算法高级教程3.10.2OnlineBipartiteMatching.pptx

算法高级教程3.10.2OnlineBipartiteMatching.pptx

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,OnlineBipartiteMatching,Amatchmakerandnboysaregatheredinaroom.ngirlsappear,oneatatime.Eachgirlhasalistofboyswhoareacceptabletoher,whichsherevealstothematchmakerassheappears.Thematchmakerimmediatelymatchesthenewgirltooneoftheboysonherlist,ifanyofthemareavailable.Thegoalistomaximizethenumberofmatches.,2,ImportanceofMatching,ResourceAllocationSchedulingMemoryManagementRoutingRobotMotionPlanningExploringanunknownterrainFindingadestinationComputationalFinance,Subroutineinmanyotheralgorithms.,GivenasinputabipartitegraphG=（U,V,E）inwhicheachvertexuU（girls）arrivesinonlinefashion,deviseanalgorithmthatmatchesu（girl）toone（boy）ofitspreviouslyunmatchedneighboursinV.Thematchinghastobeimmediateandisirrevocable,oncemade.Theobjectiveistomaximizethesizeoftheresultingmatching.AssumethattheinputgraphGhasaperfectmatchingi.e.amatchingofsizen.Wedenoteaperfectmatchingbyafunctionm:

UV.Hence,ac-competitivealgorithmmustreturnamatchingofsizeatleastcn.,4,DeterministicAlgorithm,whenuarrivesassignittoasomeunmatchedneighbour.Lemma3.10.1.Theabovealgorithmhasacompetitiveratioof1/2.Proof.Ifavertexu1isnotpresentintheresultingmatchingM,thenitdoesnothaveanunmatchedneighbour,ifnot,wewouldhavematchedu1tothatneighbour.Hence,theresultingmatchingmustbemaximal.Foreveryedgeu,m（u）,eithervertexuorm（u）ispresentinM.So,atleastn/2verticesarematchedandhencethealgorithmhasacompetitiveratioof1/2.,5,Onlinebipartitematching,V（boys）,U（girls）,Onlinebipartitematching,V（boys）,U（girls）,Onlinebipartitematching,V（boys）,U（girls）,Onlinebipartitematching,V（boys）,U（girls）,Onlinebipartitematching,V（boys）,U（girls）,Onlinebipartitematching,V（boys）,U（girls）,Onlinebipartitematching,V（boys）,U（girls）,Wecanprovethatanydeterministicalgorithmcannotdobetterthantheobviousalgorithm.Anadversarycanlimitthesizeofmatchington/2inthefollowingway:

Letthefirstn/2verticesthatarrivehaveedgestoalltheverticesinV.Clearly,theadversarycandeterminetheverticesinVthatwillbematched.Letthenextn/2verticesthatarrivecontainedgesonlytothoseverticesinVwhicharealreadymatched.Theinputgraphhasaperfectmatchingbutthesecondhalfoftheverticesarenotmatched;henceouranalysisistightforthedeterministiccase.,13,Onlinebipartitematching,n/2,:

n/2:

V（boys）,U（girls）,Onlinebipartitematching,n/2,:

n/2:

V（boys）,U（girls）,Onlinebipartitematching,:

:

n/2,V（boys）,U（girls）,RandomizedAlgorithm,Foreachpossibleinput,calculatetheexpectationoftheanswerandtaketheworstexpectedvalueamongalltheinputs.Inourcase,aninputinstancewouldbespecifiedbyagraphGalongwithanarrivalorder.Sotheinputspacewouldcontainallpossible（G,）pairs.Themostnaturalwaytointroducerandomnesswouldbetomatchutooneoftheunmatchedneighbourspickedrandomly.Thisalgorithmperformsbetterthanthedeterministicalgorithm;howevertheimprovementisnotsubstantial.,17,Lemma3.10.2.Arandomizedalgorithmthatpicksanunmatchedneighbouruniformlyandrandomlyhasacompetitiveratioofatmost1/2+O（logn）/n.Proof.Toseewhythisisthecase,considerthefollowinginput.Letu1,.,unUandv1,.,vnV.Thereisanedgebetweenuiandviforalli.EveryvertexinthefirsthalfofU=u1,.,un/2isconnectedtoeveryvertexinthesecondhalfofV=vn/2,.,vnasshownbelow:

18,Onlinebipartitematching,:

:

n/2,n/2,VU,u1,un/2un/2+1,:

:

v（boys）（girls）,1,vn/2vn/2+1,vnun,Tightinstanceforthenaverandomizedalgorithm,Theverticesarriveintheorderoftheirindices.Intuitively,thealgorithmfailstoperformwellonthisinputsinceitmatchestoomanyusfromthefirsthalftothevsofthesecondhalf.Thefirstn/2verticesfromUaredefinitelyinthematchingsinceallofthemgetatleastoneunmatchedneighbourwhentheyarrive.EachuifromthesecondhalfofUcanbematchedtovi,ifviisnotalreadymatched.Whatistheprobabilitythatthishappens?

20,LetusfindtheexpectednumberofverticesthatarematchedinthefirsthalfofV.Whenu1arrives,itcanpickeitherv1orvn/2,.,vn.Sotheprobabilityofv1gettingmatchedis1/（n/2+1）.Similarly,whenu2arrives,theprobabilitythatv2getsmatchedis:

21,Similarly,probabilityofv3gettingmatchedislessthan1/（n/21）andingeneralavertexviinthefirsthalfofVhaslessthan1/（n/2i+2）probabilityofbeinginthematching.LetEvbetheexpectednumberofverticesfromv1,.,vn/2inthematching.,Hence,lessthanO（logn）unmatchedneighboursareexpectedtobeavailabletothesecondhalfofU,whichprovesourclaim.22,Rankingalgorithm,Aslightlydifferentrandomizedalgorithm,namedRankingKarp,performsmuchbetter.Thealgorithmisasfollows:

Ranking（）Initialization:

Pickarandompermutation（ranking）oftheverticesinVForeachuUthatarrives: