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Ⅰ Foundations 3

Introduction 3

1 The Role of Algorithms in Computing 5

1.1 Algorithms 5

1.2 Algorithms as a technology 10

2 Getting Started 15

2.1 Insertion sort 15

2.2 Analyzing algorithms 21

2.3 Designing algorithms 27

3 Growth of Functions 41

3.1 Asymptotic notation 41

3.2 Standard notations and common functions 51

4 Recurrences 62

4.1 The substitution method 63

4.2 The recursion-tree method 67

4.3 The master method 73

4.4 Proof of the master theorem 76

5 Probabilistic Analysis and Randomized Algorithms 91

5.1 The hiring Problem 91

5.2 Indicator random variables 94

5.3 Randomized algorithms 99

5.4 Probabilistic analysis and further uses of indicator random variables 106

Ⅱ Sorting and Order Statistics 123

Introduction 123

6 Heapsort 127

6.1 Heaps 127

6.2 Maintaining the heap property 130

6.3 Building a heap 132

6.4 The heapsort algorithm 135

6.5 Priority queues 138

7 Quicksort 145

7.1 Description of quicksort 145

7.2 Performance of quicksort 149

7.3 A randomized version of quicksort 153

7.4 Analysis of quicksort 155

8 Sorting in Linear Time 165

8.1 Lower bounds for sorting 165

8.2 Counting sort 168

8.3 Radix sort 170

8.4 Bucket sort 174

9 Medians and Order Statistics 183

9.1 Minimum and maximum 184

9.2 Selection in expected linear time 185

9.3 Selection in Worst-case linear time 189

Ⅲ Data Structures 197

Introduction 197

10 Elementary Data Structures 200

10.1 Stacks and queues 200

10.2 Linked lists 204

10.3 Implementing pointers and objects 209

10.4 Represeting rooted trees 214

11 Hash Tables 221

11.1 Direct-address tables 222

11.2 Hash tables 224

11.3 Hash functions 229

11.4 Open addressing 237

11.5 Perfect hashing 245

12 Binary Search Trees 253

12.1 What is a binary search tree? 253

12.2 Querying a binary search tree 256

12.3 Insertion and deletion 261

12.4 Randomly built binary search trees 265

13 Red-Black Trees 273

13.1 Properties of red-black trees 273

13.2 Rotations 277

13.3 Insertion 280

13.4 Deletion 288

14 Augmenting Data Structures 302

14.1 Dynamic order statistics 302

14.2 How to augment a data structure 308

14.3 Interval trees 311

Ⅳ Advanced Design and Analysis Techniques 321

Introduction 321

15 Dynamic Programming 323

15.1 Assembly-line scheduling 324

15.2 Matrix-chain multiplication 331

15.3 Elements of dynamic programming 339

15.4 Longest common subsequence 350

15.5 Optimal binary search trees 356

16 Greedy Algorithms 370

16.1 An activity-Selection Problem 371

16.2 Elements of the greedy Strategy 379

16.3 Huffman codes 385

16.4 Theoretical foundations for greedy methods 393

16.5 A task-scheduling problem 399

17 Amortized Analysis 405

17.1 Aggregate analysis 406

17.2 The accounting method 410

17.3 The Potential method 412

17.4 Dynamic tables 416

Ⅴ Advanced Data Structures 431

Introduction 431

18 B-Trees 434

18.1 Definition of B-trees 438

18.2 Basic operations on B-trees 441

18.3 Deleting a key form a B-tree 449

19 Binomial Heaps 445

19.1 Binomial trees and binomial heaps 457

19.2 Operations on binomial heaps 461

20 Fibonacci Heaps 476

20.1 Structure of Fibonacci heaps 477

20.2 Mergeable-heap Operations 479

20.3 Decreasing a key and deleting a node 489

20.4 Bounding the maximum degree 493

21 Data Structures for Disjoint Sets 498

21.1 Disjoint-set Operations 498

21.2 Linked-list representation of disjoint sets 501

21.3 Disjoint-set forests 505

21.4 Analysis of union by rank with path compression 509

Ⅵ Graph Algorithms 525

Introduction 525

22 Elementary Graph Algorithms 527

22.1 Representations of graphs 527

22.2 Breadth-first search 531

22.3 Depth-fist search 540

22.4 Topological sort 549

22.5 Strongly connected components 552

23 Minimum Spanning Trees 561

23.1 Growing a minimum spanning tree 562

23.2 The algorithms of kruskal and prim 567

24 Single-Source Shortest Paths 580

24.1 The Bellman-Ford algorithm 588

24.2 Single-source shortest paths in directed acyclic graphs 592

24.3 Dijkstra s algorithm 595

24.4 Difference constraints and shortest paths 601

24.5 Proofs of shortest-paths properties 607

25 All-Pairs Shortest Paths 620

25.1 Shortest Paths and matrix multiplication 622

25.2 The Floyed-Warshall algorithm 629

25.3 Johnson s algorithm for sparse graphs 636

26 Maximum Flow 643

26.1 Flow networks 644

26.2 The Ford-Fulkerson method 651

26.3 Maximum bipartite matching 664

26.4 Push-relabel algorithms 669

26.5 The relabel-to-front algorithm 681

Ⅶ Selected Topics 701

Introduction 701

27 Sorting Networks 704

27.1 Comparison networks 704

27.2 The zero-one Principle 709

27.3 A bitonic sorting network 712

27.4 A merging network 716

27.5 A sorting network 719

28 Matrix Operations 725

28.1 Properties of matrices 725

28.2 Strassen s algorithm for matrix multiplication 735

28.3 Solving systems of linear equations 742

28.4 Inverting matrices 755

28.5 Symmetric positive-definite matrices and least-squares approximation 760

29 Linear Programming 770

29.1 Standard and slack forms 777

29.2 Formulating problems as linear programs 785

29.3 The simplex algorithm 790

29.4 Duality 804

29.5 The initial basic feasible solution 811

30 Polynomials and the FFT 822

30.1 Representation of polynomials 824

30.2 The DFT and FFT 830

30.3 Efficient FFT implementations 839

31 Number-Theoretic Algorithms 849

31.1 Elementary number-theoretic notions 850

31.2 Greatest common divisor 856

31.3 Modular arithmetic 862

31.4 Solving modular linear equations 869

31.5 The Chinese remainder theorem 873

31.6 Powers of an element 876

31.7 The RSA public-Key cryptosystem 881

31.8 Primality testing 887

31.9 Integer factorization 896

32 String Matching 906

32.1 The naive String-matching algorithm 909

32.2 The Rabin-Karp algorithm 911

32.3 String matching with finite automata 916

32.4 The Knuth-Morris-Pratt algorithm 923

33 Computational Geometry 933

33.1 Line-segment properties 934

33.2 Determining whether any pair of segments intersects 940

33.3 Finding the convex hull 947

33.4 Finding the closest pair of points 957

34 NP-Completeness 966

34.1 Polynomial time 971

34.2 Polynomial-time verification 979

34.3 NP-completeness and reducibility 984

34.4 NP-completeness proofs 995

34.5 NP-complete problems 1003

35 Approximation Algorithms 1022

35.1 The vertex-cover problem 1024

35.2 The traveling-salesman problem 1027

35.3 The set-covering problem 1033

35.4 Randomization and linear programming 1039

35.5 The subset-sum problem 1043

Ⅷ Appendix: Mathematical Background 1057

Introduction 1057

A Summations 1058

A.1 Summation formulas and Properties 1058

A.2 Bounding summations 1062

B Sets.Etc 1070

B.1 Sets 1070

B.2 Relations 1075

B.3 Functions 1077

B.4 Graphs 1080

B.5 Trees 1085

C Counting and Probability 1094

C.1 Counting 1094

C.2 Probability 1100

C.3 Discrete random variables 1106

C.4 The geometric and binomial distributions 1112

C.5 The tails of the binomial distribution 1118

Bibliography 1127

Index 1145