We include both the interface file and the implementation, as well as a Makefile showing how to build and run the test program.
The Makefile includes some extra arguments to gcc to turn on the TEST_MAIN
macro and supply the extra information needed to run gcov. If you type make
test, it will make and run testCounter, and then run gcov to verify that we
did in fact hit all lines of code in the program.
416
/*
* Abstract counter type.
*
* You can increment it, decrement it, and test for zero.
*
* Increment and decrement operations return 1 if successful,
* 0 if the operation would cause underflow or overflow.
*/
typedef struct counter Counter;
/* make a new counter starting at 0 */
Counter *counterCreate(void);
/* destroy a counter */
void counterDestroy(Counter *);
/* return 1 if counter is 0, 0 otherwise */
int counterIsZero(const Counter *);
/* increment a counter, returns 1 if successful, 0 if increment would cause overflow */
int counterIncrement(Counter *);
/* decrement a counter, returns 1 if successful, 0 if decrement would cause underflow */
int counterDecrement(Counter *);
examples/unitTest/counter.h
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<assert.h>
"counter.h"
#include <stdint.h>
#define COUNTER_MAX (UINT64_MAX)
struct counter {
uint64_t value;
};
/* make a new counter starting at 0 */
Counter *
counterCreate(void)
{
Counter *c;
417
c = malloc(sizeof(Counter));
assert(c);
c->value = 0;
}
return c;
/* destroy a counter */
void
counterDestroy(Counter *c)
{
free(c);
}
/* return 1 if counter is 0, 0 otherwise */
int
counterIsZero(const Counter *c)
{
return c->value == 0;
}
/* increment a counter, returns 1 if successful, 0 if increment would cause overflow */
int
counterIncrement(Counter *c)
{
if(c->value == COUNTER_MAX) {
return 0;
} else {
c->value++;
return 1;
}
}
/* decrement a counter, returns 1 if successful, 0 if decrement would cause underflow */
int
counterDecrement(Counter *c)
{
if(c->value == 0) {
return 0;
} else {
c->value--;
return 1;
}
}
418
#ifdef TEST_MAIN
int
main(int argc, char **argv)
{
Counter *c;
/* black box testing */
c = counterCreate();
assert(counterIsZero(c));
assert(counterIncrement(c)
assert(!counterIsZero(c));
assert(counterIncrement(c)
assert(!counterIsZero(c));
assert(counterDecrement(c)
assert(!counterIsZero(c));
assert(counterDecrement(c)
assert(counterIsZero(c));
assert(counterDecrement(c)
assert(counterIsZero(c));
assert(counterIncrement(c)
assert(!counterIsZero(c));
/* 0 */
== 1);
/* 1 */
== 1);
/* 2 */
== 1);
/* 1 */
== 1);
/* 0 */
== 0);
/* 0 */
== 1);
/* 1 */
counterDestroy(c);
/* white box testing */
c = counterCreate();
assert(c->value == 0);
assert(counterIncrement(c)
assert(c->value == 1);
assert(counterIncrement(c)
assert(c->value == 2);
assert(counterDecrement(c)
assert(c->value == 1);
assert(counterDecrement(c)
assert(c->value == 0);
assert(counterDecrement(c)
assert(c->value == 0);
assert(counterIncrement(c)
assert(c->value == 1);
/* 0 */
== 1);
/* 1 */
== 1);
/* 2 */
== 1);
/* 1 */
== 1);
/* 0 */
== 0);
/* 0 */
== 1);
/* 1 */
/* force counter value to COUNTER_MAX to test for overflow protection */
c->value = COUNTER_MAX;
/* COUNTER_MAX */
assert(counterIncrement(c) == 0);
/* COUNTER_MAX */
assert(c->value == COUNTER_MAX);
419
assert(counterDecrement(c) == 1);
assert(c->value == COUNTER_MAX-1);
assert(counterIncrement(c) == 1);
assert(c->value == COUNTER_MAX);
/* COUNTER_MAX-1 */
/* COUNTER_MAX */
counterDestroy(c);
return 0;
}
#endif
examples/unitTest/counter.c
CC=c99
CFLAGS=-g3 -pedantic -Wall
all: seqprinter
seqprinter: main.o sequence.o
$(CC) $(CFLAGS) -o $@ $^
test: seqprinter
./seqprinter
# these rules say to rebuild main.o and sequence.o if sequence.h changes
main.o: main.c sequence.h
sequence.o: sequence.c sequence.h
clean:
$(RM) -f seqprinter *.o
examples/ADT/sequence/Makefile
6.5.2
Test harnesses
Here are some older notes on testing using a test harness that does some basic
tricks like catching segmentation faults so that a program can keep going even if
one test fails.
6.5.2.1
Module interface
The module will be a stack for storing integers.
Let’s start with the interface, which we’ll put in a file called stack.h:
6.5.2.1.1
stack.h
420
/*
* This is an "opaque struct"; it discourages people from looking at
* the inside of our structure. The actual definiton of struct stack
* is contained in stack.c.
*/
typedef struct stack *Stack;
/* constructor and destructor */
Stack stack_create(void);
void stack_destroy(Stack);
/* returns 0 on allocation error */
/* push a new element onto the stack */
void stack_push(Stack , int new_element);
/* return 1 if the stack is empty, 0 otherwise */
int stack_isempty(Stack);
/* remove and return top element of stack */
/* returns STACK_EMPTY if stack is empty */
#define STACK_EMPTY (-1)
int stack_pop(Stack);
Our intent is that an Stack acts like a stack— we push things onto it using
stack_push, and then pull them off again in reverse order using stack_pop.
Ideally, we don’t ever pop the stack when it’s empty (which we can detect using
stack_isempty), but if we do, we have stack_pop return something well-defined.
6.5.2.2
Test code
Let’s write some test code to try this out. Because our initial stack implementation may be exceptionally bug-ridden, we’ll use a test harness that provides
macros for detecting and intercepting segmentation faults and similar disasters.
The various testing wrappers are defined in the files tester.h and tester.c,
from the chapter on testing; you should feel free to use it for your own purposes.
I’ve added line numbers in comments to all the TEST lines so we can find them
again later.
6.5.2.2.1
#include
#include
#include
#include
test-stack.c
<stdio.h>
<setjmp.h>
<signal.h>
<unistd.h>
#include <stdlib.h>
421
#include "stack.h"
#include "tester.h"
#define STRESS_TEST_ITERATIONS (1000000)
int
main(int argc, char **argv)
{
Stack s;
int i;
tester_init();
/* first we need to build one */
TRY { s = stack_create(); } ENDTRY;
/* 25 */ TEST_ASSERT(s != 0);
/* now we'll try pushing and popping a bit */
TRY { stack_push(s, 1); } ENDTRY;
TRY { stack_push(s, 2); } ENDTRY;
TRY { stack_push(s, 3); } ENDTRY;
/*
/*
/*
/*
/*
/*
/*
/*
/*
/*
32
33
34
35
36
37
38
39
40
41
*/
*/
*/
*/
*/
*/
*/
*/
*/
*/
TEST(stack_isempty(s), 0);
TEST(stack_pop(s), 3);
TEST(stack_isempty(s), 0);
TEST(stack_pop(s), 2);
TEST(stack_isempty(s), 0);
TEST(stack_pop(s), 1);
TEST(stack_isempty(s), 1);
TEST(stack_pop(s), STACK_EMPTY);
TEST(stack_isempty(s), 1);
TEST(stack_pop(s), STACK_EMPTY);
/* can we still push after popping too much? */
TRY { stack_push(s, 4); } ENDTRY;
/* 45 */ TEST(stack_isempty(s), 0);
/* 46 */ TEST(stack_pop(s), 4);
/* 47 */ TEST(stack_isempty(s), 1);
/* 48 */ TEST(stack_pop(s), STACK_EMPTY);
/* 49 */ TEST(stack_isempty(s), 1);
/* let's do some stress testing */
/* we won't use TEST for this because we might get too much output */
TRY {
for(i = 0; i < STRESS_TEST_ITERATIONS; i++) {
422
stack_push(s, i);
}
for(i = 0; i < STRESS_TEST_ITERATIONS; i++) {
stack_push(s, 957);
if(stack_pop(s) != 957) {
/* 60 */ FAIL("wanted 957 but didn't get it");
abort();
}
}
for(i = STRESS_TEST_ITERATIONS - 1; i >= 0; i--) {
if(stack_isempty(s)) {
/* 66 */ FAIL("stack empty too early");
abort();
}
if(stack_pop(s) != i) {
/* 70 */ FAIL("got wrong value!");
abort();
}
}
} ENDTRY; /* 74 */
/* 76 */ TEST(stack_isempty(s), 1);
TRY { stack_destroy(s); } ENDTRY;
tester_report(stdout, argv[0]);
return tester_result();
}
There is a lot of test code here. In practice, we might write just a few tests to
start off with, and, to be honest, I didn’t write all of this at once. But you can
never have too many tests— if nothing else, they give an immediate sense of
gratification as the number of failed tests drops.
6.5.2.3
Makefile
• Finally, we’ll write a Makefile:
6.5.2.3.1
Makefile
CC=gcc
CFLAGS=-g3 -Wall -ansi -pedantic
all:
test: test-stack
423
./test-stack
@echo OK!
test-stack: test-stack.o tester.o stack.o
$(CC) $(CFLAGS) -o $@ $^
test-stack.o: stack.h tester.h
stack.o: stack.h
Note that we don’t provide a convenient shortcut for building test-stack
without running it. That’s because we want to run the test code every single
time.
6.5.3
Stub implementation
Of course, we still can’t compile anything, because we don’t have any implementation. Let’s fix that. To make it easy to write, we will try to add as little as
possible to what we already have in stack.h:
6.5.3.1
stack.c
#include <stdlib.h>
#include "stack.h"
struct stack { int dummy; };
Stack stack_create(void) { return malloc(sizeof(struct stack)); }
void stack_destroy(Stack s) { free(s); }
void stack_push(Stack s, int elem) { ; }
int stack_pop(Stack s) { return STACK_EMPTY; }
int stack_isempty(Stack s) { return 1; }
Will this work? Of course not. There’s hardly any code! But maybe it will
compile if we run make test:
$ make test
gcc -g3 -Wall -ansi -pedantic
gcc -g3 -Wall -ansi -pedantic
gcc -g3 -Wall -ansi -pedantic
gcc -g3 -Wall -ansi -pedantic
./test-stack
test-stack.c:32: TEST FAILED:
test-stack.c:33: TEST FAILED:
test-stack.c:34: TEST FAILED:
test-stack.c:35: TEST FAILED:
test-stack.c:36: TEST FAILED:
test-stack.c:37: TEST FAILED:
-c -o test-stack.o test-stack.c
-c -o tester.o tester.c
-c -o stack.o stack.c
-o test-stack test-stack.o tester.o stack.o
stack_isempty(s) -> 1 but expected 0
stack_pop(s) -> -1 but expected 3
stack_isempty(s) -> 1 but expected 0
stack_pop(s) -> -1 but expected 2
stack_isempty(s) -> 1 but expected 0
stack_pop(s) -> -1 but expected 1
424
test-stack.c:45: TEST FAILED: stack_isempty(s) -> 1 but expected 0
test-stack.c:46: TEST FAILED: stack_pop(s) -> -1 but expected 4
test-stack.c:60: wanted 957 but didn't get it
test-stack.c:74: Aborted (signal 6)
./test-stack: errors 8/17, signals 1, FAILs 1
make[1]: *** [test] Error 8
Hooray! It compiles on the first try! (Well, not really, but let’s pretend it did.)
Unfortunately, it only passes any tests at all by pure dumb luck. But now we
just need to get the code to pass a few more tests.
6.5.4
Bounded-space implementation
Here’s a first attempt at a stack that suffers from some artificial limits. We
retain the structure of the original broken implementation, we just put a few
more lines of code in and format it more expansively.
6.5.4.1
stack.c
#include <stdlib.h>
#include "stack.h"
#define MAX_STACK_SIZE (100)
struct stack {
int top;
int data[MAX_STACK_SIZE];
};
Stack
stack_create(void)
{
struct stack *s;
}
s = malloc(sizeof(*s));
s->top = 0;
return s;
void
stack_destroy(Stack s)
{
free(s);
}
425
void
stack_push(Stack s, int elem)
{
s->data[(s->top)++] = elem;
}
int
stack_pop(Stack s)
{
return s->data[--(s->top)];
}
int
stack_isempty(Stack s)
{
return s->top == 0;
}
Let’s see what happens now:
$ make test
gcc -g3 -Wall -ansi -pedantic
-c -o test-stack.o test-stack.c
gcc -g3 -Wall -ansi -pedantic
-c -o tester.o tester.c
gcc -g3 -Wall -ansi -pedantic
-c -o stack.o stack.c
gcc -g3 -Wall -ansi -pedantic -o test-stack test-stack.o tester.o stack.o
./test-stack
test-stack.c:40: TEST FAILED: stack_isempty(s) -> 0 but expected 1
test-stack.c:41: TEST FAILED: stack_pop(s) -> 409 but expected -1
test-stack.c:47: TEST FAILED: stack_isempty(s) -> 0 but expected 1
test-stack.c:48: TEST FAILED: stack_pop(s) -> 0 but expected -1
test-stack.c:49: TEST FAILED: stack_isempty(s) -> 0 but expected 1
test-stack.c:74: Segmentation fault (signal 11)
test-stack.c:76: TEST FAILED: stack_isempty(s) -> 0 but expected 1
free(): invalid pointer 0x804b830!
./test-stack: errors 6/17, signals 1, FAILs 0
make[1]: *** [test] Error 6
There are still errors, but we get past several initial tests before things blow up.
Looking back at the line numbers in test-stack.c, we see that the first failed
test is the one that checks if the stack is empty after we pop from an empty stack.
The code for stack_isempty looks pretty clean, so what happened? Somewhere
s->top got set to a nonzero value, and the only place this can happen is inside
stack_pop. Aha! There’s no check in stack_pop for an empty stack, so it’s
decrementing s->top past 0. (Exercise: why didn’t the test of stack_pop fail?)
426
6.5.5
First fix
If we’re lucky, fixing this problem will make the later tests happier. Let’s try a
new version of stack_pop. We’ll leave everything else the same.
int
stack_pop(Stack s)
{
if(stack_isempty(s)) {
return STACK_EMPTY;
} else {
return s->data[--(s->top)];
}
}
And now we get:
$ make test
gcc -g3 -Wall -ansi -pedantic
-c -o test-stack.o test-stack.c
gcc -g3 -Wall -ansi -pedantic
-c -o tester.o tester.c
gcc -g3 -Wall -ansi -pedantic
-c -o stack.o stack.c
gcc -g3 -Wall -ansi -pedantic -o test-stack test-stack.o tester.o stack.o
./test-stack
test-stack.c:74: Segmentation fault (signal 11)
test-stack.c:76: TEST FAILED: stack_isempty(s) -> 0 but expected 1
./test-stack: errors 1/17, signals 1, FAILs 0
make[1]: *** [test] Error 1
Which is much nicer. We are still failing the stress test, but that’s not terribly
surprising.
6.5.6
Final version
After some more tinkering, this is what I ended up with. This version uses a
malloc’d data field, and realloc’s it when the stack gets too big.
6.5.6.1
stack.c
#include <stdlib.h>
#include "stack.h"
struct stack {
int top;
int size;
int *data;
};
/* first unused slot in data */
/* number of slots in data */
/* stack contents */
427
#define INITIAL_STACK_SIZE (1)
#define STACK_SIZE_MULTIPLIER (2)
Stack
stack_create(void)
{
struct stack *s;
s = malloc(sizeof(*s));
if(s == 0) return 0;
s->top = 0;
s->size = INITIAL_STACK_SIZE;
s->data = malloc(s->size * sizeof(*(s->data)));
if(s->data == 0) return 0;
}
/* else everything is ok */
return s;
void
stack_destroy(Stack s)
{
free(s->data);
free(s);
}
void
stack_push(Stack s, int elem)
{
if(s->top == s->size) {
/* need more space */
s->size *= STACK_SIZE_MULTIPLIER;
s->data = realloc(s->data, s->size * sizeof(*(s->data)));
if(s->data == 0) {
abort();
/* we have no other way to signal failure :-( */
}
}
/* now there is enough room */
s->data[s->top++] = elem;
}
int
stack_pop(Stack s)
{
428
}
if(stack_isempty(s)) {
return STACK_EMPTY;
} else {
return s->data[--(s->top)];
}
int
stack_isempty(Stack s)
{
return s->top == 0;
}
At last we have a version that passes all tests:
$ make test
gcc -g3 -Wall
gcc -g3 -Wall
gcc -g3 -Wall
gcc -g3 -Wall
./test-stack
OK!
6.5.7
-ansi
-ansi
-ansi
-ansi
-pedantic
-c -o test-stack.o test-stack.c
-pedantic
-c -o tester.o tester.c
-pedantic
-c -o stack.o stack.c
-pedantic -o test-stack test-stack.o tester.o stack.o
Moral
Writing a big program all at once is hard. If you can break the problem down
into little problems, it becomes easier. “Test first” is a strategy not just for
getting a well-tested program, but for giving you something easy to do at each
step— it’s usually not too hard to write one more test, and it’s usually not too
hard to get just one test working. If you can keep taking those small, easy steps,
eventually you will run out of failed tests and have a working program.
6.5.8
/*
*
*
*
*
*
*
*
*
*
*
Appendix: Test macros
Test macros.
Usage:
#include
#include
#include
#include
<setjmp.h>
<stdio.h>
<signal.h>
<unistd.h>
testerInit();
-- Initialize internal data structures.
429
* testerReport(FILE *, "name"); -- Print report.
* testerResult();
-- Returns # of failed tests.
*
* TRY { code } ENDTRY;
*
* Wraps code to catch seg faults, illegal instructions, etc. May not be
* nested.
* Prints a warning if a signal is caught.
* To enforce a maximum time, set alarm before entering.
*
* TEST(expr, expected_value);
*
* Evaluates expr (which should yield an integer value) inside a TRY.
* Prints a warning if evaluating expr causes a fault or returns a value
* not equal to expected_value.
*
* TEST_ASSERT(expr)
*
* Equivalent to TEST(!(expr), 0)
*
* You can also cause your own failures with FAIL:
*
* TRY {
*
x = 1;
*
if(x == 2) FAIL("why is x 2?");
* } ENDTRY;
*
* To limit the time taken by a test, call tester_set_time_limit with
* a new limit in seconds, e.g.
*
* tester_set_time_limit(1);
* TRY { while(1); } ENDTRY;
*
* There is an initial default limit of 10 seconds.
* If you don't want any limit, set the limit to 0.
*
*/
/* global data used by macros */
/* nothing in here should be modified directly */
extern struct tester_global_data {
jmp_buf escape_hatch;
/* jump here on surprise signals */
int escape_hatch_active;
/* true if escape hatch is usable */
int tests;
/* number of tests performed */
int errors;
/* number of tests failed */
int signals;
/* number of signals caught */
430
int expr_value;
int setjmp_return;
int try_failed;
int user_fails;
int time_limit;
} TesterData;
/*
/*
/*
/*
/*
expression value */
return value from setjmp */
true if last try failed */
number of calls to FAIL */
time limit for TRY */
/* set up system; call this before using macros */
void testerInit(void);
/* prints a summary report of all errors to f, prefixed with preamble */
/* If there were no errors, nothing is printed */
void testerReport(FILE *f, const char *preamble);
/* returns number of errors so far. */
int testerResult(void);
/* set a time limit t for TRY, TEST, TEST_ASSERT etc. */
/* After t seconds, an ALARM signal will interrupt the test. */
/* Set t = 0 to have no time limit. */
/* Default time limit is 10 seconds. */
void tester_set_time_limit(int t);
const char *testerStrsignal(int);
/* internal hack; don't use this */
/* gruesome non-syntactic macros */
#define TRY \
TesterData.try_failed = 0; \
alarm(TesterData.time_limit); \
if(((TesterData.setjmp_return = setjmp(TesterData.escape_hatch)) == 0) \
&& (TesterData.escape_hatch_active = 1) /* one = is correct*/)
#define ENDTRY else { \
fprintf(stderr, "%s:%d: %s (signal %d)\n", \
__FILE__, __LINE__, \
testerStrsignal(TesterData.setjmp_return), \
TesterData.setjmp_return); \
TesterData.signals++; \
TesterData.try_failed = 1; \
} \
alarm(0); \
TesterData.escape_hatch_active = 0
/* another atrocity */
#define TEST(expr, expected_value) \
TesterData.tests++; \
TesterData.errors++; /* guilty until proven innocent */ \
431
TRY { TesterData.expr_value = (expr); \
if(TesterData.expr_value != expected_value) { \
fprintf(stderr, "%s:%d: TEST FAILED: %s -> %d but expected %d\n", \
__FILE__, __LINE__, __STRING(expr), \
TesterData.expr_value, expected_value); \
} else { \
TesterData.errors--; \
} \
} \
ENDTRY; \
if(TesterData.try_failed) \
fprintf(stderr, "%s:%d: TEST FAILED: %s caught signal\n", \
__FILE__, __LINE__, __STRING(expr))
#define TEST_ASSERT(expr) TEST((expr) != 0, 1)
#define FAIL(msg) \
(fprintf(stderr, "%s:%d: %s\n", __FILE__, __LINE__, (msg)), \
TesterData.user_fails++, \
TesterData.try_failed = 1)
examples/testHarness/tester.h
#define _GNU_SOURCE
#include
#include
#include
#include
/* get strsignal def */
<stdio.h>
<signal.h>
<string.h>
<setjmp.h>
#include "tester.h"
struct tester_global_data TesterData;
const char *
testerStrsignal(int sig)
{
return strsignal(sig);
}
static void
tester_sighandler(int signal)
{
if(TesterData.escape_hatch_active) {
TesterData.escape_hatch_active = 0;
longjmp(TesterData.escape_hatch, signal);
}
}
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void
testerInit(void)
{
TesterData.escape_hatch_active = 0;
TesterData.tests = 0;
TesterData.errors = 0;
TesterData.signals = 0;
TesterData.user_fails = 0;
}
signal(SIGSEGV, tester_sighandler);
signal(SIGILL, tester_sighandler);
signal(SIGFPE, tester_sighandler);
signal(SIGALRM, tester_sighandler);
signal(SIGBUS, tester_sighandler);
signal(SIGABRT, tester_sighandler);
void
testerReport(FILE *f, const char *preamble)
{
if(TesterData.errors != 0 || TesterData.signals != 0) {
fprintf(f, "%s: errors %d/%d, signals %d, FAILs %d\n",
preamble,
TesterData.errors,
TesterData.tests,
TesterData.signals,
TesterData.user_fails);
}
}
int
testerResult(void)
{
return TesterData.errors;
}
void
tester_set_time_limit(int t)
{
TesterData.time_limit = t;
}
examples/testHarness/tester.c
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6.6
Algorithm design techniques
6.6.1
Basic principles of algorithm design
The fundamental principle of algorithm design was best expressed by the mathematician George Polya: “If there is a problem you can’t solve, then there is
an easier problem you can solve: find it.” For computers, the situation is even
better: if there is any technique to make a problem easier even by a tiny bit, then
you can repeat the technique—possibly millions or even billions of times—until
the problem becomes trivial.
For example, suppose we want to find the maximum element of an array of n
ints, but we are as dumb as bricks, so it doesn’t occur to us to iterate through
the array keeping track of the largest value seen so far. We might instead be
able to solve the problem by observing that the maximum element is either (a)
the last element, or (b) the maximum of the first n − 1 elements, depending on
which is bigger. Figuring out (b) is an easier version of the original problem, so
we are pretty much done once we’ve realized we can split the problem in this
way. Here’s the code:
/* returns maximum of the n elements in a */
int
max_element(int a[], int n)
{
int prefix_max;
assert(n > 0);
}
if(n == 1) {
return a[0];
} else {
prefix_max = max_element(a, n-1);
if(prefix_max < a[n-1]) {
return a[n-1];
} else {
return prefix_max;
}
}
Note that we need a special case for a 1-element array, because the empty prefix
of such an array has no maximum element. We also assert that the array
contains at least one element, just to avoid mischief.
One problem with this algorithm (at least when coding in C) is that the recursion
may get very deep. Fortunately, there is a straightforward way to convert the
recursion to a loop. The idea is that instead of returning a value from the
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recursive call, we put it in a variable that gets used in the next pass through the
loop. The result is
/* returns maximum of the n elements in a */
int
max_element(int a[], int n)
{
int i;
/* this replaces n-1 from the recursive version */
int prefix_max;
assert(n > 0);
prefix_max = a[0];
/* this is the i == 0 case */
for(i = 1; i < n; i++) {
if(prefix_max < a[i]) {
prefix_max = a[i]; /* was return a[n-1] */
}
/* else case becomes prefix_max = prefix_max, a noop */
}
}
/* at the end we have to return a value for real */
return prefix_max;
6.6.2
Specific techniques
Algorithm design often requires both creativity and problem-specific knowledge,
but there are certain common techniques that appear over and over again. The
following classification is adapted from Anany Levitin, Introduction to the Design
& Analysis of Algorithms, Addison-Wesley, 2003.
Brute force Try all possible solutions until you find the right one.
Divide and conquer Split the problem into two or more subproblems, solve
the subproblems recursively, and then combine the solutions.
Decrease and conquer Reduce the problem to a single smaller problem, solve
that problem recursively, and then use that solution to solve the original
problem.
Transform and conquer Either (a) transform the input to a form that makes
the problem easy to solve, or (b) transform the input into the input to
another problem whose solution solves the original problem.
Use space Solve the problem using some auxiliary data structure.
Dynamic programming Construct a table of solutions for increasingly large
subproblems, where each new entry in the table is computed using previous
entries in the table.
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Greedy method Run through your problem one step at a time, keeping track
of the single best solution at each step. Hope sincerely that this will not
lead you to make a seemingly-good choice early with bad consequences
later.
Some of these approaches work better than others—it is the role of algorithm
analysis (and experiments with real computers) to figure out which are likely to
be both correct and efficient in practice. But having all of them in your toolbox
lets you try different possibilities for a given problem.
6.6.3
Example: Finding the maximum
Though this classification is not completely well-defined, and is a bit arbitrary
for some algorithms, it does provide a useful list of things to try in solving a
problem. Here are some examples of applying the different approaches to a
simple problem, the problem of finding the maximum of an array of integers.
Brute force For index i, test if A[i] is greater than or equal to every element
in the array. When you find such an A[i], return it. For this algorithm,
T (n) = n · Θ(n) = Θ(n2 ) if implemented in the most natural way.
Divide and conquer If A has only one element, return it. Otherwise, let
m1 be the maximum of A[1] . . . A[n/2], and let m2 be the maximum of
A[n/2 + 1] . . . A[n]. Return the larger of m1 and m2 . The running time is
given by $T(n) = 2T (n/2) + Θ(1) = Θ(n).
Decrease and conquer If A has only one element, return it. Otherwise, let
m* be the maximum of A[2] . . . A[n]. Return the larger ofPA[0] and m. Now
n
the running time is given by T (n) = T (n − 1) + Θ(1) = i=1 Θ(1) = Θ(n).
Transform and conquer Sort the array, then return A[n]. Using an optimal
comparison-based sort, this takes Θ(n log n) + Θ(1) = Θ(n log n) time. The
advantage of this approach is that you probably don’t have to code up the
sorting routine yourself, since most libraries include sorting.
Use space Insert all elements into a balanced binary search tree, then return
the rightmost element. Cost is Θ(n log n) to do n insertions, plus Θ(log n)
to find the rightmost element, for a total of Θ(n log n). Sorting is equivalent
and probably easier.
Dynamic programming Create an auxiliary array B with indices 1 to n. Set
B[1] = A[1]. As i goes from 2 to n, set B[i] to the larger of B[i − 1] and
A[i], so that B[i] is always the maximum among A[1] . . . A[i]. Return B[n].
Cost: Θ(n). As is often the case, one can reduce the space to O(1) by
throwing away parts of B that we aren’t going to look at again.
Greedy method Let m = A[1]. For each element A[i] in A[2 . . . n], if A[i] > m,
set m to A[i]. Return the final value of m. Cost: Θ(n). This algorithm is
pretty much identical to the previous one.
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6.6.4
Example: Sorting
The sorting problem asks, given as input an array A of n elements in arbitrary
order, to produce as output an array containing the same n elements in nondecreasing order, i.e. with A[i] ≤ A[i + 1] for all i. We can apply each of the
techniques above to this problem and get a sorting algorithm (though some are
not very good).
Brute force For each of the n! permutations of the input, test if it is sorted by
checking A[i] ≤ A[i + 1] for all i. Cost if implemented naively: n! · Θ(n) =
Θ((n + 1)!). This algorithm is known as deterministic monkeysort or
deterministic bogosort. It also has a randomized variant, where the
careful generation of all n! permutations is replaced by shuffling. The
randomized variant is easier to code and runs at about the same speed
as the deterministic variant, but does not guarantee termination if the
shuffling is consistently unlucky.
Divide and conquer Sort A[1 . . . bn/2c and A[bn/2 + 1c . . . n] separately, then
merge the results (which takes Θ(n) time and Θ(n) additional space if
implemented in the most straightforward way). Cost: T (n) = 2T (n/2) +
Θ(n) = Θ(n log n) by the Master Theorem. This method gives mergesort,
one of the fastest general-purpose sorting algorithms. The merge can be
avoided by carefully splitting the array into elements less than and elements
greater than some pivot, then sorting the two resulting piles; this gives
quicksort. The performance of quicksort is often faster than mergesort in
practice, but its worst-case performance (when the pivot is chosen badly)
is just as bad as the result of insertion sort, which we will look at next.
Decrease and conquer Remove A[n], sort the remainder, then insert A[n] in
the appropriate place. This algorithm is called insertion sort. The final
insertion step requires finding the right place (which can be done fairly
quickly if one is clever) but then moving up to n − 1 elements to make
room for A[n]. Total cost is given by T (n) = T (n − 1) + Θ(n) = T (n2 ).
Transform and conquer I’m not aware of any good general transform-andconquer approach to sorting (there are some bad ones), but in some cases
one can transform seemingly general sorting problem (e.g. sorting strings)
into specialized sorting problems that permit faster solutions (e.g. sorting
small integers).
Use space Insert the elements into a balanced binary search tree, then read
them out from left to right. Another version: insert them into a heap.
Both take Θ(n log n) time, but are more complicated to implement than
mergesort or quicksort unless you have binary search tree or heap code
lying around already.
Dynamic programming Insertion sort may be seen as an example of this.
Greedy method Find the smallest element, mark it as used, and output it.
Repeat until no elements are left. The result is selection sort), which runs
in a respectable but suboptimal Θ(n2 ) time.
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6.7
Bit manipulation
Sometimes it is convenient to consider a block of chars as really being a block
of bits. This requires using C’s bit operators to get at individual bits.
Here are some simple macros for extracting a particular bit from a char array,
thought of as a large vector of bits. These assume that the bytes are stored in
little-endian order, which means that the least significant bytes come first (see
Endianness). This may produce odd results if you feed them a char * that has
been converted from a larger integer type.
#define BITS_PER_BYTE (8)
/* extract the n-th bit of x */
#define GET_BIT(x, n) ((((x)[(n) / BITS_PER_BYTE]) & (0x1 << ((n) % BITS_PER_BYTE))) != 0)
/* set the n-th bit of x to 1 */
#define SET_BIT(x, n) ((x)[(n) / BITS_PER_BYTE]) |= (0x1 << ((n) % BITS_PER_BYTE))
/* set the n-th bit of x to 0 */
#define RESET_BIT(x, n) ((x)[(n) / BITS_PER_BYTE]) &= ~(0x1 << ((n) % BITS_PER_BYTE))
If you want to get multiple bits, use the right-shift operator to shift them over
to the right end of the word and then mask with bitwise AND. For example:
#define BITS_PER_BYTE (8)
/* this rather nasty expression constructs an all-ones byte */
#define BYTE_MASK ((1 << BITS_PER_BYTE) - 1)
/* extract the n-th byte from a word */
#define GET_BYTE(x, n) (((x) >> BITS_PER_BYTE * (n)) & BYTE_MASK)
/* extract n bits starting at position i from x */
#define GET_BITS(x, i, j) (((x) >> (i)) & ((1 << n) - 1))
/* another definition of GET_BIT */
#define GET_BIT2(x, n) GET_BITS(x, n, 1)
Many much more sophisticated techniques for doing bit-fiddling can be found at
http://www.jjj.de/bitwizardry/bitwizardrypage.html.
6.8
Persistence
When a C program exits, all of its global variables, local variables, and heapallocated blocks are lost. Its memory is reclaimed by the operating system,
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erased, and handed out to other programs. So what happens if you want to keep
data around for later?
To make this problem concrete, let’s suppose we want to keep track of a hit
counter for web pages. From time to time, the user will run the command
count_hit number where number is an integer value in the range 0 to 99, say.
(A real application would probably be using urls, but let’s keep things as simple
as possible.) We want count_hit to print the number of times the page with the
given number has been hit, i.e. 1 the first time it is called, 2 the next time, etc.
Where can we store the counts so that they will survive to the next execution of
count_hit?
6.8.1
A simple solution using text files
The simplest solution is probably to store the data in a text file. Here’s a
program that reads a file hits, increments the appropriate value, and the writes
out a new version. To reduce the chances that data is lost (say if count_hit
blows up halfway through writing the file), the new values are written to a new
file hit~, which is then renamed to hit, taking the place of the previous version.
#include <stdio.h>
#include <stdlib.h>
#define NUM_COUNTERS (100)
/* number of counters we keep track of */
#define COUNTER_FILE "/tmp/hit" /* where they are stored */
#define NEW_COUNTER_FILE COUNTER_FILE "~" /* note use of constant string concatenation */
int
main(int argc, char **argv)
{
int c;
int i;
int counts[NUM_COUNTERS];
FILE *f;
if(argc < 2) {
fprintf(stderr, "Usage: %s number\n", argv[0]);
exit(1);
}
/* else */
c = atoi(argv[1]);
if(c < 0 || c >= NUM_COUNTERS) {
fprintf(stderr, "Counter %d not in range 0..%d\n", c, NUM_COUNTERS - 1);
exit(2);
}
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f = fopen(COUNTER_FILE, "r");
if(f == 0) {
perror(COUNTER_FILE);
exit(3);
}
/* read them in */
for(i = 0; i < NUM_COUNTERS; i++) {
fscanf(f, "%d", &counts[i]);
}
fclose(f);
printf("%d\n", ++counts[c]);
/* write them back */
f = fopen(NEW_COUNTER_FILE, "w");
for(i = 0; i < NUM_COUNTERS; i++) {
fprintf(f, "%d\n", counts[i]);
}
fclose(f);
rename(NEW_COUNTER_FILE, COUNTER_FILE);
}
return 0;
examples/persistence/textFile.c
If you want to use this, you will need to create an initial file /tmp/hit with
NUM_COUNTERS zeroes in it.
Using a simple text file like this is the easiest way to keep data around, since
you can look at the file with a text editor or other tools if you want to do things
to it. But it means that the program has to parse the file every time it runs. We
can speed things up a little bit (and simplify the code) by storing the values in
binary.
6.8.2
Using a binary file
Here’s a version that stores the data as a binary file of exactly sizeof(int) * NUM_COUNTERS
bytes. It uses the stdio routines fread and fwrite to read and write the file.
These are much faster than the loops in the previous program, since they can
just slap the bytes directly into counts without processing them at all.
The program also supplies and extra flag b to fopen. This is ignored on Unix-like
machines but is needed on Windows machines to tell the operating system that
440
the file contains binary data (such files are stored differently from text files on
Windows).
#include <stdio.h>
#include <stdlib.h>
#define NUM_COUNTERS (100)
/* number of counters we keep track of */
#define COUNTER_FILE "/tmp/hit" /* where they are stored */
#define NEW_COUNTER_FILE COUNTER_FILE "~" /* note use of constant string concatenation */
int
main(int argc, char **argv)
{
int c;
int counts[NUM_COUNTERS];
FILE *f;
if(argc < 2) {
fprintf(stderr, "Usage: %s number\n", argv[0]);
exit(1);
}
/* else */
c = atoi(argv[1]);
if(c < 0 || c >= NUM_COUNTERS) {
fprintf(stderr, "Counter %d not in range 0..%d\n", c, NUM_COUNTERS - 1);
exit(2);
}
f = fopen(COUNTER_FILE, "rb");
if(f == 0) {
perror(COUNTER_FILE);
exit(3);
}
/* read them in */
fread(counts, sizeof(*counts), NUM_COUNTERS, f);
fclose(f);
printf("%d\n", ++counts[c]);
/* write them back */
f = fopen(NEW_COUNTER_FILE, "wb");
fwrite(counts, sizeof(*counts), NUM_COUNTERS, f);
fclose(f);
441
rename(NEW_COUNTER_FILE, COUNTER_FILE);
}
return 0;
examples/persistence/binaryFile.c
Again, you’ll have to initialize /tmp/hit to use this; in this case, you want it to
contain exactly 400 null characters. On a Linux machine you can do this with
the command dd if=/dev/zero of=/tmp/hit bs=400 count=1.
The advantage of using binary files is that reading and writing them is both
simpler and faster. The disadvantages are (a) you can’t look at or update the
binary data with your favorite text editor any more, and (b) the file may no
longer be portable from one machine to another, if the different machines have
different endianness or different values of sizeof(int). The second problem we
can deal with by converting the data to a standard word size and byte order
before storing it, but then we lose some advantages of speed.
6.8.3
A version that updates the file in place
We still may run into speed problems if NUM_COUNTERS is huge. The next program
avoids rewriting the entire file just to update one value inside it. This program
uses the fseek function to position the cursor inside the file. It opens the file
using the "r+b" flag to fopen, which means to open an existing binary file for
reading and writing.
#include <stdio.h>
#include <stdlib.h>
#define NUM_COUNTERS (100)
/* number of counters we keep track of */
#define COUNTER_FILE "/tmp/hit" /* where they are stored */
int
main(int argc, char **argv)
{
int c;
int count;
FILE *f;
if(argc < 2) {
fprintf(stderr, "Usage: %s number\n", argv[0]);
exit(1);
}
/* else */
c = atoi(argv[1]);
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if(c < 0 || c >= NUM_COUNTERS) {
fprintf(stderr, "Counter %d not in range 0..%d\n", c, NUM_COUNTERS - 1);
exit(2);
}
f = fopen(COUNTER_FILE, "r+b");
if(f == 0) {
perror(COUNTER_FILE);
exit(3);
}
/* read counter */
fseek(f, sizeof(int) * c, SEEK_SET);
fread(&count, sizeof(int), 1, f);
printf("%d\n", ++count);
/* write it back */
fseek(f, sizeof(int) * c, SEEK_SET);
fwrite(&count, sizeof(int), 1, f);
fclose(f);
}
return 0;
examples/persistence/binaryFileFseek.c
Note that this program is not only shorter than the last one, but it also avoids
allocating the counts array. It also is less likely to run into trouble with running
out of space during writing. If we ignore issues of concurrency, this is the best
we can probably do with just stdio.
6.8.4
An even better version using mmap
We can do even better using the mmap routine, available in all POSIX-compliant
C libraries. POSIX, which is short for Portable Standard Unix, is supported by
essentially all Unix-like operating systems and NT-based versions of Microsoft
Windows. The mmap routine tells the operating system to “map” a file in the
filesystem to a region in the process’s address space. Reading bytes from this
region will read from the file; writing bytes to this region will write to the file
(although perhaps not immediately). Even better, if more than one process
calls mmap on the same file at once, they will share the memory region, so that
updates made by one process will be seen immediately by the others (with some
caveats having to do with how concurrent access to memory actually works on
real machines).
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Here is the program using mmap:
#include
#include
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<sys/types.h>
<sys/stat.h>
<fcntl.h>
<sys/mman.h>
/* For mmap.
I think mman is short for "memory management." */
#define NUM_COUNTERS (100)
/* number of counters we keep track of */
#define COUNTER_FILE "/tmp/hit" /* where they are stored */
#define NEW_COUNTER_FILE COUNTER_FILE "~" /* note use of constant string concatenation */
int
main(int argc, char **argv)
{
int c;
int *counts;
int fd;
if(argc < 2) {
fprintf(stderr, "Usage: %s number\n", argv[0]);
exit(1);
}
/* else */
c = atoi(argv[1]);
if(c < 0 || c >= NUM_COUNTERS) {
fprintf(stderr, "Counter %d not in range 0..%d\n", c, NUM_COUNTERS - 1);
exit(2);
}
/* open and map the file */
fd = open(COUNTER_FILE, O_RDWR);
if(fd < 0) {
perror(COUNTER_FILE);
exit(3);
}
counts = mmap(0, sizeof(*counts) * NUM_COUNTERS, PROT_READ|PROT_WRITE, MAP_SHARED, fd, 0
if(counts == 0) {
perror(COUNTER_FILE);
exit(4);
}
printf("%d\n", ++counts[c]);
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/* unmap the region and close the file just to be safe */
munmap(counts, sizeof(*counts) * NUM_COUNTERS);
close(fd);
}
return 0;
examples/persistence/binaryFileMmap.c
Now the code for actually incrementing counts[c] and writing it to the file
is trivial. Unfortunately, we have left stdio behind, and have to deal with
low-level POSIX calls like open and close to get at the file. Still, this may be
the most efficient version we can do, and becomes even better if we plan to do
many updates to the same file, since we can just keep the file open.
6.8.5
Concurrency and fault-tolerance issues: ACIDity
All of the solutions described so far can fail if you run two copies of count_hits
simultaneously. The mmap solution is probably the least vulnerable to failures,
as the worst that can happen is that some update is lost if the same locations is
updated at exactly the same time. The other solutions can fail more spectacularly;
simultaneous writes to /tmp/hit~ in the simple text file version, for example,
can produce a wide variety of forms of file corruption. For a simple web page hit
counter, this may not be a problem. If you are writing a back-end for a bank,
you probably want something less vulnerable.
Database writers aim for a property called ACIDity from the acronym ACID
= Atomicity, Consistency, Isolation, and Durability. These are defined
for a system in which the database is accessed via transactions consisting of
one or more operations. An example of a transaction might be ++counts[c],
which we can think of as consisting of two operations: reading counts[c], and
writing back counts[c]+1.
Atomicity means that either every operation in a transaction is performed or
none is. In practice, this means if the transaction fails any partial progress must
be undone.
Consistency means that at the end of a transaction the database is in a “consistent”
state. This may just mean that no data has been corrupted (e.g. in the text
data file we have exactly 100 lines and they’re all integer counts), or it may also
extend to integrity constraints enforce by the database (e.g. in a database of
airline flights, the fact that flight 2937 lands at HVN at 22:34 on 12/17 implies
that flight 2937 exists, has an assigned pilot, etc.).
Isolation says that two concurrent transactions can’t detect each other; the
partial progress of one transaction is not visible to others until the transaction
commits.
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Durability means that the results of any committed transaction are permanent.
In practice this means there is enough information physically written to a disk
to reconstruct the transaction before the transaction finished.
How can we enforce these requirements for our hit counter? Atomicity is not
hard: if I stop a transaction after a read but before the write, no one will be the
wiser (although there is a possible problem if only half of my write succeeds).
Consistency is enforced by the fseek and mmap solutions, since they can’t change
the structure of the file. Isolation is not provided by any of our solutions,
and would require some sort of locking (e.g. using flock) to make sure that
only one program uses the file at a time. Durability is enforced by not having
count_hits return until the fclose or close operation has succeeded (although
full durability would require running fsync or msync to actually guarantee data
was written to disk).
Though it would be possible to provide full ACIDity with enough work, this is
a situation where using an existing well-debugged tool beats writing our own.
Depending on what we are allowed to do to the machine our program is running
on, we have many options for getting much better handling of concurrency. Some
standard tools we could use are:
• gdbm. This is a minimal hash-table-on-disk library that uses simplistic
locking to get isolation. The advantage of this system is that it’s probably
already installed on any Linux machine. The disadvantage is that it doesn’t
provide much functionality beyond basic transactions.
• Berkeley DB is a fancier hash-table-on-disk library that provides full
ACIDity but not much else. There is a good chance that some version of
this is also installed by default on any Linux or BSD machine you run into.
• Various toy databases like SQLite or MySQL provide tools that look very
much like serious databases with easy installation and little overhead.
These are probably the solutions most people choose, especially since
MySQL is integrated tightly with PHP and other Web-based scription
languages. Such a solution also allows other programs to access the table
without having to know a lot of details about how it is stored, because the
SQL query language hides the underlying storage format.
• Production-quality databases like PostgreSQL, SQL Server, or Oracle
provide very high levels of robustness and concurrency at the cost of
requiring non-trivial management and possibly large licensing fees. This is
what you pick if you really are running a bank.
7
What next?
Congratulations! You now know everything there is to know about programming
in C. Now what do you do?
My recommendation would be the following: learn C++, since you know 75% of
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it already, and you will be able to escape from some (but not all) of the annoying
limitations of C. And learn a scripting language you can be comfortable with,
for writing programs quickly where performance isn’t the main requirement.
7.1
What’s wrong with C
In this class, we had to work around fundamental limitations in C on several
occasions.
C doesn’t have a garbage collector Many modern program languages will
detect and free unreachable data for you automatically. C doesn’t, so the
programmer has to spend a lot of time worrying about when and by whom
data allocated with malloc will be passed to free. Not only does this
create many possibilities for error, but it also means that certain kinds
of data structures in which a single component of the data structure is
pointed to by an unpredictable number of other components are difficult
to write in C, since it’s hard to tell when it is safe to free a component.
Garbage-collected languages avoid all of these problems at a slight cost
in performance. Though there exists a garbage collector for C/C++
http://www.hboehm.info/gc/, it isn’t 100% portable and may not work as
well as a built-in collector. C doesn’t support any kind of polymorphism
Polymorphism is when a function can work on more than one data
type. The closest C can do is either parameterized macros (see
MacrosMacros.html)), heavy use of void * and function pointers as in
qsort, or various nasty hacks where code is automatically generated
with type names filled in from a base template (see MacrosMacros.html)
again). Most modern programming languages have some sort of support
for polymorphism, allowing you to write, for example, a generic sorting
routine without resorting to void *-like departures from the type system.
C doesn’t have exceptions
Exceptions are a mechanism for doing non-standard returns from a function
when something blows up, which get caught using an “exception handler”
that is often separate from the code handling the normal return values
and which can often be used to catch exceptions from a variety of sources.
Instead, C requires function writers to invent and document an ad-hoc
protocol for indicating bad outcomes for every function they write, and
requires function users to remember to test for bad return values. Most
programmers are too lazy to do this all the time, leading to undetected
run-time errors. Most modern programming languages fix this. C doesn’t
support object-oriented programming very well
“Object-oriented” is a buzzword with many possible meanings (but see
http://c2.com/cgi/wiki?HeInventedTheTerm). However, at minimum it
means that in addition to supporting polymorphism (described above),
your language should support strong encapsulation (controlling who can
get at the internal data of an object) and inheritance (allowing one abstract
447
data type to be defined by extending another). You can fake most of these
things in C if you try hard enough (for example, using function pointers),
but it is always possible to muck around with internal bits of things just
because of the unlimited control C gives you over the environment. This
can quickly become dangerous in large software projects. C provides only
limited support for avoiding namespace collisions
In a large C program, it’s impossible to guarantee that my
eat_leftovers function exported from leftovers.c doesn’t conflict with your eat_leftovers function in cannibalism.c. A mediocre
solution is to use longer names: leftovers_eat_leftovers vs
cannibalism_eat_leftovers, and one can also play games with
function pointers and global struct variables to allow something like
leftovers.eat_leftovers vs cannibalism.eat_leftovers.
Most
modern programming languages provide an explicit package or namespace
mechanism to allow the programmer to control who sees what names
where.
7.2
What C++ fixes
On the above list, C++ fixes everything except the missing garbage collector.
If you want to learn C++, you should get a copy of The C++ Programming
Language, by Bjarne Stroustrup, which is the definitive reference manual. But
you can get a taste of it from several on-line tutorials:
• C++ tutorial for C users, by Eric Brasseur. Exactly what it says. Introduces C++ features not found in C in order of increasing complexity.
• Some other on-line tutorials that assume little or no prior programming
experience:
– http://www.cplusplus.com/doc/tutorial/
– http://www.cprogramming.com/tutorial.html
7.3
Other C-like languages
C syntax has become the default for new programming languages targeted at a
general audience. Some noteworthy examples of C-like languages are Java (used
in Android), Objective-C (used in OSX and iOS), and C# (used in Windows).
Each of these fix some of the misfeatures of C (including the lack of a garbage
collector and bounds checks on arrays) while retaining much of the flavor of
C. Which to choose probably depends on what platform you are interested in
developing for.
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7.4
Scripting languages
Much current programming is done in so-called scripting languages like
Python, Perl, PHP, JavaScript, Visual Basic, Tcl, etc. These are generally
interpreted languages similar to Lisp or Scheme under the hood, with dynamic
typing (type information is carried along with values, so type errors are detected
only at runtime but polymorphism is provided automatically), garbage collectors, and support for many advanced programming features like objects and
anonymous functions. What distinguishes scripting languages from the Lisp-like
languages is that the syntax is generally more accessible to newcomers and
the language runtime usually comes with very large libraries providing built-in
tools for doing practical programming tasks like parsing odd input formats
and interfacing to databases and network services. The result is that common
programming tasks can be implemented using very few lines of code, at a cost
in performance that ranges from slight to horrendous depending on what you
are doing.
Let’s look at an example in two common scripting languages, Perl and Python.
Here are some solutions to an old assignment, which find all the palindromes on
stdin and report the first non-matching character for any non-palindrome.
The original C version looks like this:
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
/* Palindrome detector.
*
* For each line of the input, prints PALINDROME if it is a palindrome
* or the index of the first non-matching character otherwise.
*
* Note: does not handle lines containing nulls.
*/
/* read a line of text from stdin
* and return it (without terminating newline) as a freshly-malloc'd block.
* Caller is responsible for freeing this block.
* Returns 0 on error or EOF.
*/
char *
getLine(void)
{
char *line;
/* line buffer */
int n;
/* characters read */
int size;
/* size of line buffer */
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int c;
size = 1;
line = malloc(size);
if(line == 0) return 0;
n = 0;
while((c = getchar()) != '\n' && c != EOF) {
while(n >= size - 1) {
size *= 2;
line = realloc(line, size);
if(line == 0) return 0;
}
line[n++] = c;
}
}
if(c == EOF && n == 0) {
/* got nothing */
free(line);
return 0;
} else {
line[n++] = '\0';
return line;
}
#define IS_PALINDROME (-1)
/* returns IS_PALINDROME if s is a palindrome,
* or index of first unmatched character otherwise. */
int
testPalindrome(const char *s)
{
int n;
/* length of s */
int i;
n = strlen(s);
/* we only have to check up to floor(n/2) */
for(i = 0; i < n/2; i++) {
if(s[i] != s[n-1-i]) {
return i;
}
}
/* else */
450
}
return IS_PALINDROME;
int
main(int argc, char **argv)
{
char *line;
int mismatch;
while((line = getLine()) != 0) {
mismatch = testPalindrome(line);
if(mismatch == IS_PALINDROME) {
puts("PALINDROME");
} else {
printf("%d\n", mismatch);
}
}
}
free(line);
return 0;
examples/scripting/palindrome.c
This version is written in Perl (http://www.perl.org):
#!/usr/bin/perl
# For each line in stdin, print PALINDROME if it is a palindrome, or index of
# the first non-matching character otherwise.
while(<>) {
chomp;
# remove trailing newline
if($_ eq reverse $_) {
print "PALINDROME\n";
} else {
for $i (0..length($_) - 1) {
if(substr($_, $i, 1) ne substr($_, length($_) - $i - 1, 1)) {
print $i, "\n";
last;
}
}
}
}
451
examples/scripting/palindrome.pl
The things to notice about Perl is that the syntax is deliberately very close to
C (with some idiosyncratic extensions like putting $ on the front of all variable
names), and that common tasks like reading all input lines get hidden inside
default constructions like while(<>) and the $_ variable that functions with no
arguments like chomp operate on by default. This can allow for very compact
but sometimes very incomprehensible code.
Here’s a version in Python (https://www.python.org/):
#!/usr/bin/python
"""For each line in stdin, print PALINDROME if it is a palindrome, or index of
the first non-matching character otherwise."""
import sys
for line in sys.stdin:
line = line.rstrip('\n')
# remove trailing newline
if line == line[::-1]:
print("PALINDROME")
else:
mismatches = [ i for i in range(len(line)) if line[i] != line[-(i+1)] ]
print(min(mismatches))
examples/scripting/palindrome.py
Here the syntax is a little more alien if you are used to C: Python doesn’t use
curly braces for block structure, using indentation instead. The code above
uses some other odd features of the language, such as the ability to take “slices”
of sequence variables like strings (the expression line[::-1] means “take all
elements of line starting from the obvious default starting point (the empty
string before the first colon) to the obvious default ending point (the empty
string before the second colon) stepping backwards one character at a time
(the -1)), a feature the language adopted from array-processing languages like
MatLab; and the ability to do list comprehensions (the large expression assigned
to mismatches), a feature that Python adopted from Haskell and that Haskell
adopted from set theory.
What these gain in short code length they lose in speed; run times on
/usr/share/dict/words in the Zoo are
C
Perl
Python
0.107s
0.580s
2.052s
Note that for Perl and Python some of the cost is the time to start the interpreter
452
and parse the script, but factors of 10–100 are not unusual slowdowns when
moving from C to a scripting language. The selling point of these languages
is that in many applications run time is not as critical as ease and speed of
implementation.
As an even shorter example, if you just want to print all the palindromes in a
file, you can do that from the command line in one line of Perl, e.g:
$ perl -ne 'chomp; print $_, "\n" if($_ eq reverse $_)' < /usr/share/dict/words
8
Assignments
8.1
8.1.1
Assignment 1, due Thursday 2015-01-29, at 11:00pm
Bureaucratic part
Make sure that you sign up for an account on the Zoo at http://zoo.cs.yale.edu/
accounts.html. If you already have an account, you still need to check the CPSC
223 box so that you can turn in assignments. It’s best to do this as soon as
possible.
You do not need to develop your solution on the Zoo, but you will need to turn
it in there, and it will be tested using the compiler on the Zoo.
8.1.2
A rotten cipher
For this assignment, you are to implement an encoder for a polyalphabetic
substitution cipher vaguely inspired by the Enigma machines used by Germany
during World War 2. Unlike the Enigma machine, this cipher doesn’t provide
much security, so you can probably tell your non-US-national friends about it
without violating US export control laws.26
Each letter 'A' through 'Z' or 'a' through 'z' is encrypted by shifting it some
number of positions forward in the alphabet, wrapping around at the end.
The number of positions is determined by an offset that changes over time. The
initial shift is 17 positions. After encoding an uppercase letter, the shift is
increased by 5. After encoding a lowercase letter, the shift is increased by 3. To
avoid overflow on long texts, it’s probably a good idea to store the offset modulo
26.
An uppercase letter is always encoded by an uppercase letter, and a lowercase
letter is always encoded by a lowercase letter. All other characters are passed
through intact.
26 Not
intended as legal advice.
453
Below is an example of encoding a famous Napoleonic palindrome using this
cipher:
Plaintext:
Offset:
Ciphertext:
8.1.3
A b l e
17 22 25 2
R x k g
w
5
b
a
8
i
s I e r e I
11 14 19 22 25 2
d W x n d K
s
7
z
a w E l b a
10 13 16 21 24 1
k j U g z b
Your task
For this assignment, you are to write a program encode.c that takes a plaintext
from stdin, encodes it using the above algorithm, and writes the result to
stdout.
For example, given the input
"Stop, thief!" cried Tom, arrestingly.
"You'll never take me alive!" replied the criminal, funereally.
Your program should output
"Jpnr, yptsw!" woihj Ccd, uorhycucygw.
"Zud'xa fztfv akxu fa znndp!" fvjiihj ctt umgnmuky, vnjdtjiwzp.
8.1.4
Hints
• You should assume that you are using the standard Latin 26-letter alphabet.
• You may assume that the characters 'A' through 'Z' and 'a' through 'z'
are represented using continuous ranges of integers, so that the expression
c - 'A' gives the position of c in the alphabet, provided c is an uppercase
character, and counting A as 0. This means that your program will not
be portable to machines that use EBCDIC or some other exotic character
representation.
• To test if a character is uppercase or lowercase, one option would be to put
#include <ctype.h> in your program and use the isupper and islower
macros. Note that these may behave oddly if you have set a locale that
uses a different alphabet. It may be safer to make your own tests.
8.1.5
Testing your assignment
Sample inputs and outputs can be found in /c/cs223/Hwk1/testFiles on the
Zoo. Note that some of these files contain non-printing characters that may
have odd effects if you send them to your screen. The safest way to test if your
program produces the same output as the sample output is probably to use cmp,
for example:
454
$ ./encode < test.in > tmp
$ cmp tmp test.out
If tmp and test.out contain the same characters, cmp will say nothing. Otherwise it will tell you the first position where they differ.
If you want to see what characters are in a binary file, trying using od -t x1z,
as in
$ echo hi > file
$ cat file
hi
$ od -t x1z file
0000000 68 69 0a
0000003
8.1.6
>hi.<
Submitting your assignment
Submit your assignment using the command:
/c/cs223/bin/submit 1 encode.c
You can test that your program compiles (and passes a few basic tests) using
the command:
/c/cs223/bin/testit 1 encode
This runs the test script in /c/cs223/Hwk1/test.encode on your submitted
assignment. You can also run this script by hand to test the version of encode.c
in your current working directory.
The unsympathetic robo-grading script used to grade this assignment may or
may not use the same tests as this command, so you should make sure your
program works on other inputs as well. You may also want to look at the style
grading checklist to see that you haven’t committed any gross atrocities against
readability, in case a human being should happen to look at your code.
You can submit your assignment more than once, but any late penalties will be
assessed based on the last submission. For more details about the submit script
and its capabilities, see here.
8.1.7
Sample solution
/*
* Encode text on stdin by alphabet rotation with shifting offset.
*
* Initially, each character 'A'..'Z' or 'a'..'z' is rotated 17 positions.
*
* After encoding an uppercase letter, the offset is increased by 5 (mod 26).
455
*
* After encoding a lowercase letter, the offset is increased by 3 (mod 26).
*
* These parameters are set using the INITIAL_OFFSET, UPPERCASE_STEP, and LOWERCASE_STEP
* constants defined below.
*
*/
#include <stdio.h>
#define INITIAL_OFFSET (17)
#define UPPERCASE_STEP (5)
#define LOWERCASE_STEP (3)
#define MODULUS ('z' - 'a' + 1)
int
main(int argc, char **argv)
{
int offset = INITIAL_OFFSET;
int c;
while((c = getchar()) != EOF) {
if(('a' <= c) && (c <= 'z')) {
putchar(((c - 'a') + offset) % MODULUS + 'a');
offset = (offset + LOWERCASE_STEP) % MODULUS;
} else if(('A' <= c) && (c <= 'Z')) {
putchar(((c - 'A') + offset) % MODULUS + 'A');
offset = (offset + UPPERCASE_STEP) % MODULUS;
} else {
putchar(c);
}
}
}
return 0;
examples/2015/hw/1/encode.c
8.2
8.2.1
Assignment 2, due Wednesday 2015-02-04, at 11:00pm
Opening a safe
The Hollywood Hackable Safe Company has announced a new line of
electronically-controlled safes with a C API to permit maximum opening
456
speed while still protecting your valuable loot. This interface is defined
in the following file, which can also be found on the Zoo in the directory
/c/cs223/Hwk2/sourceFiles:
/*
* API for safes made by the
* Hollywood Hackable Safe Company LLC.
*/
typedef struct safe Safe;
/* opaque data type for a safe */
/*
* Returns the number of tumblers on a safe.
* If this is n, the possible tumbler indices will be 0 through n-1.
* */
int numTumblers(Safe *s);
/*
* Returns the number of positions of each tumbler.
* If this is n, the possible tumbler positions will be 0 through n-1.
*/
int numPositions(Safe *s);
/* Return codes for tryCombination */
#define SAFE_BAD_COMBINATION (-1)
#define SAFE_SELF_DESTRUCTED (-2)
/*
* Try a combination.
*
* This should be an array of numTumbler(s) ints.
*
* Returns contents of safe (a non-negative int) if combination is correct
* and safe has not yet self-destructed.
*
* Returns SAFE_BAD_COMBINATION if combination is incorrect
* and safe has not yet self-destructed.
*
* Returns SAFE_SELF_DESTRUCTED if safe has self-destructed.
*
* Note: may modify combination.
*/
int tryCombination(Safe *s, int *combination);
examples/2015/hw/2/safe.h
The noteworthy function in this API is tryCombination, which takes a pointer
457
to a safe and an array of ints representing the combination, and returns either
the contents of the safe (an int), the special code SAFE_BAD_COMBINATION if the
combination is incorrect, or the special code SAFE_SELF_DESTRUCTED if the safe
blew up after seeing too many bad combinations. Note that tryCombination
does not declare its second argument to be const and may not leave it intact.
The additional functions allow you to obtain important information about the
safe, like how many tumblers it has and what values these tumblers can be set
to. The behavior of a safe given a combination with the wrong number of values
or values outside the permitted range is undefined.
Your task is to write a function openSafe that will open a safe, if possible, by
trying all possible combinations. Note that if the safe self-destructs before you
can try all the possibilities, this task may not in fact be possible. Your openSafe
function should return SAFE_SELF_DESTRUCTED in this case. Your function
should be defined in a file openSafe.c and should match the declaration in this
file:
/* Include safe.h before this file to get the definition of Safe. */
/*
* Open a safe and return the value returned by tryCombination,
* or SAFE_SELF_DESTRUCTED if the safe self-destructed.
*/
int openSafe(Safe *s);
examples/2015/hw/2/openSafe.h
It is recommended that you put the lines below in your openSafe.c file to ensure
consistency with these declarations:
#include "safe.h"
#include "openSafe.h"
You may put additional functions in openSafe.c if that would be helpful. You
should declare these static to avoid the possibility of namespace conflicts.
In addition to safe.h and openSafe.h, /c/cs223/Hwk2/sourceFiles also contains a main.c file that can be compiled together with openSafe.c to generate a
program that can be called from the command line. This program generates a
safe with a pseudorandom combination based on parameters specified on the
command line, runs your openSafe routine on it, and prints the value that
openSafe returns. You should not rely on your function being tested with this
particular program.
8.2.2
Submitting your assignment
Submit your assignment as usual with
/c/cs223/bin/submit 2 openSafe.c
458
You do not need to submit any other files (and the test script will ignore them
if you do).
You can test that your program compiles and passes a few basic tests with the
command
/c/cs223/bin/testit 2 openSafe
This runs the test script in /c/cs223/Hwk2/test.openSafe on your submitted assignment. You can also run this script by hand to test the version of
openSafe.c in your current working directory.
8.2.3
Valgrind
You may need to allocate storage using malloc to complete this assignment. If
you do so, you should make sure that you call free on any block you allocate
inside your openSafe function before the function returns. The test.openSafe
script attempts to detect storage leaks or other problems resulting from misuse of
these routines by running your program with valgrind. You can also use valgrind
yourself to track down the source of errors, particularly if you remember to
compile with debugging info turned on using the -g3 option to gcc. The script
/c/cs223/bin/vg gives a shortcut for running valgrind with some of the more
useful options.
8.2.4
Sample solution
#include <stdlib.h>
#include <assert.h>
#include "safe.h"
#include "openSafe.h"
/* set combination to all zeros */
static void
zeroCombination(int n, int *combination)
{
int i;
}
for(i = 0; i < n; i++) {
combination[i] = 0;
}
/* non-destructive version of tryCombination */
static int
nondestructiveTryCombination(Safe *s, const int *combination)
459
{
int
int
int
int
*copy;
result;
n;
i;
/* duplicate of combination */
/* result of tryCombination */
/* number of tumblers */
n = numTumblers(s);
copy = (int *) malloc(sizeof(int) * n);
for(i = 0; i < n; i++) {
copy[i] = combination[i];
}
result = tryCombination(s, copy);
free(copy);
}
return result;
/* update combination to next value */
static void
nextCombination(int n, int base, int *combination)
{
int i;
}
/* we are essentially incrementing an n-digit number in given base */
/* this means setting any digit that overflows to 0 and continuing */
/* until we get a digit we can increment without carrying */
for(i = 0; i < n && ++(combination[i]) >= base; i++) {
combination[i] = 0;
}
int
openSafe(Safe *s)
{
int *combination;
int n;
int base;
int result;
/*
/*
/*
/*
counter for combinations */
number of tumblers */
number of positions */
result of tryCombination */
/* allocate space */
n = numTumblers(s);
460
base = numPositions(s);
combination = malloc(sizeof(int) * n);
assert(combination);
for(zeroCombination(n, combination);
(result = nondestructiveTryCombination(s, combination)) == SAFE_BAD_COMBINATION;
nextCombination(n, base, combination));
}
free(combination);
return result;
examples/2015/hw/2/openSafe.c
8.3
8.3.1
Assignment 3, due Wednesday 2015-02-11, at 11:00pm
Quadratic letter sequences
Given a string s, an quadratic letter sequence for s is defined by giving
non-negative integer coefficients c0 , c1 , c2 , where at least one of c1 and c2 is not
zero, and computing the sequence of letters s[c0 + c1 · i + c2 · i2 ] for i = 0, 1, 2, . . ..
These can be used to hide secret message inside larger texts. For example, the
famous Napoleonic palindrome “Able was I ere I saw Elba” hides the word “bIb”
at positions 1, 9, and 23, which are generated by c0 = 1, c1 = 5 and c2 = 3:
i
c0 + c1 i + c2 i2
0
1
2
1=1+5·0+3·0
9 = 1 + 5 · 1 + 3 · 12
23 = 1 + 5 · 2 + 3 · 22
Similarly, we can use quadratic letter sequences to reveal secret messages hidden
in the lyrics of K-pop songs:
$ ./qls hail satan < gangnam-style-excerpt.txt
470 3 5 hail
14 10 30 satan
14 56 7 satan
or even examine Act 1 of The Tempest to help resolve the Shakespeare authorship
question:27
27 Stratfordians, Oxfordians, and other conspiracy theorists might object that these results
depend critically on the precise formatting of the text. We counter this objection by observing
that we used the Project Gutenberg e-text of The Tempest, which, while not necessarily the
461
$ ./qls "Bacon" "de Vere" "Marlowe" "Stanley" "that Stratford dude" < tempest-act-one.txt
120 387 777 Bacon
120 542 906 Bacon
120 851 850 Bacon
120 1592 726 Bacon
120 1607 472 Bacon
120 2461 95 Bacon
120 2729 50 Bacon
120 3225 215 Bacon
120 3420 284 Bacon
120 4223 330 Bacon
120 4534 76 Bacon
120 5803 29 Bacon
143 46 161 Bacon
143 268 727 Bacon
143 684 1434 Bacon
[... 280 more lines of Bacon omitted ...]
19959 1178 87 Bacon
5949 239 465 Marlowe
8.3.2
Your task
Write a program qls.c that takes a text on stdin and searches for quadratic
letter sequences that start with the strings given in argv. Your program should
output all such quadratic letter sequences that it finds, using the format
printf("%d %d %d %s\n", [...]);
where [...] should be replaced by appropriate expressions to give c0 , c1 , c2 ,
and the string found.
If a string appears more than once at the start of a quadratic letter sequence,
your program should print all occurrences. The order your output lines appear
in is not important, since the test script sorts them into a canonical order. Do
whatever is convenient.
Your program should be reasonably efficient, but you do not need to get carried
away looking for a sophisticated algorithm for this problem. Simply testing all
plausible combinations of coefficients should be enough.
Because neither K-pop songs nor Elizabethan plays use null characters, you may
assume that no null characters appear in your input.
most favored by academic Shakespeare scholars, is the easiest version to obtain on-line. We
consider it further evidence of Sir Francis Bacon’s genius that not only was he able to subtly
encode his name throughout his many brilliant plays, but he was even able to anticipate the
effects of modern spelling and punctuation on this encoding.
462
You may also assume that any search strings will contain at least two characters,
in order to keep the number of outputs finite.
8.3.3
Submitting your assignment
Submit your assignment as usual with
/c/cs223/bin/submit 3 qls.c
You can run some basic tests on your submitted solution with
/c/cs223/bin/testit 3 qls
The test program is also available as /c/cs223/Hwk3/test.qls. Sample inputs
and outputs can be found in /c/cs223/Hwk3/testFiles. The title of each file
contains the test strings used, separated by - characters. Before comparing the
output of your program to the output files, you may find it helpful to run it
through sort, e.g.
./qls hail satan < hail-satan.in | sort > test.out
diff test.out hail-satan.out
8.3.4
Sample solution
/*
* Search for quadratic letter sequences starting with words from argv on stdin.
*
* A quadratic letter sequence of length n in s is a sequence of characters
*
*
s[c0 + c1*i + c2*i*i]
*
* where c0, c1, c2 are all >= 0, at least one of c1 and c2 is > 0,
* and i ranges over 0, 1, 2, ..., n-1.
*
* For each QLS found, prints c0, c1, c2, and the target string to stdout.
*/
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<string.h>
<assert.h>
#define NUM_COEFFICIENTS (3)
/* how many coefficients to pass around */
/*
* Return true iff we get a match in s for t with coefficients c
*
463
* Behavior is undefined if coefficients would send us off the end of s.
*/
static int
qlsMatch(const char *s, const char *t, int c[NUM_COEFFICIENTS])
{
int i;
for(i = 0; t[i] != '\0'; i++) {
if(s[c[0] + c[1] * i + c[2] * i * i] != t[i]) {
/* no match */
return 0;
}
}
}
return 1;
/*
* Search for quadratic letter sequences in s starting with t
* and print results to stdout.
*/
static void
qlsSearch(const char *s, const char *t)
{
int c[NUM_COEFFICIENTS]; /* coefficients */
int lenS; /* length of s */
int lenT; /* length of t */
int maxI; /* maximum value for i (this is lenT-1) */
lenS = strlen(s);
lenT = strlen(t);
maxI = lenT-1;
/* try all possible c[0] that will let us finish before lenS */
for(c[0] = 0; c[0] + maxI < lenS; c[0]++) {
/* if s[c[0]] isn't right, c[1] and c[2] can't fix it */
if(s[c[0]] == t[0]) {
/* try all feasible c[1] */
for(c[1] = 0; c[0] + c[1] * maxI < lenS; c[1]++) {
/* try all feasible c[2], but start at 1 if c[1] == 0 */
for(c[2] = (c[1] == 0); c[0] + c[1] * maxI + c[2] * maxI * maxI < lenS; c[2]
/* now see if we get a match */
if(qlsMatch(s, t, c)) {
printf("%d %d %d %s\n", c[0], c[1], c[2], t);
}
}
464
}
}
}
}
/* used internally by getContents; initial size of buffer */
#define INITIAL_BUFFER_SIZE (16)
/*
* Return a single string holding all characters from stdin.
*
* This is malloc'd data that the caller should eventually free.
*/
static char *
getContents(void)
{
size_t size;
size_t len;
char *text;
int c;
size = INITIAL_BUFFER_SIZE;
len = 0;
text = malloc(size);
assert(text);
while((c = getchar()) != EOF) {
/* grow the buffer if full */
if(len >= size) {
size *= 2;
text = realloc(text, size);
assert(text);
}
}
text[len++] = c;
/* cleanup */
text = realloc(text, len+1);
assert(text);
text[len] = '\0';
}
return text;
465
int
main(int argc, char **argv)
{
int i;
char *s;
s = getContents();
for(i = 1; i < argc; i++) {
qlsSearch(s, argv[i]);
}
free(s);
return 0;
}
examples/2015/hw/3/qls.c
8.4
8.4.1
Assignment 4, due Wednesday 2015-02-18, at 11:00pm
An ASCII art compositor
For this assignment you are to write a program that takes from stdin a sequence
of instructions for pasting ASCII art pictures together, reads those pictures from
files, and writes the combined picture to stdout.
Each instruction is of the form row column filename, suitable for reading
with scanf("%d %d %s", &row, &col, filename);, where row and col are
declared as ints and filename is a suitably large buffer of chars. Such an
instruction means to paste the contents of file filename into the picture with
each character shifted row rows down and column columns to the right of its
position in file filename. When pasting an image, all characters other than space
(' ', or ASCII code 32) overwrite any characters from earlier files at the same
position. Spaces should be treated as transparent, having no effect on the final
image.
For example, suppose that the current directory contains these files:
# # #
\==========/
\......../
examples/2015/hw/4/ship
/\
466
/vv\
/vvvv\
||
examples/2015/hw/4/tree
* * *
____|_|_|_____
|_____________|
|___HAPPY_____|
|__BIRTHDAY___|
|_____________|
examples/2015/hw/4/cake
Then this is what we should get from executing the command:
$ echo "1 1 ship 3 5 ship 3 19 tree 7 2 ship 13 4 ship 4 22 tree 5 6 cake" | ./compositor
# # #
\==========/
\......#.# #
/\
\==========/
/vv\/\
\....*.*.*
/vvv/vv\
____|_|_|_____|/vvvv\
|_____________|
||
\===|___HAPPY_____|
\..|__BIRTHDAY___|
|_____________|
# # #
\==========/
\......../
examples/2015/hw/4/example.out
8.4.2
Submitting your assignment
For this assignment, you may submit whatever source files you like, along with
a file Makefile that will generate the program compositor when make is called
with no arguments (see the instructions for using make.)
You can test your submitted assignment using the public test script with
/c/cs223/bin/testit 4 public
You may also test your unsubmitted assignment in the current working directory
with
467
/c/cs223/Hwk4/test.public
The test script is intended mostly to guard against trivial errors in output format
and is not necessarily exhaustive.
8.4.3
8.4.3.1
Notes
Input
For parsing the commands on stdin, we recommend using scanf. You can test
for end of file by checking if scanf correctly parsed all three arguments, as in
int row;
int col;
char filename[BUFFER_SIZE];
while(scanf("%d %d %s", &row, &col, filename) == 3) {
/* do something with this file */
}
You may assume that row and col are always non-negative.
Your program should exit with a non-zero error code if it cannot open a file
for reading. Because scanf’s %s conversion specifier only reads up to the next
whitespace character, you may assume that filenames do not contain whitespace.
You may also assume that no filename appearing in the input will require more
than 2048 bytes to store (including the terminal null character).28
You may assume that the input contains no null characters.
8.4.3.2
Output
Your output should include newline and space characters to put the composited
characters in the appropriate rows and columns. It should not include any more
of such characters than are absolutely necessary.
For example, there should never be a space at the end of a line (even if there
is a space at the end of a line in one of the input files). Similarly, there should
not be any blank lines at the end of your output. You may, however, find it
necessary to add a newline to the end of the last line to avoid having the output
end in the middle of a line.
8.4.3.3
General
You may assume that the final picture is not so big that you can’t store a row
or column number for one of its characters in an int.
28 Normally this is a dangerous thing to assume, but this assignment is complicated enough
already.
468
8.4.4
Sample solution
I wrote two versions of this. The first used a jagged array to represent an image,
but I decided I didn’t like it and did another version using a sorted linked list of
points. This second version is shown below.
/*
* Alternate version of ASCII art thing using a queue.
*/
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
/*
* Idea of this data structure is that we have a sorted array
* of pixels, where each pixel specifies a row, column, and character
* to put in that position. The sort order is row then column.
*
* This is organized as a queue in the sense that we can push
* new pixels on to the end of it, although as it happens we
* never actually dequeue anything.
*/
struct pixel {
int row;
int col;
char value;
};
struct queue {
size_t top;
/* number of elements */
size_t size; /* number of allocated slots */
struct pixel *pixels; /* pixel values, sorted by row then column */
};
#define QUEUE_INITIAL_SIZE (16)
/* create new empty queue */
struct queue *
queueCreate(void)
{
struct queue *q;
q = malloc(sizeof(struct queue));
assert(q);
469
q->top = 0;
q->size = QUEUE_INITIAL_SIZE;
q->pixels = malloc(sizeof(struct pixel) * q->size);
assert(q->pixels);
}
return q;
/* clean up queue */
void
queueDestroy(struct queue *q)
{
free(q->pixels);
free(q);
}
/* add a new pixel to queue */
void
queuePush(struct queue *q, struct pixel p)
{
while(q->top >= q->size) {
q->size *= 2;
q->pixels = realloc(q->pixels, sizeof(struct pixel) * q->size);
assert(q->pixels);
}
}
q->pixels[q->top++] = p;
/* returns malloc'd data, free with queueDestroy */
struct queue *
queueRead(const char *filename)
{
FILE *f;
struct queue *q;
struct pixel p;
int c;
q = queueCreate();
f = fopen(filename, "r");
if(f == 0) {
perror(filename);
exit(1);
}
470
p.row = p.col = 0;
while((c = getc(f)) != EOF) {
switch(c) {
case '\n':
p.row++;
p.col = 0;
break;
case ' ':
p.col++;
break;
default:
p.value = c;
queuePush(q, p);
p.col++;
break;
}
}
fclose(f);
}
return q;
/* write pixels in queue to stdout */
void
queueWrite(const struct queue *q)
{
int outputRow = 0;
int outputCol = 0;
int i;
for(i = 0; i < q->top; i++) {
while(outputRow < q->pixels[i].row) {
putchar('\n');
outputRow++;
outputCol = 0;
}
while(outputCol < q->pixels[i].col) {
putchar(' ');
outputCol++;
}
putchar(q->pixels[i].value);
outputCol++;
}
471
}
/* end last row */
putchar('\n');
/*
* Merge two queues, creating a new, freshly-allocated queue.
* New queue is sorted. If there are pixels in both left
* and right with the same row and column, the one from right
* overwrites the one from left.
*/
struct queue *
queueMerge(const struct queue *left, const struct queue *right)
{
int l = 0;
int r = 0;
struct queue *q;
q = queueCreate();
while(l < left->top && r < right->top) {
if(left->pixels[l].row < right->pixels[r].row) {
queuePush(q, left->pixels[l++]);
} else if(left->pixels[l].row == right->pixels[r].row) {
if(left->pixels[l].col < right->pixels[r].col) {
queuePush(q, left->pixels[l++]);
} else if(left->pixels[l].col == right->pixels[r].col) {
/* right wins but both increment */
queuePush(q, right->pixels[r++]);
l++;
} else {
/* right is earlier */
queuePush(q, right->pixels[r++]);
}
} else {
/* right is earlier */
queuePush(q, right->pixels[r++]);
}
}
/* clean out whichever tail is still nonempty */
while(l < left->top) {
queuePush(q, left->pixels[l++]);
}
while(r < right->top) {
472
}
}
queuePush(q, right->pixels[r++]);
return q;
/* in-place offset by r rows and c columns */
void
queueOffset(struct queue *q, int r, int c)
{
int i;
}
for(i = 0; i < q->top; i++) {
q->pixels[i].row += r;
q->pixels[i].col += c;
}
/* max filename size as promised in assignment text */
#define BUFFER_SIZE (2048)
int
main(int argc, char **argv)
{
struct queue *merged;
/* holding place for result of merge */
struct queue *left;
/* accumulated picture */
struct queue *right;
/* new picture */
int row;
/* row offset for new picture */
int col;
/* column offset for new picture */
char filename[BUFFER_SIZE]; /* filename for new picture */
if(argc != 1) {
fprintf(stderr, "Usage: %s\n", argv[0]);
return 1;
}
for(left = queueCreate(); scanf("%d %d %s", &row, &col, filename) == 3; left = merged) {
right = queueRead(filename);
queueOffset(right, row, col);
merged = queueMerge(left, right);
}
queueDestroy(left);
queueDestroy(right);
473
queueWrite(left);
queueDestroy(left);
return 0;
}
examples/2015/hw/4/compositor.c
Here is a Makefile.
8.5
8.5.1
Assignment 5, due Wednesday 2015-02-25, at 11:00pm
Build a Turing machine!
A Turing machine is a hypothetical computational device that consists of a
little trolley, called the controller, that rolls around on an infinite paper tape
on which it can write letters. The controller itself has a small number of states
that affect its behavior, and a program that tells it what to do when it is in a
particular state and sees a particular letter.
In C terms, we can think of a Turing machine as consisting of an infinite array of
chars (representing the tape), an integer index into the array (representing the
current position of the controller), and an int (representing the current state of
the controller). At each step, the machine
1. Looks at the symbol s on the current tape cell.
2. Looks up its state q and symbol s in the table representing its program.
This table will specify an action, which consists of three parts:
a. A new symbol to write to the current tape cell.
b. A direction - or + to move.
c. A new state to switch to.
If the new state is 0, the machine halts and takes no more steps.
For this assignment, you are to write a Turing machine simulator. The program
for machine will be supplied in argv; argv[i] will give the behavior of the
machine when in state i, where the first three characters specify what to do
if the machine sees an 'a', the second three characters specify what to do if
the machine sees a 'b', and so on. Each of these three-letter groups will look
like (new-symbol,direction,new-state), where new-symbol is the new symbol to
write (a lower-case letter), direction is the direction to move ('+', meaning one
position to the right, or '-', meaning one position to the left) and new-state is
the new state to go to (a digit). Your program should run this program starting
in state 1 on some location in the middle of a tape initially made up entirely of
'a' characters, and continue until the program enters the special halting state 0.
It should then print the number of steps until this occurred.
474
You may assume that the program is argv is complete in the sense that it
includes rules for any combination of state and symbol you will encounter while
executing it. You are not required to detect if this assumption is violated.
8.5.2
Example
The program
b+2a-0 a-1a-1
gives instructions for what to do in state 1 (b+2a-0) and state 2 (a-1a-1). In
state 1, if the controller reads an a, the triple b+2 means that it should write
b, move right (+), and switch to state 2. If instead it reads a b, the triple a-0
means that it should write a, move left (-), and halt (0). In state 2, the machine
always writes a, moves left, and switches to state 1.
Below is a depiction of this machine’s execution. It passes through 4 states
(including both the initial state and the final halting state) using a total of 3
steps. The controller and its current state is shown above its current position on
the tape at each point in time. To avoid having to put in infinitely long lines,
only the middle three tape cells are shown.
1
aaa
2
aba
1
aba
0
aaa
8.5.3
Your task
You should submit a Makefile and whatever source files are needed to generate a
program ./turing when make is called with no arguments. The turing program
should simulate a Turing machine as described above and print the number of
steps that it takes until it halts in decimal format, followed by a newline. It
should not produce any other output. For example, using the program above,
your program should print 3:
$ ./turing b+2a-0 a-1a-1
3
475
For more complex programs you may get different results. Here is a 3 state, 3
symbol program that runs for a bit longer:
$ ./turing b+2a-0c-3 b-3c+2b-2 b-1a+2c-1
92649163
You may assume that tape symbols can always be represented by lowercase
letters, that states can always be represented by single digits, and that argv is
in the correct format (although it may be worth including a few sanity checks in
your program just in case).
Not all Turing machine programs will halt. Your program is not required to
detect if the Turing machine it is simulating will halt eventually or not (although
it should notice if it does halt).
8.5.4
Submitting your assignment
Submit all files needed to build your program as usual using /c/cs223/bin/submit
5 filename.
There is a public test script in /c/cs223/Hwk5/test.public. You can run this
on your submitted files with /c/cs223/bin/testit 5 public.
8.5.5
/*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Sample solution
Simple Turing machine simulator.
Tape holds symbols 0 (default) through 2.
Controller programming is specified in argv:
argv[i] gives transitions for state i as six characters.
Each triple of characters is <action><direction><new-state>
where <action> is one of:
a,b,c: write this value to tape
<direction> is one of:
-: go left
+: go right
.: stay put
476
* The three pairs give the transition for reading 0, 1, 2 from tape.
*
* State 0 is the halting state.
*
* On halting, prints number of transitions followed by contents
* of all tape cells that have ever been visited by the
* finite-state controller.
*/
#include
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<string.h>
<assert.h>
<sys/types.h>
struct configuration {
unsigned int state;/*
size_t leftmost;
/*
size_t rightmost; /*
size_t current;
/*
size_t tapeLength; /*
char *tape;
/*
};
state of head */
leftmost cell visited */
rightmost cell visited */
current cell */
current allocated space for tape */
contents of cells */
/* increase the size of the tape and recenter contents in middle */
void
configurationExpand(struct configuration *c)
{
size_t newTapeLength;
char *oldTape;
char *newTape;
size_t i;
ssize_t offset;
newTapeLength = 4*c->tapeLength;
newTape = malloc(newTapeLength);
assert(newTape);
for(i = 0; i < newTapeLength; i++) {
newTape[i] = 0;
}
/* copy old tape */
offset = newTapeLength / 2 - c->current;
for(i = c->leftmost; i <= c->rightmost; i++) {
477
}
newTape[i + offset] = c->tape[i];
oldTape = c->tape;
c->tape = newTape;
c->tapeLength = newTapeLength;
c->current += offset;
c->leftmost += offset;
c->rightmost += offset;
}
free(oldTape);
#define INITIAL_TAPE_LENGTH (16)
struct configuration *
configurationCreate(void)
{
struct configuration *c;
size_t i;
c = malloc(sizeof(struct configuration));
assert(c);
c->state = 1;
c->tapeLength = INITIAL_TAPE_LENGTH;
c->leftmost = c->rightmost = c->current = c->tapeLength / 2;
c->tape = malloc(c->tapeLength);
assert(c->tape);
for(i = 0; i < c->tapeLength; i++) {
c->tape[i] = 0;
}
}
return c;
void
configurationDestroy(struct configuration *c)
{
free(c->tape);
free(c);
}
#define SYMBOL_BASE ('a')
#define STATE_BASE ('0')
478
/* used for debugging mostly */
void
configurationPrint(const struct configuration *c)
{
size_t i;
for(i = c->leftmost; i < c->current; i++) {
putchar(' ');
}
putchar(STATE_BASE + c->state);
putchar('\n');
}
for(i = c->leftmost; i <= c->rightmost; i++) {
putchar(SYMBOL_BASE + c->tape[i]);
}
putchar('\n');
int
main(int argc, char **argv)
{
struct configuration *c;
char cellValue;
const char *transition;
size_t steps;
if(argc == 1) {
fprintf(stderr, "Usage: %s transitions\n", argv[0]);
return 1;
}
c = configurationCreate();
steps = 0;
while(c->state != 0) {
steps++;
/* execute the next transition */
assert(c->state < argc);
cellValue = c->tape[c->current];
assert(0 <= cellValue);
assert(3*(cellValue+1) <= strlen(argv[c->state]));
transition = argv[c->state] + 3*c->tape[c->current];
479
c->tape[c->current] = transition[0] - SYMBOL_BASE;
switch(transition[1]) {
case '-':
if(c->current == 0) {
configurationExpand(c);
}
c->current--;
if(c->current < c->leftmost) {
c->leftmost = c->current;
}
break;
case '+':
if(c->current == c->tapeLength - 1) {
configurationExpand(c);
}
c->current++;
if(c->current > c->rightmost) {
c->rightmost = c->current;
}
break;
case '.':
/* do nothing */
break;
default:
fprintf(stderr, "Bad direction '%c'\n", transition[2]);
exit(2);
break;
}
c->state = transition[2] - STATE_BASE;
#ifdef PRINT_CONFIGURATION
configurationPrint(c);
#endif
}
/* print number of steps */
printf("%zu\n", steps);
configurationDestroy(c);
}
return 0;
480
examples/2015/hw/5/turing.c
CC=gcc
CFLAGS=-std=c99 -Wall -pedantic -g3
all: turing
compositor: turing.o
$(CC) $(CFLAGS) -o $@ $^
clean:
$(RM) turing *.o
examples/2015/hw/5/Makefile
8.6
8.6.1
Assignment 6, due Wednesday 2015-03-25, at 11:00pm
Sinking ships
For this assignment, you are to implement a data structure for playing a game
involving ships placed in a large square grid. Each ship occupies one more more
squares in either a vertical or horizontal line, and has a name that consists of a
single char other than a period (which will be used to report the absence of a
ship). Ships have a bounded maximum length; attempts to place ships longer
than this length have no effect.
All type and constant definitions for the data type, and all function declarations,
are given in the file ships.h, which is shown below, and which you can also find
in /c/cs223/Hwk6/sourceFiles/ships.h. The playing field is represented by
a struct field (which you get to define). A new struct field is created by
fieldCreate, and when no longer needed should be destroyed by fieldDestroy.
These data types from ships.h control ship naming and placement. Note that
uint32_t is defined in stdint.h (which is also included by inttypes.h. You
will need to include one of these files before ships.h to get this definition.
typedef uint32_t coord;
struct position {
coord x;
coord y;
};
struct ship {
struct position topLeft;
int direction;
unsigned int length;
/* coordinates of top left corner */
/* HORIZONTAL or VERTICAL */
/* length of ship */
481
};
char name;
/* name of ship */
Actual placement is done using the fieldPlaceShip function, declared as follows:
void fieldPlaceShip(struct field *f, struct ship s);
A ship of length m placed horizontally with its top left corner at position (x, y)
will occupy positions (x, y) through (x+m−1, y). If instead it is placed vertically,
it will occupy positions (x, y) through (x, y + m − 1). If any of these coordinates
exceed the maximum coordinate COORD_MAX (defined in ships.h), the ship will
not be placed. The ship will also not be placed if its name field is equal to
NO_SHIP_NAME or if the length exceeds MAX_SHIP_LENGTH.
If the new ship will occupy any position as a ship previously placed in the field,
the previous ship will be removed. It is possible for many ships to be removed
at once in this way.
The fieldAttack function can be used to remove a ship at a particular location
without placing a new ship. It returns the name of the removed ship, if any, or
NO_SHIP_NAME if there is no ship at that location.
Finally, the fieldCountShips returns the number of ships still present in the
field.
Your job is to write an implementation of these functions, which you should
probably put in a file ships.c. You must also supply a Makefile, which,
when make is called with no arguments, generates a test program testShips
from your implementation and the file testShips.c that we will provide. You
should not count on precisely this version of testShips.c being supplied; your
implementation should work with any main program that respects the interface
in ships.h.
8.6.2
Things to watch out for
You should write your implementation so that it will continue to work if the
typedef for coord, or the definitions of the constants COORD_MAX, NO_SHIP_NAME,
SHIP_MAX_LENGTH, HORIZONTAL, or VERTICAL change. You may, however, assume
that coord is an unsigned integer type and the COORD_MAX is the largest value
that can be represented by this type.
If it helps in crafting your implementation, you may assume that
MAX_SHIP_LENGTH will alway be a reasonably small constant.
You do
not need to worry about implementing a data structure that will handle huge
ships efficiently. On the other hand, COORD_MAX as defined in the default
ships.h is 23 2 − 1, so you will need to be able to deal with a field with at
least 26 4 possible locations, a consideration you should take into account when
choosing a data structure to represent a field.
482
8.6.3
The testShips program
The supplied testShips program creates a field, processes commands from
stdin, then destroys the field when it reaches EOF or a command that it can’t
parse. Each command is either of the form
+ x y vertical length name
which means to call fieldPlaceShip for a new ship at coordinates (x,y), with
direction HORIZONTAL if isVertical is false and VERTICAL otherwise, and length
length and name name.
The command
- x y
calls fieldAttack on coordinates (x,y).
Here is a small input file:
+
+
+
-
0
1
3
1
0 0 3 a
2 0 4 b
0 1 3 c
0
After processing the first line, the contents of the field should look like this:
aaa..
.....
.....
.....
.....
After processing the second line:
aaa..
.....
.bbbb
.....
.....
The third line places a ship vertically that intersects one of the previous ships,
sinking it:
aaac.
...c.
...c.
.....
.....
Had ship c been shifted one position to the left, it would have sunken a as well.
Finally, the last line drops a bomb at (1, 0), sinking a:
483
...c.
...c.
...c.
.....
.....
The input files used by test.public can be found in /c/cs223/Hwk6/testFiles.
Some of these were generated randomly using the script /c/cs223/Hwk6/makeRandom,
which you should feel free to use for your own nefarious purposes.
Because the interface in ships.h gives no way to find out what ships are currently
in the field, the test program will not actually produce pictures like the above.
Instead, it prints after each command a line giving the name of the ship sunken by
fieldAttack (or NO_SHIP_NAME if no ship is sunk or fieldPlaceShip is called)
and the number of ships left in the field following the attack. So the user must
imagine the carnage as the 100000 ships in randomSparseBig.in somehow leave
only 25336 survivors in randomSparseBig.out, demonstrating the importance
of strict navigational rules in real life.
Figure 8: Why you should respect people’s space
484
8.6.4
Submitting your assignment
Submit your assignment as usual with
/c/cs223/bin/submit 6
You can run the public test script in /c/cs223/Hwk6/test.public on your
submitted files with
/c/cs223/bin/testit 6 public
8.6.5
Provided source files
#define HORIZONTAL (0)
#define VERTICAL (1)
/* place ship horizontally */
/* place ship vertically */
#define MAX_SHIP_LENGTH (17) /* length of longest ship (width is always 1) */
#define NO_SHIP_NAME ('.') /* what to return when hitting no ship */
/*
* Type for coordinates, and their maximum possible value.
*
* Include <stdint.h> before this header file
* to get the definition of uint32_t
* and its maximum value UINT32_MAX.
*/
typedef uint32_t coord;
#define COORD_MAX (UINT32_MAX)
/*
* Non-opaque structs for passing around positions and ship placements.
*/
struct position {
coord x;
coord y;
};
struct ship {
struct position topLeft;
int direction;
unsigned int length;
char name;
};
/*
/*
/*
/*
coordinates of top left corner */
HORIZONTAL or VERTICAL */
length of ship */
name of ship */
/*
485
* Create a playing field for holding ships.
*/
struct field *fieldCreate(void);
/*
* Free all space associated with a field.
*/
void fieldDestroy(struct field *);
/*
* Place a ship in a field with given placement and name.
*
* If placement.length is less than one or greater than MAX_SHIP_LENGTH,
* or if some part of the ship would have a coordinate greater than COORD_MAX,
* or if the ship's name is NO_SHIP_NAME,
* the function returns without placing a ship.
*
* Placing a new ship that intersects any previously-placed ships
* sinks the previous ships, removing them from the field.
*/
void fieldPlaceShip(struct field *f, struct ship s);
/*
* Attack!
*
* Drop a shell at given position.
*
* Returns NO_SHIP_NAME if attack misses (does not intersect any ship).
*
* Otherwise returns name of ship hit.
*
* Hitting a ship sinks it, removing it from the field.
*/
char fieldAttack(struct field *f, struct position p);
/*
* Return number of ships in the field.
*/
size_t fieldCountShips(const struct field *f);
examples/2015/hw/6/ships.h
#include
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<assert.h>
<stdint.h>
<inttypes.h>
486
#include "ships.h"
#define PLACE_SHIP ('+')
#define ATTACK ('-')
/* command to place a new ship */
/* command to attack a location */
int
main(int argc, char **argv)
{
struct field *f; /* where we keep our ships */
int command;
/* command char */
struct ship s;
/* ship we are placing */
struct position p; /* location to attack */
int sank;
/* ship we sank */
if(argc != 1) {
fprintf(stderr, "Usage: %s\n", argv[0]);
return 1;
}
f = fieldCreate();
while((command = getchar()) != EOF) {
switch(command) {
case PLACE_SHIP:
if(scanf("%" SCNu32 " %" SCNu32 "%d %u %c ", &s.topLeft.x, &s.topLeft.y, &s.
/* not enough args */
fprintf(stderr, "Not enough enough args to %c\n", PLACE_SHIP);
return 1;
}
/* else */
/* fix the direction to match actual definitions */
s.direction = s.direction ? VERTICAL : HORIZONTAL;
fieldPlaceShip(f, s);
sank = NO_SHIP_NAME;
break;
case ATTACK:
if(scanf("%" SCNu32 " %" SCNu32 " ", &p.x, &p.y) != 2) {
fprintf(stderr, "Not enough enough args to %c\n", ATTACK);
return 1;
}
/* else */
487
sank = fieldAttack(f, p);
break;
default:
/* bad command */
fprintf(stderr, "Bad command %c\n", command);
return 1;
break;
}
printf("%c %zu\n", sank, fieldCountShips(f));
}
fieldDestroy(f);
}
return 0;
examples/2015/hw/6/testShips.c
8.6.6
Sample solution
#include
#include
#include
#include
<stdlib.h>
<assert.h>
<string.h>
<stdint.h>
#include "ships.h"
/* basic hash table */
struct field {
size_t size;
size_t occupancy;
struct elt **table;
};
struct elt {
struct elt *next;
struct ship ship;
};
/* number of slots in table */
/* number of elements in table */
/* hash table, malloc'd */
/* pointer to next element in linked list */
/* ship in this element */
/* picked more or less at whim from http://planetmath.org/goodhashtableprimes */
#define X_HASH_FACTOR (201326611)
#define Y_HASH_FACTOR (3145739)
488
static size_t
hash(struct position p)
{
return X_HASH_FACTOR * p.x + Y_HASH_FACTOR * p.y;
}
#define DEFAULT_INITIAL_SIZE (8)
/* like fieldCreate, but argument gives initial size */
static struct field *
fieldCreateInternal(size_t initialSize)
{
struct field *f;
size_t i;
f = malloc(sizeof(struct field));
assert(f);
f->size = initialSize;
f->occupancy = 0;
f->table = malloc(sizeof(struct elt *) * f->size);
for(i = 0; i < f->size; i++) {
f->table[i] = 0;
}
}
return f;
struct field *
fieldCreate(void)
{
return fieldCreateInternal(DEFAULT_INITIAL_SIZE);
}
/* destroy contents of f but don't free f itself */
static void
fieldDestroyContents(struct field *f)
{
size_t i;
struct elt *e;
struct elt *next;
for(i = 0; i < f->size; i++) {
489
}
}
for(e = f->table[i]; e != 0; e = next) {
next = e->next;
free(e);
}
free(f->table);
void
fieldDestroy(struct field *f)
{
fieldDestroyContents(f);
free(f);
}
/* when to grow field */
#define MAX_ALPHA (1)
/*
* Helper for fieldPlaceShip.
*
* This skips all the sanity-checking in fieldPlaceShip,
* and just performs the hash table insertion.
*/
static void
fieldInsertShip(struct field *f, struct ship s)
{
size_t h;
/* hashed coordinates */
struct elt *e; /* new element to insert */
h = hash(s.topLeft) % f->size;
e = malloc(sizeof(struct elt));
assert(e);
}
e->ship = s;
e->next = f->table[h];
f->table[h] = e;
f->occupancy++;
void
fieldPlaceShip(struct field *f, struct ship s)
{
490
struct
struct
struct
size_t
field *f2;
elt *e;
position pos;
i;
/* test if we can just throw this away */
if(s.name == NO_SHIP_NAME
|| s.length == 0
|| s.length > MAX_SHIP_LENGTH
|| (s.direction == HORIZONTAL && s.topLeft.x > COORD_MAX - (s.length - 1))
|| (s.direction == VERTICAL
&& s.topLeft.y > COORD_MAX - (s.length - 1))
)
{
return;
}
/* else */
if(f->occupancy >= f->size * MAX_ALPHA) {
/* grow the field */
f2 = fieldCreateInternal(f->size * 2);
/* copy to new field */
for(i = 0; i < f->size; i++) {
for(e = f->table[i]; e != 0; e = e->next) {
/* skip testing for occupancy or intersections */
fieldInsertShip(f2, e->ship);
}
}
/* transplant new field into old field */
fieldDestroyContents(f);
*f = *f2;
}
free(f2);
/* check for intersections */
pos = s.topLeft;
for(i = 0; i < s.length; i++) {
if(s.direction == HORIZONTAL) {
pos.x = s.topLeft.x + i;
} else {
pos.y = s.topLeft.y + i;
}
fieldAttack(f, pos);
491
}
}
/* call helper to do the actual hash table insertion */
fieldInsertShip(f, s);
/*
* Helper for fieldAttack.
*
* If there is a ship with topLeft at given position, return pointer
* to location in hash table that points to it (either table entry
* or next component).
*
* If not, return null.
*/
static struct elt **
fieldShipAt(struct field *f, struct position p)
{
struct elt **prev; /* previous pointer */
for(prev = &f->table[hash(p) % f->size]; *prev != 0; prev = &((*prev)->next)) {
if((*prev)->ship.topLeft.x == p.x && (*prev)->ship.topLeft.y == p.y) {
return prev;
}
}
}
/* didn't find anything */
return 0;
/*
* Attack!
*
* Drop a shell at given position.
*
* Returns 0 if attack misses (does not intersect any ship).
*
* Otherwise returns name of ship hit,
* which should be freed by caller when no longer needed.
*
* Hitting a ship sinks it, removing it from the field.
*/
char
fieldAttack(struct field *f, struct position p)
{
struct position p2;
492
int i;
int direction;
struct elt **prev;
struct elt *freeMe;
char name;
for(direction = 0; direction <= 1; direction++) {
for(i = 0; i < MAX_SHIP_LENGTH && i <= (direction == HORIZONTAL ? p.x : p.y); i++) {
if(direction == HORIZONTAL) {
p2.x = p.x - i;
p2.y = p.y;
} else {
p2.x = p.x;
p2.y = p.y - i;
}
prev = fieldShipAt(f, p2);
if(prev) {
/* if we sink anybody, it will be this ship */
/* but maybe it doesn't reach */
/* or points in the wrong direction */
if((*prev)->ship.length > i && (*prev)->ship.direction == direction) {
/* got it */
freeMe = *prev;
*prev = freeMe->next;
name = freeMe->ship.name;
free(freeMe);
f->occupancy--;
}
}
}
}
return name;
} else {
/* didn't get it */
/* maybe try again in other direction */
break;
}
/* didn't get anything */
return NO_SHIP_NAME;
493
/*
* Return number of ships in the field.
*/
size_t
fieldCountShips(const struct field *f)
{
return f->occupancy;
}
examples/2015/hw/6/ships.c
8.7
8.7.1
Assignment 7, due Wednesday 2015-04-01, at 11:00pm
Solitaire with big cards
For this assignment you are to implement a strategy for playing a card game
involving moving cards (represented by uint64_ts) down through a sequence of
n piles. The interface to your strategy is given in the file strategy.h, shown
below:
/*
* Interface for card-playing strategy.
*
* The deal function supplies a new card to the strategy. Each possible card will only be d
*
* The play function should return a card that has been dealt previously but not yet played.
* If asked for a card when the hand is empty, its behavior is undefined.
*/
#include <stdint.h>
typedef uint64_t Card;
/* representation of a card */
/* opaque type for strategy data */
typedef struct strategy Strategy;
/* set up a new strategy for numPiles many piles */
Strategy *strategyCreate(int numPiles);
/* clean up all space used by a strategy */
void strategyDestroy(Strategy *);
/* add a card to the current hand */
void strategyDeal(Strategy *, Card);
494
/* play a card from pile k */
Card strategyPlay(Strategy *, int k);
examples/2015/hw/7/strategy.h
Initially, the player has n piles, numbered 1 through n. The strategyDeal
function is called to indicate that a new card has been dealt to pile n. The
strategyPlay function is called to indicate that a card should be moved from
pile k to pile k-1; this function should return the card to move. Cards moved to
pile 0 leave the game and are not used again. Each card is unique: once a card is
dealt, the same card will never be dealt again during the same play of the game.
The choice of when to deal and when to play from pile is controlled by some
external entity, which at some point will stop and compute the smallest card in
each pile. The goal of the strategy is to make these smallest cards be as large as
possible, giving priority to the highest-numbered piles: given two runs of the
game, the better-scoring one is the one that has the larger smallest card in pile
n, or, if both have the same smallest card in pile n, the one that has the larger
smallest card in pile n − 1, and so forth. A tie would require that both runs end
with the same smallest card in every pile. An empty pile counts as UINT64_MAX
for this purpose (although note that a strategy has no control over which piles
are empty).
Your job is to implement a strategy that produces the best possible result
for any sequence of calls to strategyDeal and strategyPlay. Your strategy
implementation will most likely need to keep track of which cards are available in
each pile, as this information is not provided by the caller. Your strategyPlay
function should only make legal moves: that is, it should only play cards that are
actually present in the appropriate pile. You may assume that strategyPlay is
never called on an empty pile.
Your implementation should consist of a file strategy.c and any supporting source and header files that you need other than strategy.h, which we
have provided for you. You should also supply a file Makefile that generates a program testStrategy when make is called with no arguments, using your implementation and the testStrategy.c file that you can find in
/c/cs223/Hwk7/sourceFiles/testStrategy.c.
8.7.2
Explanation of the testing program
The testStrategy program implements one of four rules for when you can play
from each pile. The arguments to testStrategy are a character indicating which
rule to apply, the number of cards to deal (which can be pretty big), and the
number of piles (which is much more limited, because testStrategy.c tracks
the pile each card is in using a char to save space). The actual cards dealt are
generated deterministically and will be the same in every execution with the
same arguments. The test files in /c/cs223/Hwk7/testFiles give the expected
495
output when testStrategy is run with the arguments specified in the filename
(after removing the - characters); this will always be the value, in hexadecimal,
of the smallest card in each pile, starting with the top pile.
For example, running the harmonic rule h with 1000 cards and 4 piles (not
counting the 0 pile) gives the output
$ ./testStrategy h 1000 4
5462035faf0d6fa1
501ebb6268d39af3
25732b5fee7c8ad7
301e0f608d124ede
This output would appear in a filename h-1000-4, if this particular combination
of parameters were one of the test cases.
8.7.3
Submitting your assignment
Submit your assignment as usual with /c/cs223/bin/submit 7. You should
submit your source file(s), your Makefile, and any other files needed to build your
program other than strategy.h and testStrategy.c, which will be supplied
by the test script. You can test your submission using the public test script
in /c/cs223/Hwk7/test.public using the command /c/cs223/bin/testit 7
public.
8.7.4
Sample solution
I implemented a heap with uint64_t elements as a separate module (heap.h,
heap.c) and then used in in a main module strategy.c that allocates a separate
heap for each pile and manages the translation between strategyDeal and
strategyPlay and the heap functions. The Makefile is pretty much the usual.
8.8
8.8.1
Assignment 8, due Wednesday 2015-04-08, at 11:00pm
An ordered set
For this assignment, you are to implement an ordered set data type for holding
null-terminated strings. The interface to this data type is given in the file
orderedSet.h, shown below.
/*
* Ordered set data structure.
*/
/* Make a new empty set */
496
struct orderedSet *orderedSetCreate(void);
/* Destroy a set */
void orderedSetDestroy(struct orderedSet *);
/* How many elements in this set? */
size_t orderedSetSize(const struct orderedSet *);
/* Insert a new element. Has no effect if element is already present. */
void orderedSetInsert(struct orderedSet *, const char *);
/* Delete an element. Has no effect if element is not already present. */
void orderedSetDelete(struct orderedSet *, const char *);
/* Return a new ordered set containing all elements e
* for which predicate(arg, x) != 0.
* The predicate function should be applied to the elements in increasing order. */
struct orderedSet *orderedSetFilter(const struct orderedSet *, int (*predicate)(void *arg, c
examples/2015/hw/8/orderedSet.h
In addition to the usual create and destroy functions, an ordered set supports
inserting and deleting elements, counting the number of distinct elements in the
set, and filtering the set based on a predicate function passed in as an argument.
This filtering operation does not modify the input set, but instead generates a
new ordered set containing only those elements on which the predicate returns a
nonzero value.
The filtering operation is where most of the excitement happens; because the
predicate function takes an argument of type void * that is also passed to the
filter function, it is possible for the predicate to compute an arbitrary function
on the elements of the set as a side-effect of processing each element to decide
whether to put it in the output set. This allows predicates to be abused to
perform all sorts of computations, including printing out all elements of the set
or computing a hash of all the strings in the set concatenated together. These
features are used by the test program testOrderedSet.c that we have provided.
To ensure that these traversals give consistent results, it is required that when
your implementation of orderedSetFilter is executed, it calls the predicate
function exactly once on each element of the set in increasing order as determined
by strcmp.
8.8.2
The testOrderedSet wrapper
The test program is a fairly thin wrapper over the implementation that allows
you to call the various functions using one-line commands on standard input.
A command is given as the first character of the line, and the rest of the line
497
contains the argument to the command if needed. The + and - commands add
or remove an element from the set, respectively, while the p, s, and h commands
print the contents of the set, the size of the set, and a hash of the set (these
commands ignore any argument). The f command removes all elements of the
set that do not contain a particular substring.
Here is a simple input to the program that inserts four strings, filters out the
ones that don’t contain ee, then prints various information about the results.
+feed
+the
+bees
+please
fee
s
h
p
This should produce the output
2
15082778b3db8cb3
bees
feed
8.8.3
Submitting your assignment
Submit, with the usual /c/cs223/bin/submit 8 filename, your Makefile
and any supporting files needed to build the program testOrderedSet from
testOrderedSet.c and orderedSet.h when make is called with no arguments.
These last two files will be provided by the test script and you do not need to
submit them.
You can test your submission against the public test script in /c/cs223/Hwk8/test.public
with /c/cs223/bin/testit 8.
8.8.4
Sample solution
There were a lot of ways to do this. For the sample solution, I decided to do
something unusual, and store the set as a hash table. This is not ordered, but
since the only operation that requires the set to be ordered is orderedSetFilter,
which will take Ω(n) time no matter how you implement it, the O(n log n) cost
to call qsort to sort the elements as needed does not add much overhead.
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
498
#include <stdint.h>
#include <string.h>
#include "orderedSet.h"
/* We'll use a hash table with linear probing.
* This is not actually ordered, but the only operations that
* depend on order a linear-time anyway, so we can afford to sort as needed */
struct orderedSet {
size_t n;
/* number of elements */
size_t size; /* size of the table */
char **table; /* hash table */
};
#define INITIAL_SIZE (16)
#define MAX_ALPHA (0.75)
/* Make a new empty set with given size */
static struct orderedSet *
orderedSetCreateInternal(size_t size)
{
struct orderedSet *s;
s = malloc(sizeof(*s));
assert(s);
s->n = 0;
s->size = size;
s->table = calloc(s->size, sizeof(char *));
}
return s;
struct orderedSet *
orderedSetCreate(void)
{
return orderedSetCreateInternal(INITIAL_SIZE);
}
/* Destroy a set */
void
orderedSetDestroy(struct orderedSet *s)
{
size_t i;
for(i = 0; i < s->size; i++) {
499
}
}
if(s->table[i]) {
free(s->table[i]);
}
free(s->table);
free(s);
/* How many elements in this set? */
size_t
orderedSetSize(const struct orderedSet *s)
{
return s->n;
}
static size_t
hash(const char *s)
{
size_t h;
/* usual crummy hash function */
for(h = 0; *s; h = h * 97 + *s++);
}
return h;
static char *
strMalloc(const char *s)
{
char *s2;
s2 = malloc(strlen(s)+1);
strcpy(s2, s);
}
return s2;
/* Insert and element without doing size check or malloc */
/* Frees element if already present */
static void
orderedSetInsertInternal(struct orderedSet *s, char *elt)
{
size_t h;
assert(elt);
500
/* skip over non-empty slots with different values */
for(h = hash(elt) % s->size; s->table[h] && strcmp(s->table[h], elt); h = (h+1) % s->siz
}
/* check if not already present */
if(s->table[h] == 0) {
s->table[h] = elt;
s->n++;
} else {
free(elt);
}
/* Insert a new element. Has no effect if element is already present. */
void
orderedSetInsert(struct orderedSet *s, const char *elt)
{
size_t h;
struct orderedSet *s2;
if(s->n >= s->size * MAX_ALPHA) {
/* rebuild the table */
s2 = orderedSetCreateInternal(s->size * 2);
/* copy all the elements */
for(h = 0; h < s->size; h++) {
if(s->table[h]) {
orderedSetInsertInternal(s2, s->table[h]);
}
}
}
}
/* free the table and then do a brain transplant */
free(s->table);
*s = *s2;
free(s2);
orderedSetInsertInternal(s, strMalloc(elt));
/* Delete an element. Has no effect if element is not already present. */
void
orderedSetDelete(struct orderedSet *s, const char *elt)
{
size_t h;
char *later;
501
/* skip over non-empty slots with different values */
for(h = hash(elt) % s->size; s->table[h] && strcmp(s->table[h], elt); h = (h+1) % s->siz
/* if we reached a nonempty slot, it must be our target */
if(s->table[h] != 0) {
/* remove the initial element */
free(s->table[h]);
s->table[h] = 0;
s->n--;
}
}
/* remove and reinsert any elements up to the next hole, in case they wanted to be e
for(h = (h+1) % s->size; s->table[h] ; h = (h+1) % s->size) {
later = s->table[h];
s->table[h] = 0;
s->n--;
orderedSetInsertInternal(s, later);
}
static int
compare(const void *s1, const void *s2)
{
return strcmp(*((const char **) s1), *((const char **) s2));
}
/* Return a new ordered set containing all elements e
* for which predicate(arg, x) != 0.
* The predicate function should be applied to the elements in increasing order. */
struct orderedSet *
orderedSetFilter(const struct orderedSet *s, int (*predicate)(void *arg, const char *), void
{
size_t h;
const char **a;
/* temporary array to sort */
size_t top;
/* where to put things in a */
size_t i;
struct orderedSet *s2;
a = malloc(sizeof(const char *) * s->size);
assert(a);
top = 0;
for(h = 0; h < s->size; h++) {
if(s->table[h]) {
502
}
}
a[top++] = s->table[h];
qsort(a, top, sizeof(const char *), compare);
s2 = orderedSetCreate();
for(i = 0; i < top; i++) {
if(predicate(arg, a[i])) {
orderedSetInsert(s2, a[i]);
}
}
free(a);
}
return s2;
examples/2015/hw/8/orderedSet.c
Makefile{.makefile include=examples/20135/hw/8/Makefile}
8.9
8.9.1
Assignment 9, due Wednesday 2015-04-15, at 11:00pm
Finding a cycle in a maze
For this problem, you are given a rectangular maze consisting of wall squares
(represented by 0) and path squares (represented by 1). Two path squares are
considered to be adjacent if they are at most one square away orthogonally or
diagonally; in chess terms, two path squares are adjacent if a king can move
from one to the other in one turn. The input to your program is a maze in which
the graph consisting of all path squares is connected and contains at most one
cycle, where a cycle is a sequence of distinct squares s1 , s2 , . . . , sk where each si
is adjacent to si+1 and sn is adjacent to s1 . Your job is to write a program maze
that finds this cycle if it exists, and marks all of its squares as cycle squares
(represented by 2).
For example, here is a picture of a 200-by-100 maze that contains a small cycle:
and here is the same maze with the cycle highlighted in white:
8.9.2
Input and output format
The input to your program should be taken from stdin, in a restricted version
of raw PGM format, an old image format designed to be particularly easy to
503
Figure 9: 200 by 100 maze
Figure 10: 200 by 100 maze, showing cycle
504
parse. The input file header will be a line that looks like it was generated by the
printf conversion string "P5 %d %d 255\n", where the first int value is the
width of the image in columns and the second is the height of the image in rows;
the same conversion string can be given to scanf to parse this line. Following
the newline will be a long sequence of bytes, each representing one pixel of the
image, with each row following immediately after the previous one. These bytes
will be either 0 or 1 depending on whether that position in the maze is a wall or
a path.
The output to your program should be in the same format, with the difference
that now some of the bytes in the image data may be 2, indicating the cycle.
If there is no cycle, the output should be identical to the input. Your program
is not required to detect or respond in any particular way to input mazes that
violate the format or do not contain a connected graph of path squares, although
you are encouraged to put in reasonable error checking for your own benefit
during testing.
For example, the maze depicted above is stored in the file 200-100-4.in.pgm;
the corresponding output is stored in the file 200-100-4.out.pgm. Other sample
inputs and outputs can be found in /c/cs223/Hwk9/testFiles.
This file format is hard to read with the naked eye, even after loading into a text
editor. The script /c/cs223/Hwk9/toPng will generate a PNG file that doubles
the pixel size and rescales the 0, 1, 2 pixel values to more reasonable values for
display. This can be called as /c/cs223/Hwk9/toPng filename.pgm to produce
a new file filename.pgm.png. This works best if filename.pgm is already in a
directory you can write to. PNG files can be displayed using most web browsers
and image manipulation tools.
8.9.3
Submitting and testing your program
Submit whatever files you need to build maze (including a Makefile that generates maze when called with no arguments) using /c/cs223/bin/submit 9.
You can apply the public test script in /c/cs223/Hwk9/test.public to your
submitted files using /c/cs223/bin/testit 9 public.
8.9.4
Sample solution
This uses breadth-first search, which makes the search a bit simpler than depthfirst search but requires some more effort to compute the cycle. The program
also includes code for generating random mazes.
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<assert.h>
<math.h>
505
#include <limits.h>
struct direction {
signed char x;
signed char y;
};
#define DIRECTIONS (8)
const
{
{
{
{
{
{
{
{
};
struct
-1, -1
-1, 0
-1, 1
0, -1
0, 1
1, -1
1, 0
1, 1
direction directions[DIRECTIONS] = {
},
},
},
},
},
},
},
}
struct position {
int x;
int y;
};
const struct position NO_POSITION = { -1, -1 };
static inline int
eqPosition(struct position p, struct position q)
{
return p.x == q.x && p.y == q.y;
}
#define WALL (0)
#define PATH (1)
#define CYCLE (2)
struct square {
int contents;
struct position parent; /* used by search routine */
};
struct maze {
struct position size;
struct square *a;
};
/* rows = size.x, columns = size.y */
/* packed array of squares */
506
/* look up a position in a maze */
#define Mref(m, pos) ((m)->a[(pos).y * (m)->size.x + (pos).x])
#define Mget(m, pos) (assert((pos).x >= 0 && (pos).y >= 0 && (pos).x < (m)->size.x && (pos).
/* add direction to source to get target */
/* returns 1 if target is in range */
int
offset(const struct maze *m, struct position *target, struct position source, struct directi
{
target->x = source.x + dir.x;
target->y = source.y + dir.y;
}
return target->x >= 0 && target->y >= 0 && target->x < m->size.x && target->y < m->size.
/* free a maze */
void
destroyMaze(struct maze *m)
{
free(m->a);
free(m);
}
/* load a maze in restricted PGM format */
struct maze *
loadMaze(FILE *f)
{
struct maze *m;
struct position i;
m = malloc(sizeof(*m));
assert(m);
fscanf(f, "P5 %d %d 255\n", &m->size.x, &m->size.y);
m->a = malloc(sizeof(struct square) * m->size.y * m->size.x);
for(i.y = 0; i.y < m->size.y; i.y++) {
for(i.x = 0; i.x < m->size.x; i.x++) {
Mref(m, i).contents = getchar();
assert(Mref(m, i).contents == 0 || Mref(m, i).contents == 1);
}
}
return m;
507
}
void
saveMaze(struct maze *m, FILE *f)
{
struct position i;
fprintf(f, "P5 %d %d 255\n", m->size.x, m->size.y);
}
for(i.y = 0; i.y < m->size.y; i.y++) {
for(i.x = 0; i.x < m->size.x; i.x++) {
putc(Mref(m, i).contents, f);
}
}
/* how many neighbors of position are PATH? */
int
countNeighbors(const struct maze *m, struct position p)
{
struct position q;
int i;
int count = 0;
for(i = 0; i < DIRECTIONS; i++) {
if(offset(m, &q, p, directions[i]) && Mget(m, q).contents == PATH) {
count++;
}
}
}
return count;
struct position
randomPosition(const struct maze *m)
{
struct position r;
r.x = rand() % m->size.x;
r.y = rand() % m->size.y;
}
return r;
#define PATIENCE_MULTIPLIER (4)
508
/* generate a random connected maze with no cycles */
struct maze *
generateMaze(struct position size)
{
struct maze *m;
struct position r;
struct position i;
size_t countdown;
/* how long to run before we get tired of not making progress */
size_t maxCountdown; /* value to reset countdown to when we make progress */
m = malloc(sizeof(struct maze));
assert(m);
m->size = size;
m->a = malloc(sizeof(struct square) * m->size.x * m->size.y);
assert(m->a);
/* start with all WALL */
for(i.y = 0; i.y < m->size.y; i.y++) {
for(i.x = 0; i.x < m->size.x; i.x++) {
Mref(m, i).contents = WALL;
}
}
/* place a PATH on a random square */
r = randomPosition(m);
Mref(m, r).contents = PATH;
maxCountdown = PATIENCE_MULTIPLIER * size.x * size.y * log(size.x * size.y);
for(countdown = maxCountdown; countdown > 0; countdown--) {
/* pick a random square */
r = randomPosition(m);
/* add if we have exactly one neighbor already in the maze */
if(Mget(m, r).contents == WALL && countNeighbors(m, r) == 1) {
Mref(m, r).contents = PATH;
}
}
}
/* reset countdown */
countdown = maxCountdown;
return m;
509
/* create a cycle by adding one extra PATH square
* that connects two existing squares */
void
mazeAddCycle(struct maze *m)
{
struct position r;
do {
r = randomPosition(m);
} while(Mget(m, r).contents != WALL || countNeighbors(m, r) != 2);
}
Mref(m, r).contents = PATH;
/* Search for a cycle of PATH nodes.
* If found, mark all nodes on the cycle as CYCLE. */
void
mazeSearchForCycle(struct maze *m)
{
struct position root;
/* root of tree */
struct position current; /* what we just popped */
struct position parent ; /* current's parent */
struct position neighbor; /* neighbor to push */
struct position ancestor; /* for filling in CYCLE */
int i;
struct position *queue;
size_t head;
/* where to dequeue */
size_t tail;
/* where to enqueue */
/* this is probably more space than we need */
queue = malloc(sizeof(struct position) * m->size.x * m->size.y);
assert(queue);
head = tail = 0;
/* clear out bookkeeping data */
for(current.y = 0; current.y < m->size.y; current.y++) {
for(current.x = 0; current.x < m->size.x; current.x++) {
Mref(m, current).parent = NO_POSITION;
}
/* probably not necessary but will avoid trouble
* if somebody calls this twice */
if(Mget(m, current).contents != WALL) {
Mref(m, current).contents = PATH;
}
510
}
/* find a root */
/* we don't care what this is, but it can't be a WALL */
do {
root = randomPosition(m);
} while(Mget(m, root).contents != PATH);
/* push root */
Mref(m, root).parent = root;
queue[tail++] = root;
/* now perform the BFS */
/* if we ever find a neighbor that is already in the tree and not our parent,
* we have found our cycle */
while(head < tail) {
current = queue[head++];
parent = Mget(m, current).parent;
/* push all neighbors not already in tree */
/* if one is in the tree, we win */
for(i = 0; i < DIRECTIONS; i++) {
if(offset(m, &neighbor, current, directions[i]) && Mget(m, neighbor).contents ==
/* is it already in the tree? */
if(!eqPosition(Mget(m, neighbor).parent, NO_POSITION)) {
/* we win */
/* cycle consists of all ancestors of neighbor and current
* up to common ancestor */
for(ancestor = neighbor; !eqPosition(ancestor, root); ancestor = Mget(m,
Mref(m, ancestor).contents = CYCLE;
}
/* also mark root */
Mref(m, root).contents = CYCLE;
/* now work up from current */
for(ancestor = current; !eqPosition(ancestor, root); ancestor = Mget(m,
if(Mget(m, ancestor).contents == PATH) {
/* add to the cycle */
Mref(m, ancestor).contents = CYCLE;
} else {
/* this is the common ancestor, which is not root */
/* mark all proper ancestors as PATH */
do {
ancestor = Mget(m, ancestor).parent;
Mref(m, ancestor).contents = PATH;
511
} while(!eqPosition(ancestor, root));
/* can't just break, too many loops */
goto doneWithSearch;
}
}
}
}
}
} else {
Mref(m, neighbor).parent = current;
queue[tail++] = neighbor;
}
doneWithSearch:
free(queue);
}
int
main(int argc, char **argv)
{
struct maze *m;
struct position size = { 80, 60 };
int seed;
switch(argc) {
case 1:
/* sample solution for the assignment */
m = loadMaze(stdin);
mazeSearchForCycle(m);
saveMaze(m, stdout);
destroyMaze(m);
break;
case 4:
/* generate a new test image */
/* usage is ./maze width height seed */
/* if seed is negative, use absolute value and don't put in cycle */
size.x = atoi(argv[1]);
size.y = atoi(argv[2]);
seed = atoi(argv[3]);
srand(seed < 0 ? -seed : seed);
m = generateMaze(size);
if(seed >= 0) { mazeAddCycle(m); }
saveMaze(m, stdout);
destroyMaze(m);
512
}
}
break;
default:
fprintf(stderr, "Usage %s or %s width height seed\n", argv[0], argv[0]);
return 1;
return 0;
examples/2015/hw/9/maze.c
And the Makefile.
9
Common C coding and debugging issues
Here are some notes from a helpful Zoo denizen about debugging programs in
223. (Note: these have been edited slightly from the original.)
Date: Thu, 10 Feb 2005 06:02:23 -0500 (EST)
From: James Terry <james.c.terry@yale.edu>
Subject: 223 coding feedback
Hi Jim,
Several of your students for 223 were up late last night in the Zoo
working on their assignments, and they seemed to be getting hung up on
some coding issues. They were pretty frustrated with some standard
language/debugging issues, so I helped them get the type-checker and
Valgrind to stop yelling at them. I noticed some recurring problems and I
thought I'd pass them on to you. They're pretty standard mistakes, and
I've made most of them myself at some point, either in your class or in
Stan's. It occurred to me that there might be more confused people than
were around last night, and they'd probably appreciate it if someone told
them about these sort of things. I'm not trying to intrude on how you
teach the class; I just thought this feedback would be helpful and I
wasn't sure that it would find its way to you otherwise. I'm sure you've
already taught them several of these, and I understand that sometimes
students just don't pay attention. Still, these seem like good points to
hammer down:
Recurring debugging/coding problems:
1. If you want a debugger/Valgrind to give you line numbers, you must
compile with debugging info turned on, i. e. using the -g[level] flag.
2. On the Zoo, pointers and int's are 4 bytes; char's are 1. (Some
513
people didn't seem to realize that a char* is 4 bytes rather than 1.)
3. I think it would be helpful if you explained why, when using
realloc(), it's a good idea to increase the allocated size
multiplicatively rather than additively. Besides, everyone loves the
"tearing down the hotel" metaphor. :)
4. If they use call-by-reference, they had better make sure that they
keep the same reference. So if they pass in a pointer as an argument to a
function, they shouldn't call malloc() or realloc() on that function.
(Mention the double pointer as another option.) Most people will make
this mistake eventually if no one warns them about it. When I was
learning C, I sort of viewed malloc() and realloc() as magical
memory-increasing functions; that is to say, I didn't think very hard
about the meaning of assigning a pointer to malloc()'s return value. I
suspect some of your students would benefit from having the details
spelled out. (Or spelled out again, if you've already done that.)
5. It's possible to get through a lot (but not all) of the CS major
without learning basic Unix shell syntax, but that's really just wasted
time. Pipes, backgrounding, man, scp, and grep really help even at the
intro level. I realize the purpose of the class isn't to teach Unix, but
in past years I think there was a TA help session on these things. They
don't need to know how to write their own Emacs modes, but the basics
would definitely be helpful.
6. malloc/free -- If Valgrind/gdb reports a problem inside of malloc() or
free(), chances are that the student has *not* discovered a bug in gcc.
(I just heard how one of Zhong's students' proved the correctness of the
libraries for his thesis; that's pretty cool.) Explain why you can't
malloc() twice on the same pointer. Explain how with multidimensional
pointers, you must malloc/free each dimension separately. Drill down the
one-to-one correspondence between malloc'ing and free'ing.
7. Null characters: It's not obvious to newbies that some library functions
require them, particularly null-terminated strings. Tell them that
char*'s must be null terminated in order for <string.h>
functions to work.
8. Off-by-one errors: Tell people that when all else fails, take a hard
look at their comparison operators; i. e. make sure that > shouldn't
really be a >=.
9. This is probably another thing for a help session or workshop, but I
feel almost everyone could benefit from basic software engineering
methodology. Stylistic awkwardness I noticed:
--Using a mess of if-then-else's instead of nested control
structures.
--Using while-loops with iterators that get initialized right
before the beginning of the loop and get incremented with each iteration,
when they could be using for-loops.
--Doing the setup work for a loop right before the beginning of
the loop and then at the end of every iteration, instead of at the
514
beginning of every iteration. Conversely: doing the cleanup work at the
beginning of every iteration and then after the loop has completed.
10. Tell them to use assert(). (Frequently.) When you cover binary
search, using placement of debugging statements in code in order to pin
down an error might be an instructive example.
11. Tell them to use either printf statements or a debugger to debug. I
think they can figure out how to do this on their own, they just need to
be told it's a good idea.
Hopefully some of these suggestions will be helpful. I won't be offended
if you don't pass them on to your students, and I understand if you put a
higher teaching priority on non-coding subjects, but all the things I
mentioned were things I wish someone had told me. I'd be happy to run a
help session on this stuff if you and the TAs are too busy, but otherwise
I wouldn't presume.
Best,
Jim
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