Algorithmics of nonuniformity: tools and paradigms

591 Pages·2019·4.572 MB·English

by  Hofri, Micha, Mahmoud, Hosam

Algorithmics of
Nonuniformity
Tools and Paradigms
Discrete Mathematics and Its Applications
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Donald L. Kreher
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Combinatorics of Compositions and Words
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Handbook of Geometric Constraint Systems Principles
Meera Sitharam, Audrey St. John, and Jessica Sidman,
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Algorithmics of Nonuniformity: Tools and Paradigms
Micha Hofri and Hosam Mahmoud
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Algorithmics of
Nonuniformity
Tools and Paradigms
Micha Hofri
Worcester Polytechnic Institute
Hosam Mahmoud
George Washington University
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Library of Congress Cataloging-in-Publication Data
Names: Hofri, Micha, author. | Mahmoud, Hosam M. (Hosam Mahmoud), 1954-
author.
Title: Algorithmics of nonuniformity : tools and paradigms / Micha Hofri and
Hosam Mahmoud.
Description: Boca Raton : CRC Press, Taylor & Francis Group, 2018. | Includes
bibliographical references and index.
Identifiers: LCCN 2018006956 | ISBN 9781498750714
Subjects: LCSH: Combinatorial analysis. | Computer algorithms. | Data
structures (Computer science) | Probabilities
Classification: LCC QA164 .H6427 2018 | DDC 511/.6--dc23
LC record available at https://lccn.loc.gov/2018006956
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ToEllenand Fari

Contents
Preface xiii
ListofSymbolsandNotation xvii
1 Introduction 1
1.1 Computingmachinesandmodels . . . . . . . . . . . . . . . . . . 1
1.1.1 Computerelements . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Turingmachines . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Pseudocode . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Asymptoticnotation . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Counting 13
2.1 Generatingfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Multivariategeneratingfunctionsandspecialnumbers . . . 17
2.1.2 Theprincipleofinclusionandexclusion . . . . . . . . . . . 20
2.2 Stirlingnumbers:Combinatorialinterpretation . . . . . . . . . . . 24
2.2.1 Stirlingnumbersofthefirstkind . . . . . . . . . . . . . . . 25
2.2.2 Stirlingnumbersofthesecondkind . . . . . . . . . . . . . 26
2.2.3 Stirlingnumbersandpowers . . . . . . . . . . . . . . . . . 27
2.3 Expansionofgeneratingfunctions . . . . . . . . . . . . . . . . . . 29
2.3.1 Directexpansionoffunctions . . . . . . . . . . . . . . . . 29
2.3.2 Lagrangeinversiontheorem . . . . . . . . . . . . . . . . . 34
2.4 Generatingfunctionsinprobability . . . . . . . . . . . . . . . . . 36
2.4.1 Convolutionofrandomvariables . . . . . . . . . . . . . . . 38
2.5 Generatingfunctionsinthesolutionofrecurrences . . . . . . . . . 40
vii
viii Contents
2.6 Notesandsources . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 SymbolicCalculus 49
3.1 Admissibleoperations . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Thesumandproductrules . . . . . . . . . . . . . . . . . . 51
3.1.2 Labeledcombinatorialoperations . . . . . . . . . . . . . . 56
3.2 Applicationsofthesymboliccalculus . . . . . . . . . . . . . . . . 60
3.2.1 Compositionsofintegers . . . . . . . . . . . . . . . . . . . 60
3.2.2 Positionaltreecounting . . . . . . . . . . . . . . . . . . . 64
3.2.3 Planetreecounting . . . . . . . . . . . . . . . . . . . . . . 65
3.2.4 Rootedorientedtrees . . . . . . . . . . . . . . . . . . . . . 66
3.3 Notesandsources . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 LanguagesandTheirGeneratingFunctions 71
4.1 Regularlanguages . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Finite-stateautomata . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Finite-stateautomataandregularlanguages . . . . . . . . . . . . . 80
4.4 Generatingfunctionsandtheirregularlanguages . . . . . . . . . . 85
4.4.1 Wordequations . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.2 Solutionstowordequations . . . . . . . . . . . . . . . . . 86
4.5 Countingregularlanguages . . . . . . . . . . . . . . . . . . . . . 88
4.5.1 Amatricialalternativetowordequations . . . . . . . . . . 93
4.5.2 Admissibilityconsiderations . . . . . . . . . . . . . . . . . 97
4.6 Notesandsources . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5 ProbabilityinAlgorithmics 105
5.1 Randomvariables . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.1.1 Independenceofdiscreterandomvariables . . . . . . . . . 110
5.1.2 Probabilityspacesforsequencesofrandomvariablesarising
incombinatorialobjects . . . . . . . . . . . . . . . . . . . 112
5.1.3 Illustrationviaruns . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Characteristicfunctions . . . . . . . . . . . . . . . . . . . . . . . 118
Contents ix
5.3 Mixeddistributions . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4.1 Booleinequality . . . . . . . . . . . . . . . . . . . . . . . 123
5.4.2 Chebyshevinequality . . . . . . . . . . . . . . . . . . . . . 125
5.4.3 Markovinequality . . . . . . . . . . . . . . . . . . . . . . 126
5.4.4 Gaussinequality . . . . . . . . . . . . . . . . . . . . . . . 127
5.4.5 Schwarzinequality . . . . . . . . . . . . . . . . . . . . . . 129
5.5 Modesofprobabilisticconvergence . . . . . . . . . . . . . . . . . 131
5.6 Someclassicresultsfromprobabilitytheory . . . . . . . . . . . . . 138
5.6.1 Weakandstronglaws . . . . . . . . . . . . . . . . . . . . 139
5.6.2 Furtherconvergencetheorems . . . . . . . . . . . . . . . . 143
5.7 Centrallimittheorems . . . . . . . . . . . . . . . . . . . . . . . . 144
5.8 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.9 Generatingrandomnumbers . . . . . . . . . . . . . . . . . . . . . 150
5.9.1 Theprobabilityintegraltransform . . . . . . . . . . . . . . 152
5.10 Notesandsources . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6 FunctionalTransforms 161
6.1 Mellintransform . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.1.1 PropertiesoftheMellintransform . . . . . . . . . . . . . . 161
6.1.2 Harmonicsums . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2 Poissonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2.1 Algebraicdepoissonization—uniformdistribution. . . . . . 175
6.2.2 Algebraicdepoissonization—arbitrarydistributions . . . . . 178
6.2.3 AsymptoticsofthePoissontransform . . . . . . . . . . . . 181
6.3 Notesandsources . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7 NonuniformPólyaUrnSchemes 187
7.1 ClassicPólyaurns . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.2 Tenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.3 Pólyaurnswithballactivity . . . . . . . . . . . . . . . . . . . . . 190