Design and Analysis of Algorithms. A contemporary Perspective

395 Pages·2019·3.48 MB·English

by  Sandeep Sen

Design and Analysis of Algorithms
Thetextintroducesreaderstodifferentparadigmsofcomputinginadditiontothetraditional
approach of discussing fundamental computational problems and design techniques in the
random access machine model. Alternate models of computation including parallel,
cache-sensitive design and streaming algorithms are dealt in separate chapters to underline
thesignificantroleoftheunderlyingcomputationalenvironmentinthealgorithmdesign. The
treatment is made rigorous by demonstrating new measures of performances along with
matchinglowerboundarguments.
The importance of greedy algorithms, divide-and-conquer technique and dynamic
programming is highlighted by additional applications to approximate algorithms that come
withguarantees.Inadditiontoseveralclassicaltechniques,thebookencouragesliberaluseof
probabilistic analysis and randomized techniques that have been pivotal for many recent
advancesinthisarea. Thereisalsoachapterintroducingtechniquesfordimensionreduction
which is at the heart of many interesting applications in data analytics as well as statistical
machinelearning. Whilethesetechniqueshavebeenknownforawhileinothercommunities,
theiradoptionintomainstreamcomputersciencehasbeenrelativelyrecent.
Concepts are discussed with the help of rigorous mathematical proofs, theoretical
explanations and their limitations. Problems have been chosen from a diverse landscape
including graphs, geometry, strings, algebra and optimization. Some exposition of
approximation algorithms has also been included, which has been a very active area of
research in algorithms. Real life applications and numerical problems are spread throughout
the text. The reader is expected to test her understanding by trying out a large number of
exerciseproblemsaccompanyingeverychapter.
Thebookassumesfamiliaritywithbasicdatastructures,tofocusonmorealgorithmicaspects
andtopicsofcontemporaryimportance.
Sandeep Sen has been a faculty member in the Department of Computer Science and
Engineering, Indian Institute of Technology Delhi, India, since 1991 where he is currently a
professor. His research spans randomized algorithms, computational geometry, dynamic
graph algorithms and models of computation. He is a Fellow of the Indian National Science
AcademyandtheIndianAcademyofSciences.
Amit Kumar is a faculty member in the Department of Computer Science and Engineering,
Indian Institute of Technology Delhi, India. His research lies in the area of combinatorial
optimization, with emphasis on problems arising in scheduling, graph theory and
clustering.HeisaFellowofIndianAcademyofSciences,andisarecentrecipientoftheShanti
SwarupBhatnagarAwardforMathematicalSciences.

Design and Analysis of Algorithms
Sandeep Sen
Amit Kumar
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Firstpublished2019
PrintedinIndia
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-PublicationData
Names: Sen,Sandeep,author. |Kumar,Amit,1976-author.
Title: Designandanalysisofalgorithms/SandeepSen,AmitKumar.
Description: NewYork,NY,USA:UniversityPrintingHouse,2019. |Includes
bibliographicalreferencesandindex.
Identifiers: LCCN2019002080|ISBN9781108496827(hardback: alk. paper)|
ISBN9781108721998(paperback: alk. paper)
Subjects: LCSH:Algorithms.
Classification: LCCQA9.58.S4542019|DDC005.1–dc23
LCrecordavailableathttps://lccn.loc.gov/2019002080
ISBN978-1-108-49682-7Hardback
ISBN978-1-108-72199-8Paperback
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To the loving memory of my parents, Sisir Sen and Krishna Sen who nourished and
inspired my academic pursuits and all my teachers who helped me imbibe the beauty
andintricaciesofvarioussubjects
–SandeepSen
Tomyparents
–AmitKumar

Content
ListofFigures xv
ListofTables xix
Preface xxi
Acknowledgments xxv
1 ModelandAnalysis 1
1.1 ComputingFibonacciNumbers 1
1.2 FastMultiplication 3
1.3 ModelofComputation 4
1.4 RandomizedAlgorithms: AShortIntroduction 6
1.4.1 Adifferentflavorofrandomizedalgorithms 8
1.5 OtherComputationalModels 10
1.5.1 Externalmemorymodel 10
1.5.2 Parallelmodel 11
FurtherReading 12
ExerciseProblems 13
2 BasicsofProbabilityandTailInequalities 16
2.1 BasicsofProbabilityTheory 16
2.2 TailInequalities 21
2.3 GeneratingRandomNumbers 26
2.3.1 Generatingarandomvariateforanarbitrarydistribution 26
viii Contents
2.3.2 Generatingrandomvariablesfromasequentialfile 27
2.3.3 Generatingarandompermutation 29
FurtherReading 31
ExerciseProblems 31
3 Warm-upProblems 34
3.1 Euclid’sAlgorithmfortheGreatestCommonDivisor(GCD) 34
3.1.1 ExtendedEuclid’salgorithm 35
3.1.2 Applicationtocryptography 36
3.2 FindingthekthSmallestElement 37
3.2.1 Choosingarandomsplitter 38
3.2.2 Medianofmedians 39
3.3 SortingWords 41
3.4 MergeableHeaps 43
3.4.1 Mergingbinomialheaps 44
3.5 ASimpleSemi-dynamicDictionary 45
3.5.1 Potentialmethodandamortizedanalysis 46
3.6 LowerBounds 47
FurtherReading 50
ExerciseProblems 50
4 OptimizationI:BruteForceandGreedyStrategy 54
4.1 HeuristicSearchApproaches 55
4.1.1 Gametrees* 57
4.2 AFrameworkforGreedyAlgorithms 60
4.2.1 Maximumspanningtree 64
4.2.2 Findingminimumweightsubset 64
4.2.3 Aschedulingproblem 65
4.3 EfficientDataStructuresforMinimumSpanningTreeAlgorithms 66
4.3.1 AsimpledatastructureforUnion–Find 68
4.3.2 Afasterscheme 69
4.3.3 Theslowestgrowingfunction? 71
4.3.4 Puttingthingstogether 72
4.3.5 Pathcompressiononly* 73
4.4 GreedyinDifferentWays 74
4.5 CompromisingwithGreedy 76