, TABLE OF CONTENTS v
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✦ ✦
✦ Table of Contents
Preface ix
Chapter 1. Computer Science: The Mechanization of Abstraction 1
1.1. What This Book Is About 3
1.2. What This Chapter Is About 6
1.3. Data Models 6
1.4. The C Data Model 13
1.5. Algorithms and the Design of Programs 20
1.6. Some C Conventions Used Throughout the Book 22
1.7. Summary of Chapter 1 23
1.8. Bibliographic Notes for Chapter 1 24
Chapter 2. Iteration, Induction, and Recursion 25
2.1. What This Chapter Is About 27
2.2. Iteration 27
2.3. Inductive Proofs 34
2.4. Complete Induction 44
2.5. Proving Properties of Programs 52
2.6. Recursive Definitions 59
2.7. Recursive Functions 69
2.8. Merge Sort: A Recursive Sorting Algorithm 75
2.9. Proving Properties of Recursive Programs 84
2.10. Summary of Chapter 2 87
2.11. Bibliographic Notes for Chapter 2 88
Chapter 3. The Running Time of Programs 89
3.1. What This Chapter Is About 89
3.2. Choosing an Algorithm 90
3.3. Measuring Running Time 91
3.4. Big-Oh and Approximate Running Time 96
3.5. Simplifying Big-Oh Expressions 101
3.6. Analyzing the Running Time of a Program 109
3.7. A Recursive Rule for Bounding Running Time 116
3.8. Analyzing Programs with Function Calls 127
3.9. Analyzing Recursive Functions 132
3.10. Analysis of Merge Sort 136
3.11. Solving Recurrence Relations 144
3.12. Summary of Chapter 3 154
3.13. Bibliographic Notes for Chapter 3 155
Chapter 4. Combinatorics and Probability 156
4.1. What This Chapter Is About 156
4.2. Counting Assignments 157
4.3. Counting Permutations 160
4.4. Ordered Selections 167
,vi TABLE OF CONTENTS
4.5. Unordered Selections 170
4.6. Orderings With Identical Items 178
4.7. Distribution of Objects to Bins 181
4.8. Combining Counting Rules 184
4.9. Introduction to Probability Theory 187
4.10. Conditional Probability 193
4.11. Probabilistic Reasoning 203
4.12. Expected Value Calculations 212
4.13. Some Programming Applications of Probability 215
4.14. Summary of Chapter 4 220
4.15. Bibliographic Notes for Chapter 4 221
Chapter 5. The Tree Data Model 223
5.1. What This Chapter Is About 223
5.2. Basic Terminology 224
5.3. Data Structures for Trees 231
5.4. Recursions on Trees 239
5.5. Structural Induction 248
5.6. Binary Trees 253
5.7. Binary Search Trees 258
5.8. Efficiency of Binary Search Tree Operations 268
5.9. Priority Queues and Partially Ordered Trees 271
5.10. Heapsort: Sorting with Balanced POTs 280
5.11. Summary of Chapter 5 284
5.12. Bibliographic Notes for Chapter 5 285
Chapter 6. The List Data Model 286
6.1. What This Chapter Is About 286
6.2. Basic Terminology 287
6.3. Operations on Lists 291
6.4. The Linked-List Data Structure 293
6.5. Array-Based Implementation of Lists 301
6.6. Stacks 306
6.7. Implementing Function Calls Using a Stack 312
6.8. Queues 318
6.9. Longest Common Subsequences 321
6.10. Representing Character Strings 327
6.11. Summary of Chapter 6 334
6.12. Bibliographic Notes for Chapter 6 335
Chapter 7. The Set Data Model 337
7.1. What This Chapter Is About 337
7.2. Basic Definitions 338
7.3. Operations on Sets 342
7.4. List Implementation of Sets 351
7.5. Characteristic-Vector Implementation of Sets 357
7.6. Hashing 360
7.7. Relations and Functions 366
7.8. Implementing Functions as Data 373
7.9. Implementing Binary Relations 380
, TABLE OF CONTENTS vii
7.10. Some Special Properties of Binary Relations 386
7.11. Infinite Sets 396
7.12. Summary of Chapter 7 401
7.13. Bibliographic Notes for Chapter 7 402
Chapter 8. The Relational Data Model 403
8.1. What This Chapter Is About 403
8.2. Relations 404
8.3. Keys 411
8.4. Primary Storage Structures for Relations 414
8.5. Secondary Index Structures 419
8.6. Navigation among Relations 423
8.7. An Algebra of Relations 428
8.8. Implementing Relational Algebra Operations 436
8.9. Algebraic Laws for Relations 440
8.10. Summary of Chapter 8 449
8.11. Bibliographic Notes for Chapter 8 450
Chapter 9. The Graph Data Model 451
9.1. What This Chapter Is About 451
9.2. Basic Concepts 452
9.3. Implementation of Graphs 459
9.4. Connected Components of an Undirected Graph 466
9.5. Minimal Spanning Trees 478
9.6. Depth-First Search 484
9.7. Some Uses of Depth-First Search 495
9.8. Dijkstra’s Algorithm for Finding Shortest Paths 502
9.9. Floyd’s Algorithm for Shortest Paths 513
9.10. An Introduction to Graph Theory 521
9.11. Summary of Chapter 9 526
9.12. Bibliographic Notes for Chapter 9 527
Chapter 10. Patterns, Automata, and Regular Expressions 529
10.1. What This Chapter Is About 530
10.2. State Machines and Automata 530
10.3. Deterministic and Nondeterministic Automata 536
10.4. From Nondeterminism to Determinism 547
10.5. Regular Expressions 556
10.6. The UNIX Extensions to Regular Expressions 564
10.7. Algebraic Laws for Regular Expressions 568
10.8. From Regular Expressions to Automata 571
10.9. From Automata to Regular Expressions 582
10.10. Summary of Chapter 10 588
10.11. Bibliographic Notes for Chapter 10 589
Chapter 11. Recursive Description of Patterns 591
11.1. What This Chapter Is About 591
11.2. Context-Free Grammars 592
11.3. Languages from Grammars 599
11.4. Parse Trees 602
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✦ ✦
✦ Table of Contents
Preface ix
Chapter 1. Computer Science: The Mechanization of Abstraction 1
1.1. What This Book Is About 3
1.2. What This Chapter Is About 6
1.3. Data Models 6
1.4. The C Data Model 13
1.5. Algorithms and the Design of Programs 20
1.6. Some C Conventions Used Throughout the Book 22
1.7. Summary of Chapter 1 23
1.8. Bibliographic Notes for Chapter 1 24
Chapter 2. Iteration, Induction, and Recursion 25
2.1. What This Chapter Is About 27
2.2. Iteration 27
2.3. Inductive Proofs 34
2.4. Complete Induction 44
2.5. Proving Properties of Programs 52
2.6. Recursive Definitions 59
2.7. Recursive Functions 69
2.8. Merge Sort: A Recursive Sorting Algorithm 75
2.9. Proving Properties of Recursive Programs 84
2.10. Summary of Chapter 2 87
2.11. Bibliographic Notes for Chapter 2 88
Chapter 3. The Running Time of Programs 89
3.1. What This Chapter Is About 89
3.2. Choosing an Algorithm 90
3.3. Measuring Running Time 91
3.4. Big-Oh and Approximate Running Time 96
3.5. Simplifying Big-Oh Expressions 101
3.6. Analyzing the Running Time of a Program 109
3.7. A Recursive Rule for Bounding Running Time 116
3.8. Analyzing Programs with Function Calls 127
3.9. Analyzing Recursive Functions 132
3.10. Analysis of Merge Sort 136
3.11. Solving Recurrence Relations 144
3.12. Summary of Chapter 3 154
3.13. Bibliographic Notes for Chapter 3 155
Chapter 4. Combinatorics and Probability 156
4.1. What This Chapter Is About 156
4.2. Counting Assignments 157
4.3. Counting Permutations 160
4.4. Ordered Selections 167
,vi TABLE OF CONTENTS
4.5. Unordered Selections 170
4.6. Orderings With Identical Items 178
4.7. Distribution of Objects to Bins 181
4.8. Combining Counting Rules 184
4.9. Introduction to Probability Theory 187
4.10. Conditional Probability 193
4.11. Probabilistic Reasoning 203
4.12. Expected Value Calculations 212
4.13. Some Programming Applications of Probability 215
4.14. Summary of Chapter 4 220
4.15. Bibliographic Notes for Chapter 4 221
Chapter 5. The Tree Data Model 223
5.1. What This Chapter Is About 223
5.2. Basic Terminology 224
5.3. Data Structures for Trees 231
5.4. Recursions on Trees 239
5.5. Structural Induction 248
5.6. Binary Trees 253
5.7. Binary Search Trees 258
5.8. Efficiency of Binary Search Tree Operations 268
5.9. Priority Queues and Partially Ordered Trees 271
5.10. Heapsort: Sorting with Balanced POTs 280
5.11. Summary of Chapter 5 284
5.12. Bibliographic Notes for Chapter 5 285
Chapter 6. The List Data Model 286
6.1. What This Chapter Is About 286
6.2. Basic Terminology 287
6.3. Operations on Lists 291
6.4. The Linked-List Data Structure 293
6.5. Array-Based Implementation of Lists 301
6.6. Stacks 306
6.7. Implementing Function Calls Using a Stack 312
6.8. Queues 318
6.9. Longest Common Subsequences 321
6.10. Representing Character Strings 327
6.11. Summary of Chapter 6 334
6.12. Bibliographic Notes for Chapter 6 335
Chapter 7. The Set Data Model 337
7.1. What This Chapter Is About 337
7.2. Basic Definitions 338
7.3. Operations on Sets 342
7.4. List Implementation of Sets 351
7.5. Characteristic-Vector Implementation of Sets 357
7.6. Hashing 360
7.7. Relations and Functions 366
7.8. Implementing Functions as Data 373
7.9. Implementing Binary Relations 380
, TABLE OF CONTENTS vii
7.10. Some Special Properties of Binary Relations 386
7.11. Infinite Sets 396
7.12. Summary of Chapter 7 401
7.13. Bibliographic Notes for Chapter 7 402
Chapter 8. The Relational Data Model 403
8.1. What This Chapter Is About 403
8.2. Relations 404
8.3. Keys 411
8.4. Primary Storage Structures for Relations 414
8.5. Secondary Index Structures 419
8.6. Navigation among Relations 423
8.7. An Algebra of Relations 428
8.8. Implementing Relational Algebra Operations 436
8.9. Algebraic Laws for Relations 440
8.10. Summary of Chapter 8 449
8.11. Bibliographic Notes for Chapter 8 450
Chapter 9. The Graph Data Model 451
9.1. What This Chapter Is About 451
9.2. Basic Concepts 452
9.3. Implementation of Graphs 459
9.4. Connected Components of an Undirected Graph 466
9.5. Minimal Spanning Trees 478
9.6. Depth-First Search 484
9.7. Some Uses of Depth-First Search 495
9.8. Dijkstra’s Algorithm for Finding Shortest Paths 502
9.9. Floyd’s Algorithm for Shortest Paths 513
9.10. An Introduction to Graph Theory 521
9.11. Summary of Chapter 9 526
9.12. Bibliographic Notes for Chapter 9 527
Chapter 10. Patterns, Automata, and Regular Expressions 529
10.1. What This Chapter Is About 530
10.2. State Machines and Automata 530
10.3. Deterministic and Nondeterministic Automata 536
10.4. From Nondeterminism to Determinism 547
10.5. Regular Expressions 556
10.6. The UNIX Extensions to Regular Expressions 564
10.7. Algebraic Laws for Regular Expressions 568
10.8. From Regular Expressions to Automata 571
10.9. From Automata to Regular Expressions 582
10.10. Summary of Chapter 10 588
10.11. Bibliographic Notes for Chapter 10 589
Chapter 11. Recursive Description of Patterns 591
11.1. What This Chapter Is About 591
11.2. Context-Free Grammars 592
11.3. Languages from Grammars 599
11.4. Parse Trees 602
