We propose a new concurrent skip list algorithm distinguish ed by a combination of simplicity and scalability The algorit hm employs optimistic synchronization searching without acquiring locks followed by short lockbased validation before adding or ID: 23105
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--------------- 157813152225+1Fig.1.Askiplistwithmaximumheightof4.Thenumberbeloweachnode(i.e.,arrayofnextpointers)isthekeyofthatnode,with 1and+1asthekeysfortheleftandrightsentinelnodesrespectively.2BackgroundAskiplist[7]isalinkedlistthatissortedbykey,andinwhichnodesareassignedarandomheight,uptosomemaximumheight,wherethefrequencyofnodesofaparticularheightdecreasesexponentiallywiththeheight.Anodeinaskiplisthasnotjustonesuccessor,butanumberofsuccessorsequaltoitsheight:eachnodestoresapointertothenextnodeinthelistofeachheightuptoitsown.Forexample,anodeofheight3storesthree\nextpointers",onetothenextnodeofheight1,onetothenextnodeofheight2,andonetothenextnodeofheight3.Figure1illustratesaskiplistinwhichthekeysareintegers.Wethinkofaskiplistashavingseverallayersoflists,andwetalkaboutthepredecessorandsuccessorofanodeateachlayer.Thelistateachlayer,otherthanthebottomlayer,isasublistofthelistatthelayerbeneathit.Becausethereareexponentiallyfewernodesofgreaterheights,wecan ndakeyquicklybysearching rstathigherlayers,skippingoverlargenumbersofshorternodesandprogressivelyworkingdownwarduntilanodewiththedesiredkeyisfound,orelsethebottomlayerisreached.Thus,theexpectedtimecomplexityofskip-listoperationsislogarithmicinthelengthofthelist.Itisconvenienttohaveleftsentinelandrightsentinelnodes,atthebeginningandendofthelistsrespectively.Thesenodeshavethemaximumheight,andinitially,whentheskiplistisempty,therightsentinelisthesuccessoroftheleftsentinelateverylayer.Theleftsentinel'skeyissmaller,andtherightsentinel'skeyisgreater,thananykeythatmaybeaddedtotheset.Searchingtheskiplistthusalwaysbeginsattheleftsentinel.3OurAlgorithmWepresentourconcurrentskip-listalgorithminthecontextofanimplementa-tionofasetobjectsupportingthreemethods,add,removeandcontains:add(v)addsvtothesetandreturnstruei vwasnotalreadyintheset;remove(v)removesvfromthesetandreturnstruei vwasintheset;andcontains(v)returnstruei visintheset.Weshowthatourimplementationislinearizable[5];thatis,everyoperationappearstotakeplaceatomicallyatsomepoint(the2 33intfindNode(intv,34Nodepreds[],35Nodesuccs[])f36intlFound= 1;37Nodepred=&LSentinel;38for(intlayer=MaxHeight 1;39layer0;40layer )f41Nodecurr=pred �nexts[layer];42while(v�curr �key)f43pred=curr;curr=pred �nexts[layer];44g45if(lFound== 1&&v==curr �key)f46lFound=layer;47g48preds[layer]=pred;49succs[layer]=curr;50g51returnlFound;52gFigure1.3.ThefindNodehelperfunctionsuccsofnodepointers,andsearchesexactlyasinasequentialskiplist,startingatthehighestlayerandproceedingtothenextlowerlayereachtimeitencountersanodewhosekeyisgreaterthanorequaltov.Thethreadrecordsinthepredsarraythelastnodewithakeylessthanvthatitencounteredateachlayer,andthatnode'ssuccessor(whichmusthaveakeygreaterthanorequaltov)inthesuccsarray.Ifit ndsanodewiththesought-afterkey,findNodereturnstheindexofthe rstlayeratwhichsuchanodewasfound;otherwise,itreturns 1.Forsimplicityofpresentation,wehavefindNodecontinuetothebottomlayerevenifit ndsanodewiththesought-afterkeyatahigherlevel,soalltheentriesinbothpredsandsuccsarraysare lledinafterfindNodeterminates(seeSection3.4foroptimizationsusedintherealimplementation).NotethatfindNodedoesnotacquireanylocks,nordoesitretryincaseofcon\rictingaccesswithsomeotherthread.Wenowconsidereachoftheoperationsinturn.3.1TheaddoperationTheaddoperation,showninFigure1.4,callsfindNodetodeterminewhetheranodewiththekeyisalreadyinthelist.Ifso(lines59{66),andthenodeisnotmarked,thentheaddoperationreturnsfalse,indicatingthatthekeyisalreadyintheset.However,ifthatnodeisnotyetfullylinked,thenthethreadwaitsuntilitis(becausethekeyisnotintheabstractsetuntilthenodeisfullylinked).Ifthenodeismarked,thensomeotherthreadisintheprocessofdeletingthatnode,sothethreaddoingtheaddoperationsimplyretries.4 54booladd(intv)f55inttopLayer=randomLevel(MaxHeight);56Nodepreds[MaxHeight],succs[MaxHeight];57while(true)f58intlFound=findNode(v,preds,succs);59if(lFound= 1)f60NodenodeFound=succs[lFound];61if(!nodeFound �marked)f62while(!nodeFound �fullyLinked)f;g63returnfalse;64g65continue;66g67inthighestLocked= 1;68tryf69Nodepred,succ,prevPred=null;70boolvalid=true;71for(intlayer=0;72valid&&(layertopLayer);73layer++)f74pred=preds[layer];75succ=succs[layer];76if(pred=prevPred)f77pred �lock.lock();78highestLocked=layer;79prevPred=pred;80g81valid=!pred �marked&&!succ �marked&&82pred �nexts[layer]==succ;83g84if(!valid)continue;86NodenewNode=newNode(v,topLayer);87for(intlayer=0;88layertopLayer;89layer++)f90newNode �nexts[layer]=succs[layer];91preds[layer] �nexts[layer]=newNode;92g94newNode �fullyLinked=true;95returntrue;96g97finallyfunlock(preds,highestLocked);g98gFigure1.4.Theaddmethod5 Ifnonodewasfoundwiththeappropriatekey,thenthethreadlocksandvalidatesallthepredecessorsreturnedbyfindNodeuptotheheightofthenewnode(lines69{84).Thisheight,denotedbytopNodeLayer,isdeterminedattheverybeginningoftheaddoperationusingtherandomLevelfunction.3Validation(lines81{83)checksthatforeachlayeritopNodeLayer,preds[i]andsuccs[i]arestilladjacentatlayeri,andthatneitherismarked.Ifvalidationfails,thethreadencounteredacon\rictingoperation,soitreleasesthelocksitacquired(inthefinallyblockatline97)andretries.IfthethreadsuccessfullylocksandvalidatestheresultsoffindNodeuptotheheightofthenewnode,thentheaddoperationisguaranteedtosucceedbecausethethreadholdsallthelocksuntilitfullylinksitsnewnode.Inthiscase,thethreadallocatesanewnodewiththeappropriatekeyandheight,linksitin,setsthefullyLinked\ragofthenewnode(thisisthelinearizationpointoftheaddoperation),andthenreturnstrueafterreleasingallitslocks(lines86{97).ThethreadwritingnewNode-�nexts[i]istheonecaseinwhichathreadmodi esthenexts eldforanodeithasnotlocked.ItissafebecausenewNodewillnotbelinkedintothelistatlayeriuntilthethreadsetspreds[i]-�nexts[i]tonewNode,afteritwritesnewNode-�nexts[i].3.2TheremoveoperationTheremoveoperationshowninFigure1.5,likewisecallsfindNodetodeterminewhetheranodewiththeappropriatekeyisinthelist.Ifso,thethreadcheckswhetherthenodeis\okaytodelete"(Figure1.6),whichmeansitisfullylinked,notmarked,anditwasfoundatitstoplayer.4Ifthenodemeetstheserequire-ments,thethreadlocksthenodeandveri esthatitisstillnotmarked.Ifso,thethreadmarksthenode,whichlogicallydeletesit(lines111{121);thatis,themarkingofthenodeisthelinearizationpointoftheremoveoperation.Therestoftheprocedureaccomplishesthe\physical"deletion,removingthenodefromthelistby rstlockingitspredecessorsatalllayersuptotheheightofthedeletednode(lines124{138),andsplicingthenodeoutonelayeratatime(lines140{142).Tomaintaintheskip-liststructure,thenodeissplicedoutofhigherlayersbeforebeingsplicedoutoflowerones(though,toensurefreedomfromdeadlock,asdiscussedinSection4,thelocksareacquiredintheoppositeorder,fromlowerlayersup).Asintheaddoperation,beforechanginganyofthedeletednode'spredecessors,thethreadvalidatesthatthosenodesareindeedstillthedeletednode'spredecessors.ThisisdoneusingtheweakValidatefunction,whichisthesameasvalidateexceptthatitdoesnotfailifthesuccessor3ThisfunctionistakenfromLea'salgorithmtoensureafaircomparisonintheex-perimentspresentedinSection5.Itreturns0withprobability34,iwithprobability2 (i+2)fori2[1;30],and31withprobability2 32.4Anodefoundnotinitstoplayerwaseithernotyetfullylinked,ormarkedandpartiallyunlinked,atsomepointwhenthethreadtraversedthelistatthatlayer.Wecouldhavecontinuedwiththeremoveoperation,butthesubsequentvalidationwouldfail.6 101boolremove(intv)f102NodenodeToDelete=null;103boolisMarked=false;104inttopLayer= 1;105Nodepreds[MaxHeight],succs[MaxHeight];106while(true)f107intlFound=findNode(v,preds,succs);108if(isMarkedjj109(lFound= 1&&okToDelete(succs[lFound],lFound)))f111if(!isMarked)f112nodeToDelete=succs[lFound];113topLayer=nodeToDelete �topLayer;114nodeToDelete �lock.lock();115if(nodeToDelete �marked)f116nodeToDelete �lock.unlock();117returnfalse;118g119nodeToDelete �marked=true;120isMarked=true;121g122inthighestLocked= 1;123tryf124Nodepred,succ,prevPred=null;125boolvalid=true;126for(intlayer=0;127valid&&(layertopLayer);128layer++)f129pred=preds[layer];130succ=succs[layer];131if(pred=prevPred)f132pred �lock.lock();133highestLocked=layer;134prevPred=pred;135g136valid=!pred �marked&&pred �nexts[layer]==succ;137g138if(!valid)continue;140for(intlayer=topLayer;layer0;layer )f141preds[layer] �nexts[layer]=nodeToDelete �nexts[layer];142g143nodeToDelete �lock.unlock();144returntrue;145g146finallyfunlock(preds,highestLocked);g147g148elsereturnfalse;149g150gFigure1.5.Theremovemethod7 152boolokToDelete(Nodecandidate,intlFound)f153return(candidate �fullyLinked154&&candidate �topLayer==lFound155&&!candidate �marked);156gFigure1.6.TheokToDeletemethod158boolcontains(intv)f159Nodepreds[MaxHeight],succs[MaxHeight];160intlFound=findNode(v,preds,succs);161return(lFound= 1162&&succs[lFound] �fullyLinked163&&!succs[lFound] �marked);164gFigure1.7.Thecontainsmethodismarked,sincethesuccessorinthiscaseshouldbethenodetoberemovedthatwasjustmarked.Ifthevalidationfails,thenthethreadreleasesthelocksontheoldpredecessors(butnotthedeletednode)andtriesto ndthenewpredecessorsofthedeletednodebycallingfindNodeagain.However,atthispointithasalreadysetthelocalisMarked\ragsothatitwillnottrytomarkanothernode.Aftersuccessfullyremovingthedeletednodefromthelist,thethreadreleasesallitslocksandreturnstrue.Ifnonodewasfound,orthenodefoundwasnot\okaytodelete"(i.e.,wasmarked,notfullylinked,ornotfoundatitstoplayer),thentheoperationsimplyreturnsfalse(line148).Itiseasytoseethatthisiscorrectifthenodeisnotmarkedbecauseforanykey,thereisatmostonenodewiththatkeyintheskiplist(i.e.,reachablefromtheleftsentinel)atanytime,andonceanodeisputinthelist(whichitmusthavebeentobefoundbyfindNode),itisnotremoveduntilitismarked.However,theargumentistrickierifthenodeismarked,becauseatthetimethenodeisfound,itmightnotbeinthelist,andsomeunmarkednodewiththesamekeymaybeinthelist.However,asweargueinSection4,inthatcase,theremusthavebeensometimeduringtheexecutionoftheremoveoperationatwhichthekeywasnotintheabstractset.3.3ThecontainsoperationFinally,weconsiderthecontainsoperation,showninFigure1.7,whichjustcallsfindNodeandreturnstrueifandonlyifit ndsaunmarked,fullylinkednodewiththeappropriatekey.Ifit ndssuchanode,thenitisimmediatefromthede nitionthatthekeyisintheabstractset.However,asmentionedabove,ifthenodeismarked,itisnotsoeasytoseethatitissafetoreturnfalse.WearguethisinSection4.8 3.4ImplementationIssuesWeimplementedthealgorithmintheJavaTMprogramminglanguage,inor-dertocompareitwithDougLea'snonblockingskip-listimplementationinthejava.util.concurrentpackage.Thearraystackvariablesinthepseudocodearereplacedbythread-localvariables,andweusedastraightforwardlockimple-mentation(wecouldnotusethebuilt-inobjectlocksbecauseouracquireandreleasepatterncouldnotalwaysbeexpressedusingsynchronizedblocks).Thepseudocodepresentedwasoptimizedforsimplicity,noteciency,andtherearenumerousobviouswaysinwhichitcanbeimproved,manyofwhichweappliedtoourimplementation.Forexample,ifanodewithanappropriatekeyisfound,theaddandcontainsoperationsneednotlookfurther;theyonlyneedtoascertainwhetherthatnodeisfullylinkedandunmarked.Ifso,thecontainsoperationcanreturntrueandtheaddoperationcanreturnfalse.Ifnot,thenthecontainsoperationcanreturnfalse,andtheaddoperationeitherwaitsbeforereturningfalse(ifthenodeisnotfullylinked)orelsemustretry.Theremoveoperationdoesneedtosearchtothebottomlayerto ndallthepredecessorsofthenodetobedeleted,however,onceit ndsandmarksthenodeatsomelayer,itcansearchforthatexactnodeatlowerlayersratherthancomparingkeys.5Thisiscorrectbecauseonceathreadmarksanode,nootherthreadcanunlinkit.Also,inthepseudocode,findNodealwaysstartssearchingfromthehighestpossiblelayer,thoughweexpectmostofthetimethatthehighestlayerswillbeempty(i.e.,haveonlythetwosentinelnodes).Itiseasytomaintainavari-ablethattracksthehighestnonemptylayerbecausewheneverthatchanges,thethreadthatcausesthechangemusthavetheleftsentinellocked.Thiseaseisincontrasttothenonblockingversion,inwhicharacebetweenconcurrentremoveandaddoperationsmayresultintherecordedheightoftheskiplistbeinglessthantheactualheightofitstallestnode.4CorrectnessInthissection,wesketchaproofforourskip-listalgorithm.Therearefourpropertieswewanttoshow:thatthealgorithmimplementsalinearizableset,thatitisdeadlock-free,thatthecontainsoperationiswait-free,andthattheunderlyingdatastructuremaintainsacorrectskip-liststructure,whichwede nemorepreciselybelow.4.1LinearizabilityFortheproof,wemakethefollowingsimplifyingassumptionaboutinitialization:Nodesareinitializedwiththeirkeyandheight,theirnextsarraysareinitializedtoallnull,andtheirfullyLinkedandmarked eldsareinitializedtofalse.5Comparingkeysisexpensivebecause,tomaintaincompatibilitywithLea'simple-mentation,comparisoninvokesthecompareTomethodoftheComparableinterface.9 Furthermore,weassumeforthepurposesofreasoningthatnodesareneverreclaimed,andthereisaninexhaustiblesupplyofnewnodes(otherwise,wewouldneedtoaugmentthealgorithmtohandlerunningoutofnodes).We rstmakethefollowingobservations:Thekeyofanodeneverchanges(i.e.,key=kisstable),andthemarkedandfullyLinked eldsofanodeareneversettofalse(i.e.,markedandfullyLinkedarestable).Thoughinitiallynull,nexts[i]isneverwrittentonull(i.e.,nexts[i]=nullisstable).Also,athreadwritesanode'smarkedornexts eldsonlyifitholdsthenode'slock(withtheoneexceptionofanaddoperationwritingnexts[i]ofanodebeforelinkingitinatlayeri).Fromtheseobservations,andbyinspectionofthecode,itiseasytoseethatinanyoperation,aftercallingfindNode,wehavepreds[i]-�keyvandsuccs[i]-�.87;挡keyvforalli,andsuccs[i]-�.87;挡key�.87;挡vfori�.87;挡lFound(thevaluereturnedbyfindNode).Also,forathreadinremove,nodeToDeleteisonlysetonce,andthatunlessthatnodewasmarkedbysomeotherthread,thisthreadwillmarkthenode,andthereafter,untilitcompletestheoperation,thethread'sisMarkedvariablewillbetrue.WealsoknowbyokToDeletethatthenodeisfullylinked(andindeedthatonlyfullylinkednodescanbemarked).Furthermore,validationandtherequirementtolocknodesbeforewritingthemensuresthataftersuccessfulvalidation,thepropertiescheckedbytheval-idation(whichareslightlydi erentforaddandremove)remaintrueuntilthelocksarereleased.Wecanusethesepropertiestoderivethefollowingfundamentallemma:Lemma1.Foranodenand0in-�.87;挡topLayer:n-�.87;挡nexts[i]=null=)n-�.87;挡keyn-&#x-276;&#x.153;nexts[i]-&#x-276;&#x.153;keyWede netherelation!isothatm!in(read\mleadstonatlayeri")ifm-&#x-276;&#x.153;nexts[i]=northereexistsm0suchthatm!im0andm0-&#x-276;&#x.153;nexts[i]=n;thatis,!iisthetransitiveclosureoftherelationthatrelatesnodestotheirimmediatesuccessorsatlayeri.Becauseanodehas(atmost)oneimmediatesuccessoratanylayer,the!irelation\follows"alinkedlistatlayeri,andinparticular,thelayer-ilistoftheskiplistconsistsofthosenodesnsuchthatLSentinel!in(plusLSentinelitself).Also,byLemma1,ifm!inandm!in0andn-&#x-276;&#x.153;keyn0-&#x-276;&#x.154;keythenn!in0.Usingtheseobservations,wecanshowthatifm!ininanyreachablestateofthealgorithm,thenm!ininanysubsequentstateunlessthereisanactionthatsplicesnoutofthelayer-ilist,thatis,anexecutionofline141.Thisclaimisprovedformallyforthelazy-listalgorithminarecentpaper[1],andthatproofcanbeadaptedtothisalgorithm.Becausenmustalreadybemarkedbeforebeingsplicedoutofthelist,andbecausethefullyLinked\ragisneversettofalse(afteritsinitialization),thisclaimimpliesthatakeycanberemovedfromtheabstractsetonlybymarkingitsnode,whichwearguedearlieristhelinearizationpointofasuccessfulremoveoperation.Similarly,wecanseethatifLSentinel!indoesnotholdinsomereachablestateofthealgorithm,thenitdoesnotholdinanysubsequentstateunlessthere10 issomeexecutionofline91withn=newNode(asdiscussedearlier,thepreviouslinedoesn'tchangethelistatlayer-ibecausenewNodeisnotyetlinkedinthen).However,theexecutionofthatlineoccurswhilenewNodeisbeinginsertedandbeforenewNodeisfullylinked.Thus,theonlyactionthataddsanodetoalistatanylevelisthesettingofthenode'sfullyLinked\rag.Finally,wearguethatifathread ndsamarkednodethenthekeyofthatnodemusthavebeenabsentfromthelistatsomepointduringtheexecutionofthethread'soperation.Therearetwocases:Ifthenodewasmarkedwhenthethreadinvokedtheoperation,thenodemusthavebeenintheskiplistatthattimebecausemarkednodescannotbeaddedtotheskiplist(onlyanewlyallocatednodecanbeaddedtotheskiplist),andbecausenotwonodesintheskiplistcanhavethesamekey,nounmarkednodeintheskiplisthasthatkey.Thus,attheinvocationoftheoperation,thekeyisnotintheskiplist.Ontheotherhand,ifthenodewasnotmarkedwhenthethreadinvokedtheoperation,thenitmusthavebeenmarkedbysomeotherthreadbeforethe rstthreadfoundit.Inthiscase,thekeyisnotintheabstractsetimmediatelyaftertheotherthreadmarkedthenode.Thisclaimisalsoprovedformallyforthesimplelazylist[1],andthatproofcanbeadaptedtothisalgorithm.4.2Maintainingtheskip-liststructureOuralgorithmguaranteesthattheskip-liststructureispreservedatalltimes.By\skip-liststructure",wemeanthatthelistateachlayerisasublistofthelistsatlowerlayers.Itisimportanttopreservethisstructure,asthecomplexityanalysisforskiplistsrequiresthisstructure.Toseethatthealgorithmpreservestheskip-liststructure,notethatlinkingnewnodesintotheskiplistalwaysproceedsfrombottomtotop,andwhileholdingthelocksonallthesoon-to-bepredecessorsofthenodebeinginserted.Ontheotherhand,whenanodeisbeingremovedfromthelist,thehigherlayersareunlinkedbeforethelowerlayers,andagain,whileholdinglocksonalltheimmediatepredecessorsofthenodebeingremoved.Thispropertyisnotguaranteedbythelock-freealgorithm.Inthatalgorithm,afterlinkinganodeinthebottomlayer,linksthenodeintherestofthelayersfromtoptobottom.Thismayresultinastateofanodethatislinkedonlyinitstopandbottomlayers,sothatthelistatthetoplayerisnotasublistofthelistatthelayerimmediatelybeneathit,forexample.Moreover,attemptstolinkinanodeatanylayerotherthanthebottomarenotretried,andhencethisstateofnonconformitytotheskip-liststructuremaypersistinde nitely.4.3Deadlockfreedomandwait-freedomThealgorithmisdeadlock-freebecauseathreadalwaysacquireslocksonnodeswithlargerkeys rst.Moreprecisely,ifathreadholdsalockonanodewithkeyvthenitwillnotattempttoacquirealockonanodewithkeygreaterthanorequaltov.Wecanseethatthisistruebecauseboththeaddandremovemethodsacquirelocksonthepredecessornodesfromthebottomlayerup,and11 SunFireTMT2000SunEnterpriseTM6500Operations: 9% add, 1% remove, 90% containsRange: 200,000500100015002000250030003500400045005000020406080ThreadsThroughputLazyLeaSeqOperations: 9% add, 1% remove, 90% containsRange: 200,00050010001500200025003000020406080ThreadsThrougputLazyLeaSeqOperations: 9% add, 1% remove, 90% containsRange: 2,000,000500100015002000250030003500020406080ThreadsThroughputLazyLeaSeqOperations: 9% add, 1% remove, 90% containsRange: 2,000,000200400600800100012001400020406080ThreadsThrougputLazyLeaSeqFig.8.Throughputinoperationspermillisecondof1,000,000operations,with9%add,1%remove,and90%containsoperations,andarangeofeither200,000or2,000,000.thekeyofapredecessornodeislessthanthekeyofadi erentpredecessornodeatalowerlayer.Theonlyotherlockacquisitionisforthenodethataremoveoperationdeletes.Thisisthe rstlockacquiredbythatoperation,anditskeyisgreaterthanthatofanyofitspredecessors.Thatthecontainsoperationiswait-freeisalsoeasytosee:itdoesnotacquireanylocks,nordoesiteverretry;itsearchesthelistonlyonce.5PerformanceWeevaluatedourskip-listalgorithmbyimplementingintheJavaprogramminglanguage,asdescribedearlier.WecomparedourimplementationagainstDougLea'snonblockingskip-listimplementationintheConcurrentSkipListMapclassofthejava.util.concurrentpackage,whichispartoftheJavaTMSE6plat-form;toourknowledge,thisisthebestwidelyavailableconcurrentskip-listimplementation.Wealsoimplementedastraightforwardsequentialskiplist,inwhichmethodsweresynchronizedtoensurethreadsafety,foruseasabaselineintheseexperiments.Wedescibesomeoftheresultsweobtainedfromtheseexperimentsinthissection.12 SunFireTMT2000SunEnterpriseTM6500Operations: 20% add, 10% remove, 70% containsRange: 200,00050010001500200025003000350040004500010203040506070ThreadsThroughputLazyLeaSeqOperations: 20% add, 10% remove, 70% containsRange: 200,0005001000150020002500020406080ThreadsThrougputLazyLeaSeqOperations: 20% add, 10% remove, 70% containsRange: 2,000,00050010001500200025003000020406080ThreadsThroughputLazyLeaSeqOperations: 20% add, 10% remove, 70% containsRange: 2,000,00020040060080010001200020406080ThreadsThrougputLazyLeaSeqFig.9.Throughputinoperationspermillisecondof1,000,000operationswith20%add,10%remove,and70%containsoperations,andrangeofeither200,000or2,000,000.Wepresentresultsfromexperimentsontwomultiprocessorsystemswithquitedi erentarchitectures.The rstsystemisaSunFireTMT2000server,whichisbasedonasingleUltraSPARCR\rT1processorcontainingeightcom-putingcores,eachwithfourhardwarestrands,clockedat1200MHz.Eachfour-strandcorehasasingle8-KBytelevel-1datacacheandasingle16-KBytein-structioncache.Alleightcoresshareasingle3-MBytelevel-2uni ed(instructionanddata)cache,andafour-wayinterleaved32-GBytemainmemory.Dataac-cesslatencyratiosareapproximately1:8:50forL1:L2:Memoryaccesses.TheothersystemisanolderSunEnterpriseTM6500server,whichcontains15sys-temboards,eachwithtwoUltraSPARCR\rIIprocessorsclockedat400MHzand2GbytesofRAMforatotalof30processorsand60GbytesofRAM.Eachprocessorhasa16-KBytedatalevel-1cacheanda16-Kbyteinstructioncacheonchip,anda8-MByteexternalcache.Thesystemclockfrequencyis80MHz.Wepresentresultsfromexperimentsinwhich,startingfromanemptyskiplist,eachthreadexecutesonemillion(1,000,000)randomlychosenoperations.Wevariedthenumberofthreads,therelativeproportionofadd,removeandcontainsoperations,andtherangefromwhichthekeyswereselected.Thekeyforeachoperationwasselecteduniformlyatrandomfromthespeci edrange.13 SunFireTMT2000SunEnterpriseTM6500Operations: 50% add, 50% remove, 0% containsRange: 200,0005001000150020002500300035004000020406080ThreadsThroughputLazyLeaOperations: 50% add, 50% remove, 0% containsRange: 200,000200400600800100012001400020406080ThreadsThrougputLazyLeaFig.10.Throughputinoperationspermillisecondof1,000,000operations,with50%addand50%removeoperations,andarangeof200,000Inthegraphsthatfollow,wecomparethethroughputinoperationspermillisecond,andtheresultsshownaretheaverageoversixrunsforeachsetofparameters.Figure8presentstheresultsofexperimentsinwhich9%oftheoperationswereaddoperations,1%wereremoveoperations,andtheremaining90%werecontainsoperations,wheretherangeofthekeyswaseithertwohundredthou-sandortwomillion.Thedi erentrangesgivedi erentlevelsofcontention,withsigni cantlyhighercontentionwiththe200,000range,comparedwiththe2,000,000range.Aswecanseefromtheseexperiments,bothourimplementationandLea'sscalewell(andthesequentialalgorithm,asexpected,isrelatively\rat).Inallbutonecase(with200,000rangeontheoldersystem),ourimplementationhasaslightadvantage.Inthenextsetofexperiments,weranwithhigherpercentagesofaddandremoveoperations,20%and10%respectively(leaving70%containsopera-tions).TheresultsareshowninFigure9.Ascanbeseen,ontheT2000system,thetwoimplementationshavesimilarperformance,withaslightadvantagetoLeainamultiprogrammedenvironmentwhentherangeissmaller(highercon-tention).Thesituationisreversedwiththelargerrange.Thisphenomenonismorenoticeableontheoldersystem:thereweseea13%advantagetoLea'simplementationonthesmallerrangewith64threads,and20%advantagetoouralgorithmwiththesamenumberofthreadswhentherangeislarger.Toexplorethisphenomenon,weconductedanexperimentwithasigni cantlyhigherlevelofcontention:halfaddoperationsandhalfremoveoperationswitharangeof200,000.TheresultsarepresentedinFigure10.Ascanbeclearlyseen,underthislevelofcontention,ourimplementation'sthroughputdegradesrapidlywhenapproachingthemultiprogrammingzone,especiallyontheT2000system.Thisdegradationisnotsurprising:Inourcurrentimplementation,whenanaddorremoveoperationfailsvalidation,orfailstoacquirealockimmediately,itsimplycallsyield;thereisnopropermechanismformanagingcontention.14 Sincetheaddandremoveoperationsrequirethatthepredecessorsseenduringthesearchphasebeunchangeduntiltheyarelocked,weexpectthatunderhighcontention,theywillrepeatedlyfail.Thus,weexpectthataback-o mechanism,orsomeothermeansofcontentioncontrol,wouldgreatlyimproveperformanceinthiscase.Toverifythatahighlevelofcon\rictisindeedtheproblem,weaddedcounterstocountthenumberofretriesexecutedbyeachthreadduringtheexperiment.Thecountersindeedshowthatmanyretriesareexecutedona64threadsrun,especiallyontheT2000.Mostoftheretriesareexecutedbytheaddmethod,whichmakessensebecausetheremovemethodmarksthenodetoberemovedbeforesearchingitspredecessorsinlowerlayers,whichpreventschangeofthesepredecessor'snextpointersbyaconcurrentaddoperation.6ConclusionsWehaveshownhowtoconstructascalable,highlyconcurrentskiplistusingaremarkablysimplealgorithm.Theprincipalopenquestioniswhetherwecanimprovethealgorithm'sperformanceathighlevelsofcontention.Onesimpleapproachistouseamoresophisticatedback-o schemewhensynchronizationcon\rictsaredetected.Anotherintriguingapproachistoallowtherandomnessoftheskip-list'sheighttobecompromisedunderhighcontention.Inthealgorithmpresentedhere,whenathreadaddsanewitem,itgivesupandretriesifitencountersasynchronizationcon\rictwhenlinkingtheitematanylevel.Inprinciple,athreadthatencountersacon\rictwhenlinkingtheitematlayer`�0couldsimplystopthere,leavingtheitemlinkedatlevelszeroto` 1,actingasifithadrandomlychosen` 1.Theresultingdatastructure,whilestructurallystillaskiplist,wouldbealittle\ratterthanitshouldbe.Whetherthisapproachise ectiveisthesubjectoffuturework.References1.Colvin,R.,Groves,L.,Luchangco,V.,andMoir,M.Formalveri cationofalazyconcurrentlist-basedset.InProceedingsofComputer-AidedVeri 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