of Computer Science ETH Zur ich 8092 Zur ich Switz er land kuhninf ethzch Thomas Moscibroda Dept of Computer Science ETH Zur ich 8092 Zur ich Switz er land moscithoinf ethzch Roger attenhof er Dept of Computer Science ETH Zur ich 8092 Zur ich Switz ID: 19212
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WhatCannotBeComputedLocally!FabianKuhnDept.ofComputerScienceETHZurich8092Zurich,Switzerlandkuhn@inf.ethz.chThomasMoscibrodaDept.ofComputerScienceETHZurich8092Zurich,Switzerlandmoscitho@inf.ethz.chRogerWattenhoferDept.ofComputerScienceETHZurich8092Zurich,Switzerlandwattenhofer@inf.ethz.chABSTRACTWegivetimelowerboundsforthedistributedapproxima-tionofminimumvertexcover(MVC)andrelatedproblemssuchasminimumdominatingset(MDS).Inkcommuni-cationrounds,MVCandMDScanonlybeapproximatedbyfactors\n(nc=k2=k)and\n(1=k=k)forsomeconstantc,wherenanddenotethenumberofnodesandthelargestdegreeinthegraph.Thenumberofroundsrequiredinordertoachieveaconstantorevenonlyapolylogarith-micapproximationratioisatleast\n( logn=loglogn)and\n(log=loglog).Byasimplereduction,thelatterlowerboundsalsoholdfortheconstructionofmaximalmatchingsandmaximalindependentsets.CategoriesandSubjectDescriptorsF.2.2[AnalysisofAlgorithmsandProblemComplex-ity]:NonnumericalAlgorithmsandProblems|computa-tionsondiscretestructures;G.2.2[DiscreteMathematics]:GraphTheory|graphal-gorithms;G.2.2[DiscreteMathematics]:GraphTheory|networkproblemsGeneralTermsAlgorithms,TheoryKeywordsapproximationhardness,distributedalgorithms,dominat-ingset,locality,lowerbounds,maximalindependentset,maximalmatching,vertexcoverTheworkpresentedinthispaperwassupported(inpart)bytheHaslerStiftung(Berne,Switzerland)andbytheNa-tionalCompetenceCenterinResearchonMobileInforma-tionandCommunicationSystems(NCCR-MICS),acentersupportedbytheSwissNationalScienceFoundationundergrantnumber5005-67322.Permissiontomakedigitalorhardcopiesofallorpartofthisworkforpersonalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforprotorcommercialadvantageandthatcopiesbearthisnoticeandthefullcitationontherstpage.Tocopyotherwise,torepublish,topostonserversortoredistributetolists,requirespriorspecicpermissionand/orafee.PODC'04,July2528,2004,St.John's,Newfoundland,Canada.Copyright2004ACM1581138024/04/0007...$5.00.1.INTRODUCTIONANDRELATEDWORKWhatcanbecomputedlocally?NaorandStockmayer[18]raisedthisforthedistributedcomputingcommunityvitalquestionoveradecadeago,andthoughthepriceoflo-calityhaspuzzledresearcherseversince,resultshavebeenrare[15].Theimportanceoflocalitystemsfromthedesireofachievingaglobalgoalbasedonlocalinformationonly.Thisisnotonlyoneofthekeychallengeswhendevelopingfastdistributedalgorithms,butitalsoimprovesfault-toleranceandallowsdetectingillegalglobalcon gurationsbycheck-inglocalconditions[1].Recentadvancesinnetworkingandever-growingdistributedsystemsdrivetheneedforathor-oughunderstandingoflocalityissues.Inthepresentpaper,westudytheimpactoflocalityintheclassicmessagepass-ingmodelwherethedistributedsystemismodelledasagraph:Nodesrepresenttheprocessorsandtwonodescancommunicateifandonlyiftheyshareanedgeinthegraph.Wepresentlowerboundsforseveraltraditionalgraphtheoryproblems,startingoutwithminimumvertexcover(MVC).AvertexcoverforagraphG=(V;E)isasubsetofnodesV0Vsuchthat,foreachedge(u;v)2E,atleastoneofthetwoincidentnodesu;vbelongstoV0.FindingavertexcoverwithminimumcardinalityisknownastheMVCproblem.Inlocalalgorithms,nodesareonlyallowedtocommuni-catewiththeirdirectneighborsinG.Afterseveralroundsofcommunicationtheyneedtocomeupwithaglobalsolution,e.g.agoodapproximationforMVC.IntuitivelyMVCcouldbeconsideredperfectlysuitedforalocalalgorithm:Anodeshouldbeabletodecidewhethertojointhevertexcoverbycommunicatingwithitsneighborsafewtimes.Verydistantnodesseemtobesuper\ruousforthisdecision.ThefactthatthereisasimplegreedyalgorithmwhichapproximatesMVCwithinafactor2intheglobalsetting,additionallyraiseshopeforanecientlocalalgorithm.Tooursurprise,however,thereisnosuchalgorithm.Inthispaper,weshowthatinkcommunicationrounds,MVCcanonlybeapproximatedbyafactor\n(nc=k2=k)foracon-stantclargerthan1=4.Thisimpliesarunningtimeof\n( logn=loglogn)inordertoachieveaconstantorevenpolylogarithmicapproximationratio.Whenmakingthere-sultdependentonthemaximumdegreeinsteadofthenumberofnodesn,theapproximationratiois\n(1=k=k)andthetimeneededtoobtainapolylogarithmicorcon-stantapproximationis\n(log=loglog).TheresultsforMVCcanbeextendedtominimumdominatingset(MDS)andotherdistributedcoveringproblems.Further,thetime lowerboundsforconstantMVCapproximationsalsoapplyfortheconstructionofmaximalmatchingsandmaximalin-dependentsets.Sincealllowerboundsholdeveninthecasesofunboundedmessagesandcompletesynchrony,thelowerboundsareatrueconsequenceoflocalitylimitations,andnotmerelyaside-e ectofcongestion,asynchrony,orlimitedmessagesize.Moreover,someofourlowerboundsarealmosttightformanycases.Inkrounds,MVCcanbeapproximatedbyafactorO(1=k)[10].Inordertoobtainapolylogarith-micapproximation,wehavetosetk=O(log=loglog),foraconstantapproximation,thenumberofroundsisk=O(log).Hence,forpolylogarithmic(andworse)approx-imationratios,ourlowerboundistightwhereasforcon-stantapproximationalgorithms,itistightuptoafactorO(loglog).FortheMDSproblem,recentresultsshowthatinkrounds,therespectivelinearprogramcanbeap-proximateduptoafactorc=pk,whereasforMDSitselfanapproximationratioofc=pklncanbeachieved[9,10].Ifmessagesizeandlocalcomputationsareunbounded,theMDSLPandMDScanbeappoximateduptofactorsO(n1=k)andO(n1=kln),respectively[10].ItiseasytoverifythatthelowerboundsforMDSalsoholdsforthefrac-tionalLPversion.ThebestknownalgorithmsformaximalmatchingandmaximalindependentsetneedtimeO(logn)[8,17].ApioneeringandseminallowerboundbyLinial[15]showsthatthenon-uniformO(logn)coloringalgorithmbyColeandVishkin[2]isasymptoticallyoptimalforthering.Thislowerboundhasbeencherishedbyresearchersasafunda-mentaladvancementinthetheoryofdistributedalgorithms.Fordi erentmodelsofdistributedcomputation,thereisanumberofotherlowerbounds[6,11],mostnotablyfortheproblemofconstructingaminimumspanningtree(MST)ofthenetworkgraph[3,4,16,20].Withtheexceptionof[3],theMSTlowerboundsapplytoamodelwheremessagesizeisbounded.In[4],thelowerboundsof[16,20]areextendedtoapproximationalgorithms.Tothebestofourknowledge,itistheonlypreviouslowerboundondistributedhardnessofapproximation.Linial'slowerbound[15]isbasedonthedrosophilamela-nogasterofdistributedcomputing,theringnetwork.FortheMVCproblem,highlysymmetricgraphssuchasringsoftenfeatureastraight-forwardsolutionwithconstantap-proximationratio.Inany-regulargraph,forexample,thealgorithmwhichincludesallnodesinthevertexcoverisalreadya2-approximationforMVC:Eachnodewillcoveratmostedges,thegraphhasn=2edges,andthereforeatleastn=2nodesneedtobeintheminimumvertexcover.Ontheotherextreme,asymmetricgraphsoftenenjoyconstant-timealgorithms,too.Inatree,choosingallinnernodesyieldsa2-approximation.Thesametrade-o existsfornodedegrees.Ifthemaximumnodedegreeislow(constant),wecantoleratetochooseallnodes,andhavebyde nitionagood(constant)approximation.Iftherearenodeswithhighdegree,thediameterofthegraphissmall,andafewcom-municationroundssucetoinformallnodesoftheentiregraph.Whatweneedisaconstructionofanottoosym-metricandnottooasymmetricgraphwithavarietyofnodedegrees!Notmanygraphswiththese\non-properties"areknownindistributedcomputing.Theproofofourlowerboundsisbasedonthetimelessin-distinguishabilityargument[7,12].Inkroundsofcommu-nication,anetworknodecanonlygatherinformationaboutnodeswhichareatmostkhopsawayandhence,onlythisinformationcanbeusedtodeterminethecomputation'sout-come.Inparticular,weshowthatafterkcommunicationroundstwoneighboringnodesseeexactlythesamegraphtopology;informallyspeaking,bothneighborsareequallyquali edtojointhevertexcover.However,inourexamplegraphs,choosingthewrongneighborwillberuinous.Theremainderofthepaperisorganizedasfollows.Sec-tion2describesthemodelofcomputation.ThelowerboundisconstructedinSection3.InSection4,weshowhowourMVClowerboundcanbeextendedtominimumdominatingset(MDS),maximalmatching(MM),andmaximalindepen-dentset(MIS).Section5concludesthepaper.2.MODELWeconsidertheclassicmessagepassingmodel,whichcon-sistsofapoint-to-pointcommunicationnetwork,describedbyanundirectedgraphG=(V;E).Nodescorrespondtoprocessorsandedgesrepresentcommunicationchannelsbe-tweenthem.Inonecommunicationround,eachnodeofthenetworkgraphcansendanarbitrarilylongmessagetoeachofitsneighbors.Localcomputationsareforfreeandini-tially,nodeshavenoknowledgeaboutthenetworkgraph.Theyonlyknowtheirownuniqueidenti er.1Inkcommu-nicationrounds,anodevmaycollecttheIDsandintercon-nectionsofallnodesuptodistancekfromv.Tv;kisde nedtobethetopologyseenbyvafterthesekrounds,i.e.Tv;kisthegraphinducedbythek-neighborhoodofvwhereedgesbetweennodesatexactlydistancekareexcluded.Thela-belling(i.e.theassignmentofIDs)ofTv;kisdenotedbyL(Tv;k).Becausemessagesizeisunbounded,thebestadeter-ministicalgorithmcandointimek,istocollectitsk-neighborhoodandbaseitsdecisionon(Tv;k;L(Tv;k)).Inotherwords,adeterministicdistributedalgorithmcanberegardedasafunctionmapping(Tv;k;L(Tv;k))tothepossi-bleoutputs.Forrandomizedalgorithms,theoutcomeofvisalsodependentontherandomnesscomputedbythenodesinTv;k.Themodelpresentedisinaccordancewiththeoneusedin[15]andtextbooks[19].Itisthestrongestpossiblemodelwhenprovinglowerboundsforlocalcomputationsbecauseitfocusesentirelyonthelocalityofdistributedproblemsandabstractsawayotherissuesarisinginthedesignofdis-tributedalgorithms(e.g.needforsmallmessages,fastlocalcomputations,etc.).Thisguaranteesthatourlowerboundsaretrueconsequencesoflocality.3.LOWERBOUNDWe rstgiveanoutlineoftheproof.ThebasicideaistoconstructagraphGk=(V;E),foreachpositiveintegerk,whichcontainsabipartitesubgraphSwithnodesetC0[C1andedgesinC0C1asshowninFigure1.SetC0consistsofn0nodeseachofwhichhas0neighborsinC1.Eachofthen001nodesinC1has1,1�0,neighborsinC0.ThegoalistoconstructGkinsuchawaythatallnodesinv2SseethesametopologyTv;kwithindistancek.Inagloballyoptimalsolution,alledgesofSmaybecoveredbynodesinC1andhence,nonodeinC0needstojointhevertexcover.1OurresultsholdforanypossibleIDspaceincludingthestandardcasewhereIDsarethenumbers1;:::;n. Inalocalalgorithm,however,thedecisionofwhetherornotanodejoinsthevertexcoverdependsonlyonitslocalview,thatis,thepair(Tv;k;L(Tv;k)).WeshowthatbecauseadjacentnodesinSseethesameTv;k,everyalgorithmaddsalargeportionofnodesinC0toitsvertexcoverinordertoendupwithafeasiblesolution.Inotherwords,weconstructagraphinwhichthesymmetrybetweentwoadjacentnodescannotbebrokenwithinkcommunicationrounds.Thisyieldssuboptimallocaldecisionsandhence,asuboptimalapproximationratio.Throughouttheproof,wewilluseC0andC1todenotethetwosetsofthebipartitesubgraphS.Ourproofisorganizedasfollows.ThestructureofGkisde nedinSubsection3.1.InSubsection3.2,weshowhowGkcanbeconstructedwithoutsmallcycles,ensuringthateachnodeseesatreewithindistancek.Subsection3.3provesthatadjacentnodesinC0andC1havethesameviewTv;kand nally,Subsection3.4derivesthelowerbounds.3.1TheClusterTreeThenodesofgraphGk=(V;E)canbegroupedintodis-jointsetswhicharelinkedtoeachotherasbipartitegraphs.Wecallthesedisjointsetsofnodesclusters.Wede nethestructureofGkusingadirectedtreeCTk=(C;A)withdoublylabelledarcs`:A! .WerefertoCTkastheclustertree,becauseeachvertexC2CrepresentsaclusterofnodesinGk.ThesizeofaclusterjCjisthenumberofnodestheclustercontains.Anarca=(C;D)2Awith`(a)=(C;D)denotesthattheclustersCandDarelinkedasabipartitegraph,suchthateachnodeu2ChasCneighborsinDandeachnodev2DhasDneighborsinC.ItfollowsthatjCjC=jDjD.Wecallaclusterleaf-clusterifitisadjacenttoonlyoneothercluster,andwecallitinner-clusterotherwise.Definition3.1.TheclustertreeCTkisrecursivelyde- nedasfollows:CT1:=(C1;A1);C1:=fC0;C1;C2;C3gA1:=f(C0;C1);(C0;C2);(C1;C3)g`(C0;C1):=(0;1);`(C0;C2):=(1;2);`(C1;C3):=(0;1)GivenCTk 1,weobtainCTkintwosteps:Foreachinner-clusterCi,addanewleaf-clusterC0iwith`(Ci;C0i):=(k;k+1).Foreachleaf-clusterCiofCTk 1with(Ci0;Ci)2Aand`(Ci0;Ci)=(p;p+1),addk 1newleaf-clustersC0jwith`(Ci;C0j):=(j;j+1)forj=0:::k;j6=p+1.Further,wede nejC0j=n0forallCTk.Figure1showsCT2.TheshadedsubgraphcorrespondstoCT1.Thelabelsofeacharca2Aareoftheform`(a)=(l;l+1)forsomel2f0;:::;kg.Further,settingjC0j=n0uniquelydeterminesthesizeofallotherclusters.Inordertosimplifytheupcomingstudyoftheclustertree,weneedtwoadditionalde nitions.ThelevelofaclusteristhedistancetoC0intheclustertree(cf.Figure1).ThedepthofaclusterCisitsdistancetothefurthestleafinthesubtreerootedatC.Hence,thedepthofaclusterplusoneequalstheheightofthesubtreecorrespondingtoC.IntheexampleofFigure1,thedepthsofC0,C1,C2,andC3are3,2,1,and1,respectively.12dd3d2d0d1d0d1d3d2d1d0d3d2d2d1d0d1dLevel 0Level 1Level 2Level 332C0CS1CCFigure1:Cluster-TreeCT2.NotethatCTkdescribesthegeneralstructureofGk,i.e.itde nesforeachnodethenumberofneighborsineachclus-ter.However,CTkdoesnotspecifytheactualadjacencies.Inthenextsubsection,weshowthatGkcanbeconstructedsothateachnode'sviewisatree.3.2TheLowerBoundGraphInSubsection3.3,wewillprovethatthetopologiesseenbynodesinC0andC1areidentical.Thistaskisgreatlysimpli edifeachnode'stopologyisatree(ratherthanageneralgraph)becausewedonothavetoworryaboutcycles.ThegirthofagraphG,denotedbyg(G),isthelengthoftheshortestcycleinG.WewanttoconstructGkwithgirthatleast2k+1sothatinkcommunicationrounds,allnodesseeatree.GiventhestructuralcomplexityofGkforlargek,constructingGkwithlargegirthisnotatrivialtask.ThesolutionwepresentisbasedontheconstructionofthegraphfamilyD(r;q)asproposedin[13].Forgivenrandq,D(r;q)de nesabipartitegraphwith2qrnodesandgirthg(D(r;q))r+5.Inparticular,weshowthatforappropriaterandq,weobtainaninstanceofGkbydeletingsomeoftheedgesofD(r;q).Inthefollowing,weintroduceD(r;q)uptothelevelofdetailwhichisnecessarytounderstandourresults.Fortheinterestedreader,wereferto[13].Foranintegerr1andaprimepowerq,D(r;q)de nesabipartitegraphwithnodesetP[LandedgesEDPL.ThenodesofPandLarelabelledbyther-vectorsoverthe nite eldq,i.e.P=L=rq.Inaccordancewith[13],wedenoteavectorp2Pby(p)andavectorl2Lby[l].Thecomponentsof(p)and[l]arewrittenasfollows(forD(r;q),thevectorsareprojectedontothe rstrcoordinates):(p)=(p1;p1;1;p1;2;p2;1;p2;2;p02;2;p2;3;p3;2;:::pi;i;p0i;i;pi;i+1;pi+1;i;:::)(1)[l]=[l1;l1;1;l1;2;l2;1;l2;2;l02;2;l2;3;l3;2;:::li;i;l0i;i;li;i+1;li+1;i;:::]:(2)Notethatthesomewhatconfusingnumberingofthecompo-nentsof(p)and[l]ischoseninordertosimplifythefollowingsystemofequations.Thereisanedgebetweentwonodes(p)and[l],exactlyifthe rstr 1ofthefollowingconditionshold(fori=2;3;:::).l1;1 p1;1=l1p1l1;2 p1;2=l1;1p1l2;1 p2;1=l1p1;1li;i pi;i=l1pi 1;i(3)l0i;i p0i;i=li;i 1p1li;i+1 pi;i+1=li;ip1li+1;i pi+1;i=l1p0i;i In[13],itisshownthatforoddr3,D(r;q)hasgirthatleastr+5.Further,ifanodeuandacoordinateofaneighborvis xed,theremainingcoordinatesofvareuniquelydetermined.Thisisconcretizedinthenextlemma.Lemma3.2.Forall(p)2Pandl12q,thereisexactlyone[l]2Lsuchthatl1isthe rstcoordinateof[l]andsuchthat(p)and[l]areconnectedbyanedgeinD(r;q).Analogously,if[l]2Landp12qare xed,theneighbor(p)of[l]isuniquelydetermined.Proof.The rstr 1equationsof(3)de nealinearsystemfortheunknowncoordinatesof[l].Iftheequationsandvariablesarewritteninthegivenorder,thematrixcor-respondingtotheresultinglinearsystemofequationsisalowertriangularmatrixwithnon-zeroelementsinthediag-onal.Hence,thematrixhasfullrankandbythebasiclawsof( nite) elds,thesolutionisunique.Exactlythesameargumentationholdsforthesecondclaimofthelemma.WearenowreadytoconstructGkwithlargegirth.WestartwithanarbitraryinstanceG0koftheclustertreewhichmayhavetheminimumpossiblegirth4.AnelaborationoftheconstructionofG0kisdeferredtoSubsection3.4.Fornow,wesimplyassumethatG0kexists.BothGkandG0karebipartitegraphswithodd-levelclustersinonesetandeven-levelclustersintheother.LetmbethenumberofnodesinthelargerofthetwopartitionsofG0k.Wechooseqtobethesmallestprimepowergreaterthanorequaltom.InbothpartitionsV1(G0k)andV2(G0k)ofG0k,weuniquelylabelallnodesvwithelementsc(v)2q.Asalreadymentioned,GkisconstructedasasubgraphofD(r;q)forappropriaterandq.Wechooseqasdescribedaboveandwesetr=2k 4suchthatg(D(r;q))2k+1.Let(p)=(p1;:::)and[l]=[l1;:::]betwonodesofD(r;q).(p)and[l]areconnectedbyanedgeinGkifandonlyiftheyareconnectedinD(r;q)andthereisanedgebetweennodesu2V1(G0k)andv2V2(G0k)forwhichc(u)=p1andc(v)=l1.Finally,nodeswithoutincidentedgesareremovedfromGk.Lemma3.3.ThegraphGkconstructedasdescribedaboveisaclustertreewiththesamedegreesiasinG0k.Gkhasatmost2mq2k 5nodesandgirthatleast2k+1.Proof.Thegirthdirectlyfollowsfromtheconstruction;removingedgescannotcreatecycles.Forthedegreesbetweenclusters,considertwoneighboringclustersC0iV1(G0k)andC0jV2(G0k)inG0k.InGk,eachnodeisreplacedbyq2k 5newnodes.TheclustersCiandCjconsistofallnodes(p)and[l]whichhavetheir rstcoordinatesequaltothelabelsofthenodesinC0iandC0j,respectively.LeteachnodeinC0ihave neighborsinC0j,andleteachnodeinC0jhave neighborsinC0i.ByLemma3.2,nodesinCihave neighborsinCjandnodesinCjhave neighborsinCi,too.Remark.In[14],ithasbeenshownthatD(r;q)isdiscon-nectedandconsistsofatleastqbr+24cisomorphiccompo-nentswhichtheauthorscallCD(r;q).Clearly,thosecom-ponentsarevalidclustertreesaswellandwecoulduseoneofthemfortheanalysis.Asourasymptoticresultsremainuna ectedbythisobservation,wecontinuetouseGkasconstructedabove.3.3EqualityofViewsInthissubsection,weprovethattwoadjacentnodesinclustersC0andC1havethesameview,i.e.withindistancek,theyseeexactlythesametopologyTv;k.Consideranodev2Gk.Giventhatv'sviewisatree,wecanderiveitsview-treebyrecursivelyfollowingallneighborsofv.Theproofislargelybasedontheobservationthatcorrespondingsubtreesoccurinbothnode'sview-tree.LetCiandCjbeadjacentclustersinCTkconnectedby`(Ci;Cj)=(l;l+1),i.e.eachnodeinCihaslneighborsinCj,andeachnodeinCjhasl+1neighborsinCi.Whentraversinganode'sview-tree,wesaythatweenterclusterCj(resp.,Ci)overlinkl(resp.,l+1)fromclusterCi(resp.,Cj).Furthermore,wemakethefollowingde nitions:Definition3.4.Thefollowingnomenclaturereferstosub-treesintheview-treeofanodeinGk.MiisthesubtreeseenuponenteringclusterC0overalinki.Bi;d;isasubtreeseenuponenteringaclusterC2CnfC0goveralinki,whereCisonlevelandhasdepthd.Definition3.5.WhenenteringsubtreeBi;d;fromaclus-teronlevel 1(+1),wewriteB"i;d;(B#i;d;).Thepredicate:inB:i;d;denotesthatinsteadofi,thelabelofthelinkintothissubtreeisi 1.Thepredicate:isnecessarywhen,afterenteringCjfromCi,weimmediatelyreturntoCionlinki.Intheview-tree,theedgeusedtoenterCjconnectsthecurrentsubtreetoitsparent.Thus,thisedgeisnotavailableanymoreandthereareonlyi 1edgesremainingtoreturntoCi.Thepredicates"and#describefromwhich\direction"aclusterhasbeenentered.Astheview-treesofnodesinC0andC1havetobeabsolutelyidenticalforourprooftowork,wemustnotneglecttheseadmittedlytiresomedetails.Thefollowingexampleshouldclarifythevariousde ni-tions.Additionally,youmayrefertotheexampleofG3inFigure3intheappendix.Example3.6.ConsiderG1.LetVC0andVC1denotetheview-treesofnodesinC0andC1,respectively:VC0=B"0;1;1[B"1;0;1VC1=B"0;0;2[M1B"0;1;1=B"0;0;2[M:1B"0;0;2=B#;:1;1;1B"1;0;1=M:2M1=B";:0;1;1[B"1;0;1Westarttheproofbygivingasetofruleswhichdescribethesubtreesseenatagivenpointintheview-tree.Wecalltheserulesderivationrulesbecausetheyallowustoderivetheview-treeofanodebymechanicallyapplyingthematchingruleforagivensubtree.Lemma3.7.ThefollowingderivationrulesholdinGk:Mi= j=0:::kj6=i 1B"j;k j;1[B";:i 1;k i+1;1B"i;d;1=Ffi+1g[Dfg[M:i+1B#i;d;1=Ffi 1;k d+1g[Dfg[Mk d+1[B";:i 1;d 1;2B"i;d;=Ffi+1g[Dfi+1g[B#;:i+1;d+1; 1 whereFandDarede nedasFW:= j=0:::k d+1j=2WB"j;d 1;+1DW:= j=k d+2:::kj=2WB"j;k j;+1:Proof.We rstshowthederivationruleforMi.ItcanbeseeninExample3.6thattheruleholdsfork=1.Fortheinductionstep,webuildCTk+1fromCTkasde nedinDe nition3.1.M(k)isaninnerclusterandtherefore,onenewclusterBk+1;0;1isadded.Thedepthofallothersubtreesincreasesby1andM(k+1):= j=0:::k+1B"j;k j;1follows.IfweenterM(k+1)overlinki,therewillbeonlyi 1 1edgeslefttoreturntotheclusterfromwhichwehadenteredC0.Consequently,thelinki 1featuresthe:predicate.Theremainingrulesfollowalongthesamelines.LetCibeaclusterwithentry-linkiwhichwas rstcreatedinCTr,rk(NotethatinCTk,r=k dholdsbecauseeachsub-treeincreasesitsdepthbyoneineach\round").AccordingtothesecondbuildingruleofDe nition3.1,rnewneighbor-ingclusters(subtrees)arecreatedinCTr+1.Moreprecisely,anewclusteriscreatedforallentry-links0:::r,excepti.Wecallthesesubtrees xed-depthsubtreesF.IfthesubtreewithrootCihasdepthdinCTk,the xed-depthsubtreeshavedepthd 1.IneachCTr0;r02fr+2;:::;kg,Ciisaninner-clusterandhence,onenewneighboringclusterwithentry-linkr0iscreated.Wecallthesesubtreesdiminishing-depthsubtreesD.InCTk,eachofthesesubtreeshasgrowntodepthk r0.Wenowturnourattentiontothedi erencesbetweenthethreerules.Theystemfromtheexceptionaltreatmentoflevel1,aswellasthepredicates"and#.InB"i;d;1,thelinki+1returnstoC0,butcontainsonlyi+1 1edgesintheview-tree.InB#i;d;1,wehavetoconsidertwospecialcases.The rstoneisthelinktoC0.Foraclusteronlevel1withentry-link(fromC0)i,theequalityk=d+iholdsandtherefore,thelinktoC0isk d+1andthus,Mk d+1follows.Secondly,wewriteB";:i 1;d 1;2,becausethereisoneedgelessleadingbacktotheclusterwherewehadcomefrom.(Notethatsinceweenteredthecurrentclusterfromahigherlevel,thelinkleadingbacktowherewecamefromisi 1,insteadofi+1).FinallyinB"i;d;,weagainhavetotreatthereturninglinki+1specially.NotethatthegeneralderivationruleforB#i;d;ismissingaswewillnotneeditfortheproof.Next,wede nethenotionofr-equality.Intuitively,iftwoview-treesarer-equal,theyhavethesametopologywithindistancer.Definition3.8.LetV1= i=0:::kbiandV2= i=0:::kb0ibeview-trees;biandb0iaresubtreesenteredonlinki.Then,V1andV2arer-equalifallcorrespondingsubtreesare(r 1)-equal,V1r=V2(=bir 1=b0i;8i2f0;:::;kg:Further,allsubtreesare0-equal:Bi;d;0=Bi0;d0;0andBi;d;0=Mi0foralli;i0;d;d0;,and0.Usingthenotionofr-equality,itisnoweasytode newhatweactuallyhavetoprove.WewillshowthatinGk,VC0k=VC1holds.ThisisequivalenttoshowingthateachnodeinC0seesexactlythesametopologywithindistancekasitsneighborinC1.Wewillnowestablishseveralhelperlemmas.Lemma3.9.Let and 0besetsofsubtrees,andletVv1=B"i;d;x[ andVv2=B"i;d;y[ 0betwoview-trees.Then,forallxandyVv1r=Vv2(= r 1= 0:Proof.AssumethattherootsofthesubtreeofVv1andVv2areCiandCj.Thesubtreeshaveequaldepthandentry-linkandtheyhavethusgrownidentically.Hence,allpathswhichdonotreturntoclustersCiandCjmustbeidentical.Further,considerallpathswhich,aftershops,returntoCiandCjoverlinki+1.Aftertheseshops,theyreturntotheoriginalclusterandseeviewsV0v1andV0v2,di eringfromVv1andVv2onlyintheplacementofthe:predicate.Thisdoesnota ect and 0andtherefore,Vv1r=Vv2(=V0v1r s=V0v2^ r 1= 0;s�1:Thesameargumentcanberepeateduntilr s=0andbecauseV0v10=V0v2,thelemmafollows.Lemma3.10.Let and 0besetsofsubtrees,andletVv1=B"i;d;x[ andVv2=B"i;d0;y[ 0betwoview-trees.Then,forallxandy,andforallrmin(d;d0),Vv1r=Vv2(= r 1= 0:Proof.W.l.o.g,weassumed0d.IntheconstructionprocessofGk,therootclustersofVv1andVv2havebeencreatedinstepsk dandk d0,respectively.ByDe nition3.1,allsubtreeswithdepthdd0havegrownidenticallyinbothviews.TheremainingsubtreesofVv2wereallcreatedinstepk d0+1andhavedepthd0 1.ThecorrespondingsubtreesinVv1haveatleastthesamedepthandhence,eachpairofcorrespondingsubtreesare(d0 1)-equal.Itfollowsthatforrmin(d;d0),thesubtreesB"i;d;xandB"i;d0;yareidenticalwithindistancer.UsingthesameargumentasinLemma3.9concludestheproof.2T2T1T0d4-1d1d1VC1d4-1d1T2d3d2++1d3d2++1d3d0-13-1d2d1d0'''dVCd02T1T0TFigure2:Theview-treesVC0andVC1inG3seenuponusinglink1.Figure2showsapartoftheview-treesofnodesinC0andC1inG3.The gureshowsthatthesubtreeswithlinks0and2cannotbematcheddirectlytooneanotherbecauseofthedi erentplacementofthe 1.Itturnsoutthatthisinherentdi erenceappearsineverystepofourtheorem.However,thefollowinglemmashowsthatthesubtreesT0andT2(T00andT02)areequaluptotherequireddistanceand hence,nodesareunabletodistinguishthem.Itisthiscrucialpropertyofourclustertree,whichallowsusto\move"the:predicatebetweenlinksiandi+2andenablesustoderivethemaintheorem.Lemma3.11.Let and 0besetsofsubtreesandletVv1andVv2bede nedasVv1=M:i[B"i 2;k i;2[ Vv2=Mi[B";:i 2;k i;2[ 0:Then,foralli2f2;:::;kg,Vv1k i=Vv2(= k i 1= 0:Proof.Again,wemakeuseofLemma3.7toshowthatMiandB"i 2;k i;2are(k i 1)-equal.Theclaimthenfollowsfromthefactthatthetwosubtreesarenotdistinguishableandtheplacementofthe:predicateisirrelevant.Mi= j=0:::kj6=i 1B"j;k j;1[B";:i 1;k i+1;1B"i 2;k i;2= j=0:::i+1j6=i 1B"j;k i 1;3[ j=i+2:::kB"j;k j;3[B#;:i 1;k i+1;1Forj=f0;:::;i 2;i;:::;kg,allsubtreesareequalaccord-ingtoLemmas3.9and3.10.ItremainstobeshownthatB"i 1;k i+1;1k i 2=B#i 1;k i+1;1.Forthatpurpose,weplugB"i 1;k i+1;1andB#i 1;k i+1;1intoLemma3.7andshowtheirequalityusingthederivationrules.Let bede nedas :=Ffi 2;ig[Dfg.B"i 1;k i+1;1=Ffig[Dfg[M:i=B"i 2;k i;2[M:i[ B#i 1;k i+1;1=Ffi 2;ig[Dfg[Mi[B";:i 2;k i;2=B";:i 2;k i;2[Mi[ Again,ifMiandB"i 2;k i;2are(k i 3)-equal,wecanmovethe:predicatebecausethesubtreesareindistinguishable.Hence,wehavetoshowMik i 3=B"i 2;k i;2.Intheproof,wehavereducedVv1k i=Vv2stepwisetoanexpressionofdiminishingequalityconditions,i.e.Vv1k i=Vv2(=Mik i 1=B"i 2;k i;2(=B"i 1;k i+1;1k i 2=B#i 1;k i+1;1(=Mik i 3=B"i 2;k i;2:ThisprocesscanbecontinueduntileitherB"i 1;k i+1;10=B#i 1;k i+1;1orMi0=B"i 2;k i;2whichisalwaystrue.Finally,wearereadytoprovethemaintheorem.Theorem3.12.ConsidergraphGk.LetVC0andVC1betheview-treesoftwoadjacentnodesinclustersC0andC1,respectively.Then,VC0k=VC1.Proof.Initially,eachnodeinC0seessubtreeMandeachnodeinC1seesB;k;1(denotesthatthesubtreehasnotbeenenteredonanylink):VC0:M= j=0:::kB"j;k j;1VC1:B;k;1= j=0:::kj6=1B"j;k j;2[M1:ItfollowsVC0k=VC1(=B"1;k 1;1k 1=M1becauseallothersubtreesare(k 1)-equalbyLemma3.9.HavingreducedVC0k=VC1toB"1;k 1;1k 1=M1,wecanfurtherreduceittoM2k 2=B"2;k 2;1:M1= j=1:::kB"j;k j;1[B";:0;k;1B"1;k 1;1=B"0;k 2;2[B"1;k 2;2[Dfg[M:2k 2=Lem.3.11B";:0;k 2;2[B"1;k 2;2[Dfg[M2:ByLemmas3.9and3.10,allsubtreeare(k 2)-equal,exceptB"2;k 2;1andM2.ItseemsclearthatwecancontinuetoreduceVC0k=VC1stepbystepinthesamefashionuntilwereach0.Forthein-ductionstep,weassumeVC0k=VC1(=B"r;k r;1k r=MrforrkandproveVC0k=VC1(=B"r+1;k r 1;1k r 1=Mr+1:Mr= j=0:::kj6=r 1B"j;k j;1[B";:r 1;k r+1;1B"r;k r;1= j=0:::rB"j;k r 1;2[Dfg[M:r+1k r 1=Lem.3.11 j=0:::rj6=r 1B"j;k r 1;2[B";:r 1;k r 1;2[ j=r+2:::kB"j;k j;2[Mr+1:ApartfromMr+1(resp,.B"r+1;k r 1;1),allsubtreesare(k r 1)-equalbyLemmas3.9and3.10.SinceMr+1andB"r+1;k r 1;1aretheonlysubtreesnotbeingimmediatelymatched,theinductionstepfollows.Forr=k 1,wegetVC0k=VC1(=B"k;0;10=Mk,whichconcludestheproofbecauseB"k;0;10=Mkistrue.Remark.Asaside-e ect,theproofofTheorem3.12hashighlightedthefundamentalsigni canceofthecriticalpathP=(1;2;:::;k)inCTk.AfterfollowingpathP,theviewofanodev2C0endsupintheleaf-clusterneighboringC0andseesi+1neighbors.Followingthesamepath,anodev02C1endsupinC0andsees ij=0j 1neighbors.Thereisnowaytomatchtheseviews.ThisinherentinequalityistheunderlyingreasonforthewayGkisde ned:Itmustbeensuredthatthecriticalpathisatleastkhopslong.3.4AnalysisInthissubsection,wederivethelowerboundsontheap-proximationratioofk-localMVCalgorithms.LetOPTbeanoptimalsolutionforMVCandletALGbethesolutioncomputedbyanyalgorithm.ThemainobservationisthatadjacentnodesintheclustersC0andC1havethesameview andtherefore,everyalgorithmtreatsnodesinbothofthetwoclustersthesameway.Consequently,ALGcontainsasigni cantportionofthenodesofC0,whereastheoptimalsolutioncoverstheedgesbetweenC0andC1entirelybynodesinC1.Lemma3.13.LetALGbethesolutionofanydistributed(randomized)vertexcoveralgorithmwhichrunsforatmostkrounds.WhenappliedtoGkasconstructedinSubsection3.2intheworstcase(inexpectation),ALGcontainsatleasthalfofthenodesofC0.Proof.Letv02C0andv12C1betwoarbitrary,adja-centnodesfromC0andC1.We rstprovethelemmafordeterministicalgorithms.ThedecisionwhetheragivennodeventersthevertexcoverdependssolelyonthetopologyTv;kandthelabellingL(Tv;k).Assumethatthelabellingofthegraphischosenuniformlyatrandom.Further,letpA0andpA1denotetheprobabilitiesthatv0andv1,respectively,endupinthevertexcoverwhenadeterministicalgorithmAop-eratesontherandomlychosenlabelling.ByTheorem3.12,v0andv1seethesametopologies,thatis,Tv0;k=Tv1;k.Withourchoiceoflabels,v0andv1alsoseethesamedis-tributiononthelabellingsL(Tv0;k)andL(Tv1;k).ThereforeitfollowsthatpA0=pA1.Wehavechosenv0andv1suchthattheyareneighborsinGk.Inordertoobtainafeasiblevertexcover,atleastoneofthetwonodeshastobeinit.ThisimpliespA0+pA11andthereforepA0=pA11=2.Inotherwords,forallnodesinC0,theprobabilitytoendupinthevertexcoverisatleast1=2.Thus,bythelinearityofexpectation,atleasthalfofthenodesofC0arechosenbyalgorithmA.Therefore,foreverydeterministicalgorithmA,thereisatleastonelabellingforwhichatleasthalfofthenodesofC0areinthevertexcover.2Theargumentforrandomizedalgorithmsisnowstraight-forwardusingYao'sminimaxprinciple.Theexpectednum-berofnodeschosenbyarandomizedalgorithmcannotbesmallerthantheexpectednumberofnodeschosenbyanop-timaldeterministicalgorithmforanarbitrarilychosendis-tributiononthelabels.Lemma3.13givesalowerboundonthenumberofnodeschosenbyanyk-localMVCalgorithm.Inparticular,wehavethatE[jALGj]jC0j=2=n0=2.WedonotknowOPT,butsincethenodesofclusterC0arenotnecessarytoobtainafeasiblevertexcover,theoptimalsolutionisboundedbyjOPTjn n0.Inthefollowing,wede nei:=i;8i2f0;:::;k+1g(4)forsomevalue.Lemma3.14.Ifk+1,thenumberofnodesnofGkisnn0 1+k+1 (k+1):Proof.Therearen0nodesinC0.By(4),thenumberofnodesperclusterdecreasesforeachadditionallevelbyafactor.Hence,aclusteronlevellcontainsn0=lnodes.2Infact,sinceatmostjC0jsuchnodescanbeinthevertexcover,foratleast1=3ofthelabellings,thenumberexceedsjC0j=2.Bythede nitionofCTk,eachclusterhasatmostk+1neighboringclustersonahigherlevel.Thus,thenumberofnodesnlonlevellisupperboundedbynl(k+1)ln0l:Summingupoveralllevelslandinterpretingthesumasageometricseries,weobtainnn0k+1i=0 k+1ln01i=0 k+1l=n0+n0 k+111 k+1=n0 1+k+1 (k+1):Itremainstodeterminetherelationshipbetweenandn0suchthatGkcanberealizedasdescribedinSubsection3.2.There,theconstructionofGkwithlargegirthisbasedonasmallerinstanceG0kwheregirthdoesnotmatter.Using(4)(i.e.i:=i),wecannowtieupthislooseendanddescribehowtoobtainG0k.ThenumberofnodesperclusterdecreasesbyafactoroneachlevelofCTk.IncludingC0,CTkconsistsofk+2levels.Themaximumnumberofneighborsinsidealeaf-clusterisk.Hence,wecansetthesizesoftheclustersontheoutermostlevelk+1tobek.Thisimpliesthatthesizeofaclusteronlevellis2k+1 l.Particularly,thesizeofC00atlevel0inG0kisn00=2k+1.LetCiandCjbetwoadjacentclusterswith`(Ci;Cj)=(i;i+1).CiandCjcansimplybeconnectedbyasmanycompletebipartitegraphsKi;i+1asnecessary.Ifweassumethatk+1=2,wehaven2n0byLemma3.14.ApplyingtheconstructionofSubsection3.2,wegetn0n00hn0i2k 5,wherehn0idenotesthesmallestprimepowerlargerthanorequalton0,i.e.hn0i4n00.Puttingalltogether,wegetn0(4n00)2k 442k 44k2:(5)Theorem3.15.TherearegraphsG,suchthatinkcom-municationrounds,everydistributedalgorithmforthemin-imumvertexcoverproblemonGhasapproximationratiosatleast\nnc=k2kand\n 1=kkforsomeconstantc1=4,wherenanddenotethenum-berofnodesandthehighestdegreeinG,respectively.Proof.Wecanchoose4 1=(2k)n1=(4k2)0duetoIn-equality(5).Finally,usingLemmas3.13and3.14,theap-proximationratio isatleast n0=2n n0n0=2=2n0(k+1)=4(k+1)(n=2)1=(4k2)41+1=(2k)(k+1)2\nn1=(4k2)k:Thesecondlowerboundfollowsfrom=k+1. Theorem3.16.Inordertoobtainapolylogarithmicorevenconstantapproximationratio,everydistributedalgo-rithmfortheMVCproblemrequiresatleast\n lognloglognand\n logloglogcommunicationrounds.Proof.Wesetk= logn=loglognforanarbitraryconstant �0.Whenpluggingthisintothe rstlowerboundofTheorem3.15,wegetthefollowingapproximationratio : \rncloglogn 2logn1 loglognlognwhere\risthehiddenconstantinthe\n-notation.Forthelogarithmof ,wegetlog cloglogn 2lognlogn 12loglogn log = c 2 12loglogn log :andtherefore 2\nlog(n)c 2 12:Bychoosinganappropriate ,wecandeterminetheex-ponentoftheaboveexpression.Foreverypolylogarithmicterm (n),thereisaconstant suchthattheaboveexpres-sionisatleast (n)andhence,the rstlowerboundofthetheoremfollows.Thesecondlowerboundfollowsfromananalogouscom-putationbysettingk= log=loglog.Remark.Byde ningi:=i;i2f0;:::;kgandk+1:=k+1=2(insteadofk+1),weobtainslightlystrongerapprox-imationlowerboundsof\n nc=k2 kand\n c0=k k:(6)Theboundsof(6)clearlydonotsucetoimprovethere-sultsofTheorem3.16.4.REDUCTIONSUsingthelowerboundforvertexcover,wecanobtainlowerboundsforseveralotherclassicalgraphproblems.Inthissection,wegivetimelowerboundsfortheconstructionofmaximalmatchingsandmaximalindependentsetsaswellasfortheapproximationofminimumdominatingset.Amaximalmatching(MM)ofagraphGisamaximalsetofedgeswhichdonotsharecommonend-points.Hence,aMMisasetofnon-adjacentedgesofGsuchthatalledgesinE(G)nMMhaveacommonend-pointwithanedgeinMM.Amaximalindependentset(MIS)isamaximalsetofnon-adjacentnodes,i.e.allnodesnotintheMISareadjacenttosomenodeoftheMIS.ThebestknownlowerboundforthedistributedcomputationofaMMoraMISis\n(logn)whichholdsforrings[15].BasedonTheorem3.16,wegetthefollowingstrongerlowerbounds.Theorem4.1.TherearegraphsGonwhicheverydis-tributed,possiblyrandomizedalgorithmrequirestime\nlognloglognand\n logloglogtocomputeamaximalmatching.Thesamelowerboundsholdfortheconstructionofmaximalindependentsets.Proof.Itiswellknownthatthesetofallend-pointsoftheedgesofaMMforma2-approximationforMVC.Thissimple2-approximationalgorithmiscommonlyattributedtoGavrilandYannakakis.Thelowerboundforthecon-structionofaMMthereforedirectlyfollowsfromTheorem3.16.FortheMISproblem,considerthelinegraphL(Gk)ofGk.ThenodesofalinegraphL(G)ofGaretheedgesofG.TwonodesinL(G)areconnectedbyanedgewheneverthetwocorrespondingedgesinGareincidenttothesamenode.TheMMproblemonagraphGisequivalenttotheMISproblemonL(G).Further,iftherealnetworkgraphisG,kcommunicationroundsonL(G)canbesimulatedink+O(1)communicationroundsonG.Therefore,thetimesttocomputeaMISonL(Gk)andt0tocomputeaMMonGkcanonlydi erbyaconstant,tt0 O(1).Letn0and0denotethenumberofnodesandthemaximumdegreeofGk,respectively.ThenumberofnodesnofL(Gk)islessthann02=2,themaximumdegreeofGkislessthan20.Becausen0onlyappearsaslogn0,thepowerof2doesnothurtandthetheoremholds(logn=(logn0)).Weconcludethissectionbyconsideringtheproblemofapproximatingtheminimumdominatingset(MDS)prob-lem.AdominatingsetSisasubsetofthenodesofagraphGsuchthatallnodesofGareeitherinSortheyhaveaneighborinS.Inanon-distributedsetting,MDSinequiv-alenttothegeneralminimumsetcoverproblem3whereasMVCisaspecialcaseofsetcoverwhichcanbeapproxi-matedmuchbetter.Itisthereforenotsurprisingthatinadistributedenvironment,MDSisstrictlyharderthanMVC,too.Inthefollowing,weshowthatthisintuitivefactcanbeformalized.Theorem4.2.TherearegraphsG,suchthatinkcom-municationrounds,everydistributedalgorithmforthemini-mumdominatingsetproblemonGhasapproximationratiosatleast\nnc=k2kand\n 1=kkforsomeconstantc,wherenanddenotethenumberofnodesandthehighestdegreeinG,respectively.Proof.WeshowthateveryMVCinstancecanbeseenasaMDSinstancewiththesamelocality.LetG0=(V0;E0)beagraphforwhichwewanttosolveMVC.WeconstructthecorrespondingdominatingsetgraphG=(V;E)asfollows.ForeverynodeandforeveryedgeinG0,thereisanodeinG.Wecallnodesvn2Vcorrespondingtonodesv02V0n-nodes,andnodesve2Vcorrespondingtoedgese02E0e-nodes.Twon-nodesareconnectedbyanedgeifandonlyiftheyareadjacentinG0.Ann-nodevnandane-nodeveareconnectedexactlyifthecorrespondingnodeandedgeareincidentinG0.Therearenoedgesbetweentwoe-nodes.Clearly,thelocalitiesofG0andGarethesame,i.e.kcommunicationroundsononeofthetwographscanbesimulatedbyk+O(1)roundsontheothergraph.LetCbeafeasiblevertexcoverforG0.Weclaimthatallnodesof3Thereexistapproximationpreservingreductionsinbothdirections. GcorrespondingtonodesinCformavaliddominatingsetonG.Byde nition,alle-nodesarecovered.TheremainingnodesofGarecoveredbecauseforagivengraph,avalidvertexcoverisavaliddominatingsetaswell.Therefore,theoptimaldominatingsetonGisatmostasbigastheoptimalvertexcoveronG0.Therealsoexistsatransformationintheotherdirection.LetDbeavaliddominatingsetonG.IfDcontainsane-nodeve,wecanreplacevebyoneofitstwoneighbors.ThesizeofDremainsthesameandallthreenodescovered(dominated)byvearestillcovered.Bythis,wegetadominatingsetD0whichhasthesamesizeasDandwhichconsistsonlyofn-nodes.BecauseD0dominatesalle-nodes,thenodesofG0correspondingtoD0formavalidvertexcover.Thus,MDSonGisexactlyashardasMVConG0andthetheoremfollowsfromTheorem3.15.Corollary4.3.Toobtainapolylogarithmicorconstantapproximationratioforminimumdominatingset,therearegraphsonwhicheverydistributedalgorithmneedstime\nlognloglognand\n logloglog:Proof.ThecorollaryisadirectconsequenceofTheorem4.2andtheproofofTheorem3.16.Remark.Notethatintheabovecorollary,wegiveatimelowerboundforconstantMDSapproximationalthoughithasbeenshownthatMDScannotbeapproximatedbetterthanlnunlessNPDTIME(nO(loglogn))[5].Becauselocalcomputationisforfreeinourmodel,however,itistheoreticallypossibletogetaconstant-factorapproximationforMDS.5.CONCLUSIONSAsdistributedsystemsgrowlarger,itisbecomingincreas-inglyvitaltodesignalgorithmswhichdonotneedtomain-tainfullinformationaboutthenetwork.Unfortunately,withafewnotableexceptions[15],therehavebeenalmostnohardresults,whichwouldhaveshedlightintothetheoreti-calpossibilitiesandlimitationsoflocality-basedapproaches.Wehaveshownlocality-imposedrestrictionsontheapprox-imabilityandcomputabilityofanumberofdistributedprob-lems.Comparingwiththerespectiveupperbounds,someofourlowerboundsareneartight.Wehopeandbelievethatthevariouslowerboundsgiveninthepresentpaperwillhelptoamelioratethissituation.6.REFERENCES[1]Y.Afek,S.Kutten,andM.Yung.TheLocalDetectionParadigmanditsApplicationstoSelf-Stabilization.TheoreticalComputerScience,186(1-2):199{229,1997.[2]R.ColeandU.Vishkin.DeterministicCoinTossingwithApplicationstoOptimalParallelListRanking.InformationandControl,70(1):32{53,1986.[3]M.Elkin.AFasterDistributedProtocolforConstructingaMinimumSpanningTree.InProc.ofthe15thAnnualACM-SIAMSymposiumonDiscreteAlgorithms(SODA),pages359{368,2004.[4]M.Elkin.UnconditionalLowerBoundsontheTime-ApproximationTradeo sfortheDistributedMinimumSpanningTreeProblem.InProc.ofthe36thACMSymposiumonTheoryofComputing(STOC),2004.[5]U.Feige.AThresholdoflnnforApproximatingSetCover.JournaloftheACM(JACM),45(4):634{652,1998.[6]F.FichandE.Ruppert.Hundredsofimpossibilityresultsfordistributedcomputing.Distrib.Comput.,16(2-3):121{163,2003.[7]M.J.Fischer,N.A.Lynch,andM.S.Paterson.ImpossibilityofDistributedConsensusWithOneFaultyProcess.J.ACM,32(2):374{382,1985.[8]A.IsraeliandA.Itai.AFastandSimpleRandomizedParallelAlgorithmforMaximalMatching.InformationProcessingLetters,22:77{80,1986.[9]F.KuhnandR.Wattenhofer.Constant-TimeDistributedDominatingSetApproximation.InProc.ofthe22ndAnnualACMSymp.onPrinciplesofDistributedComputing(PODC),pages25{32,2003.[10]F.KuhnandR.Wattenhofer.DistributedCombinatorialOptimization.TechnicalReport426,ETHZurich,Dept.ofComputerScience,2003.[11]E.KushilevitzandY.Mansour.An\n(Dlog(N=D))LowerBoundforBroadcastinRadioNetworks.SIAMJournalonComputing,27(3):702{712,June1998.[12]L.Lamport,R.Shostak,andM.Pease.TheByzantineGeneralsProblem.ACMTrans.Program.Lang.Syst.,4(3):382{401,1982.[13]F.LazebnikandV.A.Ustimenko.ExplicitConstructionofGraphswithanArbitraryLargeGirthandofLargeSize.DiscreteAppliedMathematics,60(1-3):275{284,1995.[14]F.Lazebnik,V.A.Ustimenko,andA.J.Woldar.ANewSeriesofDenseGraphsofHighGirth.BulletinoftheAmericanMathematicalSociety(N.S.),32(1):73{79,1995.[15]N.Linial.LocalityinDistributedGraphAlgorithms.SIAMJournalonComputing,21(1):193{201,1992.[16]Z.Lotker,B.Patt-Shamir,andD.Peleg.DistributedMSTforConstantDiameterGraphs.InProc.ofthe20thAnnualACMSymposiumonPrinciplesofDistributedComputing(PODC),pages63{71,2001.[17]M.Luby.ASimpleParallelAlgorithmfortheMaximalIndependentSetProblem.SIAMJournalonComputing,15:1036{1053,1986.[18]M.NaorandL.Stockmeyer.WhatCanBeComputedLocally?InProc.ofthe25thAnnualACMSymp.onTheoryofComputing(STOC),pages184{193,1993.[19]D.Peleg.DistributedComputing:ALocality-SensitiveApproach.SIAM,2000.[20]D.PelegandV.Rubinovich.ANear-TightLowerBoundontheTimeComplexityofDistributedMinimum-WeightSpanningTreeConstruction.SIAMJournalonComputing,30(5):1427{1442,2000. APPENDIXd2d1d3d0d0d2d3d3d1d0d0d1d2d2d0d3d1d0d3d0d2d3d2d1d0d2d1d0d3d2d0dd-1d20d1d-1d31d0d-1d2-1d12d-1d30d-1d2-1d12d-1d3-1d43d3d2d0d-1d11d3d-1d21d0d3d2d0d-1d10d-1d2-1d43d0d3d3d2d0d-1d10d-1d12d3d2d1d0d1d2d-1d32d1d-1d0-1d32d1d0d-1d4-1d43d0d1d3d-1d21d0d3d2d0d-1d12d-1d01d2d-1d3-1d43d0d3d3d2d0d-1d10d-1d12d3d2d1d0d1d2d-1d32d1d-1d0-1d32d1d0d-1d43d-1d20d1d-1331dVC0VC1Figure3:TheClusterTreeCT3andthecorrespondingview-treesofnodesinC0andC1.TheclustertreesCT1andCT2areshadeddarkandlight,respectively.Thelabelsofthearcsoftheclustertreerepresentthenumberofneighborsofnodesofthelower-levelclusterintheneighboringhigher-levelcluster.Thelabelsofthereverselinksareomitted.Intheview-trees,anarclabelledwithistandsforiedges,allconnectingtoidenticalsubtrees.