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This page intentionally left blank AcquisitionsEditor: MattGoldstein ProjectEditor: MaiteSuarez-Rivas ProductionSupervisor: MarilynLloyd MarketingManager: MichelleBrown MarketingCoordinator: JakeZavracky ProjectManagement: WindfallSoftware Composition: WindfallSoftware,usingZzTEX Copyeditor: CarolLeyba TechnicalIllustration: DartmouthPublishing Proofreader: JenniferMcClain Indexer: TedLaux CoverDesign: JoyceCosentinoWells CoverPhoto: ©2005TimLaman/NationalGeographic.Apairofweaverbirdswork togetherontheirnestinAfrica. PrepressandManufacturing: CarolineFell Printer: CourierWestford AccessthelatestinformationaboutAddison-WesleytitlesfromourWorldWideWeb site:http://www.aw-bc.com/computing Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Addison-Wesley was aware of a trademark claim, the designations have been printedininitialcapsorallcaps. The programs and applications presented in this book have been included for their instructionalvalue.Theyhavebeentestedwithcare, butarenotguaranteedforany particularpurpose.Thepublisherdoesnotofferanywarrantiesorrepresentations,nor doesitacceptanyliabilitieswithrespecttotheprogramsorapplications. LibraryofCongressCataloging-in-PublicationData Kleinberg,Jon. Algorithmdesign/JonKleinberg,E´vaTardos.—1sted. p. cm. Includesbibliographicalreferencesandindex. ISBN0-321-29535-8(alk.paper) 1.Computeralgorithms. 2.Datastructures(Computerscience) I.Tardos,E´va. II.Title. QA76.9.A43K542005 005.1—dc22 2005000401 Copyright©2006byPearsonEducation,Inc. For information on obtaining permission for use of material in this work, please submitawrittenrequesttoPearsonEducation,Inc.,RightsandContractDepartment, 75ArlingtonStreet,Suite300,Boston,MA02116orfaxyourrequestto(617)848-7047. All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,mechanical, photocopying,recording,oranytohermediaembodimentsnowknownorhereafterto becomeknown,withoutthepriorwrittenpermissionofthepublisher.Printedinthe UnitedStatesofAmerica. ISBN0-321-29535-8 1 2 3 4 5 6 7 8 9 10-CRW-08 07 06 05 About the Authors Jon Kleinberg is a professor of Computer Science at CornellUniversity.HereceivedhisPh.D.fromM.I.T. in1996.HeistherecipientofanNSFCareerAward, anONRYoungInvestigatorAward,anIBMOutstand- ing Innovation Award, the National Academy of Sci- encesAwardforInitiativesinResearch,researchfel- lowships from the Packard and Sloan Foundations, and teaching awards from the Cornell Engineering CollegeandComputerScienceDepartment. Kleinberg’sresearchiscenteredaroundalgorithms,particularlythosecon- cernedwiththestructureofnetworksandinformation,andwithapplications to information science, optimization, data mining, and computational biol- ogy.Hisworkonnetworkanalysisusinghubsandauthoritieshelpedformthe foundationforthecurrentgenerationofInternetsearchengines. E´vaTardosisaprofessorofComputerScienceatCor- nell University. She received her Ph.D. from Eo¨tvo¨s University in Budapest, Hungary in 1984. She is a member of the American Academy of Arts and Sci- ences,andanACMFellow;sheistherecipientofan NSFPresidentialYoungInvestigatorAward,theFulk- erson Prize, research fellowships from the Guggen- heim, Packard, and Sloan Foundations, and teach- ingawardsfromtheCornellEngineeringCollegeand ComputerScienceDepartment. Tardos’s research interests are focused on the design and analysis of algorithms for problems on graphs or networks. She is most known for her work on network-flow algorithms and approximation algorithms for network problems. Her recent work focuses on algorithmic game theory, an emerging areaconcernedwithdesigningsystemsandalgorithmsforselfishusers. This page intentionally left blank Contents About the Authors v Preface xiii 1 Introduction: Some Representative Problems 1 1.1 AFirstProblem:StableMatching 1 1.2 FiveRepresentativeProblems 12 SolvedExercises 19 Exercises 22 NotesandFurtherReading 28 2 Basics of Algorithm Analysis 29 2.1 ComputationalTractability 29 2.2 AsymptoticOrderofGrowth 35 2.3 Implementing the Stable Matching Algorithm Using Lists and Arrays 42 2.4 ASurveyofCommonRunningTimes 47 2.5 AMoreComplexDataStructure:PriorityQueues 57 SolvedExercises 65 Exercises 67 NotesandFurtherReading 70 3 Graphs 73 3.1 BasicDefinitionsandApplications 73 3.2 GraphConnectivityandGraphTraversal 78 3.3 ImplementingGraphTraversalUsingQueuesandStacks 87 3.4 TestingBipartiteness:AnApplicationofBreadth-FirstSearch 94 3.5 ConnectivityinDirectedGraphs 97 viii Contents 3.6 DirectedAcyclicGraphsandTopologicalOrdering 99 SolvedExercises 104 Exercises 107 NotesandFurtherReading 112 4 Greedy Algorithms 115 4.1 IntervalScheduling:TheGreedyAlgorithmStaysAhead 116 4.2 SchedulingtoMinimizeLateness:AnExchangeArgument 125 4.3 OptimalCaching:AMoreComplexExchangeArgument 131 4.4 ShortestPathsinaGraph 137 4.5 TheMinimumSpanningTreeProblem 142 4.6 Implementing Kruskal’s Algorithm: The Union-Find Data Structure 151 4.7 Clustering 157 4.8 HuffmanCodesandDataCompression 161 ∗ 4.9 Minimum-Cost Arborescences: A Multi-Phase Greedy Algorithm 177 SolvedExercises 183 Exercises 188 NotesandFurtherReading 205 5 Divide and Conquer 209 5.1 AFirstRecurrence:TheMergesortAlgorithm 210 5.2 FurtherRecurrenceRelations 214 5.3 CountingInversions 221 5.4 FindingtheClosestPairofPoints 225 5.5 IntegerMultiplication 231 5.6 ConvolutionsandtheFastFourierTransform 234 SolvedExercises 242 Exercises 246 NotesandFurtherReading 249 6 Dynamic Programming 251 6.1 WeightedIntervalScheduling:ARecursiveProcedure 252 6.2 Principles of Dynamic Programming: Memoization or Iteration overSubproblems 258 6.3 SegmentedLeastSquares:Multi-wayChoices 261 ∗Thestarindicatesanoptionalsection.(SeethePrefaceformoreinformationabouttherelationships amongthechaptersandsections.)