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1 AN INTRODUCTION TO COMBINATORIAL PROBLEMS AND TECHNIQUES 1

1.1 The Time to Complete a Project 2

1.2 A Matching Problem 10

1.3 A Knapsack Problem 16

1.4 Algorithms and Their Efficiency 23

Historical Notes 35

Supplementary Exercises 37

Computer Projects 39

Suggested Readings 40

2 SETS,RELATIONS,AND FUNCTIONS 41

2.1 Set Operations 41

2.2 Equivalence Relations 47

2.3 Partial Ordering Relations 54

2.4 Functions 65

2.5 Mathematical Induction 76

2.6 Applications 84

Historical Notes 93

Supplementary Exercises 95

Computer Projects 98

Suggested Readings 98

3 CODING THEORY 99

3.1 Congruence 100

3.2 The Euclidean Algorithm 106

3.3 The RSA Method 113

3.4 Error-Detecting and Error-Correcting Codes 122

3.5 Matrix Codes 132

3.6 Matrix Codes that Correct All Single-Digit Errors 140

Historical Notes 147

Supplementary Exercises 149

Computer Projects 152

Suggested Readings 153

4 GRAPHS 154

4.1 Graphs and Their Representations 154

4.2 Paths and Circuits 164

4.3 Shortest Paths and Distance 181

4.4 Coloring a Graph 193

4.5 Directed Graphs and Multigraphs 202

Historical Notes 219

Supplementary Exercises 220

Computer Projects 226

Suggested Readings 227

5 TREES 228

5.1 Properties of Trees 228

5.2 Spanning Trees 238

5.3 Depth-First Search 253

5.4 Rooted Trees 266

5.5 Binary Trees and Traversals 274

5.6 Optimal Binary Trees and Binary Search Trees 287

Historical Notes 306

Supplementary Exercises 308

Computer Projects 311

Suggested Readings 312

6 MATCHING 313

6.1 Systems of Distinct Representatives 313

6.2 Matchings in Graphs 319

6.3 A Matching Algorithm 327

6.4 Applications of the Algorithm 337

6.5 The Hungarian Method 346

Historical Notes 354

Supplementary Exercises 355

Computer Projects 357

Suggested Readings 357

7 NETWORK FLOWS 358

7.1 Flows and Cuts 358

7.2 A Flow Augmentation Algorithm 369

7.3 The Max-Flow Min-CutTheorem 382

7.4 Flows and Matchings 389

Historical Notes 397

Supplementary Exercises 397

Computer Projects 400

Suggested Readings 401

8 COUNTING TECHNIQUES 402

8.1 Pascal's Triangle and the Binomial Theorem 402

8.2 Three Fundamental Principles 406

8.3 Permutations and Combinations 416

8.4 Arrangements and Selections with Repetitions 421

8.5 Probability 428

8.6 The Principle of Inclusion-Exclusion 434

8.7 Generating Permutations and r-Combinations 445

Historical Notes 452

Supplementary Exercises 453

Computer Projects 456

Suggested Readings 457

9 RECURRENCE RELATIONS AND GENERATING FUNCTIONS 458

9.1 Recurrence Relations 458

9.2 The Method of Iteration 470

9.3 Linear Difference Equations with Constant Coefficients 482

9.4 Analyzing the Efficiency of Algorithms with Recurrence Relations 494

9.5 Counting with Generating Functions 506

9.6 The Algebra of Generating Functions 513

Historical Notes 523

Supplementary Exercises 524

Computer Projects 527

Suggested Readings 528

10 COMBINATORIAL CIRCUITS AND FINITE STATE MACHINES 529

10.1 Logical Gates 529

10.2 Creating Combinatorial Circuits 538

10.3 Karnaugh Maps 546

10.4 Finite State Machines 560

Historical Notes 569

Supplementary Exercises 570

Computer Projects 573

Suggested Readings 573

A AN INTRODUCTION TO LOGICAND PROOF 574

A.1 Statements and Connectives 574

A.2 Logical Equivalence 583

A.3 Methods of Proof 587

Historical Notes 593

Supplementary Exercises 594

Suggested Readings 596

B MATRICES 597

Historical Notes 604

C THE ALGORITHMS IN THIS BOOK 607

BIBLIOGRAPHY 613

ANSWERSTO ODD-NUMBERED EXERCISES 618

PHOTO CREDITS 658

INDEX 659