• Get Start Here
• A Touch of Setup
• String Reversal
• Palindromes
• Integer Reversal
• MaxChars
• The Classic FizzBuzz
• Array Chunking
• Anagrams
• Sentence Capitalization
• Printing Steps
• Two Slided Steps - Pyramids
• Fins The Vowels
• Enter the Matrix Spiral
• Runtime Complexity
• Runtime Complexity in Practice - Fibonacci
• The Queue
• Underwater Queue Weaving
• Stack Em Up With Stacks
• Two Become One
• Linked Lists
• Fins The Midpoint
• Circular Lists
• Step Baсk From The Tail
• Building a Tree
• Tree Width with Level Width
• My Best Friend Binary Search Trees
• Validating a Binary Search Tree
• Back to JavaЫcript - Events
• Sorting With BubbleSort
• Sort By Selection
• Ask MergeSort
• Not Done Yet

• 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
  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 772

• 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

• 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

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