Analytic combinatorics

698 Pages·2005·10.478 MB·English

by  Flajolet Ph., Sedgewick R.

ANALYTIC COMBINATORICS
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⋆
(Chapters I, II, III, IV, V, VI, VII, VIII, IX )
PHILIPPE FLAJOLET & ROBERT SEDGEWICK
AlgorithmsProject DepartmentofComputerScience
INRIARocquencourt PrincetonUniversity
78153LeChesnay Princeton,NJ08540
France USA
ZerothEdition.FifthPrinting:October1,2005(correctedOctober2,2005)
(ThistemporaryversionwillexpireonNovember15,2005)
♥♥♥♥♥♥
2
InthiseditionofOctober2,2005:
— Chapters0,1,2,3,4areinclose-to-finalform.
— Chapter9isstillinpreliminaryform
— Chapter10,inpreparationisnotincluded.
Otherschapterareinintermediateform,waitingtoberevised.
Also,bothlongandshortformsofconstructionnamesarecurrentlyused,withonedestinedtobecomethe
standardeventually.Forunlabelledclasses,thedictionaryis
S SEQ, M MSET, P PSET, C CYC,
≡ ≡ ≡ ≡
whileforlabelledclasses,
S SEQ, P SET, C CYC.
≡ ≡ ≡
i
PREFACE
ANALYTICCOMBINATORICSaimsatpredictingpreciselythepropertiesoflarge
structured combinatorial configurations, through an approach based extensively on
analyticmethods.Generatingfunctionsarethecentralobjectsofthetheory.
Analyticcombinatoricsstartsfromanexactenumerativedescriptionofcombina-
torialstructuresbymeansofgeneratingfunctions,whichmaketheirfirstappearance
aspurelyformalalgebraicobjects. Next, generatingfunctionsare interpretedasan-
alytic objects, that is, as mappings of the complex plane into itself. Singularities
determinea function’scoefficients in asymptotic form and lead to precise estimates
forcountingsequences. Thischainappliestoalargenumberofproblemsofdiscrete
mathematicsrelativetowords,trees,permutations,graphs,andsoon.Asuitableadap-
tationofthemethodsalsoopensthewaytothequantitativeanalysisofcharacteristic
parametersoflargerandomstructures,viaaperturbationalapproach.
Analyticcombinatoricscanaccordinglybeorganizedbasedonthreecomponents:
SymbolicMethodsdevelopssystematicrelationsbetweensomeofthemajor
constructions of discrete mathematics and operations on generating func-
tionswhichexactlyencodecountingsequences.
ComplexAsymptoticselaboratesacollectionofmethodsbywhichonecan
extract asymptotic counting information from generating functions, once
theseareviewedasanalytictransformationsofthecomplexdomain.Singu-
laritiesthenappeartobeakeydeterminantofasymptoticbehaviour.
RandomStructuresconcernsitselfwithprobabilisticpropertiesoflargeran-
domstructures. Whichpropertiesholdwith highprobability? Whichlaws
govern randomness in large objects? In the context of analytic combina-
torics, these questionsare treated by a deformation(addingauxiliaryvari-
ables) and a perturbation(examiningthe effectof small variationsof such
auxiliaryvariables)ofthestandardenumerativetheory.
THEAPPROACHtoquantitativeproblemsofdiscretemathematicsprovidedbyan-
alyticcombinatoricscanbeviewedasanoperationalcalculusforcombinatorics.The
present book exposes this view by means of a very large number of examples con-
cerningclassicalcombinatorialstructures—mostnotably,words,trees,permutations,
andgraphs. Theeventualgoalisaneffectivewayofquantifyingmetricpropertiesof
largerandomstructures.
Givenitscapacityofquantifyingpropertiesoflargediscretestructures, Analytic
Combinatorics is susceptible to many applications, within combinatorics itself, but,
perhapsmoreimportantly,within otherareasof science where discrete probabilistic
modelsrecurrentlysurface,likestatisticalphysics,computationalbiology,orelectri-
cal engineering. Last but notleast, the analysis of algorithmsand data structuresin
ii
computersciencehasservedandstillservesasanimportantmotivationinthedevel-
opmentofthetheory.
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PartA:SymbolicMethods. ThispartspecificallyexposesSymbolicCombina-
torics,whichisaunifiedalgebraictheorydedicatedtosettingupfunctionalrelations
betweencountinggeneratingfunctions. As it turnsout, a collection of general(and
simple) theorems provide a systematic translation mechanism between combinato-
rialconstructionsandoperationsongeneratingfunctions. Thistranslationprocessis
a purely formal one. Precisely, as regards basic counting, two parallel frameworks
coexist—one for unlabelled structures and ordinary generating functions, the other
forlabelledstructuresandexponentialgeneratingfunctions. Furthermore,withinthe
theory, parameters of combinatorialconfigurations can be easily taken into account
byaddingsupplementaryvariables. Threechaptersthencomposethispart: ChapterI
dealswithunlabelledobjects;ChapterIIdevelopsinaparallelwaylabelledobjects;
ChapterIIItreatsmultivariateaspectsofthetheorysuitablefortheanalysisofparam-
etersofcombinatorialstructures.
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PartB:Complexasymptotics. ThispartspecificallyexposesComplexAsymp-
totics, which is a unified analytic theory dedicated to the process of extracting as-
ymptotic information from counting generating functions. A collection of general
(and simple) theorems provide a systematic translation mechanism between gener-
ating functions and asymptotic forms of coefficients. Four chapters compose this
part. ChapterIVservesasanintroductiontocomplex-analyticmethodsandproceeds
withthetreatmentofmeromorphicfunctions,thatis,functionswhosesingularitiesare
poles,rationalfunctionsbeingthesimplestcase. ChapterVdevelopsapplicationsof
rationalandmeromorphicasymptoticsofgeneratingfunctions,withnumerousappli-
cations related to words and languages, walks and graphs, as well as permutations.
Chapter VI develops a general theory of singularity analysis that applies to a wide
varietyofsingularitytypes, such as square-rootorlogarithmic,andhasapplications
to trees as well as to other recursively defined combinatorial classes. Chapter VII
presents applications of singularity analysis to 2-regular graphs and polynomials,
treesofvarioussorts,mappings,context-freelanguages,walks,andmaps. Itcontains
inparticularadiscussionoftheanalysisofcoefficientsofalgebraicfunctions. Chap-
terVIIIexploressaddlepointmethods,whichareinstrumentalinanalysingfunctions
withaviolentgrowthatasingularity,aswellasmanyfunctionswithonlyasingularity
atinfinity(i.e.,entirefunctions).
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Part C: Random Structures. This part includes Chapter IX dedicated to the
analysisofmultivariategeneratingfunctionsviewedasdeformationandperturbation
of simple (univariate) functions. As a consequence, many important characteristics
ofclassicalcombinatorialstructurescanbepreciselyquantifiedindistribution.Chap-
ter??isanepilogue,whichoffersabriefrecapitulationofthemajorasymptoticprop-
ertiesofdiscretestructuresdevelopedinearlierchapters.
iii
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PartD:Appendices. AppendixAsummarizessomekeyelementaryconceptsof
combinatorics and asymptotics, with entries relative to asymptotic expansions, lan-
guages, and trees, amongst others. Appendix B recapitulates the necessary back-
ground in complex analysis. It may be viewed as a self-contained minicourse on
the subject, with entries relative to analytic functions, the Gamma function, the im-
plicitfunctiontheorem,andMellintransforms. AppendixCrecallssomeofthebasic
notionsofprobabilitytheorythatareusefulinanalyticcombinatorics.
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THIS BOOKismeanttobereader-friendly. Eachmajormethodisabundantlyil-
lustratedbymeansofconcreteexamples1treatedindetail—therearescoresofthem,
spanningfromafractionofapagetoseveralpages—offeringacompletetreatmentofa
specificproblem.Theseareborrowednotonlyfromcombinatoricsitselfbutalsofrom
neighbouringareasofscience. Withaviewofaddressingnotonlymathematiciansof
variedprofilesbutalsoscientistsofotherdisciplines,AnalyticCombinatoricsisself-
contained,includingampleappendicesthatrecapitulatethenecessarybackgroundin
combinatoricsandcomplexfunctiontheory.ArichsetofshortNotes—therearemore
than250ofthem—areinsertedinthetext2andcanprovideexercisesmeantforself-
studyorforstudents’practice, aswellasintroductionsto thevastbodyof literature
that is available. We have also made everyeffortto focus on core ideas ratherthan
technicaldetails,supposingacertainamountofmathematicalmaturitybutonlybasic
prerequisitesonthepartofourgentlereaders. Thebookisalsomeanttobestrongly
problem-oriented,andindeeditcanberegardedasamanual,orevenahugealgorithm,
guidingthe readerto the solution of a very large variety of problemsregardingdis-
cretemathematicalmodelsofvariedorigins. Inthisspirit,manyofourdevelopments
connectnicelywithcomputeralgebraandsymbolicmanipulationsystems.
COURSES can be (and indeed have been) based on the book in various ways.
ChaptersI–IIIonSymbolicMethodsserveasasystematicyetaccessibleintroduction
to the formalside of combinatorialenumeration. As such it organizestransparently
some of the rich materialfoundin treatises3 like those of Bergeron-Labelle-Leroux,
Comtet,Goulden-Jackson,andStanley.ChaptersIV–VIIIrelativetoComplexAsymp-
toticsprovidealargesetofconcreteexamplesillustratingthepowerofclassicalcom-
plexanalysisandofasymptoticanalysisoutsideoftheirtraditionalrangeofapplica-
tions.Thismaterialcanthusbeusedincoursesofeitherpureorappliedmathematics,
providing a wealth of nonclassical examples. In addition, the quiet but ubiquitous
presenceofsymbolicmanipulationsystemsprovidesanumberofillustrationsofthe
power of these systems while making it possible to test and concretely experiment
with a great many combinatorial models. Symbolic systems allow for instance for
fast random generation, close examination of non-asymptotic regimes, efficient ex-
perimentationwithanalyticexpansionsandsingularities,andsoon.
1Examplesaremarkedby“EXAMPLE (cid:3)”.
2Notesareindicatedby(cid:3) (cid:1). ···
3 ···
Referencesaretobefoundinthebibliographysectionattheendofthebook.
iv
Our initial motivation when starting this project was to build a coherent set of
methodsusefulintheanalysisofalgorithms,adomainofcomputersciencenowwell-
developedandpresentedinbooksbyKnuth,Hofri,andSzpankowski,inthesurveyby
Vitter–Flajolet,aswellasinourearlierIntroductiontotheAnalysisofAlgorithmspub-
lishedin1996. Thisbookcanthenbeusedasasystematicpresentationsofmethods
thathaveprovedimmenselyusefulinthisarea;seeinparticulartheArtofComputer
ProgrammingbyKnuthforbackground.Studiesinstatisticalphysics(vanRensburg,
andothers),statistics(e.g.,DavidandBarton)andprobabilitytheory(e.g.,Billingsley,
Feller),mathematicallogic(Burris’book),analyticnumbertheory(e.g.,Tenenbaum),
computationalbiology(Waterman’stextbook),aswellasinformationtheory(e.g.,the
booksbyCover–Thomas,MacKay,andSzpankowski)pointtomanystartlingconnec-
tionswith yetotherareasof science. Thebookmay thusbe usefulasa supplemen-
taryreferenceonmethodsandapplicationsincoursesonstatistics,probabilitytheory,
statistical physics, finite model theory, analytic number theory, information theory,
computeralgebra,complexanalysis,oranalysisofalgorithms.
Acknowledgements. This book would be substantially different and much less in-
formative without Neil Sloane’s Encyclopedia of Integer Sequences, Steve Finch’s
MathematicalConstants, Eric Weisstein’s MathWorld, and the MacTutor History of
MathematicssitehostedatStAndrews. Allare(oratleasthavebeenatsomestage)
freely available on the Internet. Bruno Salvy and Paul Zimmermann have devel-
oped algorithms and libraries for combinatorial structures and generating functions
that are based on the MAPLE system for symbolic computations and have proven
to be extremely useful. We are deeply grateful to the authors of the free software
Unix,Linux,Emacs,X11,TEXandLATEXaswellastothedesignersofthesymbolic
manipulationsystem MAPLE forcreatinganenvironmentthathasprovedinvaluable
to us. We also thankstudentsin coursesat Barcelona, Berkeley(MSRI), Bordeaux,
Caen,Paris(E´colePolytechnique,E´coleNormale,University),Princeton,Santiagode
Chile,Udine,andViennawhosefeedbackhasgreatlyhelpedusprepareabetterbook.
Thanksfinally to numerouscolleagues for their feedback. In particular, we wish to
acknowledge the support, help, and interaction provided at an incredibly high level
bymembersoftheAnalysisofAlgorithms(AofA)community,withaspecialmention
forHsien-KueiHwang,SvanteJanson,DonKnuth,GuyLouchard,AndrewOdlyzko,
DanielPanario,HelmutProdinger,BrunoSalvy,Miche`leSoria,WojtekSzpankowski,
BrigitteValle´e,andMarkWilson.StanBurrisandBrigitteValle´eespeciallyhavepro-
videdcommentsthathaveledustorevisethepresentationof severalsectionsofthis
book, while Svante Janson and Lo¨ıc Turbanhave providedextremelydetailed feed-
backonseveralchapters,enablingustocorrectmanyerrors. Finally,supportofour
homeinstitutions(INRIAandPrincetonUniversity)aswellasvariousgrants(French
government,EuropeanUnionandtheALCOMProject,NSF)havecontributedtomak-
ingourcollaborationpossible.
PHILIPPEFLAJOLET&ROBERT SEDGEWICK
October2,2005
Contents
AninvitationtoAnalyticCombinatorics 1
PartA. SYMBOLICMETHODS 13
I. CombinatorialStructuresandOrdinaryGeneratingFunctions 15
I.1. Symbolicenumerationmethods 16
I.2. Admissibleconstructionsandspecifications 23
I.3. Integercompositionsandpartitions 37
I.4. Wordsandregularlanguages 47
I.5. Treestructures 61
I.6. Additionalconstructions 76
I.7. Perspective 83
II. LabelledStructuresandExponentialGeneratingFunctions 87
II.1. Labelledclasses 88
II.2. Admissiblelabelledconstructions 91
II.3. Surjections,setpartitions,andwords 98
II.4. Alignments,permutations,andrelatedstructures 110
II.5. Labelledtrees,mappings,andgraphs 115
II.6. Additionalconstructions 125
II.7. Perspective 135
III. CombinatorialParametersandMultivariateGeneratingFunctions 139
III.1. Anintroductiontobivariategeneratingfunctions(BGFs) 141
III.2. Bivariategeneratingfunctionsandprobabilitydistributions 144
III.3. InheritedparametersandordinaryMGFs 151
III.4. InheritedparametersandexponentialMGFs 163
III.5. Recursiveparameters 170
III.6. Completegeneratingfunctionsanddiscretemodels 175
III.7. Additionalconstructions 187
III.8. Extremalparameters 202
III.9. Perspective 206
PartB. COMPLEXASYMPTOTICS 209
IV. ComplexAnalysis,RationalandMeromorphicAsymptotics 211
IV.1. Generatingfunctionsasanalyticobjects 212
IV.2. Analyticfunctionsandmeromorphicfunctions 216
IV.3. Singularitiesandexponentialgrowthofcoefficients 226
v
vi CONTENTS
IV.4. Closurepropertiesandcomputablebounds 236
IV.5. Rationalandmeromorphicfunctions 242
IV.6. Localizationofsingularities 250
IV.7. Singularitiesandfunctionalequations 261
IV.8. Perspective 273
V. ApplicationsofRationalandMeromorphicAsymptotics 277
V.1. Regularspecificationandlanguages 278
V.2. Latticepathsandwalksontheline. 291
V.3. Thesupercriticalsequenceanditsapplications 304
V.4. Functionalequations:positiverationalsystems 311
V.5. Pathsingraphs,automata,andtransfermatrices. 318
V.6. Additionalconstructions 337
V.7. Notes 342
VI. SingularityAnalysisofGeneratingFunctions 345
VI.1. Introduction 346
VI.2. Coefficientasymptoticsforthebasicscale 350
VI.3. Transfers 357
VI.4. Firstexamplesofsingularityanalysis 362
VI.5. Inversionandimplicitlydefinedfunctions 366
VI.6. Singularityanalysisandclosureproperties 370
VI.7. Multiplesingularities 382
VI.8. TauberiantheoryandDarboux’smethod 384
VI.9. Notes 389
VII. ApplicationsofSingularityAnalysis 391
VII.1. The“exp–log”schema 392
VII.2. Simplevarietiesoftrees 398
VII.3. Positiveimplicitfunctions 414
VII.4. Theanalysisofalgebraicfunctions 419
VII.5. Combinatorialapplicationsofalgebraicfunctions 437
VII.6. Notes 451
VIII. SaddlePointAsymptotics 455
VIII.1. Preamble:Landscapesofanalyticfunctionsandsaddlepoints 456
VIII.2. Overviewofthesaddlepointmethod 459
VIII.3. Largepowers 469
VIII.4. Fourcombinatorialexamples 475
VIII.5. Admissibility 486
VIII.6. Combinatorialaveragesanddistributions 492
VIII.7. Variationsonthethemeofsaddlepoints 500
VIII.8. Notes 508
PartC. RANDOMSTRUCTURES 511
IX. MultivariateAsymptoticsandLimitDistributions 513
IX.1. Limitlawsandcombinatorialstructures 515
IX.2. Discretelimitlaws 521
IX.3. Combinatorialinstancesofdiscretelaws 528
CONTENTS vii
IX.4. Continuouslimitlaws 539
IX.5. Quasi-powersandGaussianlimits 544
IX.6. Perturbationofmeromorphicasymptotics 549
IX.7. Perturbationofsingularityanalysisasymptotics 562
IX.8. Perturbationofsaddlepointasymptotics 582
IX.9. Locallimitlaws 586
IX.10. Largedeviations 590
IX.11. Non-Gaussiancontinuouslimits 593
IX.12. Multivariatelimitlaws 605
IX.13. Notes 606
PartD. APPENDICES 609
AppendixA. AuxiliaryElementaryNotions 611
AppendixB. BasicComplexAnalysis 629
AppendixC. ComplementsofProbabilityTheory 655
Bibliography 665
Index 681