From Mathematics to Generic Programming

311 Pages·2014·2.418 MB·English

by  Alexander A. Stepanov [Александр Александрович Степанов], Daniel E. Rose

From Mathematics to
Generic Programming
This page intentionally left blank
From Mathematics to
Generic Programming
Alexander A. Stepanov
Daniel E. Rose
UpperSaddleRiver,NJ(cid:15)Boston(cid:15)Indianapolis(cid:15)SanFrancisco
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LibraryofCongressCataloging-in-PublicationData
Stepanov,AlexanderA.
Frommathematicstogenericprogramming/AlexanderA.Stepanov,DanielE.Rose.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-0-321-94204-3(pbk.:alk.paper)
1. Genericprogramming(Computerscience)—Mathematics. 2. Computeralgorithms. I.Rose,
DanielE.II.Title.
QA76.6245.S742015
005.1'1—dc23
2014034539
Copyright©2015PearsonEducation,Inc.
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irdprinting,August2015
Contents
Acknowledgments ix
AbouttheAuthors xi
Authors’Note xiii
1 WhatThisBookIsAbout 1
1.1 ProgrammingandMathematics 2
1.2 AHistoricalPerspective 2
1.3 Prerequisites 3
1.4 Roadmap 4
2 TheFirstAlgorithm 7
2.1 EgyptianMultiplication 8
2.2 ImprovingtheAlgorithm 11
2.3 ThoughtsontheChapter 15
3 AncientGreekNumberTheory 17
3.1 GeometricPropertiesofIntegers 17
3.2 SiftingPrimes 20
3.3 ImplementingandOptimizingtheCode 23
3.4 PerfectNumbers 28
3.5 ThePythagoreanProgram 32
3.6 AFatalFlawintheProgram 34
3.7 ThoughtsontheChapter 38
4 Euclid’sAlgorithm 41
4.1 AthensandAlexandria 41
4.2 Euclid’sGreatestCommonMeasureAlgorithm 45
4.3 AMillenniumwithoutMathematics 50
4.4 TheStrangeHistoryofZero 51
4.5 RemainderandQuotientAlgorithms 53
4.6 SharingtheCode 57
4.7 ValidatingtheAlgorithm 59
4.8 ThoughtsontheChapter 61
v
vi Contents
5 TheEmergenceofModernNumberTheory 63
5.1 MersennePrimesandFermatPrimes 63
5.2 Fermat’sLittleTheorem 69
5.3 Cancellation 72
5.4 ProvingFermat’sLittleTheorem 77
5.5 Euler’sTheorem 79
5.6 ApplyingModularArithmetic 83
5.7 ThoughtsontheChapter 84
6 AbstractioninMathematics 85
6.1 Groups 85
6.2 MonoidsandSemigroups 89
6.3 SomeTheoremsaboutGroups 92
6.4 SubgroupsandCyclicGroups 95
6.5 Lagrange’sTheorem 97
6.6 TheoriesandModels 102
6.7 ExamplesofCategoricalandNon-categoricalTheories 104
6.8 ThoughtsontheChapter 107
7 DerivingaGenericAlgorithm 111
7.1 UntanglingAlgorithmRequirements 111
7.2 RequirementsonA 113
7.3 RequirementsonN 116
7.4 NewRequirements 118
7.5 TurningMultiplyintoPower 119
7.6 GeneralizingtheOperation 121
7.7 ComputingFibonacciNumbers 124
7.8 ThoughtsontheChapter 127
8 MoreAlgebraicStructures 129
8.1 Stevin,Polynomials,andGCD 129
8.2 GöttingenandGermanMathematics 135
8.3 NoetherandtheBirthofAbstractAlgebra 140
8.4 Rings 142
8.5 MatrixMultiplicationandSemirings 145
8.6 Application: SocialNetworksandShortestPaths 147
8.7 EuclideanDomains 150
8.8 FieldsandOtherAlgebraicStructures 151
8.9 ThoughtsontheChapter 152
9 OrganizingMathematicalKnowledge 155
9.1 Proofs 155
9.2 TheFirstTheorem 159
Contents vii
9.3 EuclidandtheAxiomaticMethod 161
9.4 AlternativestoEuclideanGeometry 164
9.5 Hilbert’sFormalistApproach 167
9.6 PeanoandHisAxioms 169
9.7 BuildingArithmetic 173
9.8 ThoughtsontheChapter 176
10 FundamentalProgrammingConcepts 177
10.1 AristotleandAbstraction 177
10.2 ValuesandTypes 180
10.3 Concepts 181
10.4 Iterators 184
10.5 IteratorCategories,Operations,andTraits 185
10.6 Ranges 188
10.7 LinearSearch 190
10.8 BinarySearch 191
10.9 ThoughtsontheChapter 196
11 PermutationAlgorithms 197
11.1 PermutationsandTranspositions 197
11.2 SwappingRanges 201
11.3 Rotation 204
11.4 UsingCycles 207
11.5 Reverse 212
11.6 SpaceComplexity 215
11.7 Memory-AdaptiveAlgorithms 216
11.8 ThoughtsontheChapter 217
12 ExtensionsofGCD 219
12.1 HardwareConstraintsandaMoreEfficientAlgorithm 219
12.2 GeneralizingStein’sAlgorithm 222
12.3 Bézout’sIdentity 225
12.4 ExtendedGCD 229
12.5 ApplicationsofGCD 234
12.6 ThoughtsontheChapter 234
13 AReal-WorldApplication 237
13.1 Cryptology 237
13.2 PrimalityTesting 240
13.3 TheMiller-RabinTest 243
13.4 TheRSAAlgorithm: HowandWhyItWorks 245
13.5 ThoughtsontheChapter 248
viii Contents
14 Conclusions 249
FurtherReading 251
A Notation 257
B CommonProofTechniques 261
B.1 ProofbyContradiction 261
B.2 ProofbyInduction 262
B.3 ThePigeonholePrinciple 263
C C++forNon-C++Programmers 265
C.1 TemplateFunctions 265
C.2 Concepts 266
C.3 DeclarationSyntaxandTypedConstants 267
C.4 FunctionObjects 268
C.5 Preconditions,Postconditions,andAssertions 269
C.6 STLAlgorithmsandDataStructures 269
C.7 IteratorsandRanges 270
C.8 TypeAliasesandTypeFunctionswithusinginC++11 272
C.9 InitializerListsinC++11 272
C.10LambdaFunctionsinC++11 272
C.11ANoteaboutinline 273
Bibliography 275
Index 281
Acknowledgments
We would like to thank all the people who contributed to making this book a
reality. OurmanagementatA9.comactivelysupportedthisprojectfromthebe-
ginning.BillStasiorinitiatedthecreationofthecoursethisbookisbasedon,and
selectedthetopicfromamongseveraloptionsweoffered. BrianPinkertonnot
onlyattendedthecourse,butalsostronglyencouragedourideaofturningthe
materialintoabook. WealsowouldliketothankMatMarcus,whocollaborated
withAlexonasimilarcourseatAdobein2004–2005.
e other members of the Fundamental Data Structures and Algorithms
forSearchteamplayedimportantrolesthroughouttheprocess. AnilGangolli
helpedshapethecontentofthecourse, RyanErnstprovidedmuchofthepro-
gramminginfrastructure,andParamjitOberoigaveinvaluablefeedbackduring
thewritingstage. Wehaveenjoyedworkingwithallofthemandaregratefulfor
theirinput.
Wearegratefultooureditors, PeterGordonandGregDoench, andtothe
teamofexpertsassembledbyAddison-Wesley,includingmanagingeditorJohn
Fuller, production editor Mary Kesel Wilson, copyeditor Jill Hobbs, and com-
positor/LaTeXexpertLoriHughesforalltheirworkinturningourroughmanu-
scriptintoapolishedbook.
Finally,we’dliketothankthemanyfriends,familymembers,andcolleagues
whoreadearlierdrasofthebookand/orgaveuscomments,corrections,sug-
gestions,advice,orotherhelp: GašperAžman,JohnBanning,CynthiaDwork,
Hernan Epelman, Ryan Ernst, Anil Gangolli, Susan Gruber, Jon Kalb, Robert
Lehr, Dmitry Leshchiner, Tom London, Mark Manasse, Paul McJones, Nico-
lasNicolov,GorNishanov,ParamjitOberoi,SeanParent,FernandoPelliccioni,
JohnReiser,RobertRose,StefanVargyas,andAdamYoung. ebookismuch
betterasaresultoftheircontributions.
Severalcarefulreaderssentcorrectionstoerrorsthatoccurredinthefirstand
secondprintingsofthisbook,orinthedigitaleditions. ankstothefollowing
peopleforreportingtheseerrors: AbutalibAghayev,VladimirBurenkov,Greg
ix