PrTxAxB6gxAxBforeveryinputxAxB2XAXBDenotebyCkgthecostofthebestkrounderrorprotocolforgandsimilarlyde neCAkCBkThecase0egCk0gdenotesLasVegaserrorlessprotocolsFinal ID: 160717
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aquadraticgapbetweenLasVegasandDeterminism,andallowingerror,thegapcanbeexponential.Weusesimpleinformationtheoreticandprobabilisticargumentstostrengthenthelowerboundof[4]intwoways.Firstweimprovetheir(k�1)-rounddeterministiclowerboundongkto (n)(forkn=logn),thusshowingthatrandomnesscanbecheaperbyafactorofk=log2nfork-roundprotocols.Thisresultalsoprovidesthelargestgapknownfork&#x-292;logninthedeterministicmodel-thepreviousonewasobtainedin[4]viacountingarguments.ThefactthatthesimulationofMcGeoch[10]isconstructive,givesthesamegapinthearbitrarypartitionmodelforanexplicitfunction,resolvinganopenquestionof[4].Second,weprovethattheprobabilisticupperboundaboveisnotveryfarfromoptimal{wegivean (n=k2)lowerbound,establishinganexponentialgapintheprob-abilisticsettingbetweenkand(k�1)-roundprotocolsforanexplicitlygivenfunction.Theexistanceofsuchfunctions(withsomewhatlargergap)wasprovedbyHalstenbergandReischuk[6],viacomplicatedcountingarguments.Theonlypreviousexponentialgapforanexplicitfunctionwasshownfork=2byYao[20].Westressthesimplicityofourprooftechnique,incontrasttothatof[6].WehaverecentlylearnedthesimilartechniqueswereusedbySmirnov[16]toprovean (n=(k(logn)k))lowerboundongk,whichismuchweakerthanourbound,butestablishanexponentialgapaswell.Finally,weusethecommunicationcomplexitycharacterizationofcircuitdepthofKarchmerandWigderson[8]toestablishgkasa\complete"problemformonotonedepth-kBooleancircuits.(ThisresultwasindependentlydiscoveredbyYanakakis[18]).Thusasimpledeterministicreductionenablestoderivethemonotoneconstant-depthhierarchyofKalwe,Paul,PippengerandYannakakis[7]fromtheconstant-roundhi-erarchyof[4].(Thereversedirectionwasprovenin[7]).Wespeculatethatournewprobabilisticlowerboundmayservetoextendthemonotonecircuithierarchyresulttodepthabovelogn,viaprobabilisticreductions(aswasdonebyRazandWigderson[15]).1.2.TheMulti-PartyModel.Chandra,FurstandLipton[2]devisedthemulti-partycommunicationcomplexitymodel.HeretplayersP1;P2;:::;PtaretryingtocomputeaBooleanfunctiong(x1;x2;:::;xt),wherexi2f0;1gni.(Untilnowallworkinthismodelconsideredequallengthinputs,i.e.ni=nforalli).Thetwististhat2 Pr[T(xA;xB)6=g(xA;xB)]foreveryinput(xA;xB)2XAXB.DenotebyCk(g)thecostofthebestk-round-errorprotocolforg,andsimilarlydeneCA;k;CB;k.Thecase=0(e.g.Ck0(g))denotesLasVegas(errorless)protocols.Finally,ifweleaveTadeterministicprotocol,andchoosetheinputuniformlyatrandom,wecandenethe-errordistributionalcomplexityDk(g)tobethecostofthebestk-roundprotocolforwhichPr[T(xA;xB)6=g(xA;xB)],underthisdistribution.Thefollowinglemmasareuseful.Lemma2.1.[20]Foreveryg;�0Dk2(g)2Ck(g).Lemma2.2.Forallconstants1 3�0�0Ck0(g)=O(Ck(g)).2.2.Results.LetVA;VBbetwodisjointsets(ofvertices)withjVAj=jVBj=nandV=VA[VB.LetFA=ffA:VA!VBg,FB=ffB:VB!VAgandf=(fA;fB):V!Vdenedbyf(v)=(fA(v)v2VAfB(v)v2VB.Foreachk0denef(k)(v)byf(0)(v)=v;f(k+1)(v)=f(f(k)(v)).Letv02VA.Thefunctionswewillbeinterestedincomputingisgk:FAFB!Vdenedbygk(fA;fB)=f(k)(v0).Remarks:Inthefollowingtheoremsnotethatthenumberofinputbitstoeachplayerisnlogn,andthattheyholdforeveryvalueofk.WealsonotethatonecanmakegkaBooleanfunctionbytaking(say)theparityoftheoutputvertex.AllourupperandlowerboundsapplytothisBooleanfunctionaswell.Theorem2.3.[14]CA;k(gk)=O(klogn).Theorem2.4.CB;k(gk)= (n�klogn).Theorem2.5.CB;k1=3(gk)=O((n=k)logn),CB;k0(gk)=O((n=k)log2n).Theorem2.6.CB;k1=3(gk)= (n k2�klogn).Remark:Attheendoftheproofsoftheorems2and4weexplainwhythe\ugly"term�klognintheselowerboundscanbeessentiallyignored.Intheremainderweshowthe\completeness"ofgkformonotonedepthkcircuits.Letgk=gk;ntostressthateachplayergetsnvertices.Definition:ForabooleanfunctionhdeneLd(h)tobethesizeoftheminimalmonotoneformulaofdepthdandunboundedfaninthatcomputesh.DeneLSd(k;n)tobethemaximumofLd(h)overallfunctionshthatcanbecomputedbymonotonecircuitsofunboundedfanindepthkandtotalsizen.DeneLFd(k;n)tobethemaximumofLd(h)overallfunctionshthatcanbecomputedbyaformulaofdepthk4 ProofofTheorems2.3and2.5.CA;k(gk)klognfollowseasily,sinceinroundttherightplayerknowsf(vt�1)=vtandcansendtheselognbitstothesecondplayer.Theideainbeatingthedeterminstic (n)lowerboundwhenthewrongplayerBstartsisasfollows:FirstBchoosesarandomsubsetUVBwithjUj=10n=k,andsendstoAffB(u):u2Ug.NowitisA'sturnandtheystartsendingeachotherv1;v2;:::asabove,butlaggingoneround\behindschedule".However,withprobability2=3,oneofthevi'swillbeinU,whichallowsthemtosavetworounds,and\nishontime".ThisgivesCB;k(gk)=O((k+n=k)logn).ThisalgorithmcanbemadeLas-VegaswithanextrafactorofO(logn)inthecomplexity.ProofofTheorems2.4and2.6Letf=(fA;fB)2FAFBbetheinput.LetT0beadeterministick-roundprotocolforgkinwhichBsendstherstmessage.Notethatatanyroundt1,ifitisB0sturntospeak,thenvt�1=f(t�1)(v0)2VA,andviceversa.ItwillbeconvenienttoreplaceT0byaprotocolTinwhichinanyroundt1,wereplacethemessagembythemessage(m;vt�1).Byinductionont,thisisalwayspossiblefortheplayerwhoseturnitis.Inparticular,itimpliesthatlognbitsaresentperround.ThusifT0usedCbits,TusesC+klognbits.WewillassumeTusesn 2bits,(willbechosenlater),andobtainacontradiction.EverynodezoftheprotocoltreeTcanbelabeledbytherectangleFzAFzBofinputsarrivingatz.BythestructureofT,ifzisatlevelt1(therootisatlevel0),thenv0;v1;:::;vt�1aredeterminedinFzAFzB.WeshallassumetheinputischosenuniformlyatrandomfromFAFB,soinfactweshallboundfrombelowthedistributionalcomlexity.Thustheprobabilityofarrivingatzis(FzAFzB),andgiventhattheinputarrivedatz,itisuniformlydistributedinFzAFzB.Themainlemmabelowintuitivelyshowsthatiftheinputarrivedatzandtherectangleatzhasniceproperties,thenwithhigh(enough)probabilitytheinputwillproceedtoachildwofzwhichisequallynice.NicemeansthatbothFzA;FzBarelargeenough,andthattheplayernotholdingvt�1hasverylittleinformationonvt=f(vt�1).Denotebyczthetotalnumberofbitssentbytheplayersbeforearrivingatz.AssumewithoutlossofgeneralitythatAspeaksatz.LetfzAandfzBberandomvariablesuniformlydistributedoverFzAandFzB,respectively.RecallthatTuses 2nbits,andletsatisfy=Maxf4p ;400g.Deneztobeniceifitsatises:6 ifthealgorithmgivesonebit(sayparity)oftheanswer,itiscorrectwithprobability1 2+2p .ConclusionofTheorem2.4Take=10�4.TherootofTisnice,sobythemainlemmaandinductionwehaveapositiveprobability(2�k)ofreachinganiceleaf,contradictingthefactthattheprotocolnevererrs.ThisprovesonlyCB;k(gk)= (n�klogn),sinceweaugmentedanarbitraryT0toaniceprotocolT.Gettingridofthe�klognterm,achievingthelowerboundCB;k(gk)= (n)(whichisstrongerwhenkn logn)requiresamoredelicateargumentthatwesketchbelow.Theideaistofollowthesamestepsoftheproofwiththefollowingchanges.(1)WestaywiththeoriginalprotocolT0,aswecannota ordtheplayerssendinglognbitsperroundasintheniceprotocolT.(2)Westillxthevertexvt�1bytheplayersendingthemessageatroundt,butavoidpayinglognbitsforthisinformationbyremovingthisvertexfromouruniverse.ThustheinformationIismeasuredrelativetoasmallersetofpointersateveryround.(3)Weproveaweakermainlemma,whichisclearlysucientinthedeterministiccase,namelythateverynicenodezhasatleastonenicechildw.Thedetailsarelefttotheinterestedreader.ConclusionofTheorem2.6.Pick=10�4k�2.Thustheprobabilityofnotreachinganiceleafisk1 25k=1 25,andtheprobabilitythattheprotocolanswerscorrectlyislessthan1 25+(1 2+2 5p k)0:95.ThuswegetfromLemmas2.1and2.2thatDB;k1=20(gk)= (n k2�klogn).Thisboundis (n k2)forallk(n logn)1=3.Forlargerkonecanusethetriviallowerboundkwhichappliestoeveryk-roundprotocol.Notethatwhenkn1=3thistrivialboundislargerthann=k2.ProofofTheorem2.7:Asmentionedabove,theleftinequalitywasprovenin[7],soweproveonlytherightinequality.TheproofisbasedontheKarchmer-Wigdersonchar-acterizationofcircuitdepthintermsofcommunicationcomplexity,whichcanbestatedasfollows.ForeverymonotonefunctionhonnvariableswithmintermsMin(h)andmaxtermsMax(h)deneacommunicationsearchproblemRmhMin(h)Max(h)[n]inwhichplayerAgetsamintermS2Min(h),playerBgetsamaxtermT2Max(h),andtheirtaskistondanelementinS\T.ThenmonotoneformulaeforhandprotocolsforRmharein1-1correspondenceviathesimplesyntacticidenticationof_gateswithplayerA'smovesand^gateswithplayerB'smoves.Inparticular,depthcorrespondstothenumberofrounds,andlogarithmofthesizetothecommunication8 ProofofTheorem3.2Werstrecallafundamentallemmafrom[11]regardingthedistributionofhashvaluesgivenlittleinformationonthehashfunctionandtheargument.Lemma3.3.[11]LetH=fh:I!OgbeacollectionofuniversalhashfunctionsfromdomainIintorangeO.LetAI,BO,CHandp=jBj=jOj.ThenjPr[h(x)2Bjx2A;h2C]�pjq pjHj=(jAjjCj)Restrictthevalueofjtobej2[p n].Thusweconsiderh:f0;1gn!f0;1gp nwhichisstillauniversalhashfunction.AssumeM3(u)p n=5.Thismeansthatthereisanewprotocol(independentofj)tocomputez=h(y)inwhichP3sendsp n=5bits,andthenplayersP1andP2cancomputeeachbitofzseparately,usingaltogethern=5bits.Let(h;y)bechosenuniformlyatrandom.Simpleaveragingshowsthattherearemessagesm1;m2;m3ofP1;P2;P3,respectively,whichunderthisdistributionsatisfyPr[m1]2�n=4,Pr[m2]2�n=4,andPr[m3]2�p n=4.Asm1corespondstoasubsetCofallhashfunctions(inputstoP1),andm2correspondstoasubsetAofallinputstoP2,wecanuseLemma3.3withjHj=jCj2n=4,1=jAj2�3n=4,B=fzgandp=2�p ntoobtainPr[h(y)=zjm1;m2;m3]2p n=4Pr[h(y)=zjm1;m2]2p n=42�p n=2=2�p n=41:Letf:f0;1gm!f0;1gnbeanarbitraryfunction,andforanym0mdenegf:f0;1gm0f0;1gm�m0f0;1glogn!f0;1gbygf(x1;x2;x3)=f(x1x2)x3,wheredenotesconcatenation.Thenexttheoremgivestherelationshipofsize-depthtrade-o sincircuitsto3-roundobliviousprotocols.Theorem3.4.Iffabovecanbecomputedbyacircuitoffan-in2,sizeO(n)anddepthO(logn),thenMs(gf)=O(n=loglogn).ProofofTheorem3.4Letf:f0;1gm!f0;1gnbecomputedbyacircuitCofsizeO(n)anddepthO(logn).ByaresultofValiant[17],thereares=O(n=loglogn)wiresinC,e1,e2;:::,eswiththefollowingproperty.Foreveryinputx2f0;1gm,andeveryj2[n];f(x)jisdeterminedbythevaluese1(x);:::;es(x)onthesewires,togetherwiththevaluesofxi;i2SjwithjSjjp n.Tocomputegf,notethatP3hasaccesstox=(x1x2)10 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