Algorithms on Trees and Graphs.pdf
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Algorithms on Trees and Graphs.pdf
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AlgorithmsonTreesandGraphsSpringer-VerlagBerlinHeidelbergGmbHGabrielValienteAlgorithmsonTreesandGraphsWith157FiguresSpringerprofDr.GabrielValienteTechnicalUniversityofCataloniaDepartmentofSoftwareJordiGirona,1-3(ModuleC6)08034BarcelonaSpainLibraryofCongressCataloging-in-PublicationDataappliedforDieDeutscheBibliothek-CIP-EinheitsaufnahmeValiente,Gabriel:
Algorithmsontreesandgraphs/GabrielValienteISBN978-3-642-07809-5ISBN978-3-662-04921-1(eBook)DOI10.1007/978-3-662-04921-1ACMComputingClassification(1998)E2,G.2.2,GA,D.IISBN978-3-642-07809-5Thisworkissubjecttocopyright.Alirightsarereserved.whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation.reprinting,reuseofillustrations,recitation.broadcasting,reproductiononmicrofilmorinanyolherway,andstorageindatabanks.DuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,inilscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-VerlagBerlinHeidelbergGmbH.ViolalionsareliableforprosecutionundertheGermanCopyrightLaw.Springer-VerlagBerlinHeidelberg2002OriginallypublishedbySpringer-VerlagBerlinHeidelbergNewYorkin2002Softcoverreprintofthehardcover1stedition2002Theuseofdesignations,trademarks.etc.inthispublicationdoesnotimply.evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandreguJationsandthereforefreeforgeneraluse.CoverDesign:
KunkelLopka.HeidelbergTypesetting:
Computertafilmbyauthor5dataPrintedanacid-freepaperSPIN1085518045/3142PS5432IOToDaina,Kristians,andDavisPrefaceGraphalgorithms,along-establishedsubjectinmathematicsandcom-putersciencecurricula,arealsoofmuchinteresttodisciplinessuchascomputationalmolecularbiologyandcomputationalchemistry.Thisbookgoesbeyondtheclassicalgraphproblemsofshortestpaths,spanningtrees,flowsinnetworks,andmatchingsinbipartitegraphs,andaddressesfurtheralgorithmicproblemsofpracticalapplicationoftreesandgraphs.Muchofthematerialpresentedinthebookisonlyavailableinthespecializedresearchliterature.Thebookisstructuredaroundthefundamentalproblemofisomor-phism.Treeisomorphismiscoveredinmuchdetail,togetherwiththerelatedproblemsofsubtreeisomorphism,maximumcommonsubtreeisomorphism,andtreecomparison.Graphisomorphismisalsocov-eredinmuchdetail,togetherwiththerelatedproblemsofsubgraphisomorphism,maximalcommonsubgraphisomorphism,andgrapheditdistance.Buildingblocksforsolvingsomeoftheseisomorphismproblemsarealgorithmsforfindingmaximalandmaximumcliques.Mostintractablegraphproblemsofpracticalapplicationarenotevenapproximabletowithinaconstantbound,andseveraloftheisomorphismproblemsaddressedinthisbookarenoexception.Thebookcanthusbeseenasacompaniontorecenttextsonapproxima-tionalgorithms15,344,butalsoasacomplementtoprevioustextsoncombinatorialandgraphalgorithms69,78,107,127,130,203,232,236,229,255,271,304,321,336,367.Thebookisconceivedonthegroundoffirst,introducingsimplealgorithmsfortheseproblemsinordertodevelopsomeintuitionbe-foremovingontomorecomplicatedalgorithmsfromtheresearchlit-erature,andsecond,stimulatinggraduateresearchontreeandgraphVIIIPrefacealgorithmsbyproviding,togetherwiththeunderlyingtheory,asolidbasisforexperimentationandfurtherdevelopment.Algorithmsarepresentedonanintuitivebasis,followedbyade-tailedexpositioninaliterateprogrammingstyle.Correctnessproofsarealsogiven,togetherwithaworst-caseanalysisofthealgorithms.Further,fullC+implementationofallthealgorithmsusingtheLEDAlibraryofefficientdatastructuresandalgorithmsisgiventhroughoutthebook.Theseimplementationsincluderesultcheckingofimplementationcorrectnessusingcorrectnesscertificates.ThechoiceofLEDA,whichisbecomingadefactostandardforgraduatecoursesongraphalgorithmsthroughouttheworld,isnotcasual,becauseitallowsthestudent,lecturer,researcher,andprac-titionertocomplementalgorithmicgraphtheorywithactualimple-mentationandexperimentation,buildinguponathoroughlibraryofefficientimplementationsofmoderndatastructuresandfundamentalalgorithms.Aninteractivedemonstrationincludinganimationsofalltheal-gorithmsusingLEDAisgiveninanappendixandisavailableonawebsite,athttp:
/www.lsi.upc.esrvaliente/algorithm/,togetherwiththesourcecodeofallthealgorithms.Theinteractivedemonstrationalsoincludesvisualcheckersofimplementationcor-rectness.StructureThebookisdividedintofourparts.PartIisofanintroductoryna-tureandconsistsoftwochapters.Chap.1includesareviewofba-sicgraph-theoreticalnotionsandresultsusedthroughoutthebook,abriefprimerofliterateprogramming,andanexpositionoftheimple-mentationcorrectnessapproachbyresultcheckingusingcorrectnesscertificates.Chap.2isdevotedexclusivelytothefundamentalalgorithmictech-niquesusedinthebook:
backtracking,branch-and-bound,divide-and-conquer,anddynamicprogramming.Thesetechniquesareillustratedbymeansofarunningexample:
algorithmsforthetreeeditdistanceproblem.PrefaceIXPartIIalsoconsistsoftwochapters.Chap.3addressesthemostcommonmethodsfortraversinggeneral,rootedtrees:
depth-firstpre-fixleftmost(preorder),depth-firstprefixrightmost,depth-firstpostfixleftmost(postorder),depth-firstpostfixrightmost,breadth-firstleft-most(top-down),breadth-firstrightmost,andbottom-uptraversal.Treedrawingisalsodiscussedasanapplicationoftreetraversalmeth-ods.Chap.4addressesseveralisomorphismproblemsonorderedandunorderedtrees:
treeisomorphism,subtreeisomorphism,andmaxi-mumcommonsubtreeisomorphism.Computationalmolecularbiol-ogyisalsodiscussedasanapplicationofthedifferentisomorphismproblemsontrees.PartIIIconsistsofthreechapters.Chap.5addressesthemostcommonmethodsfortraversinggraphs:
depth-firstandbreadth-firsttraversal,whichrespectivelygeneralizedepth-firstprefixleftmost(preorder)andbreadth-firstleftmost(top-down)treetraversal.Left-mostdepth-firsttraversalofanundirectedgraph,aparticularcaseofdepth-firsttraversal,isalsodiscussed.Isomorphismoforderedgraphsisalsodiscussedasanapplicationofgraphtraversalmethods.Chap.6addressestherelatedproblemsoffindingcliques,inde-pendentsets,andvertexcoversintreesandgraphs.Multiplealign-mentofproteinsequencesincomputationalmolecularbiologyisalsodiscussedasanapplicationofcliquealgorithms.Chap.7addressesseveralisomorphismproblemsongraphs:
graphisomorphism,graphautomorphism,subgraphisomorphism,andmax-imalcommonsubgraphisomorphism.Chemicalstructuresearchisalsodiscussedasanapplicationofthedifferentgraphisomorphismproblems.PartIVconsistsofthreeappendices,followedbybibliographicref-erencesandanindex.AppendixAgivesanoverviewofLEDA,in-cludingasimpleC+representationoftreesasLEDAgraphs,andaC+implementationofradixsortusingLEDA.TheinteractivedemonstrationofgraphalgorithmspresentedthroughoutthebookisputtogetherinAppendixB.Finally,AppendixCcontainsacompleteindextoallprogrammodulesdescribedinthebook.XPrefaceAudienceThisbookissuitableforuseinupperundergraduateandgraduatelevelcoursesonalgorithmicgraphtheory.Itcanalsobeusedasasup-plementarytextinbasicundergraduateandgraduate-levelcoursesonalgorithmsanddatastructures,andincomputationalmolecularbiol-ogyandcomputationalchemistrycoursesaswell.Somebasicknowl-edgeofdiscretemathematics,datastructures,algorithms,andpro-grammingattheundergraduatelevelisassumed.AcknowledgmentsThisbookisbasedonlecturestaughtattheTechnicalUniversityofCatalonia,Barcelonabetween1996and2002,andtheUniversityofLatvia,Rigabetween2000and2002.NumerouscolleaguesattheTechnicalUniversityofCataloniahaveinfluencedtheapproachtodatastructuresandalgorithmsontreesandgraphsexpressedinthisbook.Inparticular,theauthorwouldliketothankJoseL.Balcazar,RafelCasas,JordiCortadella,JosepDiaz,ConradoMartinez,XavierMesseguer,RobertoNieuwenhuis,FernandoOrejas,JordiPetit,SalvadorRoura,andMariaSerna,tonamejustafew.Ithasbeenapleasuretoshareteachingandresearchexperienceswiththemoverthelastseveralyears.TheauthorwouldalsoliketothankRicardoBaeza-Yates,FrancescRosse116,andStevenSkiena,fortheirstandingsupportandencour-agement,andHans-JorgKreowski,forsupportingbasicandappliedresearchongraphalgorithmswithinthefieldofgraphtransformation.IthasbeenapleasurefortheauthortoworkouteditorialmatterstogetherwithAlfredHofmann,IngeborgMayer,andPeterStraBerofSpringer-Verlag.SpecialthanksgototheTechnicalUniversityofCataloniaforfundingthesabbaticalyearduringwhichthisbookwaswritten,andtotheInstituteofMathematicsandComputerScience,UniversityofLatvia,inparticulartoJanisBarzdinsandRusinsFreivalds,forhost-,ingthesabbaticalvisit.July2002GabrielValienteContentsPartI.Introduction1.Introduction.31.1TreesandGraphs.31.2LiterateProgramming.231.3ImplementationCorrectness.281.4RepresentationofTreesandGraphs.34Summary.49BibliographicNotes.49ReviewProblems.50Exercises.512.AlgorithmicTechniques.552.1TheTreeEditDistanceProblem.552.2Backtracking.702.3Branch-and-Bound.802.4Divide-and-Conquer.852.5DynamicProgramming.94Summary.107BibliographicNotes.107ReviewProblems.108Exercises.110PartII.AlgorithmsonTrees3.TreeTraversal.1133.1PreorderTraversalofaTree.1133.2
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