Introduction to algorithms

The latest edition of the essential text and professional reference, with substantial new material on such topics as vEB trees, multithreaded algorithms, dynamic programming, and edge-based flow.

Some books on algorithms are rigorous but incomplete; others cover masses of material but lack rigor. Introduction to Algorithms uniquely combines rigor and comprehensiveness. The book covers a broad range of algorithms in depth, yet makes their design and analysis accessible to all levels of readers. Each chapter is relatively self-contained and can be used as a unit of study. The algorithms are described in English and in a pseudocode designed to be readable by anyone who has done a little programming. The explanations have been kept elementary without sacrificing depth of coverage or mathematical rigor.

The first edition became a widely used text in universities worldwide as well as the standard reference for professionals. The second edition featured new chapters on the role of algorithms, probabilistic analysis and randomized algorithms, and linear programming. The third edition has been revised and updated throughout. It includes two completely new chapters, on van Emde Boas trees and multithreaded algorithms, substantial additions to the chapter on recurrence (now called "Divide-and-Conquer"), and an appendix on matrices. It features improved treatment of dynamic programming and greedy algorithms and a new notion of edge-based flow in the material on flow networks. Many exercises and problems have been added for this edition. The international paperback edition is no longer available; the hardcover is available worldwide.

Author Notes

Thomas H. Cormen received a Ph. D. from MIT in 1992. He is an associate professor at Dartmouth College.

Cormen is one of the authors of Introduction to Algorithms.

(Bowker Author Biography)

Preface	p. xiii
I Foundations
Introduction	p. 3
1 The Role of Algorithms in Computing	p. 5
1.1 Algorithms	p. 5
1.2 Algorithms as a technology	p. 11
2 Getting Started	p. 16
2.1 Insertion sort	p. 16
2.2 Analyzing algorithms	p. 23
2.3 Designing algorithms	p. 29
3 Growth of Functions	p. 43
3.1 Asymptotic notation	p. 43
3.2 Standard notations and common functions	p. 53
4 Divide-and-Conquer	p. 65
4.1 The maximum-subarray problem	p. 68
4.2 Strassen's algorithm for matrix multiplication	p. 75
4.3 The substitution method for solving recurrences	p. 83
4.4 The recursion-tree method for solving recurrences	p. 88
4.5 The master method for solving recurrences	p. 93
4.6 Proof of the master theorem	p. 97
5 Probabilistic Analysis and Randomized Algorithms	p. 114
5.1 The hiring problem	p. 114
5.2 Indicator random variables	p. 118
5.3 Randomized algorithms	p. 122
5.4 Probabilistic analysis and further uses of indicator random variables	p. 130
II Sorting and Order Statistics
Introduction	p. 147
6 Heapsort	p. 151
6.1 Heaps	p. 151
6.2 Maintaining the heap property	p. 154
6.3 Building a heap	p. 156
6.4 The heapsort algorithm	p. 159
6.5 Priority queues	p. 162
7 Quicksort	p. 170
7.1 Description of quicksort	p. 170
7.2 Performance of quicksort	p. 174
7.3 A randomized version of quicksort	p. 179
7.4 Analysis of quicksort	p. 180
8 Sorting in Linear Time	p. 191
8.1 Lower bounds for sorting	p. 191
8.2 Counting sort	p. 194
8.3 Radix sort	p. 197
8.4 Bucket sort	p. 200
9 Medians and Order Statistics	p. 213
9.1 Minimum and maximum	p. 214
9.2 Selection in expected linear time	p. 215
9.3 Selection in worst-case linear time	p. 220
III Data Structures
Introduction	p. 229
10 Elementary Data Structures	p. 232
10.1 Stacks and queues	p. 232
10.2 Linked lists	p. 236
10.3 Implementing pointers and objects	p. 241
10.4 Representing rooted trees	p. 246
11 Hash Tables	p. 253
11.1 Direct-address tables	p. 254
11.2 Hash tables	p. 256
11.3 Hash functions	p. 262
11.4 Open addressing	p. 269
11.5 Perfect hashing	p. 277
12 Binary Search Trees	p. 286
12.1 What is a binary search tree?	p. 286
12.2 Querying a binary search tree	p. 289
12.3 Insertion and deletion	p. 294
12.4 Randomly built binary search trees	p. 299
13 Red-Black Trees	p. 308
13.1 Properties of red-black trees	p. 308
13.2 Rotations	p. 312
13.3 Insertion	p. 315
13.4 Deletion	p. 323
14 Augmenting Data Structures	p. 339
14.1 Dynamic order statistics	p. 339
14.2 How to augment a data structure	p. 345
14.3 Interval trees	p. 348
IV Advanced Design and Analysis Techniques
Introduction	p. 357
15 Dynamic Programming	p. 359
15.1 Rod cutting	p. 360
15.2 Matrix-chain multiplication	p. 370
15.3 Elements of dynamic programming	p. 378
15.4 Longest common subsequence	p. 390
15.5 Optimal binary search trees	p. 397
16 Greedy Algorithms	p. 414
16.1 An activity-selection problem	p. 415
16.2 Elements of the greedy strategy	p. 423
16.3 Huffman codes	p. 428
16.4 Matroids and greedy methods	p. 437
16.5 A task-scheduling problem as a matroid	p. 443
17 Amortized Analysis	p. 451
17.1 Aggregate analysis	p. 452
17.2 The accounting method	p. 456
17.3 The potential method	p. 459
17.4 Dynamic tables	p. 463
V Advanced Data Structures
Introduction	p. 481
18 B-Trees	p. 484
18.1 Definition of B-trees	p. 488
18.2 Basic operations on B-trees	p. 491
18.3 Deleting a key from a B-tree	p. 499
19 Fibonacci Heaps	p. 505
19.1 Structure of Fibonacci heaps	p. 507
19.2 Mergeable-heap operations	p. 510
19.3 Decreasing a key and deleting a node	p. 518
19.4 Bounding the maximum degree	p. 523
20 van Emde Boas Trees	p. 531
20.1 Preliminary approaches	p. 532
20.2 A recursive structure	p. 536
20.3 The van Emde Boas tree	p. 545
21 Data Structures for Disjoint Sets	p. 561
21.1 Disjoint-set operations	p. 561
21.2 Linked-list representation of disjoint sets	p. 564
21.3 Disjoint-set forests	p. 568
21.4 Analysis of union by rank with path compression	p. 573
VI Graph Algorithms
Introduction	p. 587
22 Elementary Graph Algorithms	p. 589
22.1 Representations of graphs	p. 589
22.2 Breadth-first search	p. 594
22.3 Depth-first search	p. 603
22.4 Topological sort	p. 612
22.5 Strongly connected components	p. 615
23 Minimum Spanning Trees	p. 624
23.1 Growing a minimum spanning tree	p. 625
23.2 The algorithms of Kruskal and Prim	p. 631
24 Single-Source Shortest Paths	p. 643
24.1 The Bellman-Ford algorithm	p. 651
24.2 Single-source shortest paths in directed acyclic graphs	p. 655
24.3 Dijkstra's algorithm	p. 658
24.4 Difference constraints and shortest paths	p. 664
24.5 Proofs of shortest-paths properties	p. 671
25 All-Pairs Shortest Paths	p. 684
25.1 Shortest paths and matrix multiplication	p. 686
25.2 The Floyd-Warshall algorithm	p. 693
25.3 Johnson's algorithm for sparse graphs	p. 700
26 Maximum Flow	p. 708
26.1 Flow networks	p. 709
26.2 The Ford-Fulkerson method	p. 714
26.3 Maximum bipartite matching	p. 732
26.4 Push-relabel algorithms	p. 736
26.5 The relabel-to-front algorithm	p. 748
VII Selected Topics
Introduction	p. 769
27 Multithreaded Algorithms	p. 772
27.1 The basics of dynamic multithreading	p. 774
27.2 Multithreaded matrix multiplication	p. 792
27.3 Multithreaded merge sort	p. 797
28 Matrix Operations	p. 813
28.1 Solving systems of linear equations	p. 813
28.2 Inverting matrices	p. 827
28.3 Symmetric positive-definite matrices and least-squares approximation	p. 832
29 Linear Programming	p. 843
29.1 Standard and slack forms	p. 850
29.2 Formulating problems as linear programs	p. 859
29.3 The simplex algorithm	p. 864
29.4 Duality	p. 879
29.5 The initial basic feasible solution	p. 886
30 Polynomials and the FFT	p. 898
30.1 Representing polynomials	p. 900
30.2 The DFT and FFT	p. 906
30.3 Efficient FFT implementations	p. 915
31 Number-Theoretic Algorithms	p. 926
31.1 Elementary number-theoretic notions	p. 927
31.2 Greatest common divisor	p. 933
31.3 Modular arithmetic	p. 939
31.4 Solving modular linear equations	p. 946
31.5 The Chinese remainder theorem	p. 950
31.6 Powers of an element	p. 954
31.7 The RSA public-key cryptosystem	p. 958
31.8 Primality testing	p. 965
31.9 Integer factorization	p. 975
32 String Matching	p. 985
32.1 The naive string-matching algorithm	p. 988
32.2 The Rabin-Karp algorithm	p. 990
32.3 String matching with finite automata	p. 995
32.4 The Knuth-Morris-Pratt algorithm	p. 1002
33 Computational Geometry	p. 1014
33.1 Line-segment properties	p. 1015
33.2 Determining whether any pair of segments intersects	p. 1021
33.3 Finding the convex hull	p. 1029
33.4 Finding the closest pair of points	p. 1039
34 NP-Completeness	p. 1048
34.1 Polynomial time	p. 1053
34.2 Polynomial-time verification	p. 1061
34.3 NP-completeness and reducibility	p. 1067
34.4 NP-completeness proofs	p. 1078
34.5 NP-complete problems	p. 1086
35 Approximation Algorithms	p. 1106
35.1 The vertex-cover problem	p. 1108
35.2 The traveling-salesman problem	p. 1111
35.3 The set-covering problem	p. 1117
35.4 Randomization and linear programming	p. 1123
35.5 The subset-sum problem	p. 1128
VIII Appendix: Mathematical Background
Introduction	p. 1143
A Summations	p. 1145
A.1 Summation formulas and properties	p. 1145
A.2 Bounding summations	p. 1149
B Sets, Etc.	p. 1158
B.1 Sets	p. 1158
B.2 Relations	p. 1163
B.3 Functions	p. 1166
B.4 Graphs	p. 1168
B.5 Trees	p. 1173
C Counting and Probability	p. 1183
C.1 Counting	p. 1183
C.2 Probability	p. 1189
C.3 Discrete random variables	p. 1196
C.4 The geometric and binomial distributions	p. 1201
C.5 The tails of the binomial distribution	p. 1208
D Matrices	p. 1217
D.1 Matrices and matrix operations	p. 1217
D.2 Basic matrix properties	p. 1222
Bibliography	p. 1231
Index	p. 1251

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