Towards Polynomial Lower Bounds for Dynamic Problems Mihai P atrascu ATT Labs ABSTRACT - PDF document

highestknowndynamiclowerboundinthecell-probemodelremains (lgN),foranyexplicitproblem[16].1.2TheMultiphaseProblemOurrstcontributionisaneasycriterionforarguingthataproblemmaynotadmitsolutionsinpolylogarithmictime.Letusdenethefollowingmultiphaseproblem,whichises-sentiallyadynamicversionofsetdisjointness:PhaseI.Wearegivenksets,S1;:::;Sk[n].WemaypreprocessthesetsintimeO(nk
).PhaseII.WearegivenanothersetT[n],andhavetimeO(n
)toreadandupdatememorylocationsfromthedatastructureconstructedinPhaseI.PhaseIII.Finally,wearegivenanindexi2[k]andmust,intimeO(),answerwhetherSiisdisjointfromT.Observethatthesizesofthesetsarenotprescribed(ex-ceptthatjSij;jTjn).TheinputinPhaseIconsistsofO(n
k)bits,theinputofPhaseIIofO(n)bits,andthein-putofPhaseIIIofO(lgn)bits(onewordintheWordRAMmodel).Thus,therunningtimeallowedineachphaseisproportionaltothemaximuminputsizeinthatphase,withastheproportionalityfactor.Wesetforththefollowingconjectureaboutthehardnessofthemultiphaseproblem,expressedintermsofourunifyingparameter.Conjecture1.Thereexistconstants �1and�0suchthatthefollowingholds.Ifk=(n ),anysolutiontothemultiphaseproblemintheWordRAMmodelrequires= (n).Webrie ymotivatethesettingofparametersinthecon-jecture.Certainly,n,sincePhaseIIIcansimplyexam-inealltheelementsofSiandT.Furthermore,itseemssome-whatquestionabletoassumeanythingbeyond= (p n),sincetheconjecturewouldfailforrandomsets(assetsofdensitymuchhigherthanp nintersectwithhighprobabil-ity).Weonlyassume= (n)forgenerality.Theconjectureissaferforkn
.Indeed,PhaseIIcantrytocomputetheintersectionofTwithallsetsSi(abooleanvectorofkentries),makingPhaseIIItrivial.ThisisachievednaivelyintimeO(k
n),butfasteralgorithmsarepossibleviafastmatrixmultiplication.However,ifPhaseIIonlyhastimen
k,itcannothopetooutputavectorofsizekwithallanswers.Ontheotherhand,wedonotwantktobetoohigh:weaskthatkn ,sincewewantapolynomiallowerboundintermsofthetotalinputsize(thatis,weneedn=(kn) (1)).Implications.ThevalueofthisconjectureliesintheeasyreductionsfromittotheproblemslistedinSection1.1.Inparticular,weobtain:Theorem2.Ifthemultiphaseproblemishard(inthesenseofConjecture1),thenforeveryproblemlistedinx1.1,thereexistsaconstant"�0suchthattheproblemcannotbesolvedwitho(N")timeperoperationando(N1+")pre-processingtime.Proofsketch.TheproofappearsinAppendixA.Here,webrie yillustratetheeaseofthesereductionsbyprovingthehardnessofErickson'sproblem.Assumewehaveafastdatastructureforthisproblem.Toconstructasolutionforthemultiphaseproblem,eachphaserunsasfollows:I.ThedatastructureisaskedtopreprocessamatrixMofknbooleanvalues,whereM[i][j]=1ij2Si.II.Foreachelementj2T,incrementcolumnjinM.III.Giventheindexi,incrementrowiinM,andthenaskforthemaximumvalueinthematrix.ReportthatSiintersectsTithemaximumvalueis3.ObservethatamaximumvalueofM[i][j]=3canonlyhappenif:(1)theelementwasoriginallyone,meaningj2Si;(2)thecolumnwasincrementedinPhaseII,meaningj2T;(3)therowwasincrementedinPhaseIII,indicatingSiwasthesetofinterest.Thus,M[i][j]=3iSi\T6=;.TheinputsizeforErickson'sproblemisO(nk)bits.IfthepreprocessingisdoneinO((nk)1+")andeachoperationissupportedinO((nk)")time,thentherunningtimeinthemultiphaseproblemwillbe:O((nk)1+")forPhaseI;O(n
(nk)")inPhaseII;andO((nk)")inPhaseIII.Thus,wehaveasolutionwith=(nk)".ThiscontradictsConjecture1for( +1)".Thus,wehaveshownalowerboundof �(nk)=( +1)
= (N=( +1))forErickson'sproblem. 1.3On3SUM-HardnessThe3SUMproblemasks,givenasetSofnnumbers,tonddistinctx;y;z2Ssuchthatx+y=z.TheproblemcanbesolvedeasilyinO(n2)time,anditisalong-standingconjecturethatthisisessentiallythebestpossible(seebe-low).Justlikeprogressondynamiccell-probelowerboundshasbeentooslowtoimpactmanynaturaldynamicproblems,progressonlowerboundsforoinealgorithms(or,inpartic-ular,circuitlowerbounds)isunlikelytoanswermanyofourpressingquestionsverysoon.Instead,itwouldbeofgreatinteresttoargue,basedonsomewidely-believedhardnessassumption,thatnaturalalgorithmicproblemslikendingmaximum oworcomputingtheeditdistancerequiresu-perlineartime.The3SUMconjectureisperhapsthebestproposalforthishardnessassumption,sinceitisacceptedquitebroadly,anditgivesarathersharplowerboundforaverysimpleprobleminthelowregimeofpolynomialtime.Unfortunately,thehopeofusing3SUMappearstoooptimisticwhencontrastedwiththecurrentstateof3SUMreductions.GajentaanandOvermars[10]werethersttouse3SUMhardnesstoargue (n2)lowerboundsincomputationalge-ometry,forproblemssuchasnding3collinearpoints,min-imumareatriangle,separatingnlinesegmentsbyaline,determiningwhethernrectanglescoveragivenrectangle,etc.Subsequently,furtherproblemssuchaspolygoncon-tainment[3]ortestingwhetheradihedralrotationwillcauseachaintoself-intersect[20]werealsoshowntobe3SUM-hard.Allthesereductionstalkabouttransformingthecondi-tionx+y=zintosomegeometricconditionon,e.g.,thecollinearityofpoints.Formally,suchreductionsworkeveninanalgebraicmodel,morphingthe3SUMinstanceintoaninstanceoftheotherproblembycommonarithmetic.Bycontrast,wewouldlikereductionstopurelycombinatorialquestions,talkingaboutgraphs,strings,etc.Suchproblems shewouldknowtheentireinput,andcouldannouncetheresultwithnofurthercommunication.Onemustalsoensurethat �1+.Otherwise,Alice'smessagewouldhaven
Mkbits,andcouldincludeak-bitvectorspecifyingwhetherSiintersectsT,foralli.Then,Bobcouldimmediatelyannouncetheanswer.Finally,itisessentialthatAliceonlyspeakinthebegin-ning.Otherwise,BoborCarmencouldannouncetheO(lgk)bitsofinputonAlice'sforehead,andAlicewouldimmedi-atelyannouncetheresult.Itiseasytoseethatastronglowerboundofthiscommuni-cationgamewouldimplyastrongversionofthemultiphaseconjecture:Observation8.Conjecture7impliesConjecture1.Thisholdseveninthestrongercell-probemodel,andevenifPhaseIisallowedunboundedtime.Proof.Weassumeasolutionforthemultiphasecon-jecture,andobtainacommunicationprotocol.AliceseesS1;:::;SkandT,andthuscansimulatetheactionsofPhaseIandPhaseII.Hermessagedescribesallcellswrittendur-ingPhaseII,includingtheiraddressesandcontents.Thistakesn
M=O(nw)bits,wherewisthewordsize.Subsequently,Bobwillexecutetheactionsofthealgo-rithminPhaseIII.Foreverycellread,hersttestswhetheritwasincludedinthemessagefromAlice.Ifnot,hecom-municatestheaddress(wbits)toCarmen.CarmenseesS1;:::;SkandcanthussimulatePhaseI.Therefore,sheknowsthatcontentsofallcellswrittenduringPhaseI,andcanreplytoBobwiththecontentsofallcellshewantstoread.Intotal,BobandCarmencommunicateM=O(w)bits.Assumingw=O(lgn),an (n)lowerboundonMimpliesan (n0)lowerboundon. Relationtoothercommunicationproblems.Theformu-lationofourcommunicationgameisinspiredbytheroundeliminationlemma[13,19].Inthistwo-playersetting,AlicereceivesS1;:::;SkandBobreceivesTandi2[k].Alicebe-ginsbysendingamessageofo(k)bits.Then,itispossibletoprovethatthemessagecanbeeliminated,whilexingiinawaythatincreasestheerroroftheprotocolbyo(1).Theideaisthatthemessagecanbexedapriori.AlicewillreceiveonlytherelevantSi,andshewillmanufactureS1;:::Si�1;Si+1;:::;Skinawaythatmakesthexedmes-sagebecorrect.Thisispossiblewithprobability1�o(1),asthemessageonlycontainso(1)bitsofinformationaboutSi.Unfortunately,inour3-partysetting,theinitialmessageofo(k)bitsmaydependonboththeinputsofBobandCarmen.Thus,Carmencannot,byherself,manufactureavectorofSj's(j6=i)thatisconsistentwiththemessage.However,theinformationtheoreticintuitionofthelemmaholds,anditisconceivablethatthemessageofAlicecanbeeliminatedinablack-boxfashionforanycommunicationproblemoftheappropriatedirect-sumstructure:Conjecture9.Considera3-partynumber-on-foreheadgameinwhichAliceholdsi2[k],Bobholdsy1;:::;yk2Y,andCarmenholdsx2X.Thegoalistocomputef(x;yi),forsomearbitraryf:XY!f0;1g.IfthereisaprotocolinwhichAlicebeginswithaprivatemessagetoBobofo(k)bits,followedbyMbitsofbidirec-tionalcommunicationbetweenBobandCarmen,thenthe2-partycommunicationcomplexityoffisO(M).Ingeneral,number-on-foreheadcommunicationgamesareconsidereddiculttoanalyze.Inparticular,theasymmetricsetupinourproblemappearssimilartoa3-partycommuni-cationgameproposedbyValiant[22,14].AstrongenoughlowerboundonValiant'sgamewouldruleoutlinear-size,logarithmic-depthcircuitsforsomeexplicitproblems.For-tunately,ourgamemaybeeasiertoanalyze,sincewearesatisedwithmuchweakerbounds(inValiant'ssetting,evenan (n1�")lowerboundwouldnotsuce).2.USING3SUMHARDNESS2.1Convolution3SUMTherstissuethatwemustovercomeforeectiveuseof3SUMhardnessisthefollowing\gap"intheproblem'scom-binatorialstructure:thetestx+y=zmustbeiteratedover�n3
triples,yetthe(tight)lowerboundisonlyquadratic.WedenetheConvolution-3SUMproblemasfollows:givenanarrayA[1::n],determinewhetherthereexisti6=jwithA[i]+A[j]=A[i+j].Observethatthisproblemhasamuchmorerigidstructure,asthepredicateisonlyevalu-atedO(n2)times.AnotherwaytohighlighttheadditionalstructureistonotethatConvolution-3SUMobviouslyhasanO(n2)algorithm,whereasthisislessobviousfor3SUM.Theorem10.If3SUMrequires (n2=f(n))expectedtime,Convolution-3SUMrequires �n2=f2�n
f(n)

expectedtime.Inparticular,if3SUMrequiresn2�o(1)time,thensodoesConvolution-3SUM.Furthermore,if3SUMrequiresn=lgO(1)ntime,sodoesConvolution-3SUM.Asanimmediateappli-cationofthisresult,wementionthatpluggingitintothereductionfrom[23]tondingagiven-weighttriangle,oneimmediatelyobtainsourimprovedboundfromTheorem4.Proof.Ourreductionfrom3SUMtoConvolution-3SUMistherstpointofdeparturefromalgebraicreductions:wewillusehashing.Conceptually,ourideaisfairlysimple.Assumethatwehadsomeinjectivehashmaph:S![n],whichislinearinthesenseh(x)+h(y)=h(z).Then,wecouldsimplyplaceevery3SUMelementx2SintothelocationA[h(x)].Ifthereexistx;y;z2Swithx+y=z,thenh(x)+h(y)=h(z)andthereforeA[h(x)]+A[h(y)]=A[h(x)+h(y)].Thus,thetriplewillbediscoveredbytheConvolution-3SUMalgorithm(nofalsenegatives).Ontheotherhand,thereareclearlynofalsepositives,sincethearrayAislledwithelementsfromS,soanyA[i]+A[j]=A[k]isavalidanswerto3SUM.Unfortunately,wedonothavelinearperfecthashing.In-stead,weuseafamilyofhashfunctionsintroducedbyDi-etzfelbinger[8].Thehashfunctionisdenedbypickingarandomoddintegeraonwbits,wherewisthemachinewordsize.Toobtainvaluesinrangef0;:::;2s�1g,thehashfunctionmultipliesxbytherandomoddvaluea(modulo2w)andkeepsthehighordersbitsoftheresultasthehashcode.InCnotation,thewordxismappedto(unsigned)(a*x)��(w-s).Thisfunctionwasalsousedintheupperboundfor3SUM[2],wherethefollowingcrucialpropertieswereshown:almostlinearity:Foranyxandy,eitherh(x)+h(y)=h(x+y)(mod2s),orh(x)+h(y)+1=h(x+y) Asnotedabove,theexpectednumberoffalsepositivesisO(n2=R).EachonewillhaveacostofO(n=R),givenbytheexhaustivesearchinlines7-8.Thus,therunningtimeofthealgorithm,excludingtheintersectionsinline5,isO(nR+n3 R2)inexpectation.SinceR=!(p nf(n))andR=o(n=f(n)),thistimeiso(n2=f(n)).However,weassumedtheConvolution-3SUMrequirestime (n2=f(n)),implyingthatthetotalcostoftheintersectionoperationsinline6mustbe (n2=f(n)).Wenowimplementtheseintersectionsintermsofndingtrianglesinatripartitegraph,obtainingthedesiredreduc-tion.Thegoalistogetridoftheset-shiftoperationsinlines4-5.Weaccomplishthisbybreakingashiftbysomei2[n]intoashiftbyimodp n,andashiftbybi p nc
p n.Formally,letthepartsinourgraphbeA=B=[R][p n],andC=[n].Weinterpretanelement(x;i)2AasthesetB(x)�i.TheedgesfromAtoCrepresenttheelementsofthesesets:anedgeexistsfrom(x;i)tosomej2Bij2B(x)�i.Anelement(x;i)2BisinterpretedasthesetB(x)+i
p n.TheedgesfromBtoCrepresenttheelementsofthesesets:anedgeexistsfrom(x;i)toj2Bij2B(x)+i
p n.Finally,theedgesfromAtoBrepresentthe2n
Rin-tersectionquestionsthatweaskinline6.ToaskwhetherB(y)intersectsB(x+y)�i,weaskwhetherB(y)+bi p ncp nintersectsB(x+y)�imodp n.Foreachtrianglereported,weruntheexhaustivesearchinlines7-8.WeexpectO(n2=R)triangles.Ifthisexpectationisexceededbyaconstant(thetrianglereportingalgorithmreportstoomanyelements),werehash. 2.3ReductiontotheMultiphaseProblemInthisnalstep,wereducetrianglereportingtothemulti-phaseproblem.CombinedwithTheorem10andLemma11,thisestablishesthereductionfrom3SUMtothemultiphaseproblemclaimedinTheorem3.Inthebeginning,wetakeedgesfromAtoC,andcon-structk=Rp nsetsindicatingtheneighborsofeachvertexfromA.ThesearethesetsgiveninPhaseI.WethenrunRp ncopiesofPhaseII.Eachcopycorre-spondstoavertexofB,andthesetTrepresentstheneigh-borsofthevertexinC.EachexecutionofPhaseIIstartswiththememorystateafterPhaseI.Anycellswrittendur-ingPhaseIIaresavedinaseparatehashtableforeachexecution.Finally,werunaquery(PhaseIII)foreachoftheO(nR)edgesbetweenAandB.Foreachsuchedge,weneedtotesttheintersectionoftheneighborhoodofavertexfromAwiththeneighborhoodofavertexfromB.ThisisdonebyrunningaPhaseIIIwiththeindexoftheAvertex,ontopofthePhaseIImemorystatecorrespondingtotheBvertex.ByLemma11,weonlyneedtodealwithO(n2=R)trian-glesinthegraph.WheneversomePhaseIIIqueryreturnsanintersection,weenumeratethetwosetsofO(n=R)sizeandndtheintersection(andthus,atriangle).ThetotalrunningtimeofthissearchisO(n3=R2)=o(n2=f(n)).Thus,therunningtimemustbedominatedbythemul-tiphaseproblem,andweobtainan (n2=f(n))boundforrunningPhaseI,O(Rp n)copiesofPhaseII,andO(nR)copiesofPhaseIII.Thenalobstacleistheuniverseofthesetsinthemul-tiphaseproblem.SincetherunningtimeisassumedtobeO(U
),whereeachsetcomesfromtheuniverse[U],weneedtodecreasetheuniversefromthecurrentU=ntogetasuperconstantlowerbound.Noticethatoursetsareverysparse,eachhavingO(n=R)values.ThissuggeststhatweshouldhasheachsetbyauniversalfunctiontoauniverseofU=c
(n R)2,forlargeenoughconstantc.Byuniversalityofthehashfunction,iftwosetsaredis-joint,afalseintersectionisintroducedwithsmallconstantprobability.WerepeattheconstructionwithO(lgn)hashfunctionschosenindependently.Weonlyperformtheex-haustivesearchifaqueryreturnstrueinalltheO(lgn)instances.Thismeansthattheexpectednumberoffalsepositivesonlyincreasesbyo(1),sotheanalysisoftherun-ningtimeisunchanged.ThetotalrunningtimeofeachoftheO(lgn)instancesisgivenby:PhaseI,takingtimeO(k
U)=O(Rp n
n2 R2)=O(n2:5=R).O(Rp n)executionsofPhaseII,takingtimeRp n
O(U)=O(n2:5=R).O(nR)executionsofPhaseIII,takingtimeO(nR).TobalancethecostsofO(n2:5=R)andO(nR),wesetR=n0:75,whichisintherangepermittedbyLemma11.Thisgivesalowerboundofn0:25�o(1).TorephrasetheboundinthelanguageofTheorem3,observethatk=Rp n=n1:25andN=O(n2=R2)=O(p n).Thus,k=O(N2:5),andN0:5�o(1).3.REFERENCES[1]N.AilonandB.Chazelle.Lowerboundsforlineardegeneracytesting.InProc.36thACMSymposiumonTheoryofComputing(STOC),pages554{560,2004.[2]I.Baran,E.D.Demaine,andM.Patrascu.Subquadraticalgorithmsfor3SUM.Algorithmica,50(4):584{596,2008.SeealsoWADS2005.[3]G.BarequetandS.Har-Peled.Polygon-containmentandtranslationalmin-Hausdor-distancebetweensegmentsetsare3SUM-hard.InProc.10thACM/SIAMSymposiumonDiscreteAlgorithms(SODA),page862^aAS863,1999.[4]T.M.Chan.Dynamicsubgraphconnectivitywithgeometricapplications.InProc.34thACMSymposiumonTheoryofComputing(STOC),pages7{13,2002.[5]T.M.Chan,M.Patrascu,andL.Roditty.Dynamicconnectivity:Connectingtonetworksandgeometry.InProc.49thIEEESymposiumonFoundationsofComputerScience(FOCS),pages95{104,2008.[6]C.DemetrescuandG.F.Italiano.Anewapproachtodynamicallpairsshortestpaths.JournaloftheACM,51(6):968{992,2004.SeealsoSTOC'03.[7]C.DemetrescuandG.F.Italiano.Trade-osforfullydynamictransitiveclosureonDAGs:breakingthroughtheO(n2)barrier.JournaloftheACM,52(2):147{156,2005.SeealsoFOCS'00.[8]M.Dietzfelbinger.Universalhashingandk-wiseindependentrandomvariablesviaintegerarithmeticwithoutprimes.InProc.13thSymposiumonTheoreticalAspectsofComputerScience(STACS),pages569{580,1996.