GarbageCollectioninObjectOriented DatabasesUsingTransactionalCyclic PrasanRoy1 ReferenceCounting S.Sudarshan1 S.Seshadri1S.Ashwin1 AviSilberschatz2 fprasan,seshadri,sudarshag@cse.iitb.ernet.in 1IndianInstituteofTechnology, Mumbai400076,India MurrayHill,NJ07974 sashwin@cs.wisc.edu 2BellLaboratories avi@bell-labs.com grammerfromexplicitlydeallocatingmemory.inthispaper,wepresentagarbage Garbagecollectionisimportantinobject-orienteddatabasestofreethepro- Abstract collectionalgorithm,calledtransactionalcyclicreferencecounting(tcrc),for objectorienteddatabases.thealgorithmisbasedonavariantofareferencecountingalgorithmproposedforfunctionalprogramminglanguagesthealgorithmkeeps trackofauxiliaryreferencecountinformationtodetectandcollectcyclicgarbage. Thealgorithmworkscorrectlyinthepresenceofconcurrentlyrunningtransactions, andsystemfailures.itdoesnotobtainanylongtermlocks,therebyminimizing interferencewithtransactionprocessing.itusesrecoverysubsystemlogstodetect algorithmandpresentresultsofaperformancestudyoftheimplementation. pointerupdates;thus,existingcodeneednotberewritten.finally,itexploits schemainformation,ifavailable,toreducecosts.wehaveimplementedthetcrc Objectorienteddatabases(OODBs),unlikerelationaldatabases,supportthenotionof 1objectidentity,andobjectscanrefertootherobjectsviaobjectidentiers.Requiring Introduction theprogrammertowritecodetotrackobjectsandtheirreferences,andtodeleteobjects CurrentlyatUniversityofWisconsinMadison1

thatarenolongerreferenced,iserrorproneandleadstocommonprogrammingerrors suchasmemoryleaks(garbageobjectsthatarenotreferredtofromanywhere,and haven'tbeendeleted)anddanglingreferences.whiletheseproblemsarepresentin traditionalprogramminglanguages,theeectofamemoryleakislimitedtoindividual programsthatcreateandaccessthem.automatedgarbagecollectionisessentialin Theproblembecomesmoreseriousinpersistentobjectstores,sinceobjectsoutlivethe runsofprograms,sinceallgarbageisimplicitlycollectedwhentheprogramterminates. anobjectorienteddatabasetoprotectfromtheerrorsmentionedabove.infact,the SmalltalkbindingfortheODMGobjectdatabasestandardrequiresautomatedgarbage Allobjectsthatarereachablefromthepersistentrootorfromthetransientprogram objectsandthearcsarethereferencesbetweenobjects.thegraphhasapersistentroot. collection. stateofanon-goingtransactionarelive;whiletherestaregarbage.weoftencallobject WemodelanOODBinthestandardwayasanobjectgraph,whereinthenodesarethe referencesaspointers. space;theentireoldspaceisthenreclaimed.incontrast,themarkandsweepalgorithm CopyingCollectorbased[YNY94]andMarkandSweepbased[AFG95].Thecopying collectoralgorithmtraversestheentireobjectgraphandcopiesliveobjectsintoanew Therehavebeentwoapproachestogarbagecollectioninobjectorienteddatabases: databaseanddeletesallobjectsthatareunmarked.thecopyingcollectoralgorithm marksallliveobjectsbytraversingtheobjectgraphandthentraverses(sweeps)theentire cases,butmaydestroyprogrammerspeciedclusteringresultinginworseperformance reclustersobjectsdynamically;thereclusteringcanimprovelocalityofreferenceinsome inothercases.thegarbagecollectionalgorithmsof[yny94]aswellas[afg95]handle concurrencycontrolandrecoveryissues. prohibitivelyexpensivefordatabaseslargerthanthememorysize,particularlyifthere aremanycross-pagereferences.intheworstcase,whenthebuersizeisasmallfraction ofthedatabasesizeandobjectsinapagerefertoobjectsinotherpagesonly,there Withboththeabovealgorithms,thecostoftraversingtheentireobjectgraphcanbe maybeani/oforeverypointerinthedatabase.toalleviatethisproblem,earlierwork fewpages.eachpartitionstoresinter-partitionreferences,thatisreferencestoobjects [YNY94,AFG95]hasattemptedtodividethedatabaseintopartitionsconsistingofa inthepartitionfromobjectsinotherpartitions,inapersistentdatastructure.objects referredtofromotherpartitionsaretreatedasiftheyarereachablefromthepersistent partition.eachpartitionisgarbagecollectedindependentofotherpartitions;references root,andarenotgarbagecollectedeveniftheyarenotreferredtofromwithinthe eciencyoccurringifthewholepartitiontsintothebuerspace. moreecient;thesmallerthepartition,themoreecientthetraversal,withmaximum toobjectsinotherpartitionsarenotfollowed.thus,partitioningmakesthetraversal Unfortunately,smallpartitionsincreasetheprobabilityofself-referentialcyclesof 2

by[cwz94]showedthatevensmallincreasesindatabaseconnectivitycanproducesignicantamountsofsuchgarbage.therefore,itisnotclearthatpartitionsizescanbtitionedgarbagecollectionalgorithms.previousworkhasmaintainedthatsuchcross cyclestructureswillbefew,andwill\probably"notbeaproblem.however,simulations madeverysmallwithouteitherfailingtocollectlargeamountsofgarbageoremploying special(andexpensive)techniquestodetectsuchcyclicgarbage. ofkeepingacountofthenumberofpointerspointingtoeachobject.whenthereference countoftheobjectbecomeszero,itisgarbageandeligibleforcollection.reference countinghastheattractivepropertiesoflocalizedandincrementalprocessing.unfortu- AnaturalalternativeisReferenceCounting.ReferenceCountingisbasedontheidea garbagethatcrosspartitionboundaries;suchcyclicgarbageisnotdetectedbytheparnately,basicreferencecountingcannotdealwithself-referentialcyclesofobjects;each objectcouldhaveapositivereferencecount,yetalltheobjectsinthecyclemaybe unreachablefromthepersistentroot,andthereforebegarbage.however,anumberof extensionsofthebasicreferencingcountingalgorithmtohandlecyclicdatahavebeen proposedintheprogramminglanguagecommunity,including:[bro85,bro84,pvep88]. [Bro85,Bro84]forfunctionalprogramminglanguages,whichhandlesselfreferentialcycles Morerecentworkinthisareaincludes[Lin90,MWL90,JL91]. (TCRC),basedonBrownbridge'salgorithm,whichissuitableforgarbagecollectionin ofgarbage.wepresentanalgorithm,calledtransactionalcyclicreferencecounting Inthispaper,weconsideraversionofreferencecounting,proposedbyBrownbridge anoodb.thesalientfeaturesofthetcrcalgorithmare: Itdetectsallselfreferentialcyclesofgarbageunlikebasicreferencecounting,and Itperformsaverylocalizedversionofmark-and-sweeptohandlecyclicdata,with thepartitionedgarbagecollectionalgorithms. eachmark-and-sweeplikelytoaccessfarfewerobjectsthanaglobalmark-andsweep.thusitdoesnothavetoexaminetheentiredatabasewhilecollecting Itallowstransactionstorunconcurrently,anddoesnotobtainanylongtermlocks, therebyminimizinginterferencewithtransactionprocessing. garbage,exceptintheworstcase. Itisintegratedwithrecoveryalgorithms,andworkscorrectlyinspiteofsystem Itexploitsschemainformation,ifavailable,toreducecosts.Inparticular,ifthe crashes.italsousesrecoverysubsystemlogstodetectpointerupdates;thus,existingapplicationcodeneednotberewritten. schemagraphisacyclic,nocyclicreferencesarepossibleinthedatabaseandtcrc behavesidenticallytoreferencecounting. 3

before;webelievethisisoneofthecentralcontributionsofourpaper.wealsopresent Designingacyclicreferencingcountingalgorithmwhichallowsconcurrentupdatesand handlessystemcrashesisrathernon-trivial,andtoourknowledgehasnotbeendone aproofofcorrectnessofthetcrcalgorithm. weexpectistrueinanyrealisticscenario,pointerupdatesareonlyasmallfractionof countbeingupdated.thisoverheadwillhaveonlyasmallimpactonperformanceif,as referencecounts.however,eachpointerupdatecanonlyresultinatmostonereference Aproblemoftencitedagainstreferencecountingschemesistheoverheadofupdating theoverallupdates.fortcrc,moreover,theoverheadisosetbythereducedcostof tendedabstractofthispaper[ars+97],inthefollowingways.thereisnolongeran assumptionthattransactionsfollowstrict2pl;infactthecurrentalgorithmmakesno traversalswhilecollectinggarbage. assumptionsaboutthelockingpoliciesusedbythetransaction.thereisnolongeran Thealgorithmpresentedinthispaperimprovesonthatpresentedinanearlierex- assumptionthattransactionsfollowstrictwal(thatis,boththeundoandredovaluesmustbeloggedbeforeactuallyperformingtheupdate);onlythenormal(non-strict) WALisassumedtobefollowed.Thatis,thecurrentalgorithmrequiresundovaluesto amorerestrictedlocaltraversalthantheearlieralgorithm,andisthereforepotentially beloggedbeforetheupdate,whiletheredovaluesmaybeloggedanytimebeforetheend moreecient. ofthetransaction(beforeoraftertheupdate).finally,thecurrentalgorithmperforms ofthetransaction. afterthetransactionends,andarenotrequiredtobeforcedtotheserverbeforetheend thatis,updatesmadebyatransactionrunningattheclientcanbereectedattheserver Fortheclient-serversetting,thecurrentalgorithmalsorelaxestheforcerequirement; WepresentaperformancestudyofTCRCbasedontheimplementation;thestudyclearly illustratesthebenetsoftcrc. markandsweepalgorithmonastoragemanagercalledbrahmadevelopediniitbombay. WehaveimplementedaprototypeoftheTCRCalgorithmaswellasthepartitioned 2 Brownbridge'sCyclicReferenceCountingAlgorithm OurTransactionalCyclicReferenceCountingalgorithmisbasedontheCyclicReference Counting(CRC)algorithmproposedbyBrownbridge[Bro84,Bro85],inthecontextof functionalprogramminglanguages. [Bro84,Bro85]istolabeledgesintheobjectgraphasstrongorweak.Thelabellingis donesuchthatacycleintheobjectgraphcannotconsistofstrongedgesalone{itmust ThebasicideabehindtheCyclicReferenceCounting(CRC)algorithmofBrownbridge 4

haveatleastoneweakedge.twoseparatereferencecountsforstrongandforweakedges strongedgesornot.hence,intheabsenceoffurtherinformation,thealgorithmtakesthe ingeneraltocheaplydeterminewhetherlabellinganewedgeasstrongcreatesacycleof (denotedsrefcandwrefcrespectively)aremaintainedperobject.itisnotpossible conservativeviewthatlabellinganewedgestrongcouldcreateacycleofstrongedges, andlabelsthenewedgeweak. fromitaredeleted.ifthesrefciszero,butwrefcisnon-zero,thereisachance thatsisinvolvedinaselfreferentialcycleofgarbage.ifthesrefcofanobjects objects,thesrefcaswellaswrefciszero,thensisgarbageandsandtheedges TheSRefCandWRefCareupdatedasedgesarecreatedanddeleted.Ifforan TCRCalgorithmdoesnotguaranteethislastproperty). isgreaterthanzero,thensisguaranteedtobereachablefromtheroot(however,our SRefCofSiszeroandWRefCofSisnonzero,alocalizedmarkandsweepalgorithm detectswhethersandanyoftheobjectsreachablefromsareindeedgarbage.the theonlypotentialcandidatesforbecominggarbagearesandobjectsreachablefroms.if IftheobjectgraphdidnothaveanygarbagebeforethedeletionofanedgetoS,then fromsandcoloursthemred.letusdenotetheabovesetbyr.itthencoloursgreen localizedmarkandsweepperformsatraversalfromsandidentiesallobjectsreachable everyobjectinrthathasareferencefromanobjectoutsider(detectedusingreference counts).italsocoloursgreenallobjectsreachablefromanygreenobject.duringthis atleastonestrongpointertoit.wewilldescribethispointerstrengthupdateindetail greenmarkingphasesomepointerstrengthsareupdatedtoensurethateveryobjecthas inthecontextofourtransactionalcyclicreferencecountingalgorithm.attheend,all environmentwithconcurrentupdatestoobjects,anddonotdealwithpersistentdata objectsinrnotmarkedgreenaregarbageandaredeleted. rithm,weredesignedforasingleusersystem.theycannotbeusedinamulti-user andfailures.ourcontributionslieinextendingbrownbridge'salgorithmto(a)uselogs However,priorcyclicreferencecountingalgorithms,includingBrownbridge'salgo- ofupdatestodetectchangestoobjectreferences,(b)toworkinanenvironmentwith strongpointers. atime,and(e)optimizethelocalizedmarkandsweepsignicantlybyfollowingonly transactionaborts,(d)handleabatchofupdatesatatimeratherthanoneupdateat concurrentupdates,(c)toworkonpersistentdatainthepresenceofsystemfailuresand Inthissection,wedescribeoursystemmodelandoutlinethearchitecturalassumptions 3onwhichourgarbagecollectorisbased. SystemModelandAssumptions andredorecordsarerepresentedasundo(tid,oid,oset,old-value),andredo(tid,oid, Inourmodel,transactionslogundoandredoinformationforallupdates.Undo 5

oset,new-value),wheretiddenotesatransactionidentierandoidanobjectidentier. theosetwecandetermineifthevaluethathasbeenupdatedisapointereld.these Objectcreationisloggedasobject-allocation(tid,oid).Thecommitlogisrepresentedas commit(tid);andtheabortlogisrepresentedasabort(tid).werequirethatfromtheoid wecanidentifythetypeoftheobject(perhapsbyrstfetchingtheobject),andfrom requirementsaresatisedbymostdatabasesystems. (oid)isvalidonlyifiteitherreferstoapersistentroot,orispresentinapointereldof ofanactivetransactionthatreadtheoidfromanobjectinthedatabaseorcreated anobjectinthedatabase,orisinthetransientmemory(programvariablesorregisters) Aswithanyothergarbagecollectionscheme,weassumethatanobjectidentier theobjectitrefersto.notethatthisprecludestransactionsfrompassingoidstoother transactions,andfromstoringoidsinexternalpersistentstorage. Assumption3.1ThetransactionsfollowWAL,thatis,theylogtheundovaluebefore actuallyperformingtheupdate,buttheredovaluemaybeloggedanytime(beforeorafter theupdate).2 Assumption3.2Alllogsforatransactionareforcedtodiskbeforecommitorabort databases. (force-logs-at-abortinadditiontoforce-logs-at-commit).2 Theassumptionsabovearesatisedbytypicalstoragemanagersforobject-oriented WewillnowdescribetheTransactionalCyclicReferenceCounting(TCRC)algorithm. 4Werstdescribethedatastructuresneededbythetransactionalcyclicreferencecounting algorithm. numberofstrongpointerspointingtotheobject,aweakreferencecount(wrefc)giving Associatedwitheachobject,wemaintainastrongreferencecount(SRefC)givingthe 4.1 DataStructures thenumberofweakpointerspointingtotheobject,andastrengthbitfortheobject. Eachpointeralsohasastrengthbit.Thepointerisstrongifthestrengthbitinthepointer andthestrengthbitintheobjectpointedtohavethesamevalue;otherwisethepointer isweak.thisrepresentationofstrengthusingtwobitsisanimportantimplementation weak,andallweakpointerstotheobjectstrong.allthatneedbedoneistoipthe valueofthestrengthbitintheobject. thestrengthofallpointerstoanobject,thatismakingallstrongpointerstotheobject trick,frombrownbridge[bro84,bro85].itmakesveryecienttheoperationofipping 6

whichcontainsoidsfortheobjectswhichhaveazerosrefc,i.e.nostrongpointers incidentonthem.thepersistentrootisneverputintothewrt. TheTCRCalgorithmalsomaintainsanothertable,theWeakReferenceTable(WRT), crashbyscanningtheentiredatabase.therefore,itisnotnecessarytomakeitpersistentandincurtheoverheadofloggingupdatestothesestructures.reconstructingthis informationatcrashwillhoweveraecttheavailabilityofthedatabase.iffastrecovery Alltheaboveinformationcanbeconstructedfromtheobjectgraphafterasystem isrequiredthenwecouldmakethesestructurespersistentatthecostofextralogging. Thechoiceofwhetherornottomakethisinformationpersistentcanbelefttoeach installation. informationtobeupdated/inserted/deleted.thustheirupdateswillbeundoneifthe fromthewrtareloggedaspartofthetransactionwhosepointerupdatecausedthe updateofthestrengthbitofanobjectorofapointer,andtheinsertordeleteofentries Iftheabovestructuresaremadepersistent,thenupdatestoSRefCandWRefC, thismeansthatirrespectiveofwhethertheabovestructuresaremadepersistentthese crashes. transactiondoesnotcomplete,andwillberedone(whilerepeatinghistory)ifthesystem neednotbepersistent. Apartfromtheabovestructures,wehavethefollowingnon-persistentstructures{ whetherthecolouroftheobjectisredorgreen.thistableisstoredondisksincethesize ReferenceTable(RRT);thistableassociateswith(some)objectsastrongredreference count(sredrefc),aweakredreferencecount(wredrefc),andabitthatindicates Thereisanon-persistenttablewhichisusedduringgarbagecollection:theRed ofthistablecouldbelargeintheworstcase,butupdatestothistablearenotlogged. thetemporaryreferencetable(trt),whichcontainsallthoseoidssuchthatareference totheobjectwasaddedordeletedbyanactivetransaction,ortheobjectwascreated bythetransaction.anoidintrtistaggedwiththetidofthetransactionthatis Similarto[AFG95]TCRCalsomaintainsannon-persistentin-memorytablecalled reponsibleforitsinsertionintotrt.theremaybemultipleentriesintrtforthesame oid.anobjectwhoseoidisintrtmaynotbegarbageevenifitisunreachablefrom anyotherobject,sincethetransactionmaystoreareferencetotheobjectbackinthe 4.2 database.updatestotrtarealsonotlogged. TCRCconsistsoftwodistinctalgorithms,runbydierentprocesses.Therstistheloganalyzeralgorithm(LogAnalyzer).Thesecondalgorithmistheactualgarbagecollection TheAlgorithm takenforthedurationofcollectgarbage.wedescribethetwoalgorithmsbelow. algorithm(collectgarbage).theexecutionoftheseissynchronizedbytwolatches:a loganalyzerlatchthatistakenforthedurationofloganalyzer,andagclatchthatis 7

4.2.1LogAnalyzer Thelog-analyzeralgorithmanalyzeslogrecordsgeneratedbythetransaction,andperformsvariousactionsbasedonthelogrecords.Weshallassumeitisrunaspartofthe transactionitself,andisinvokedeachtimealogrecordisappendedtothesystemlog tail,andisatomicwithrespecttotheappendingofthelogrecord. thread,andwhenatransactionappendsalogrecordtothesystemlog,itactuallyonly deliversittothelog-analyzer,whichthenappendsthelogrecordtothesystemlog. Intheactualimplementation,itispossibletorunthelog-analyzerasaseparate IftheSRefCfallstozeroafterthedecrementthentheobject'soidisputintoWRT. decrementsthewrefcorsrefcforanobjectwhenapointertotheobjectisdeleted. ProcedureAddPointer,bydefault,setsthestrengthofthepointertobeweakandincrementstheWRefCoftheobjectpointedto.Thestrengthissettoweaksothatcycles Thelog-analyzermakesuseofthefollowingprocedures.ProcedureDeletePointer (whichisalsoacquiredbythegarbagecollectionthread)toestablishaconsistentpoint makesomenewpointersstrong. ofstrongedgesarenotcreated;however,wewillseeinsection6thatwemaybeableto theloganalyzeranddependingonthetypeofthelogrecordvariousactionsasoutlined inthelog.thelatchisobtainedforthedurationoftheprocedure.thelogisanalyzedby TheprocedureLogAnalyzerworksasfollows.Firstitobtainstheloganalyzerlatch belowaretaken. Forundo/redologrecordscausedbypointerupdates,thereferencecountsforthe aectedobjectsareupdated.thisisdonebydeletepointerincaseofundologs, Forlogrecordscorrespondingtothecreationofobjects,thereferencecountsfor thetrttaggedwiththetidofthetransactionthatmadetheupdate. andaddpointerincaseofredologs.theoidoftheaectedobjectisinsertedinto thenewobjectareinitializedtozero,andtheoidofthecreatedobjectisinserted Forend-of-transaction(commitorabort)logrecords,thealgorithmrsttriesto intothewrt.theoidofthecreatedobjectisinsertedintothetrttaggedwith thetidofthetransactionthatcreatedtheobject. notinprogressandremovealltheoidentriesfortheterminatingtransactionfrom thetrtandthegclatchreleasedthereafter(recallthateachoidentryintrtis taggedwiththetidofthetransactionthatisresponsibleforitspresenceintrt). getthegclatch.ifthelatchisobtainedimmediately,thengarbagecollectionis However,ifthegclatchcannotbeobtainedimmediatelythenagarbagecollectionis Alloperationsonpointerstrengthsandreferencecountsareprotectedbyalatchon inprogressconcurrently.inthiscase,theoidentriesfortheterminatingtransaction arenotremoved,butinsteadaggedforlaterremovalbythegarbagecollector. 8

WRTandTRTarealsoprotectedbylatches. theobjectpointedto,althoughnotexplicitlymentionedinouralgorithms.accessto Property4.1ThepersistentrootisneverplacedintheWRT.Ithasnoreferencesto it.thereforeitneveroccursinthetrtorrrt.2 Thefollowingpropertiesfollowfromtheabovediscussion. when(a)thetransactionhasendedand(b)garbagecollectionisnotinprogress.2 Property4.2TheobjectsinTRTcorrespondingtoatransactionareremovedonly 4.2.2GarbageCollector Thegarbagecollectionalgorithmisactivatedperiodically(possiblydependingonavailabilityoffreespace).Thealgorithmmakesuseofthefollowingsupportfunctions. inrrt.thepseudocodeforredtraverseappearsinfigure1.redtraverseperformsa fuzzylocalizedtraversaloftheobjectgraph.itisinvokedonallobjectsinwrtthatare garbage.therestofthegarbagecollectionalgorithmisrestrictedtoonlytheobjects ProcedureRedTraversepopulatestheRRTwithobjectsitidentiesaspotential notintrt.thereafter,anobjectisvisitedbyanyoftheinvocationsifandonlyifthe objectisnotintrtandalltheobjectsthathaveastrongpointertothisobjecthave beenvisitedearlier.nolocksareobtainedontheobjectsbeingtraversed.shortterm latchesmaybeobtainedonobjectsorpagestoensurephysicalconsistency.redtraverse objectinrrtattheinstantitvisitstheobject.redtraversealsomaintainsforeach objectinrrt,twocounts:sredrefcandwredrefc,givingrespectivelythenumber marksallvisitedobjectsredandputstheminrrt. ofstrongandweakpointerstotheobjectfromallotherobjectsvisited.thesecountsare Additionally,RedTraversecachesthereferencecounts(SRefCandWRefC)ofeach Figure1,theseobjectsarecolouredblue.Fortherestofthepaper,weignorethepresence sincesuchobjectsmaybeaddedtorrtlaterinthetraversal.inthepseudocodeof thesecountsforobjectsthatarereachablebyasingleweakedgefromobjectsinrrt, maintainedontheyduringthetraversal;inordertodoso,redtraversealsomaintains ofblueobjectsinrrt,andassumethattheyareexplicitlyremovedfromrrtafterthe lastinvocationofredtraverse.theinvocationsofredtraversearecollectivelytermedred traversal. objectsinrrtgreenandupdatingsomepointerstrengthstoensurethateveryobject absenceofupdatetransactionsduringthegarbagecollectionphase,noobjectwillbein itvisitshasatleastonestrongpointerreferringtotheobject(thisensuresthatinthe ProcedureGreenTraverseperformsafuzzytraversalwiththepurposeofmarkinglive WRTthusensuringthatthereisnoworktobedoneduringthenextgarbagecollections phase.seetheorem5.11foraformalproofofthisstatement).inaddition,thepointer 9

ProcedureRedTraverse(oid) Input:oid:(oidof)theobjecttobetraversed f gproceduretraverse(oid) ifoidisnotinrrt InsertRRT(oid) Input:oid:(oidof)theobjecttobetraversed f ifoidisnotintrtf foreachpointeroid!poidf colouroidred ifpoidisnotinrrt updatesredrefcpoidandwredrefcpoid ifsredrefcpoid==srefcpoid InsertRRT(poid) dependingonthestrengthofoid!poid g g Traverse(poid) /*thisisthelaststrongpointertopoid*/ Input:oid:(oidof)theobjecttobeinsertedintoRRT gprocedureinsertrrt(oid) f colouroidblue insertoidintorrt g cachecurrentvaluesofsrefcandwrefcforoid insrefcoidandwrefcoidrespectively withsredrefcoid=wredrefcoid=0 Figure1:PseudoCodeforRedTraverse 10

Input:oid:(oidof)theobjecttobetraversed ProcedureGreenTraverse(oid) f colouroidyellow forallpointersoid!poidf gforallpointersoid!poidf ifpoidisinrrtandisnotyellow ifpoidisgreenf GreenTraverse(poid) ifsrefcpoid==0 updatereferencecountsofpoid makeoid!poidstrong(ifweak) getloganalyzerlatch removepoidfromwrt gelsef releaseloganalyzerlatch updatereferencecountsofpoid getloganalyzerlatch releaseloganalyzerlatch makeoid!poidweak(ifstrong) gcolouroidgreen g ifsrefcpoid==0 insertpoidintowrt g Figure2:PseudocodeforGreenTraverse 11

ProcedureCollectGarbage f RRT=fg acquiregclatch /*P0*/ foreachoidinwrtbutnotintrt /*alsocachesthereferencecounts RedTraverse(oid) WaitforalltransactionsinTLISTtoend TLIST=listofalltransactionsactiveatP1 /*P1*/ ofvisitedobjectsinsrefcoidandwrefcoid*/ /*P2:instantwhenalltransactionsinTLISTend*/ L0: foreachoidinrrtbutnotintrtf ifsrefcoid+wrefcoid> /*SRefCoidandWRefCoidreferto cachedreferencecounts*/ ifsrefcoid==0/*oidisinwrt*/ SRedRefCoid+WRedRefCoidf updatereferencecountsofoid invertthestrengthofallreferencestooid getloganalyzerlatch removeoidfromwrt g/*p3*/ g GreenTraverse(oid) releaseloganalyzerlatch L1:foreachoidinRRTthatisredandisinTRT foreachoidinrrtthatisred /*P4*/ GreenTraverse(oid) releasegclatch removeallaggedentriesfromtrt /*P5*/ Collect(oid) g Figure3:PseudoCodeforCollectGarbage 12

strengthupdateshavetobedoneinsuchafashionthatstrongcyclesdonotremainat theendofthegarbagecollectionphase.thepseudocodeforgreentraverseappearsin Figure2.StartingfromtheobjectProcedureGreenTraverseisinvokedon,itvisitsallthe objectsinrrtthatarereachablefromthisobjectinadepthrstmanner:itbacktracks fromanobjectafterithasvisitedallobjectsinrrtreachablefromthatobject.just beforebacktrackingfromanobject,itcolourstheobjectgreenandupdates(ifneeded) thestrengthsofthereferencesoutoftheobject.ifthereferenceistoagreenobject,itis madestrongifitisweak;otherwiseifthereferenceistoanyotherobject(possiblytoan objectnotinrrt)thenitismadeweakifitisstrong.theinvocationsofgreentraverse arecollectivelytermedgreentraversal. pointedto.italsoremovestheobjectfromwrt. oftheobject,updatingthestoredreferencecounts(srefcandwrefc)oftheobjects ProcedureCollectactuallydeletesanobject;beforedoingso,itdeletesallpointersout infigure3.below,wepresentadiscussionofthestepsinvolved.additionally,wepoint outinstancesduringtheexecution(showninthegure)thatwillbereferredintheproof. ThegarbagecollectionalgorithmisimplementedbyProcedureCollectGarbage,shown toterminate. foralltransactionsthatwereactiveatsomepointp1aftertheendoftheredtraversal invokedonallobjectsthatareinwrtbutnotintrt.aftertheredtraversal,wewait Therststepistoacquiregclatch.AtsomepointP0afterthis,RedTraverseis (non-strict)walandarbitrarylockingprotocols.wepresenttheintuitionbelow;the formalproofappearsinsection5. IncaseatransactionTdoesnotfollowstrict-2PLlocking,thefollowingscenariocan Thiswaitisnecessaryforcorrectnessofthealgorithminfaceoftransactionsfollowing inx,andthenreleasesthelockonx.next,allthereferencestoygetdeletedandallthe occur.ttakesalockonsomeobjectx,readsthereferencetoyfromsomepointereld becauseoidofyiscachedbytwhichcaninsertapointertoyfromsomeotherobject transactionsthatmadethedeletionscommit(thiscanhappenbecausethasreleased thelockonx).thismakesyunreachablefromanyliveobject;butyisnotgarbage later.thewaitensuresthatalltransactionssuchastendbeforeanalysisproceeds. insertsthatoccuredduringtheredtraversalareforcedtodiskinadditiontotheundologs forpointerdeletionsbeforeanalysisproceeds;thisisbecauseallthelogsforatransaction are(byassumption)necessarilyforcedbeforetheendofthetransaction.thismakesthe Thewaitfurtherensures,infaceof(non-strict)WAL,thatalltheredologsforpointer alatchonthetransactiontablewhilescanningit.thisissafebecauseonlythefollowing TRTconsistentwithrespecttoanyinsertionordeletionofpointersthatmighthave occuredduringtheredtraversal. endsanddoesnotappearintlist thisisacceptablebecausewewerejustgoingto mayhappenwhilethescanisinprogress:(a)sometransactionthatwasactiveatp1 ThelistoftransactionsTLISTcanbedeterminedfuzzily thatis,weneednottake 13

waitforittoendanyway;or(b)somesometransactionstartsafterp1anditsentry appearsintlist thisisacceptablebecausethiscanonlyextendthewait.p2isthe GreenTraverseisinvokedonanobjectthatisinRRTbutnotinTRTifthetotalred instantwhenallthetransactionsintlistterminate. traversalisstrictlylessthanitstotalreferencecounts(srefc+wrefc)thatare referencecounts(sredrefc+wredrefc)fortheobjectcomputedduringthered Next,wedogreentraversaltomarkgreenallthelivenodesinRRT.AfterP2, countsupdatedatomicallywithrespecttothelog-analyzer(loganalyzerlatchisusedfor theinvocation,thestrengthsofallreferencestothisobjectareinvertedandthereference proof,theseobjectsarelive:theyarereferredfromsomeobjectnotinrrt.butbefore cachedduringtheredtraversal(ascheckedinstatementl0).aswillbeshowninthe ensurethatinabsenceofupdatetransactionsnoworkwillbedonebythenextgarbage collectionphaseisover,therewillbeatleastonestrongpointertotheobject(noticethat thispurpose).theinversionofstrengthsisnecessarytoensurethatafterthisgarbage mightcausestrongcyclestobeformed.but,asprovedinlemma5.8,thesewillnot allreferencestotheobjectfromobjectsnotinrrtmustbeweak).thisisnecessaryto collectionsphase;thisisformallyprovedintheorem5.11.thisinversionofstrengths existafterthisgarbagecollectionphaseisover. aboveobjectsarealsomarkedgreen,byinvokinggreentraverse.theseinvocationsget andcanpotentiallybestoredbackinthedatabase.objectsthatarereachablefromthe alsomarkedgreensincetheirreferencesmaystillbestoredinanongoingtransaction AftertheaboveinvocationscompleteatP3,anyobjectsinRRTthatareinTRTare overatp4. GreenTraverseonredobjectsthathavebeeninsertedintoTRTsincetheywerechecked followingsimpleprocedureisenough.wemakerepeatedscansoftherrt,invoking obtainedsuchthatatthispoint,noredobjectinrrtisintrt.itturnsoutthatthe Inthepseudocode,wehaveleftunspeciedhow(atstepL1)theconsistentpointis inthepreviousscan;andterminatingwhenwecomeacrossnosuchobjectinthelatest scan.theconsistentpointcorrespondstotheinstantthelastscanstarts.thisisbecause intrtduringthelastscan acontradiction. ifsomeobjectinrrtisredandisintrtatthestartofthelastscan,thenduringthe staysintrtbecauseofproperty4.2(b).butthen,itmustbedetectedtoberedand scanitstaysredbecauseofthefactthatgreentraverseisnotinvokedintheinterim;and is,sincegclatchwasacquired)areremoved. removablebytransactionsthatcompletedsincethegarbagecollectionphasebegan(that atp5.finally,thegclatchisreleasedandalltheentriesintrtthatweretaggedas AllobjectsinRRTthatareredatP4arecollectednext.Thecollectiongetsover 14

4.2.3SupportforLogicalUndobytheRecoveryManager TheTCRCalgorithmneedssomesupportfromtherecoverymanagerintheformof supportinglogicalundostoensurecorrectness.thissupportisrequiredonlyifwechoose tomaintainthereferencecounts,pointerstrengthsandwrtpersistent.therearesome actionswhoseundoshavetobeperformedlogicallyandnotphysically.wediscussthem belowanddiscusswhatthelogicalundoshoulddoineachcase: thepointerwiththestrengthsettobeweak(evenifitwasstrongearlier).similarly, thecorrectnessofthealgorithm.therightwaytoundoapointerdeletionistoreinsert update,ifperformednaively,mayintroducestrongcyclesinthegraph,whichcanaect PointerDeletionandStrengthUpdate:Undoofapointerdeletionorstrength theundoofapointerstrengthupdate(doneincaseofsystemcrashduringthegarbage strength). collectionphase)istosetthestrengthofthepointerasweak(irrespectiveoftheoriginal entobjectswhicharelockedbythetransactions.theobjectoitselfneednotbelocked sinceonlyareferencetoitisbeingupdated.onlyshorttermlatchesarenecessaryfor rentlyupdatedbymultipletransactions(includingthegarbagecollector)throughdier- ReferenceCountsUpdate:ThereferencecountsofanobjectOcanbeconcur- maintainingphysicalconsistency.ifatransactionthatupdatedthereferencecountofan objectaborts,itshouldbelogicallyundone:theundoofareferencecountincrementis isalwaysanincrementofwrefcsinceareinsertedpointerisalwaysweak. adecrementofthesamereferencecount,whiletheundoofareferencecountdecrement Weformallystatethedenitionofagarbageobject. 5 ProofofCorrectness reachablefromthepersistentrootorfromanyobjectinthetrtorfromanyobject Denition5.1(GarbageObject) whosereference(oid)hasbeenreadbyanyactivetransaction.2 Anobjectisdenedtobegarbageifitisnot doesnotsatisfytheotherconditionsandthereforeisgarbageintheconventionalsense. NotethateventuallyanobjectthatisgarbageintheconventionalsensewillleaveTRT andanyactivetransactionthatreadareferencetoitwillterminateandthuswillbecome TheabovedenitionconsidersanobjectasliveifitisreachablefromTRT,evenifit theabovedenition. garbageinthesenseofdenition5.1.ourlemmasandproofsaresimpliedbyusing 5.1 AtanyinstantbetweenP1andP4,anobjectxinRRTisinoneofthefollowingthree states: ObjectStates 15

S1:xisreachablefromanobjectoutsideRRTbutisnotreachable,throughapath S2:xisreachable,throughapathconsistingentirelyofobjectsinRRT,fromanobject consistingonlyofobjectsinrrt,fromanyobjectthatisinrrtandalsointrt. S3:xisneitherinstateS1norinstateS2,thatisxisneitherreachablefromanyobject outsiderrtnorfromanyobjectinrrtthatisalsointrt. inrrtthatisalsointrt. Intheabove,anobjectisassumedtobereachablefromitselfthroughanullpath. leavingoidsinwrtwhichwillbetraversedbyanothergarbagecollectionphase,which iteventuallycollectsallgarbageobjects. Abadlydesignedgarbagecollectionalgorithmcouldcreateinniteworkforitself,by WeneedtoprovethatTCRCissafe:itdoesnotcollectanyliveobjects;andcomplete: TCRC;thatis,intheabsenceofupdatetransactions,thesystemeventuallyreachesa inturnleavesoidsinwrt,adinnitum.weguaranteethatthisdoesnothappenin statewheregarbagecollectionthreaddoesnomorework. Lemma5.1IfatransactionnotinTLISTactiveataninstantPstrictlybetweenP1 resultsthatfollow. Wewillmakeuseofthefollowingpropertiesofthealgorithmintheproofofthe andp2hasreadareferencetoanobjectthatisinstates3atp,thentheobjectisin states2atp2. Proof:SupposesometransactionTnotinTLISTisactiveatPandhasreadareference tosomeobjectthatisinstates3atpbutnotinstates2atp2.letxbetherstsuch objecttowhichareferenceisreadbyt.also,letp0betheinstantwhentreadsthe rstobjectythathasareferencetox. possible: Case1:ThereferencefromytoxispresentatP. ThereferencefromytoxwaspresentatP0.Eitherofthefollowingtwocasesare T.Therefore,yisinstateS2atP2.IfthereferencefromytoxispresentatP2, thatisinstates3atpbutnotinstates2atp2areferencetowhichwasreadby toywasreadbytbeforethereferencetox.recallthatxwastherstobject SincexisinstateS3atP,ymustalsobeinstateS3atP.Denitely,areference betweenpandp2,thenbyassumption3.1andproperty4.2(b)xisintrt,and henceinstates2,atp2. thenxisinstates2atp2.otherwise,ifthereferencefromytoxhasbeendeleted Case2:ThereferencefromytoxisnotpresentatP. tion3.1,thelogfortheabovedeletionmusthavebeenanalyzedbetweenp0and ThereferencefromytoxhasbeendeletedbetweenP0andP.BecauseofAssump- 16

TheaboveimpliesthatxmustbeinstateS2atP2.Provedbycontradiction.2 Property4.2(b),xmustbeinTRT,andhenceinstateS2,atP. P.P0occuredafterP1becauseTcouldnothavebeenactiveatP1.Therefore,by transactionactiveatpcouldhavereadareferencetoit. Proof:SupposesometransactionTisactiveatPandhasreadareferencetosome Lemma5.2IfanobjectisinstateS3ataninstantPbetweenP2andP4,thenno byt.also,letp0betheinstantwhentreadstherstobjectythathasareferenceto x.denitely,areferencetoywasreadbytbeforethereferencetox.recallthatx objectthatisinstates3atp.letxbetherstsuchobjecttowhichareferenceisread P.ButitwaspresentatP0. instates3atp.sincexisinstates3atp,thereferencefromytoxisnotpresentat wastherstobjectinstates3areferencetowhichwasreadbyt.therefore,yisnot P0occuredafterP1becauseTcouldnothavebeenactiveatP1.Therefore,byProperty4.2(b),xmustbeinTRTatP,andthereforecannotbeinstateS3atP.Proved ThereferencefromytoxhasbeendeletedbetweenP0andP.BecauseofAssumption3.1,thelogfortheabovedeletionmusthavebeenanalyzedbetweenP0andP. bycontradiction.2 Lemma5.3IfanobjectvisinRRTbutisnotinTRTatP2,then(a)noreference tovisupdated(insertedordeleted)betweenp0andp1;and(b)allreferencestovthat Proof:NologcorrespondingtoanupdateofareferencetovthatoccursbeforeP1is redtraversal. existbetweenp0andp1areaccountedinthetotalreferencecountsofvcachedduring endbeforep2.therefore,byassumption3.2,thelogmustbeanalyzedbeforep2.but then,byproperty4.2(b),vwouldbeintrtatp2 acontradiction. analyzedafterp0.thisisbecausethetransactionthatisresponsiblefortheupdatemust andp1,considertheinstantwhenthelogforthereferenceupdateisanalyzed.asshown above,thismustoccurbeforep0.butthen,becausethetransactionisactiveatpby Property4.2(a)visinTRTatP,andthereforeisinTRTatP2byProperty4.2(b),a SupposeanupdateofsomereferencetovtakesplaceatsomeinstantPbetweenP0 oftheunaccountedreferenceisanalyzedatsomeinstantafterp0,acontradiction.the contradiction.thisprovespart(a)ofthelemma. countsofvatp0.thisisbecauseotherwiseitmustbethatthelogfortheinsertion totalreferencecountsforvmusthaveremainedunchangedbetweenp0andtheinstant AllreferencestovthatexistbetweenP0andP1areaccountedinthetotalreference whentheyarecachedduringtheredtraversal,againbecauseofthefactthatnologis 2analyzedintheinterimthatcancausethechange.Thisimpliespart(b)ofthelemma. 17

Lemma5.4IfanobjectinRRTisinstateS1atP2,thenitisreachablefromsome inrrtatp0.now,considertheinstantpwhentherstpathtoxfromsomeobjectnot Proof:SupposethatanobjectxinstateS1atP2isnotreachablefromanyobjectnot objectnotinrrtatp0. inrrtwascreated.lettheinsertionofthereferencefromobjectytoobjectzthatisin beinstates2atp2byassumption3.1andproperty4.2(b),whichisacontradiction. entirelyofobjectsinrrt.thispathremainsintacttillp2becauseotherwisexwould RRTberesponsibleforthesame.Then,atP,thereexistsapathfromztoxconsisting endbeforep2andthereforebecauseofassumption3.2andproperty4.2,zwouldbein startedafterp1. TRTatP2.ButthenxwouldbeinstateS2atP2leadingtoacontradiction.Thus,T IfthetransactionTthatmadetheaboveinsertionexistedbeforeP1,thenitwould havereadareferencetoz.butsincezisunreachablefromanyobjectnotinrrtor fromanyobjectintrtatp0,itmustbeinstates3atp0.butthen,bylemma5.1,z (andhencex)mustbeinstates2atp2,leadingtoacontradiction.2 ThisimpliesthatPoccuredafterP1.AtsomeinstantP0betweenP1andP,Tmust garbagecollection. Lemma5.5IfanobjectinRRTisinstateS3atsomeinstantPbetweenP2andP4, Thefollowingresultstatestherestrictionsonthestatetransitionofobjectsduring atransitiontosomeotherstateimmediatelyafterp. Proof:SupposethatanobjectxinstateS3atsomeinstantPbetweenP2andP4makes thenitremainsinstates3betweenpandp4. ofstates3,noobjectinaliesoutsiderrt.moreover,atpnoobjectinaisreachable xwouldalsobereachablefromthisobjectatpandthereforenotbeinstates3.this fromanobjectoutsiderrtorfromanobjectinrrtthatisalsointrt otherwise LetthesetAcontainalltheobjectsfromwhomxisreachableatP.Bydenition impliesthatallobjectsinaareinstates3atp. A;theinstantPcorrespondstotheoccurenceoftheupdateitself,ortheanalysisof thelogfortheupdate whicheverisearlier.thetransactiontthatisresponsiblefor theupdatemusthaveobtainedareferencetoybeforeitisabletomaketheupdateor Thetransitionofstatemustbeduetoanupdateofareferencetosomeobjectyin yisinstates3atpasshownabove.provedbycontradiction.2 P.BecauseofAssumption3.2,TisactiveatP.ButthiscontradictsLemma5.2because generatealogforthesame.inotherwords,tmusthaveobtainedareferencetoybefore Lemma5.6AtP4,anobjectinRRTisrediitisinstateS3. Next,weproveaninvariantofthealgorithm. 18

ByLemma5.5,xisnotinstateS3atP2. Proof:(Onlyif)SupposethatatP4,anobjectxinRRTisredandisnotinstateS3. intrtaregreen.twocasesarepossible. leadingtoterminationofthefor-loop.atp,therefore,allobjectsinrrtthatarealso ConsidertheinstantPwhentheconditioninthestatementL1evaluatestofalse Case1:xisinstateS1atP2. Then,byLemma5.4,itisreachablefromsomeobjectnotinRRTatP0.Letthe butvisinrrt.now,twosubcasesarepossible. Case1.1:ThepathfromutoxisintactbetweenP0andP. referencefromobjectutoobjectvbealongthepathsuchthatuisnotinrrt accountedintheredreferencecountsofv.thus,thecachedtotalreference accountedintheredreferencecountsofvareaccountedinthecachedtotal referencecountsofv.sinceuisnotinrrt,thereferencefromutovisnot ByLemma5.3(b),thereferencefromutovaswellasthereferencesthatare countsofvarestrictlygreaterthantheredreferencecountsofv. SincevisinRRTbutnotinTRT,statementL0willbeexecutedforv.For andisintactatthispoint,xwillbecolouredgreen(ifnotalreadyso). beinvokedonv.sincethepathfromvtoxconsistsofonlyobjectsinrrt thereasonsstatedabove,theconditionwillbesatisedandgreentraversewill Case1.2:ThepathfromutoxisbrokenbetweenP0andP. (ymightbethesameasx).beingintrt,ymustbegreenatp.butsincethe thepathisdeletedbetweenp0andp,butthepathfromytoxisintactatp pathfromytoxisintactbetweenp0andp,theinvocationofgreentraverse ConsidertheobjectyalongthepathatP0suchthatthereferencetoyalong Case2:xisinstateS2atP2. Thatis,atP2thereexistedapath,consistingonlyofobjectsinRRT,toxfrom thatcolouredygreenmustalsohavecolouredxgreen(ifnotalreadyso). someobjectinrrtthatisalsointrt.thepathcouldhavebrokenbetweenp2 andp.therestoftheproofissimilartocase1.2above.considertheobjecty alongthepathatp2suchthatthereferencetoyalongthepathisdeletedbetween P2andP,butthepathfromytoxisintactatP(ymightbethesameasx). BeinginTRT,ymustbegreenatP.Butsincethepathfromytoxisintact TheaboveimpliesthatxisgreenatP,andthereforeatP4 acontradiction.this betweenp2andp,theinvocationofgreentraversethatcolouredygreenmustalso provesthatxmustbeinstates3atp4.2 havecolouredxgreen(ifnotalreadyso). colouringgreensomeobjectythatisinstates3atp4. Proof:(If)Supposethatagreentraversalinvokedatsomeobjectxwasresponsiblefor 19

atp4 otherwiseywouldbeinstates2atp4byassumption3.1andproperty4.2(b). atthetimeoftheinvocation. Therefore,xmustalsobeinstateS3atP4.Inparticular,xcouldnothavebeeninTRT Thepathfromxtoythatexistedatthetimeofthegreentraversalmustbeintact conditioninstatementl0musthaveevaluatedtotruewhenitwasexecutedforx.that forx.thiscanonlyoccurifareferencefromsomeobjectztoxthatexisted(asper is,theredreferencecountsforxarestrictlylessthanthecachedtotalreferencecounts Butthen,theonlywaygreentraversalcouldhavebeeninvokedonxisthatthe thecachedtotalreferencecountsofx,bylemma5.3(b))betweenp0andp1wasnot traversedintheredtraversalandthereforeisnotaccountedintheredreferencecounts ofxḃylemma5.3(a),noupdateofanyreferencetoxtakesplacebetweenp0andp1. ThisimpliesthatthereferencefromztoxexistedatP0.Moreover,thisreferencecould nothavebeendeletedbetweenp0andp4,otherwisebyassumption3.1andproperty4.2 xwouldbeintrtatp4,acontradiction.thisfurtherimpliesthatzmustbeinrrt, otherwisexwouldnotbeins3atp4. thereferencefromztoxexistedatp0anditdidnotgetdeletedbetweenp0andp1. Butthen,thereferencewouldhavebeenaccountedintheredreferencecountsofv a contradiction.2 Now,weknowthatzwasvisitedbytheredtraversalbetweenP0andP1.Also, Theorem5.7(Safety)OnlygarbageobjectsarecollectedbyCollectGarbage. Theresultsstatedaboveareputtogetherintheformofthefollowingtheorem. Proof:AtP4,allredobjectsareinstateS3byLemma5.6.BydenitionofthestateS3, theseobjectsarenotreachablefrom(a)anyobjectnotinrrt.thisimpliesthatthey arenotreachablefromthepersistentroot(whichisneverinrrtbyproperty4.1)or fromanyobjectintrtnotinrrt;(b)anyobjectintrtthatisinrrt.therefore, theseobjectareneitherreachablefromthepersistentrootnorfromanyobjectintrt. ByLemma5.2,noreferencetoanyoftheseobjectshasbeenreadbyanytransaction activeatp4. Thisimplies,bydenition,thatallobjectsthatareredatP4aregarbage.Sinceonly AllobjectsfromwhichanobjectinstateS3isreachableatP4areinstateS3atP4. theseobjectsarecollectedbycollectgarbage,thetheoremisproved.2 Forthis,wefurtherneedthefollowingresults. Lemma5.8AcycleofstrongreferencescanexistonlybetweenP2andP5. Next,weprovethatTCRCiscomplete thatis,itcollectsallgarbageeventually. Proof:CollectGarbagechangesthereferencestrengthsonlybetweenP2andP5.Transactionscanonlydeletestrongreferences theyneverchangepointerstrengthsorinsert 20

strongpointers.weassumethatthereexistnocyclesofstrongreferenceswhentheobjectsareloaded.therefore,itissucienttoprovethatifthereexistnocyclesofstrong P5.SincetherearenocyclesofstrongreferencesatP2,wecannothaveacycleofstrong referencesatp2thenthereexistnocyclesofstrongreferencesatp5. referencesthatdoesnotcontainanobjectinrrtatp5. OnlythestrengthsofreferencesfromobjectsinRRTarechangedbetweenP2and xtoyaremadeweakduringthetraversal.thus,atp5thereexistnocyclesofstrong beforeitbacktrackedfromy.then,allreferencesfromytoxaremadestrongandfrom referencescontainingonlygreenobjects. Considertwogreenobjectsxandysuchthatthegreentraversalbacktrackedfromx RRTweak.ThisfurtherguaranteesthatthereexistnocyclesofstrongreferencesatP5 musthaveredobjects. ButallredobjectsatP4arecollectedasgarbagebeforeP5.Thisprovesthatthere Also,thegreentraversalmakesallreferencesfromgreenobjectstoobjectsnotin existnocyclescontainingstrongreferencesatp5.2 AsuchthatanobjectcomesafterallobjectsthathaveastrongreferencetoitatP0. Proof:LetAbethesetofallgarbageobjectsatP0.Wexanorderontheobjectsin Lemma5.9IfanobjectisgarbageP0thenitisinstateS3atP4. ThisispossiblebecausebyLemma5.8,cyclesofstrongreferencescannotexistatP0. Thereafter,itputsanobjectinRRTifitisnotinTRTandallobjectswhichhavestrong referencestothisobjectareinrrt. RecallthatredtraversalputsinRRTallobjectsinWRTthatarenotinTRT. Case1:TheredonotexistanystrongreferencestoxatP0.ButthenxisinWRT betwocasespossible. LetxbetherstobjectintheaboveorderingthatisnotinRRTatP2.Therecan atp0andnotintrtbecauseitisgarbage.beinggarbage,itremainsthatway Case2:ThereexiststrongreferencestoxatP0.SincexisgarbageatP0,allobjects invokedonit.thus,itmustbeincludedinrrt. duringthecourseoftheredtraversal,andthereforeredtraversemustbehavebeen thathaveastrongreferencetoxmustbegarbageatp0.butthen,theymustbe ina,andmustoccurbeforexintheordering.bythechoiceofx,alltheseobjects TheaboveimpliesthatxmustbeinRRTatP2 acontradiction.thisprovesthatall mustbeinrrtatp2.also,xisnotintrtbecauseitisgarbage.therefore,it objectsinaareinrrtatp2.further,noneoftheseobjects(whicharegarbageatp0) mustbeincludedinrrtduringtheredtraversal. arereachablefromobjectsnotinrrtatp2(whichareliveatp0).moreover,noneof 21

theseobjects,beinggarbagealsoatp2,arereachablefromobjectsthatareintrtat P2.Puttingtheabovetogether,weseethatallobjectsinAareinstateS3atP2. lemma.2 Theorem5.10(Completeness)AllgarbageobjectsareeventuallycollectedbyCollectGarbage. Proof:ByLemma5.9andLemma5.6,agarbageobjectwillbecolouredredatP4in therstcollectgarbageinvokedafteritbecamegarbageandthereforewillbecollected. 2Theorem5.11(BoundedWork) concurrentlyexecutingupdatetransactions,wrtwillbeemptyatp5. Proof:Sincetherearenotransactionupdatesexecutingconcurrentlywiththegarbage IfTRTisemptyatP0theninabsenceofany Butthen,byLemma5.5,allobjectsinAmustbeinstateS3atP4.Thisprovesthe lection. collection,thestrengthofallreferencesremainssameduringthecourseofgarbagecol- traversal.sincethisstrongreferenceisnottraversedbyduringthegreentraversal,it someobjectnotinrrt otherwisexwouldhavebeenincludedinrrtduringred referencefromsomeobjectinrrt,thenxmusthaveatleastonestrongreferencefrom First,weconsiderobjectsnotinRRT.IfanobjectxnotinRRTthathasastrong inrrtisnotinwrtatp5. isnevermadeweakbythegarbagecollector.further,sinceonlyobjectsinrrtget deleted,thereferencenevergetsdeletedeither.thisimpliesthatanyobjectthatisnot onestrongreferencetothemfromothergreenobjectsatp4. Thus,allgreenobjectsexcepttheonesonwhichthetraversalisinvokedhaveatleast traversedobjecttoasyetuntraversedobjectsinrrtstrongbeforetraversingthelatter. WeconsidertheobjectsinRRTnext.Thegreentraversalmakesthereferencesfroma traversalonit,thereferencestoitfromobjectsnotinrrtbecomestrong.because whenthestrengthsofreferencestoanobjectareinvertedbeforeinvocationofgreen referencesfromoutsiderrt.byconstruction,allthesereferencesareweak.therefore, BecausenoobjectisinTRT,greentraversalisstartedonlyonobjectswhichhave thesereferencesarenevertraversedduringthegreentraversal,theyarestrongatp4. collected,resultingindeletionofreferencesfromredobjectstogreenobjects.butfrom theresultsprovedabove,thesereferencedeletionscannotputanygreenobjectintowrt. greenobjectorfromanobjectnotinrrtatp4.betweenp4andp5,redobjectsare Summarizing,everygreenobjecthasatleastonestrongreferencetoitfromanother Thus,nogreenobjectisinWRTatP5. alltheobjectsbecausetheredonotexistanyredobjectsatp5.thus,itisprovedthat noobjectisinwrtatp5inabsenceoftransactionupdates.2 Thus,nogreenobjectorobjectnotinRRTisinWRTatP5.Butthisaccountsfor 22

latterinvocation;andthereforebylemma5.11thewrtwillbeemptyattheendof thesecondinvocation. CollectGarbagetotheendofthenext,thenTRTwillbeemptyatthebeginningofthe Inparticular,ifnoupdatetransactionsexistfromthebeginningofoneinvocationof WenowseehowtouseinformationfromthedatabaseschematooptimizeTCRC.The schemagraphisadirectedgraphinwhichthethenodesaretheclassesintheschema. 6 UsingtheSchemaGraph forthepointersbetweentheactualinstancesoftheobjects. AnedgefromnodeitonodejintheschemagraphdenotesthatClassihasanattribute graphisnotinvolvedinacycle,thenneithercananedgeeintheobjectgraphforwhich Eisthetemplate. thatisareferencetoclassj.thepointersintheschemagraphthusformatemplate Welabeledgeswhicharenotpartofacycleintheschemagraphasacyclicand IfanedgeEinaschema templateedgeintheschemagraphisacyclic,thestrengthofeissettobestrong.during garbagecollection,inredtraverse,wedonotfollowstrongedgeswhosetemplateedgeis theothersascyclic.whenaddinganedgeetotheobjectgraph,ifitscorresponding andtcrcbehavesjustlikereferencecounting,reducingthecostsignicantly. acyclic.intheextremecasewheretheschemagraphisacyclic,noedgesaretraversed, 7Inthissection,weoutlinethesetofassumptionsrequiredforouralgorithmtowork correctlyinadatashippingclientserverenvironment. ExtensiontoaClientServerEnvironment Assumption7.1Thetransactionsrunonlyattheclients.Theservercandetermine actionsrunningatanyclientalwaysseethelateststateofthedatabase.2 Assumption7.2Cacheconsistencyisguaranteedamongtheclients.Thatis,thetrans- whattransactionsarearepossiblyactiveattheclientsatanygiveninstant.2 attheserver(wal).2 Assumption7.3Allundorecordsarereceivedtheserverbeforetheupdateisreected Thisallowsustothinkofthetransactionsasrunningonasingleclient. abort(force-logs-at-abortinadditiontoforce-logs-at-commit).2 Assumption7.4Alllogsforatransactionarereceivedattheserverbeforecommitor 23

dataafterithasbeensenttotheserver. andwhetherornotdataiscachedattheclient.theclientscanretaincopiesofupdated Toguaranteethatthealgorithmworkscorrectlyintheclient-serversettingwiththe Ourtechniquesarenotaectedbytheunitofdatashipping(suchaspageorobject) aboveassumptions,theonlychangerequiredinthealgorithmisthefollowinggeneralizationofproperty4.2. when(a)thetransactionhasended;(b)allupdatesbythetransactionarereectedat Property7.1TheobjectsinTRTcorrespondingtoatransactionareremovedonly theserver;and(c)garbagecollectionisnotinprogressattheserver.2 Becausethedatabasestateattheserver(wherethegarbagecollectorisrunning)isnot current,itmighthappenthatanobjectmayhaveareferencetosomenewlycreated server.assuch,someextracarehastobetakenduringtraversalsoftheobjectgraph. NotethattheLogAnalyzeraswellastheCollectGarbagealgorithmsarerunatthe objectthatisnotyetpresentattheserver.suchadanglingreferenceissimplyignored duringthetraversals. referthereaderto[rss+98]foraproof. Wecanprovethecorrectnessofthealgorithmintheclientserversettingalsoand algorithmonanobjectmanagercalledbrahmadevelopedatiitbombay.brahma WeimplementedtheTCRCalgorithmandthePartitionedMarkandSweep(PMS) 8 PerformanceEvaluation supportsconcurrenttransactionsandhasacompleteimplementationofthearies recoveryalgorithm.itprovidesextendiblehashindicesaswellasb+-treeindicesas thetid(toalloweasydeletionofallentriesofatransaction). additionalaccessmechanisms. ontheoidwhilethetrtisanin-memoryhashtableindexedseparatelyontheoidand TheWRTisimplementedasanoptionallypersistentextendiblehashtableindexed onlypersistentstructuresrequiredbypmsareoneincomingreferencelist(irl)per SRefCandWRefCarestoredinanoptionallypersistenton-diskhashtable.The partitionwhichismaintainedasapersistentb+-tree. Thereferencecounts Inparticular,weworkedonthestandardsmall-9datasetinOO7whichwasalsoused in[yny94]fortheirsimulationstudy.theoo7parametersandtheirvaluesforthis datasetaregivenintable1andareexplainedbelow.figure4illustratestheoo7 OurperformancestudyinthissectionisbasedonthestandardOO7benchmark[CDN93]. Eachmoduleconsistsofatreeofobjectscalledassemblies.Thetreeisacompletetree benchmark. withafanoutofnumassmperassmandhasnumassmlevelslevels.thelast TheOO7datasetiscomposedofanumberofmodules,speciedbyNUMMODULES. 24

Module i id type builddate manual design_root Manual text complex assemblies base assemblies 1 2 3 4 N Design Library of Composite Parts Figure4:TheOO7Benchmark 25