Introduction to Algorithms, Third Edition

1313 Pages·2010·5.21 MB·English

T H O M A S H. CORMEN
CHARLES E. L E I SERSON
R O N A L D L. RIVEST
C L I F F O R D STEIN
I N T R O D U C T I O N T O
A L G O R I T H M S
T H I R D E D I T I O N
Introduction to Algorithms
Third Edition

ThomasH.Cormen
Charles E. Leiserson
RonaldL. Rivest
CliffordStein
Introduction to Algorithms
Third Edition
TheMITPress
Cambridge,Massachusetts London, England
(cid:2)c 2009MassachusettsInstituteofTechnology
Allrightsreserved.Nopartofthisbookmaybereproducedinanyformorbyanyelectronicormechanicalmeans
(includingphotocopying,recording,orinformationstorageandretrieval)withoutpermissioninwritingfromthe
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Forinformationaboutspecialquantitydiscounts,pleaseemailspecial [email protected].
ThisbookwassetinTimesRomanandMathtimePro2bytheauthors.
PrintedandboundintheUnitedStatesofAmerica.
LibraryofCongressCataloging-in-PublicationData
Introductiontoalgorithms/ThomasH.Cormen...[etal.].—3rded.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-262-03384-8(hardcover:alk.paper)—ISBN978-0-262-53305-8(pbk.:alk.paper)
1.Computerprogramming. 2.Computeralgorithms. I.Cormen,ThomasH.
QA76.6.I5858 2009
005.1—dc22
2009008593
10 9 8 7 6 5 4 3 2
Contents
Preface xiii
I Foundations
Introduction 3
1 TheRoleofAlgorithmsinComputing 5
1.1 Algorithms 5
1.2 Algorithmsasatechnology 11
2 GettingStarted 16
2.1 Insertion sort 16
2.2 Analyzing algorithms 23
2.3 Designing algorithms 29
3 GrowthofFunctions 43
3.1 Asymptoticnotation 43
3.2 Standardnotations andcommonfunctions 53
4 Divide-and-Conquer 65
4.1 Themaximum-subarray problem 68
4.2 Strassen’salgorithm formatrixmultiplication 75
4.3 Thesubstitution methodforsolving recurrences 83
4.4 Therecursion-tree methodforsolvingrecurrences 88
4.5 Themastermethodforsolving recurrences 93
? 4.6 Proofofthemastertheorem 97
5 ProbabilisticAnalysisandRandomizedAlgorithms 114
5.1 Thehiringproblem 114
5.2 Indicator randomvariables 118
5.3 Randomizedalgorithms 122
? 5.4 Probabilistic analysis andfurtherusesofindicator randomvariables
130
vi Contents
II Sorting and OrderStatistics
Introduction 147
6 Heapsort 151
6.1 Heaps 151
6.2 Maintaining theheapproperty 154
6.3 Buildingaheap 156
6.4 Theheapsort algorithm 159
6.5 Priorityqueues 162
7 Quicksort 170
7.1 Description ofquicksort 170
7.2 Performanceofquicksort 174
7.3 Arandomized versionofquicksort 179
7.4 Analysisofquicksort 180
8 SortinginLinearTime 191
8.1 Lowerbounds forsorting 191
8.2 Countingsort 194
8.3 Radixsort 197
8.4 Bucketsort 200
9 MediansandOrderStatistics 213
9.1 Minimumandmaximum 214
9.2 Selectioninexpected lineartime 215
9.3 Selectioninworst-case lineartime 220
III DataStructures
Introduction 229
10 ElementaryDataStructures 232
10.1 Stacksandqueues 232
10.2 Linkedlists 236
10.3 Implementing pointersandobjects 241
10.4 Representing rootedtrees 246
11 HashTables 253
11.1 Direct-address tables 254
11.2 Hashtables 256
11.3 Hashfunctions 262
11.4 Openaddressing 269
? 11.5 Perfecthashing 277
Contents vii
12 BinarySearchTrees 286
12.1 Whatisabinarysearchtree? 286
12.2 Queryingabinarysearchtree 289
12.3 Insertion anddeletion 294
? 12.4 Randomlybuiltbinarysearchtrees 299
13 Red-BlackTrees 308
13.1 Propertiesofred-black trees 308
13.2 Rotations 312
13.3 Insertion 315
13.4 Deletion 323
14 AugmentingDataStructures 339
14.1 Dynamicorderstatistics 339
14.2 Howtoaugmentadatastructure 345
14.3 Intervaltrees 348
IV Advanced Designand Analysis Techniques
Introduction 357
15 DynamicProgramming 359
15.1 Rodcutting 360
15.2 Matrix-chain multiplication 370
15.3 Elementsofdynamicprogramming 378
15.4 Longestcommonsubsequence 390
15.5 Optimalbinarysearchtrees 397
16 GreedyAlgorithms 414
16.1 Anactivity-selection problem 415
16.2 Elementsofthegreedy strategy 423
16.3 Huffmancodes 428
? 16.4 Matroidsandgreedymethods 437
? 16.5 Atask-scheduling problem asamatroid 443
17 AmortizedAnalysis 451
17.1 Aggregateanalysis 452
17.2 Theaccounting method 456
17.3 Thepotential method 459
17.4 Dynamictables 463
viii Contents
V Advanced DataStructures
Introduction 481
18 B-Trees 484
18.1 DefinitionofB-trees 488
18.2 Basicoperations onB-trees 491
18.3 DeletingakeyfromaB-tree 499
19 FibonacciHeaps 505
19.1 StructureofFibonacciheaps 507
19.2 Mergeable-heap operations 510
19.3 Decreasingakeyanddeleting anode 518
19.4 Boundingthemaximumdegree 523
20 vanEmdeBoasTrees 531
20.1 Preliminaryapproaches 532
20.2 Arecursivestructure 536
20.3 ThevanEmdeBoastree 545
21 DataStructuresforDisjointSets 561
21.1 Disjoint-set operations 561
21.2 Linked-list representation ofdisjointsets 564
21.3 Disjoint-set forests 568
? 21.4 Analysisofunionbyrankwithpathcompression 573
VI GraphAlgorithms
Introduction 587
22 ElementaryGraphAlgorithms 589
22.1 Representations ofgraphs 589
22.2 Breadth-firstsearch 594
22.3 Depth-firstsearch 603
22.4 Topological sort 612
22.5 Stronglyconnected components 615
23 MinimumSpanningTrees 624
23.1 Growingaminimumspanning tree 625
23.2 ThealgorithmsofKruskalandPrim 631
Contents ix
24 Single-SourceShortestPaths 643
24.1 TheBellman-Fordalgorithm 651
24.2 Single-source shortestpathsindirected acyclicgraphs 655
24.3 Dijkstra’salgorithm 658
24.4 Differenceconstraints andshortestpaths 664
24.5 Proofsofshortest-paths properties 671
25 All-PairsShortestPaths 684
25.1 Shortestpathsandmatrixmultiplication 686
25.2 TheFloyd-Warshall algorithm 693
25.3 Johnson’s algorithm forsparsegraphs 700
26 MaximumFlow 708
26.1 Flownetworks 709
26.2 TheFord-Fulkerson method 714
26.3 Maximumbipartitematching 732
? 26.4 Push-relabel algorithms 736
? 26.5 Therelabel-to-front algorithm 748
VII Selected Topics
Introduction 769
27 MultithreadedAlgorithms 772
27.1 Thebasicsofdynamicmultithreading 774
27.2 Multithreaded matrixmultiplication 792
27.3 Multithreaded mergesort 797
28 MatrixOperations 813
28.1 Solvingsystemsoflinearequations 813
28.2 Inverting matrices 827
28.3 Symmetricpositive-definite matricesandleast-squares approximation
832
29 LinearProgramming 843
29.1 Standardandslackforms 850
29.2 Formulatingproblemsaslinearprograms 859
29.3 Thesimplexalgorithm 864
29.4 Duality 879
29.5 Theinitialbasicfeasiblesolution 886