Table of Contents

I Foundations

    Introduction 3

    1 The Role of Algorithms in Computing 5

        1.1 Algorithms 5

        1.2 Algorithms as a technology 11

    2 Getting Started 16

        2.1 Insertion sort 16

        2.2 Analyzing algorithms 23

        2.3 Designing algorithms 29

    3 Growth of Functions 43

        3.1 Asymptotic notation 43

        3.2 Standard notations and common functions 53

    4 Divide-and-Conquer 65

        4.1 The maximum-subarray problem 68

        4.2 Strassen's algorithm for matrix multiplication 75

        4.3 The substitution method for solving recurrences 83

        4.4 The recursion-tree method for solving recurrences 88

        4.5 The master method for solving recurrences 93

        4.6 Proof of the master theorem 97

    5 Probabilistic Analysis and Randomized Algorithms 114

        5.1 The hiring problem 114

        5.2 Indicator random variables 118

        5.3 Randomized algorithms 122

        5.4 Probabilistic analysis and further uses of indicator random variables 130

II Sorting and Order Statistics

    Introduction 147

    6 Heapsort 151

        6.1 Heaps 151

        6.2 Maintaining the heap property 154

        6.3 Building a heap 156

        6.4 The heapsort algorithm 159

        6.5 Priority queues 162

    7 Quicksort 170

        7.1 Description of quicksort 170

        7.2 Performance of quicksort 174

        7.3 A randomized version of quicksort 179

        7.4 Analysis of quicksort 180

    8 Sorting in Linear Time 191

        8.1 Lower bounds for sorting 191

        8.2 Counting sort 194

        8.3 Radix sort 197

        8.4 Bucket sort 200

    9 Medians and Order Statistics 213

        9.1 Minimum and maximum 214

        9.2 Selection in expected linear time 215

        9.3 Selection in worst-case linear time 220

III Data Structures

    Introduction 229

    10 Elementary Data Structures 232

        10.1 Stacks and queues 232

        10.2 Linked lists 236

        10.3 Implementing pointers and objects 241

        10.4 Representing rooted trees 246

    11 Hash Tables 253

        11.1 Direct-address tables 254

        11.2 Hash tables 256

        11.3 Hash functions 262

        11.4 Open addressing 269

        11.5 Perfect hashing 277

    12 Binary Search Trees 286

        12.1 What is a binary search tree? 286

        12.2 Querying a binary search tree 289

        12.3 Insertion and deletion 294

        12.4 Randomly built binary search trees 299

    13 Red-Black Trees 308

        13.1 Properties of red-black trees 308

        13.2 Rotations 312

        13.3 Insertion 315

        13.4 Deletion 323

    14 Augmenting Data Structures 339

        14.1 Dynamic order statistics 339

        14.2 How to augment a data structure 345

        14.3 Interval trees 348

IV Advanced Design and Analysis Techniques

    Introduction 357

    15 Dynamic Programming 359

        15.1 Rod cutting 360

        15.2 Matrix-chain multiplication 370

        15.3 Elements of dynamic programming 378

        15.4 Longest common subsequence 390

        15.5 Optimal binary search trees 397

    16 Greedy Algorithms 414

        16.1 An activity-selection problem 415

        16.2 Elements of the greedy strategy 423

        16.3 Huffman codes 428

        16.4 Matroids and greedy methods 437

        16.5 A task-scheduling problem as a matroid 443

    17 Amortized Analysis 451

        17.1 Aggregate analysis 452

        17.2 The accounting method 456

        17.3 The potential method 459

        17.4 Dynamic tables 463

V Advanced Data Structures

    Introduction 481

    18 B-Trees 484

        18.1 Definition of B-trees 488

        18.2 Basic operations on B-trees 491

        18.3 Deleting a key from a B-tree 499

    19 Fibonacci Heaps 505

        19.1 Structure of Fibonacci heaps 507

        19.2 Mergeable-heap operations 510

        19.3 Decreasing a key and deleting a node 518

        19.4 Bounding the maximum degree 523

    20 van Emde Boas Trees 531

        20.1 Preliminary approaches 532

        20.2 A recursive structure 536

        20.3 The van Emde Boas tree 545

    21 Data Structures for Disjoint Sets 561

        21.1 Disjoint-set operations 561

        21.2 Linked-list representation of disjoint sets 564

        21.3 Disjoint-set forests 568

        21.4 Analysis of union by rank with path compression 573

VI Graph Algorithms

    Introduction 587

    22 Elementary Graph Algorithms 589

        22.1 Representations of graphs 589

        22.2 Breadth-first search 594

        22.3 Depth-first search 603

        22.4 Topological sort 612

        22.5 Strongly connected components 615

    23 Minimum Spanning Trees 624

        23.1 Growing a minimum spanning tree 625

        23.2 The algorithms of Kruskal and Prim 631

    24 Single-Source Shortest Paths 643

        24.1 The Bellman-Ford algorithm 651

        24.2 Single-source shortest paths in directed acyclic graphs 655

        24.3 Dijkstra's algorithm 658

        24.4 Difference constraints and shortest paths 664

        24.5 Proofs of shortest-paths properties 671

    25 All-Pairs Shortest Paths 684

        25.1 Shortest paths and matrix multiplication 686

        25.2 The Floyd-Warshall algorithm 693

        25.3 Johnson's algorithm for sparse graphs 700

    26 Maximum Flow 708

        26.1 Flow networks 709

        26.2 The Ford-Fulkerson method 714

        26.3 Maximum bipartite matching 732

        26.4 Push-relabel algorithms 736

        26.5 The relabel-to-front algorithm 748

VII Selected Topics

    Introduction 769

    27 Multithreaded Algorithms

        27.1 The basics of dynamic multithreading 774

        27.2 Multithreaded matrix multiplication 792

        27.3 Multithreaded merge sort 797

    28 Matrix Operations 813

        28.1 Solving systems of linear equations 813

        28.2 Inverting matrices 827

        28.3 Symmetric positive-definite matrices and least-squares approximation 832

    29 Linear Programming 843

        29.1 Standard and slack forms 850

        29.2 Formulating problems as linear programs 859

        29.3 The simplex algorithm 864

        29.4 Duality 879

        29.5 The initial basic feasible solution 886

    30 Polynomials and the FFT 898

        30.1 Representing polynomials 900

        30.2 The DFT and FFT 906

        30.3 Efficient FFT implementations 915

    31 Number-Theoretic Algorithms 926

        31.1 Elementary number-theoretic notions 927

        31.2 Greatest common divisor 933

        31.3 Modular arithmetic 939

        31.4 Solving modular linear equations 946

        31.5 The Chinese remainder theorem 950

        31.6 Powers of an element 954

        31.7 The RSA public-key cryptosystem 958

        31.8 Primality testing 965

        31.9 Integer factorization 975

    32 String Matching 985

        32.1 The naive string-matching algorithm 988

        32.2 The Rabin-Karp algorithm 990

        32.3 String matching with finite automata 995

        32.4 The Knuth-Morris-Pratt algorithm 1002

    33 Computational Geometry 1014

        33.1 Line-segment properties 1015

        33.2 Determining whether any pair of segments intersects 1021

        33.3 Finding the convex hull 1029

        33.4 Finding the closest pair of points 1039

    34 NP-Completeness 1048

        34.1 Polynomial time 1053

        34.2 Polynomial-time verification 1061

        34.3 NP-completeness and reducibility 1067

        34.4 NP-completeness proofs 1078

        34.5 NP-complete problems 1086

    35 Approximation Algorithms 1106

        35.1 The vertex-cover problem 1108

        35.2 The traveling-salesman problem 1111

        35.3 The set-covering problem 1117

        35.4 Randomization and linear programming 1123

        35.5 The subset-sum problem 1128

VIII Appendix: Mathematical Background

    Introduction 1143

    A Summations 1145

        A.1 Summation formulas and properties 1145

        A.2 Bounding summations 1149

    B Sets, Etc. 1158

        B.1 Sets 1158

        B.2 Relations 1163

        B.3 Functions 1166

        B.4 Graphs 1168

        B.5 Trees 1173

    C Counting and Probability 1183

        C.1 Counting 1183

        C.2 Probability 1189

        C.3 Discrete random variables 1196

        C.4 The geometric and binomial distributions 1201

        C.5 The tails of the binomial distribution 1208

    D Matrices 1217

        D.1 Matrices and matrix operations 1217

        D.2 Basic matrix properties 1222

    Bibliography 1231

    Index

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