本书充分考虑到初学者的需要，内容、例题、习题都经过精心的挑选和组织，讲解细致，循序渐进，实例贴近日常生活或计算机应用。本书注重算法，且算法描述独立于某种具体的编程语言。教师可根据学生的层次和兴趣来灵活拓展和组织讲解内容。
本书可作为计算机专业或其他相关专业的离散数学教材或教学参考书，也可作为自学者的参考用书
第1章 组合问题与组合技术引论1
1.1 工程完成时间的问题2
1.2 匹配问题10
1.3 背包问题16
1.4 算法及其效率23
历史注记35
补充习题37
计算机题39
推荐读物40
第2章 集合、关系和函数41
2.1 集合运算41
2.2 等价关系47
*2.3 偏序关系54
2.4 函数65
2.5 数学归纳法76
2.6 应用84
历史注记93
补充习题95
计算机题98
推荐读物98
第3章 编码理论99
3.1 同余100
3.2 欧几里得算法106
3.3 RSA方法113
3.4 检错码和纠错码122
3.5 矩阵码132
3.6 单纠错矩阵码140
历史注记147
补充习题149
计算机题152
推荐读物153
第4章 图154
4.1 图及其表示154
4.2 通路和回路164
4.3 最短通路和距离181
4.4 图着色193
4.5 有向图和有向多重图202
历史注记219
补充习题220
计算机题226
推荐读物227
第5章 树228
5.1 树的性质228
5.2 生成树238
5.3 深度优先搜索253
5.4 根树266
5.5 二叉树和遍历274
5.6 最优二叉树和二叉搜索树287
历史注记306
补充习题308
计算机题311
推荐读物312
第6章 匹配313
6.1 相异代表系313
6.2 图中的匹配319
6.3 匹配算法327
6.4 算法的应用337
6.5 匈牙利方法346
历史注记354
补充习题355
计算机题357
推荐读物357
第7章 网络流358
7.1 流和割358
7.2 流增广算法369
7.3 最大流最小割定理382
7.4 流和匹配389
历史注记397
补充习题397
计算机题400
推荐读物401
第8章 计数技术402
8.1 帕斯卡三角形和二项式定理402
8.2 3个基本原理406
8.3 排列和组合416
8.4 允许重复的排列和组合421
8.5 概率428
*8.6 容斥原理434
*8.7 排列和r组合的生成445
历史注记452
补充习题453
计算机题456
推荐读物457
第9章 递推关系与生成函数458
9.1 递推关系458
9.2 迭代法470
9.3 常系数线性差分方程482
*9.4 用递推关系分析算法的效率494
9.5 用生成函数计数506
9.6 生成函数的代数513
历史注记523
补充习题524
计算机题527
推荐读物528
第10章 组合电路和有限状态机529
10.1 逻辑门529
10.2 构造组合电路538
10.3 卡诺图546
10.4 有限状态机560
历史注记569
补充习题570
计算机题573
推荐读物573
附录A 逻辑和证明简介574
A.1 命题和联结词574
A.2 逻辑等价583
A.3 证明的方法587
历史注记593
补充习题594
推荐读物596
附录B 矩阵597
历史注记604
附录C 本书中的算法607
参考文献613
奇数号习题答案618
图片来源658
Contents
1AN INTRODUCTION TO COMBINATORIAL PROBLEMS AND TECHNIQUES1
1.1 The Time to Completea 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 Pathsand Circuits 164
4.3 Shortest Paths and Distance 181
4.4 Coloringa 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-Cut Theorem 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 LOGIC AND 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
ANSWERS TO ODD-NUMBERED EXERCISES 618
PHOTO CREDITS 658
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