Ecient Updates for Continuous Skyline Computations YuL - PDF document

EcientUpdatesforContinuousSkylineYu-LingHsuehRogerZimmermannWei-ShinnKuDept.ofComputerScience,UniversityofSouthernCalifornia,LosAngeles,CA90089ComputerScienceDepartment,NationalUniversityofSingapore,Singapore117543 2Yu-LingHsuehRogerZimmermannWei-ShinnKu ESC Query Processor S1 Skyline Evaluation S2 Skyline Evaluation Data Set Request Skyline Query Results B Discarded points /cess Data Fig.1.ESCsystemframeworkskylinequeriesassumedthatdataobjectsarestatic[13,15].Otherapproachesassumedthattheskylinecomputationinvolvedonlyapartialofdynamicdi-mensions[4].Inthispaper,weaddressEcientUpdatesforContinuousSkylineoverdynamicobjects(forshort),whereobjectswithnamicdimensionsmoveinanunrestrictedmanner.Eachdimensionrepresentsaspatialornon-spatialvalue.Towardsanecientcontinuousskylinecomputa-tionthefollowingchallengesmustbeaddressed:aneectiveincrementalskylinequeryresultupdatemechanismthatisneededprovidesafastresponsetimeofreportingthecurrentqueryresults,andanecientstrategytoreducethesearchspacedimensionalityisrequired.Existingwork[6,14,19]generallycomputesanumberofdatapointsubsets,eachofwhichisexclusivelydominatedbyoneskylinepoint.Therefore,whenaskylinepointmovesorisdeleted,onlyitsexclusivelydominatedsubsetmustbescanned.Thedeterminationofsuchanexclusivedatasetisverycomputationallycomplexinhigherdimensionsanditincursaseriousburdenforthesysteminahighlydynamicenvironment.Therefore,thesesystemsareoftenunabletoprovideup-to-datequeryresultswithaquickresponsetime.Weproposealgorithmtoecientlymanagethequeryresultsbydelegatingthetime-consumingskylineupdatecomputationstoanotherindependentprocedure,whichisprocessedafterthequeryprocessorreportsthelatestskylinequeryresults.Thekeyideaistomaintainasecondskyline2)setwhichisaskylinecandidatesetpre-computedwhenatraditionalskyline(whichwereferasthe“rstskyline1)pointrequestsanupdate.Withtheknowledgeofthesecondskylineset,theskylinequeryresultcanbeupdatedwithinalimitedsearchspaceandtheexpensivecomputations(e.g.,searchingfornewsecondskylinestosubstituteapromotedsecondskylinepoint)canbedecoupledfromthe“rstskylineupdatecomputations.Figure1showstheframeworkofthesystem.Thequeryprocessorini-tiallycomputesthe“rstandsecondskylinepoints.Anyupdatesperformedonthedatasetarealsosubmittedtothequeryprocessor.First,Taskamineswhethertheupdaterequest(e.g.,insertingorremovingadatapoint)aectsthe“rstskylineset.Iftherequestpointbecomesanew1point,Taskinsertsthenew1pointintothecurrent1setandremovesthecurrentskylinepointsthataredominatedbythenew1point.Thesediscarded1points(new EcientUpdatesforContinuousSkylineComputations32points)areprocessedbyTasklatertoupdatethe2set.Incasethatanupdaterequeststemsfromaremovedormoving1point,someexclusivepointsareleftun-dominated.Thequeryprocessorsearchesfornewsubstitute1pointsonlyfromthe2set.ThequeryresultsareimmediatelyoutputassoonasTaskiscompleted.TheprocessingtimeofthesequenceofTasksisthesystemresponsetimetoaskylinequeryupdate.Taskmaintainsthe2pointswhenany2pointisinsertedorremoved.ToenhanceTask,whichinvolvestheexpensivecomputationofdeterminingexclusivedatapointswheresearchesforneworsubstitute2pointsfromtherestofthedataset,wealsoproposeanapproximateexclusivedataregionwithloweramortizedcostthanexistingtechniques[14,19].Theremainderofthispaperisorganizedasfollows.Section2describestherelatedwork.Section3presentsanddetailsourcontinuousskylinequeryprocessingdesign.Weexten-sivelyverifytheperformanceofourtechniqueinSection4and“nallyconcludewithSection5.2RelatedWorkBorzsonyietal[1]proposedthestraightforwardnon-progressiveBlock-Nested-Loop(BNL)and(DC)algorithms.TheBNLapproachre-cursivelycompareseachdatapointwiththecurrentsetofcandidateskylinepoints,whichmightbedominatedlater.BNLdoesnotrequiredataindexingandsorting.TheDCapproachdividesthesearchspaceandevaluatesthesky-linepointsfromitssub-regions,respectively,followedbymergeoperationstoevaluatethe“nalskylinepoints.Bothalgorithmsmayincurmanyiterationsandtheyareinadequateforon-lineprocessing.In[17],Tanetalpresentedtwoprogressiveprocessingalgorithms:theapproachandthemethod.encodesdimensionalvaluesofdatapointsintobitstringstospeedupthedominancecomparisons.Theindexmethodclassi“esasetofpointsintolists,whicharesortedinincreasingorderoftheminimumcoor-dinate.Indexscansthelistssynchronouslyfromthe“rstentrytothelastone.Withthepruningstrategies,thesearchspaceisreduced.Thenearestneigh-bor(NN)method[5]indexesthedatasetwithanR-tree.NNutilizesnearestneighborqueriesto“ndtheskylineresults.Theapproachrepeatsthequery-and-divideprocedureandinsertsthenewpartitionsthatarenotdominatedbysomeskylinepointintotheto-dolist.Thealgorithmterminateswhentheto-do-listisempty.In[13],abranchandboundskyline(BBS)algorithmtraversesanR-treeto“ndtheskylinepoints.AlthoughBBSoutperformstheNNapproach,theperformancecandeteriorateduetomanyunnecessarydominancechecks.Finally,manyoftherecenttechniquesaimatcontinuousskylinesupportformovingobjectsanddatastreams.Linetal.[8]presentskylinequeriesagainstthemostrecentelementstosupporton-linecomputationagainstslidingwindowsoverarapiddatastream.Morseetal.[11]proposeascalableLookOutalgorithmforupdatingthecontinuoustime-intervalskylineeciently.Sharifzadehetal.[16]introducetheconceptofSpatialSkylineQueries(SSQ). 4Yu-LingHsuehRogerZimmermannWei-ShinnKuGivenasetofdatapointsandasetofquerypoints,SSQretrievesthosepointsofwhicharenotdominatedbyanyotherpointinconsideringtheirderivedspatialattributestoquerypointsin.Formovingquerypoints,acon-tinuousskylinequeryprocessingstrategyispresentedin[4]withakinetic-baseddatastructure.However,promptqueryresponseisnotconsideredinthedesign.AsuiteofnovelskylinealgorithmsbasedonaZ-ordercurve[3]isproposedin[6].Amongthesolutions,ZUpdatefacilitatesincrementalskylineresultmain-tenancebyutilizingthepropertiesofZ-ordercurve.Otherrelatedtechniquescanbefoundintheliterature[2,19,9,12,18].However,alltheaforementionedstudiesdierfromthemaingoalofthisresearch…supportingfrequentskylinedataobjectupdatesecientlywhileprovidingaquickresponse.3ESCAlgorithm3.1TheProblemDeÞnitionofContinuousSkylineQueries 128754 1S51S 1S9876543 1S Region I9 edge of theuniverse S2 2S 1S22S Region II 6p Fig.2.1and2setsTheformalde“nitionofskylinepointsin-dimensionalspaceisadistinctobjectset,whereanytwoobjects,...,x)and,...,y)inthesetsatisfytheconditionthatifforanyk,x,thereexistsatleastonedimensionofthatsatis“es.Wesay).Thegeneralsetupoftheproblemconsistsofasetofdynamicqueryanddataobjectswithdimensions.Movingobjectscanfreelymoveinanunrestrictedandunpredictablefashion,meaningthattheirmayarbitrarilychangetheirvalues.Themajorchallengingissueofacontinuousskylinequeryistoavoidunnecessarydominancecheckingonirrelevantdatapointsforskylinequeryresultupdates.Afterobservingthealgorithm[13],wededucedthatwhenevaluatingtheskylinequeryresult,asetsecondskyline2)pointscanalwaysbeobtainedwithlittleextraworkwhile EcientUpdatesforContinuousSkylineComputations5retrievingthe“rstskyline1)points.Werefertothetraditionalskylinequeryresultasthe“rstskyline,consistingof,...,s.Thesecondskyline,...,sisde“nedasfollows:DeÞnition1:Adatapointisasecondskylinepoint.Informally,all2pointsaredominatedbyandtherestofthedatapoints(2)aredominatedbyboth1andWhena1pointisremovedoratleastonevalueofitsdimensionschanges,2pointsarenaturallyconsideredasnew1pointcandidatestosubstituteŽ.Thefeaturesofa2setareasfollows:(1)itisapre-computedsetthatcoversallthenew1candidatepoints,and(2)2isarelativelysmalldataset.Therefore,withtheknowledgeof2,thequeryprocessorcanecientlyupdatethequeryresultandprovideaquickerresponsetimetothequerypoint.AnexampleisshowninFigure2.Ifa1pointmovestoRegion,thesearchspacefortoupdatethequeryresultonlyinvolvesthe1setandthe2set.Inthiscase,remainsa1point,butitdominatesneedstoremovefromthe1setandbecomesanew2point,sincenoexisting2pointcandominateit.Duetothemovementofsearchesfornew1pointsfrom2set.Since(anexclusivedatapoint)isleftun-dominated,becomesanew1pointandisremovedfromthe2set.Thealgorithmdelegatesthenecessary2maintenance(anindependentprocedurefrom1updates)tothequeryprocessorafter1updatesarecompleted.Forexample,new2pointsmustberetrievedtosubstitute.ToavoidscanningthroughtheentiredatapointsinRegionfornew2points,weproposeanapproximateexclusivedataregion)computationincontrasttoatraditionalexclusivedataregion)computation.Basedonourobservationandanalysis,weprovidethelemmasforincrementallyupdatingtheskylinequeryresultsinthefollowingsections.Table1summarizesthesymbolsandfunctionsweusethroughoutthefollowingsections. Symbols P Numberofdataobjects d Numberofdimension 1 Firstskylinepointset(traditionalskylinequeryresultset) 2 Secondskylinepointset DataRtree Disk-basedRtreeforindexing S1Rtree Main-memoryRtreeforindexing1points S2Rtree Main-memoryRtreeforindexing2points p) Asetofdatapointsintheexclusivedataregion p) Asetofdatapointsintheapproximateexclusivedataregion (p) Asetofskylinepointsinthedominanceareaof DomArea Thedominanceareaof Table1.Symbolsandfunctions3.2SecondSkylineComputationTheexistingwork[14,19]performsthetime-consumingexclusivedatapointcomputationsfortheskylinequeryresultupdates.InFigure3,thegrayareas 6Yu-LingHsuehRogerZimmermannWei-ShinnKurepresentthetraditionalthatcontainexclusivedatapoints.Anisnotusuallypre-computedbecauseofthecomplexityofthecalculation.Incontrast,sincethe2points(new1candidates)canbeeasilycomputedbeforeany1pointissuesanupdate,thequeryprocessorisabletosatisfyaqueryrequestwiththelatestqueryresultandwithaquickerresponsetime.Tofurtherreducethesearchspaceofvisiting2pointstoupdatetheskylinequeryresult,weintroduceandde“neadominancesetforeach1pointdominancesetcontainsagroupof2pointswhicharedominatedby(denotedbytosubstitutearemovedormovingpointwhendominancerelationshiphaschanged.ForexampleinFigure3thedominancesetof.Ifisremoved,onlychecksthe2pointsin),insteadoftheentirepoints.Inthisexample,becomesanew1point,soitisremovedfromWeformallyde“neadominancesetandestablishLemma1whichstatesthatadominancesetmustcontainallthenecessary1candidatepointsasfollows: 128754 1S2 11S9876543 41S 1S 1S 2S 2S 2S42S exclusive dataregion Fig.3.Dominancesetv.s.DeÞnition2DominanceSet:dominancesetofaskylinepoint(denotedby,...s)isa2subsetwhere,and0).Each)isexclusivefromanyotherdominanceset;therefore,1),where1)=isthesizeofLemma1:Givena).Letbetheskylinepointsextractedfrom)mustcontainisasubsetofD(Proof:(Bycontradiction)Letbeapointnotincludedin).Thisisacontradiction,sinceisonlydominatedby.Therefore,itmustbein)mustcontainallpointsin InFigure3,containstwo2pointsinthesetwhichisasupersetof.Onecanobservethatsomenon-exclusive2points EcientUpdatesforContinuousSkylineComputations7)canbeassignedtodierentdominancesets.Intuitively,the1pointwiththeminimaltothequerypoint(whichhasthelargestdominancearea)maycontainthemost2points.Thus,itmightproducealoadimbalanceproblembecausethequeryprocessorneedstoperformmanydominancecheckswhenaskylinepointwithashortmoves.Toensurethateachdominancesethasevenlydistributed2points,theinsertsanon-exclusive2pointinto),wherehastheminimalvalueof)amongallother1points.Inouralgorithm,weutilizeapproachtoinitiallycomputetheskylinequeryresults.Alongwiththequeryevaluation,2pointsandthedominancesetofeach1pointarecomputedduringtheexecutionofthemodi“eddominance-checkingprocedurewhichrunsawindowquerytodetermineasetofcandidateskylinepoints.Letbethenextdiscardedentryduringtheprocessofthedominance-checkingprocedureisdominatedbysome1point).Therefore,thealgorithmproceedstoinsertintoadominancesetandexaminewhetherisa2point.Givenisaheapthatisthesetoftheexistingskylinepointswhoseentriesintersect.Sincealwaysvisitsentriesintheascendingorderoftheirwehaves.mindiste.mindist.Withthesortingofbythedescendingorder,andthevalueof(isminimalamongallother1points.Next,Lemma2isprovidedtoprovethecorrectnessofthe2extraction.Lemma2:Givenapointwhichisdominatedby,where1.Ifmustbea2point.Proof:isnotdominatedby(canneverbedominatedbyany2pointin)either,bytransitivity.Therefore,ifisnotdominatedbyany2pointinisguaranteedtobea“nal2point. ThepseudocodeisshowninAlgorithm1,wheretheadditionalconditions(Lines10-16and19-27)areinsertedintothedominance-checkingcodeforre-2pointsanddeterminingthedominancesets.Line4sortstheheapindescendingorderofthesuchthattheskylinepointswithlargerareexamined“rst.Line12obtainsthedominatingskylinepointwhichisinsertedinto)later.BasedonLemma2,Lines13-15checkwhetherisa2point.Lines20-23ensurethateach2isadatapoint.Ifisanintermediatenode,isperformedtoretrievelocalskylinepointsfromtheentry.Lines23and25insertthe“nal2pointsinto2andupdatesthe2setbydeleting2pointsthataredominatedby.To“ndsuchaset,thealgorithmperformsS2Rtree),whichisawindowquerythat“ndsthe2pointsinthedominanceareasof3.3DescriptionoftheESCAlgorithmThemainproceduresofthealgorithmincludefortheupdatesandforthe2setmaintenance.delegatesmostofexpensivecomputationsthatareirrelevantto1queryresultsto 8Yu-LingHsuehRogerZimmermannWei-ShinnKu Algorithm1ESCdominance-check( insertallentriesoftherootRintheheapisDominated=false,heapnotemptyremovetopheapentrye//theheapissortedindescendingorderofisanintermediateentry)(eachchildintersectswithintoheapendforisDominated=true;,ifisnotempty:theÞrst1pointdominating(each2skylinepointasaregulardatapointandreturnisDominatedendforendifendifendwhileisanintermediateentry)performDataRtree.BBS()thatreturnsaskylinepointsetbethedatasetthatisnotdominatedbyRtree.W)andinsertintoRtree.W)andinsertintoendifendifreturnisDominated Toimprovetheperformanceof,weintroducetheconceptofanapproximateexclusivedataregion)thathelpstoreducetheamortizedcostofthe2updates.When=2,thetraditionalisaregularrectangle.However,ahasanirregularshapeinhigherdimensions.Forexample,inFigure4(a),isaskylinepointtodelete.Theisairregularrectangleafterdeletingtheoverlappingareawiththedominanceareaof.Basedonthisobservation,wecanobtainaregularshapedonlywhenweconsidertheskylinepointswhichhaveavaluelargerthanthatofinonlyonedimension.BecausethesepointsarecompletelyoutsideŽoftheEDR,theycantrimtheentireareasthatrepresenttheupperdimensionalvalueDeÞnition3,...,x),and,...,yDomAreaDomAreaDomArea),thereexistsexactlyoneForexample,inFigure4(b),istheskylinetodeleteandthesolidrectangleboxisan,whichisaregularshaperesultingfromtrimmingtheoverlap-pingdominanceareasofutilizesthetosearchforthe2pointsbytraversingtheR-tree.Eachextractedfromtheheapischeckedwhetheritintersectswiththe.Iftrue,checkswhetherisdominatedbytheexisting2points.Whena1pointisnewlyinsertedintothesystemorwhenitmoves,needstore-groupanewdominancesetfor.Asimplesolutionistocheckevery2pointwhichcurrentlybelongstoadominancesetofsome1point EcientUpdatesforContinuousSkylineComputations9 S2iS2 (12, 7, 9)(16, 3, 18) vS2(21, 1, 13) (a)3-EDRexample S2iS2 (12, 3, 9)(4, 1, 17) (b)AEDRexampleFig.4.Traditionalandmigratethe2pointtothedominancesetofifnecessary.Instead,weprovide,(thepseudocodeispresentedinAlgorithm2)applyingthefollowingLemmathatpresentsaheuristictoavoidcheckingtheentireLemma3:Givenanew1point,re-groupthepointsin),onlyProof:Proofbyde“nition.Letbea1pointthathasthevalueof(),thevalueof()mustbesmallerthanthevalueof(mustremaininthesamedominancesetof.Therefore,itisnotnecessarytore-groupthesepointsin Algorithm2 (each),where)andremove(endifendfor algorithmisimplementedinanevent-drivenfashiontohandletheskylinequeryupdates.Themainproceduresincluderithm3)and(Algorithm4).Whenthequeryprocessorreceivesarequest(2,orregulardatapoint),it“rstperformstoex-aminewhethertherequestaectsthe1set(thequeryresult)andoutputstheupdated1pointsifthesethasbeenmodi“ed.Thencessestherestofnon-1-relatedcomputations.Intheprocedure,Line6performstheS1Rtree.dominace-descendingfunctionwherethedominancechecksaccesstheS1Rtreeinthedescendingorderoftheoftheentries.WeusethesameprincipleoftheESCdominance-checkalgorithm(discussedinSection3.2)to“ndthedominating1point(Line7)forarequestpoint.Ifbecomesanew2pointevaluatedbyisinsertedinto 10Yu-LingHsuehRogerZimmermannWei-ShinnKuLines910updatethe1setifisanew1pointanddeletetheset,whichisanexisting1setdominatedbyisobtainedbyexecutingawindowqueryS1Rtree),usingthedominanceareaofastherangeontheS1Rtree.Line11insertsthenew1pointinto1andwilllaterpassthisset1)isperformedto“nda2setforSinceallthepointsinbecomenew2points(insertedinto2inLine12),2setisupdatedlaterinbyaddingthe2set.Lines15basicallycheckallthe2points)whethertheyarestilldominatedbymovesorisremovedfromthesystem.InLine18,sincepointaftermoves)canneverdominateany1point,isaddedtothe1setdirectly.Thisisbecauseisanexclusivedatapoint,andthereforemustnotdominateanyexisting1points. Algorithm3 beanew1pointsetbeanew2pointset betheexisting2pointstoremovebethelast-updatedpointof,ifwasa1pointS1Rtree.dominace-descendingbethe1pointwiththeminimal()valueamongallother1points==false)S1Rtreeendifwasa1point)(eachS1Rtree.dominace-descending)==false)remove( endifendforendifoutputtheupdated1setandcontinue 2)procedure isamoreexpensiveprocedurethan,becauseitinvolvescomputationsto“ndasetofnew2pointstosubstituteamovingorremoved2point.Lines67areprocessedifisanew2point.Theinsertionofmaydominatesomeexisting2points;therefore,Line6“ndsthedominated2points(S2Rtree))andremovesthemfromthe2set.Similarly,inLine10,sinceeachpointin2wasoriginallya1point,thesetisdirectlyremovedfromthe2setwithoutperformingawindowquerytolookforthedominatedpoints.Thedeletionofthe2pointset 2isexecutedinLines1112andcontainsthesubstitute2points,after 2isremovedfrom2set.Finally,isperformedto“ndagroupof2pointsforeachpointin EcientUpdatesforContinuousSkylineComputations11 Algorithm4 bethelast-updatedpointof ),ifwasa2point==true))S2Rtree.dominace==false))S2Rtree)andremove(),where)wasthedominatingpointofendifendifDataRtree-AEDR 2),whereisaregulardatasetandisnotdominatedby2points. A//A 2FindDomSet(S 4ExperimentalEvaluationWeevaluatedtheperformanceofthealgorithmbycomparingitwiththewell-knownapproach[14]andthealgorithm[19].ForthecomputationsinBBS,weadoptthe(AdaptiveBranch-and-BoundSearch)[19]toavoidcomplexirregular-shapedsicallytraversestheR-treeanddetermineswhetheranintermediateintersectswiththedominanceareaofaskylinetodelete.Ifthisistrue,itfur-thercheckswhetheranyexistingskylinedominates.AllofthesealgorithmsutilizeR-treesastheunderlyingstructureforindexingthedataandskylinepoints.WeusetheSpatialIndexLibrary[7]fortheR-treeindex.Apagesizeof4Kbytesisdeployed,resultinginnodecapacitiesbetween94(=5)and204=2).1and2setsareindexedbyamain-memoryR-treetoimprovetheperformanceofthedominancechecks.Ourdatasetsaregeneratedonaterrainservicespaceof[01000]withtherandomwalkmobilitymodel[10].Eachobjectmoveswithaconstantvelocityuntilanexpirationtime.Thevelocityisthenre-placedbyanewvelocitywithanewexpirationtime.Wegeneratedfrom100,000to1,000,000normaldistributeddatapointswithadimensionintherangeof2to5.Theobjectupdateratioissetinarangefrom1%to10%.ExperimentsareconductedwithaPentium3.20GHzCPUand1GByteofmemory.Thequeryresultsareevaluatedinanevent-drivenapproach.Therefore,thequeryprocessorcallsdierentproceduresbasedoneachspeci“ceventtype.Themainmeasure-mentinthefollowingsimulationsistheresponseCPUtime(fromreceivingadataupdaterequesttothe1updatecompletiontimeortheevaluationtimeof)andtheoverallCPUtime(theevaluationtimeof).FortheoverallCPUtimealsorepre-sentstheresponsetime.Ourexperimentsuseseveralmetricstocomparethesealgorithms.Table2summarizesthedefaultparametersettingsinthefollowingsimulations. Parameter Default 100,000,500,000,1,000,000 5 2,3,4,5 10% 1%,5%,10% Table2.Simulationparameters 12Yu-LingHsuehRogerZimmermannWei-ShinnKu4.1UpdateRatioFirst,weevaluatedtheimpactoftheupdateratio.Figures5(a)and(b)showtheresponsetimeandoverallCPUtimeasafunctionofupdateratio,respectively,andFigure5(c)illustratestheI/Ocostforthethreemethods.We“xthedatacardinalityat100,000anddimensionalityat5.Theapproachachievesabetterperformancethanforallupdaterates.Thedegrada-tionofiscausedbytheexpensiveMaximumCoveragecomputationsscanningovertheprojectionlistsandtheincreaseofskylinepointsizewhichincursbiggerprojectionlists.alsooutperformsbothmethodsintermsoftheoverallCPUtime,sincetheamortizedcostofthecomputationsandexclusivedataevaluationislowerthantheothertwomethods. 100 200 300 400 500 600 10 5 1 Response CPU Time (sec) ESC DeltaSky ABBS (a)ResponseCPUtime 0 100 200 300 400 500 600 10 5 1 Overall CPU Time (sec) ESC DeltaSky ABBS (b)OverallCPUtime 0 50 100 150 200 250 10 5 1 I/O Cost ESC DeltaSky ABBS (c)I/OcostFig.5.Performancev.s.UpdateRatio(=100k,=5)4.2DimensionalityNextwereportontheimpactofthedimensionalityontheperformanceofallthreemethods.Figures6(a)(b)(c)showtheCPUoverheadsandI/Ocostv.s.thedimensionalityrangingfrom=2to5,respectively.Whenincreases,theper-formanceofallmethodsisdegradedbecausetheexclusivedatapointcomputa-tionsarecomplexandR-treesfailto“lteroutirrelevantdataentriesinhigherdimensions.Fromallthe“gures,wecanseethatoutperformsintermsoftheCPUtimeandI/Ocost. 100 200 300 400 500 600 5 4 3 2 Response CPU Time (sec) ESC DeltaSky ABBS (a)ResponseCPUtime 0 100 200 300 400 500 600 5 4 3 2 Overall CPU Time (sec) ESC DeltaSky ABBS (b)OverallCPUtime 0 50 100 150 200 250 5 4 3 2 I/O Cost ESC DeltaSky ABBS (c)I/OcostFig.6.Performancev.s.Dimensionality(=100k,=10%) EcientUpdatesforContinuousSkylineComputations134.3CardinalityFigures7(a)(b)showtheresponseandoverallCPUtimeasafunctionofthenumberofdatapoints,respectively,andFigure7(c)illustratesthecorrespondingI/Ocost.Overall,theCPUoverheadsincreaseasafunctionofthenumberofdatapoints.achievesasigni“cantreductionintermsoftheresponseCPUtimecomparedtotakesadvantageofthepre-computed2pointsretrievedbythelatestprocedureandquicklylocatesrelevantnew1candidatesforsubstitutingaremovedormoving1point.Aswecanseefromtheexperimentalresults,theadoptionofachievebetteroverallCPUperformanceandcompetitiveI/Ocostwith 0 1000 2000 3000 4000 5000 6000 7000 1000k 500k 100k Response CPU Time (sec) ESC DeltaSky ABBS (a)ResponseCPUtime 0 1000 2000 3000 4000 5000 6000 7000 1000k 500k 100k Overall CPU Time (sec) ESC DeltaSky ABBS (b)OverallCPUtime 0 200 400 600 800 1000 1200 1400 1600 1000k 500k 100k I/O Cost ESC DeltaSky ABBS (c)I/OcostFig.7.Performancev.s.Cardinality(=5,=10%)5ConclusionsInthispaper,weproposeanincrementalskylineupdateapproach.OuralgorithmachievesafasterresponsetimeandoverallCPUperformance.Withtheadoptionofthepre-computed2sets,canecientlyupdatethesky-linequeryresultsanddelegatethemostcomplexcomputationstoaseparateprocedurethatexecutesaftertheupdatesofthequeryresultsarecompleted.Anapproximateexclusivedataregion()isproposedandourexperimentscon“rmthefeasibilityofwhichhasalowamortizedcostoftheexclu-sivedataevaluationinhighdimensionalanddynamicdataenvironments.Theprocedure“rstexaminesalltheincomingdatarequestsandup-datesthe1resultifnecessaryandtheprocedureintegratesourlemmasandheuristicstoachievealowCPUoverheadandreducedI/Ocost.1.S.B¨onyi,D.Kossmann,andK.Stocker.TheSkylineOperator.InProceedingsofthe17thInternationalConferenceonDataEngineering(ICDE),Heidelberg,,pages421…430,2001.2.C.Y.Chan,P.-K.Eng,andK.-L.Tan.Strati“edcomputationofskylineswithpartially-ordereddomains.InProceedingsoftheACMSIGMODInternationalConferenceonManagementofData,Baltimore,Maryland,USA,pages203…214, 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