FilteringNogoodsLazilyinDynamicSymmetryBreakingDuringSearchJimmyH.M.Le - PDF document

FilteringNogoodsLazilyinDynamicSymmetryBreakingDuringSearchJimmyH.M.LeeandZichenZhuDepartmentofComputerScienceandEngineeringTheChineseUniversityofHongKongShatin,N.T.,HongKongThegenerationandGACenforcementofalargenumberofweaknogoodsinSymmetryBreakingDuringSearch(SBDS)iscostlyandoftennotworthwhileintermsofprunings.Inthispaper,weproposeweak-nogoodconsistency(WNC)forno-goodsandalazypropagatorforSBDS(anditsvari-ants)usingwatchedliteraltechnology.Wegivefor-malresultsonthestrengthandrelativelylowspaceandtimecomplexitiesofthelazypropagator.No- Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) 339 isasetofassignments,oneforeachvariablein.Apartialassignmentisasubsetofafullassignment.Aisafullassignmentthatsatiseseverymember.Anassignmentiff)=.Aniff.Ifanassignmentisneithersatisednorfalsied,itisunresolvednogoodisthenegationofapartialassignmentwhichisnotinanysolution.Nogoodscanalsobeexpressedinanequivalentimplicationform.Adirectednogoodisanim-plicationoftheform:::,wherethelefthandsidelhs:::andtherighthandsiderhsaredenedwithrespecttothepositionof.WecallallassignmentsintheLHSasLHSassignmentsandthenegationoftheRHSastheRHS.ThemeaningofisthattheRHSassignmentisincompatiblewiththeLHSassignments,andshouldberuledoutfromwhentheLHSistrue.Iflhsisempty,.Hereafter,directedno-goodsaresimplycallednogoodswhenthecontextisclear.Inthispaper,weconsidersearchtreeswithbinarybranch-ing,inwhicheverynon-leafnodehasexactlytwochildren.Supposeanon-leafnodeasthebranchingvariableandvalue.Theleftandrightchildrenof[f[frespectively.Wecallbranchingassignment[fbacktrackingassignment[f[fWesaythatbacktrackingtakesplace.Eachnodeassociatedwithapartialassignmentwhichisthesetofbranchingassignmentscollectedfromtheroot.IfaisinasubtreeundernodeistheistheancestornodeWeassumethattheCSP,exceptthegeneratedsymmetrybreakingnogoods,atasearchtreenodeisalwaysmadeGACusinganAC3-likeeMackworth,1977algorithmtocoordi-natethetriggeringofthepropagatorsassociatedwiththein-dividualconstraints.Consistencystrongerthanorequal)consistencyiffany-consistentCSPisalsoconsistent.Consistencystrictlystrongerthanbutnotviceversa.ConsistencyequivalenttoHereweconsidersymmetryasapropertyofthesetofso-lutions.AsolutionsymmetrysymmetryRossi,VanBeek,andWalsh,isasolution-preservingpermutationonassignments.Symmetrybreakingmethodstrongerinnodes(resp.solutions)pruningthanmethod,denotedbyresp:,whenallthenodes(resp.solutions)prunedwouldalsobeprunedby.Symmetrybreakingstrictlystrongerinnodes(resp.solutions)prun-thanmethod,denotedbyresp:resp:resp:Notethatrespectively.LexLeaderxLeaderCrawfordetal.,1996addsonelexicographi-calorderingconstraintconstraintFrischetal.,2002lex,pervariablesymmetryaccordingtoaxedvariableorder.GiventhesetofallsymmetriestoaCSP,symmetrybreak-ingduringsearch(SBDS)(SBDS)GentandSmith,2000addscon-ditionalconstraintsforeachsymmetryuponbacktracking.Consideranodeinthesearchtreewithpartialassign-,branchingvariableandvalue.Afterbacktrackingfromthenode[f,foreachsolutionsymmetrySBDSaddsthefollowingconditionalconstrainttothenode[fmeaningthatoncehasbeensearched,itssym-metricpartialassignmentforanyinthesymmetrysetunderthissubtreeshouldnotbesearchedatall.Notethattheaddedconstraintisanogood.Weak-NogoodConsistencyThewatchedliteraltechniquetechniqueMoskewiczetal.,2001isex-tendedtoimplementingpropagatorsforCSPsCSPsGent,Jeffer-son,andMiguel,2006.Whenanassignment(asaliteral)watched,thesatisfactionoftheassignmenttriggerspropagatorintheAC-3likealgorithm.ToenforceGACforanogood,onlytwoassignmentsthatarenotsatisedneedtobewatchedatchedMoskewiczetal.,2001.Uponeachbacktrack-ing,SBDSwouldaddnewnogoodsintotheconstraintstore.Maintainingsuchalargesetofnogoodsiscostlysincethetriggerstotheirpropagatorsareoftenfruitless.Weproposetowatchonlyoneassignmenttotradepruningpowerforef-ciencybyenforcingnogoodwithaweakerconsistency.Anogood:::weak-nogoodconsistent(WNC)ifff;m^jj;m;D)=AnogoodisWNCifeither(1)oneoftheLHSassignmentsisnotsatisedor(2)allLHSassignmentsaresatisedanditsRHSassignmentisfalsied.WenowcomparetheconsistencylevelofGACandWNC.Theorem1.GACWNConanogood.Proof.WerstprovethatGACimpliesWNC.AnogoodisGACifeither(a)itistrueor(b)ithasatleasttwounresolvedassignments.For(a),eitheroneoftheLHSassignmentsisfalsied,orallLHSassignmentsaresatisedanditsRHSassignmentisfalsied.BothcasessatisfythetwoconditionsofWNCrespectivelyandthenogoodisalsoWNC.For(b),oneoftheunresolvedassignmentsmustbeintheLHS.ThusoneoftheLHSassignmentsisnotsatisedandcondition(1)ofWNCissatised.ThenogoodisWNC.ToproveGACisstrictlystrongerthanWNC,considerthe=1)=1)=1)withdomains)=)=;D)=.ThenogoodisWNCsince=1)isnotsatisedyetbutnotGACsince1hasnosupport. WNClosespruningopportunitiesinwhichanunresolvedLHSassignmentcanbeenforcedtobetruewhenallotherassignmentsinthisnogoodareallsatised.Experimentalre-sultsconrmhoweverthatsuchcaseshappennotveryoften.ToenforceWNCforasymmetrybreakingnogood,wepro-posealazypropagator,LazyNgProp,inAlgorithm1.Thepropagatorcanendwithtwodifferentoutputs:ENTAILEDstoresthecurrentpartialassignment.Withoutlossofgenerality,weassumethatvariablesinareininput;:::;xforeaseofexplanation.Wealsoassume 340 thatapropagatorisimmediatelytriggeredoncewhenitisrstaddedtotheconstraintstore. Algorithm1LazyNgProp Require::currentpartialassignment:symmetry:variableofthewatchedLHSassignment:valueofthewatchedLHSassignment:variableoftheRHS:valueoftheRHS:positionoftheassignmentinwhosesymmetricassignmentisbeingwatched:thelengthoftheLHSforor+1;\f)=(ENTAILEDwatch(~ENTAILED Uponeachbacktracking,wecreateapropagatorforeachgeneratedsymmetrybreakingnogoodandmakeavailablethe,thelengthoftheLHSandtheRHS=~theassociatednogoodtothepropagator.Thepointerisini-tializedto-1meaningthatthepropagatordoesnotwatchanyassignmentwhenitisinitiallytriggered.Thefor-loopinlines1-9startsfromthecurrentwatchedassignmenttolookforthenextunresolvedassignmenttowatch.Oncefound,thenogoodisWNCandthepropagatorcanexit(line9)withreturned.If,however,anyLHSassignmentisfoundfalsied,thenogoodisentailedandcanberemovedfromtheconstraintstore(line3,exitwithENTAILED).Ifnounre-solvedassignmentsarefound,allLHSassignmentsaresatis-edandpruningiseffectedusingtheRHS(line10)andagainthenogoodisentailed(line11).AscanbeseeninAlgorithm1,LazyNgPropwatchesonlyoneassignmentatatime.SymmetricassignmentsintheLHSofnogoodsarealsoonlycomputedondemandbutnotea-gerly.Inaddition,thecurrentpartialassignmentcanbesharedbyallpropagatorsandbacktrackable.Theorem2.LazyNgPropenforcesWNCforanogood.Proof.Ifthepropagatorforanogoodline9,ithasanunresolvedassignmentbeingwatched.Fromline1,weknowisintheLHSof.Thuscondi-tion(1)ofWNCissatisedandisWNC.ThepropagatorcanreturnENTAILEDatline3or11.Inline3,condition(1)ofWNCissatised.Inline11,condition(2)issatisedsinceallLHSassignmentsaresatisedandtheRHSassignmentisfalsied.Again,isWNC. Theorem3.GivenaCSP=(X;D;C.ThespacecomplexityofLazyNgPropforanogoodis,andthetimecomplexityisProof.LazyNgProprecordsonlythesymmetry(justanameorpointer),theLHSwatchedassignment,theRHSassign-ment,theLHSwatchedassignment'scorrespondingassign-mentpositionandthelengthoftheLHS.Thetotalspacecomplexityis.ThemaximumnumberofassignmentsineachnogoodisandwescantheassignmentsonlyonceLazyNgProp.Thetimecomplexityis Theorem4.GivenaCSP=(X;D;Casetofsymmetries.ThetotalspaceandtimecomplexityofSBDSutilizingLazyNgProptoenforceWNConallnogoodsateachsearchtreenodeisj�1)+j�respectively.Proof.ThetotalnumberofnogoodsinSBDSisatmostj�andthelengthofapartialas-signmentcanbeatmost.Whileeachlteringalgorithmspacecomplexity,thetotalspacecomplexityisj�1)+LazyNgPropscanstheLHSassignmentsincrementallyduringtheAC3-likealgorithmbymaintaining.Themax-imumnumberofsuchassignmentsis.ThusthetotaltimecomplexityforLazyNgProptohandleallthenogoodsinSBDSisj� GACvsWNCinSBDSandItsVariantsInadditiontoReSBDSandLReSBDS,PartialSBDS(ParS-arS-Fleneretal.,2002isSBDSbutdealswithonlyagivensubsetofallsymmetries.LDSBLDSBMearsetal.,2013isafurtherdevelopmentofshortcutSBDSSBDSGentandSmith,whichhandlesonlyactivesymmetriesandtheircom-positions.WNCcanalsobeusedinParSBDS,LDSB,ReS-BDSandLReSBDSsincetheygeneratesymmetrybreakingnogoodsinasimilarfashion.Inthissection,wecomparethenodeandsolutionpruningpowerbetweenGACandWNCinSBDSanditsvariantsre-spectively.WedenotethemethodsenforcingWNCwithsub-WNC,andthoseenforcingGACwithoutasubscript.Inthefollowing,weassumeallmethodsusethesamestaticvariableandvalueorderingheuristics.Theorem5.WNCandSBDSWNCProof.SupposeSBDSWNCleavestwosymmetricsolutions.Supposefurtheristheirdeepestcommonances-tornodeandissearchedearlierthan.Uponbacktrack-ingfromtheleftsubtreetotherightat,thenogoodwhichprunesthesymmetricsolutionofmustbeposted.SinceLazyNgPropcanonlypostponesomepruningopportunitiestodeepernodes,isprunedbythenogoodeventually.ThusWNChasthesolutionpruningpowerasSBDS.Astheentiresymmetrygroupisposted,SBDSwouldneverleaveanysymmetricsubtrees.WhileSBDSWNCcanpost-ponethepruningopportunityintodeepernodes,itleavessomesymmetricpartsandpruneslessnodes.SBDSWNC 341 backtrackmoreandgeneratemorenogoods,buttheseno-goodsbreaknonewsymmetries(sinceallsymmetriesarebrokeninSBDS).ThusthesearchtreeofSBDSWNCthatofSBDS. Inpartialsymmetrybreaking,suchasParSBDSandReS-BDS,onlyasubsetofallsymmetriesisposted.SinceWNCisweaker,therearesymmetricsubtreesthatareprunedbyGACbutnotbyWNC.Inturn,WNCsearchesandalsobacktracksmore,thussometimespostingmorenogoodsthanGAC.Suchnogoodscanpotentiallybreakcompositionsymmetriesthatarenotposted.Asaresult,itisdifculttocomparetheoreti-callythestrengthofGACandWNConParSBDS,ReSBDSandLReSBDSrespectively,butisinterestingfuturework.Theorem6.GACforunconditionalnogoods.Proof.Unconditionalnogoodsaresimplydisequalitycon-straints.WNCandGAChandlethemequally. Theorem7.(resp.)LDSBWNCProof.AllsymmetrybreakingnogoodsinLDSBareuncon-ditionalsinceLDSBhandlesonlyactivesymmetriesandtheircompositions.ResultsfollowdirectlyfromTheorem6. GeneralizedWeak-incNGsConsistencyLeeandZhu(2014b)showthatthesetofsymmetrybreak-ingnogoodsaddedbySBDSanditsvariantsforonesymme-tryatasearchnodeareincreasingnogoods.TheyalsogivetheincNGsglobalconstraintandalteringalgorithmwhichisstrongerthanGAConeachnogood.NowweproposeaweakerconsistencyfortheincNGsconstraint.AsetofdirectednogoodsincreasingeasingLeeandZhu,ifthenogoodscanformasequence;:::;ngsuchthat(i)foranyy;tand(ii)nonogoodsareimpliedbyanother.Anogoodlowerthaniffji,andhigherthanAsetofincreasingnogoods=;:::;nggeneralizedweak-incNGscon-sistent(GWIC)ifff;t^jj;D)==;t;D)=.Thus,isGWICifeither(1)oneoftheLHSas-signmentsofnogoodisnotsatisedandforeachlowerthan'sLHSassignmentsaresatisedanditsRHSassignmentisfalsiedor(2)theLHSassignmentsofallnogoodsaresatisedandallRHSassignmentsarefalsied.GWIChasthefollowingtheorems.Theorem8.Givenincreasingnogoods=;:::;ngGWIConisequivalenttoWNConeachh;tProof.WerstprovethatifisGWIC,allindividualno-goodsareWNC.Onecaseisthatthereexistssuchthatsatisescondition(1)ofGWIC.Nowallnogoodslowersatisfycondition(2)ofWNC,andtheremainingno-goodswhoseLHSssubsumelhssatisfycondition(1)ofWNC.ThusallnogoodsareWNC.Anothercaseisthatisescondition(2)ofGWIC,whichimpliesthatallnogoodssatisfycondition(2)ofWNC.Secondly,weproveifallindividualnogoodsareWNC,isGWIC.Supposewecanndanogoodwhichisthelowestnogoodinandsatisescondition(1)ofWNC.Allnogoodslowerthanmustsatisfycondition(2)ofWNCsincealltheirLHSassignmentsmustbesatised.Thuscon-dition(1)ofGWICissatised.Ifwecannotndsuchanallnogoodssatisfycondition(2)ofWNC.Nowcondition(2)ofGWICissatised.Therefore,isGWICiffallindividualnogoodsareWNC. Theorem9.GACGWIConanincNGsconstraint.Proof.Givenincreasingnogoods=;:::;ng.LeeandZhu(2014b)showthatGAConisstrictlystrongerthanGAConindividualnogoods(),whichisstrictlystrongerthanWNContheindividualnogoods()byTheorem1.Inaddition,WNContheindividualnogoods()isequivalenttoGWIConbyTheorem8.Resultfollows. Asequenceofincreasingnogoods=;:::;ngforasymmetrycanbeencodedcompactly.Theincreas-ingpropertyguaranteesthatlhslhsforallji.ThustheLHSassignmentsofallnogoodsareavailableinlhs.Supposeisaddedatanancestornodeisthepartialassignmentof.Wemusthavelhs.Therefore,wecanalwaysconstruct(byap-plyingsymmetrytransformation)lhsfromthepartialassignmentatanydescendantnodeof.Wealsocollectallthebacktrackingassignments(refertoSection2)fromroottoasearchnode,andcallit.Eachbacktrackingassignmentisalsoassociatedwithits,whichisthelengthofthepar-tialassignmentassociatedwiththenodewherebacktrackingtakesplace.Wecanconstruct(byapplyingsymmetrytrans-rhshs;tfromthebacktrackingas-signmentsin.Usingtogether,wecanreconstructallnogoodsinusingsymmetrySection3showsthatwatchingoneunresolvedassignmentintheLHSisenoughtoenforceWNConanogood.Anim-portantconsequenceoflhslhsforalljiisthatwatchinganassignmentinlhswatchingthesameassignmentinlhs;:::;lhs.Ingeneral,whenwewatchanassignmentthatrstappearsonlylhs,wearealsowatchingthesameassignmentinlhs;:::;lhs.Thus,wecanwatchonlyoneun-resolvedassignmentforourlazypropagatorfor.Wescanthesymmetricassignmentsofthecurrentpartialassignmentateachsearchnodetondtherstunresolvedassignmenttowatch.Duringscanning,whenweencounteranassignment,therearethreepossibilities.(a)issatised.WeshouldmakeuseoftolookforallnogoodswithatrueLHS,andenforcetheRHSsofalldiscoverednogoodstobetruetoef-fectprunings.Thenwecontinueourpreviousscanningtothenextassignment.(b)isfalsied.Wecanstopscanningsinceisentailed.(c)isunresolved.Westopscanningandwatch,sinceisnowGWIC.TheincNGsconstraintgrowssincenogoodsareaddeddynamicallyuponbacktracking.Itcanhappenthat,atone 342 searchnode,allsymmetricequivalentsofthoseinthecurrentaretrue.Therearenoassignmentstowatch.Insuchacase,thepropagatorshouldstillbetriggereduponbacktracking(whenisupdated),whichiswhenanewno-goodisaddedtothepropagator.Wenowpresentourlazypropagator,LazyincNGsProp,fortheincNGsconstraintasshowninAlgorithm2.Thepropaga-torcanreturntwodifferentresults:ENTAILEDstoresthecurrentpartialassignment.storesallback-trackingassignmentsandtheirdepthasapairfromrootuptothecurrentsearchnode.Ifweviewasarrays,theirindicesstartfrom0.Withoutlossofgenerality,weassumethatvariablesinareininputorder;:::;x Algorithm2LazyincNGsProp Require::currentpartialassignment:allbacktrackingassignmentsandtheirdepthfromrootuptothecurrentnode:symmetry:variableofthewatchedLHSassignment:valueofthewatchedLHSassignment:positionoftheassignmentinwhosesymmetricassignmentisbeingwatched=0:positionoftherstassignmentinwhosesym-metricassignmenthasnotbeenenforcedbythisconstraintj�+1foror+1)=(ENTAILEDwatch+1j� Algorithm3 for;d)=(+1 Foreachgivensymmetry,anincNGsglobalconstraintwouldbepostedattherootnode.WecreateapropagatorLazyincNGsPropandmakeavailablethesymmetrytothepropagator.Thispropagatoristriggeredbythewatchedas-.Iftherearenoassignmentstowatch,itistriggeredwhenisupdated.Thepointeralwayspointstothewatchedassignment,andisinitializedto-1towatchnothing.Ifthepropagatoristriggeredby'supdate,andnoassignmentsarewatchedandisnotupdatedfromthelastpropagation(line1),thismeanstheLHSsofallincreasingnogoodsaresatisedand(line2)iscalledtopruneallsymmetricequivalentsofthebacktrackingassignmentsinstartingfrompointerwhosecorrespondingnogoodsarenotenforcedyet.Afterdoingthat,theconstraintisGWICandthepropagatorcanexit(line15).Ifthepropagatoristrig-geredbythewatchedassignment'supdate,thefor-loopinlines4-13looksforthenextunresolvedassign-menttowatchasinAlgorithm1.TheonlythingtotakecareisifanassignmentintheLHSissatised,(line13)iscalledtopruneallsymmetricequivalentsofthebacktrack-ingassignmentsinstartingfrompointerwhoseLHSsaresatisedandcorrespondingnogoodsarenotenforcedyet.AscanbeseeninAlgorithm2,LazyincNGsPropwatchesonlyoneassignmentatatime,andalsogetstriggeredwhenisupdated.Allsymmetricassignmentsarecomputedondemandbutnoteagerly.Notethatshouldbeback-trackableandcanbesharedbyallpropagators.Theorem10.LazyincNGsPropenforcesGWICforaninc-NGsglobalconstraint.Proof.Thepropagatorreturnsatline12or15.Line12isreachedwhenthereisanunresolvedassignmentbeingwatched.NowallnogoodswhoseLHSsdonotcontainaretruesincetheirLHSassignmentsaresatisedandtheirRHSsareenforcedatline2or13by.Forallno-goodscontainingmustbeintheirLHSsaccord-ingtoline4.Thuscondition(1)ofGWICissatisedandtheincreasingnogoodsisGWIC.Line15meansallLHSassign-mentsofcurrentnogoodsaresatisedandallRHSassign-mentsarefalsied.Nowallnogoodsaretrue.Condition(2)ofGWICissatisedandthisincreasingnogoodsisGWIC.IfthepropagatorreturnsENTAILEDatline6,thereexistsansuchthatitsLHScontainsandalllowernogoodshavetheirLHSassignmentssatisedandRHSassignmentfalsied.Thus,condition(1)ofGWICissatisedandthisincreasingnogoodsisGWIC. Theorem11.GivenaCSP=(X;D;C.ThespacecomplexityofLazyincNGsPropforaninc-NGsglobalconstraintis,andthetimecomplexityisj�Proof.LazyincNGsPropneedstorecordthesymmetry,theLHSwatchedassignment,twopointers.Theto-talspacecomplexityis.Thereareatmostj�numberofsymmetricassignmentstogenerateandj�toprune.Thetimecom-plexityisj� ExperimentalResultsThissectiongivesthreeexperiments,allwithmatrixsymme-tries(variablesymmetries).Ourmethodworksforarbitrarysymmetries,butbetterasthenumberofsymmetrygrows.Ma-trixsymmetriesarecommoninmanyCSPmodelsandtheirnumbersareexponentialwithproblemsize.Werstsolvethebenchmarksusingtheefcientandwidelyusedstaticmethod 343 Table1:ErrorCorrectingCode-LeeDistance(allsolutions) n;c;b DoubleLex LexLeader ft ft 4,4,8 32,469839,25142.19 7,863267,81515.14 5,2,10 8741,5714.73 34,6597.48 5,6,4 710,731725,83716.39 269,841354,1848.1 5,6,5 1,441,2245,508,192116.88 451,3031,918,57945.17 5,6,6 297,47611,709,068303.4 82,7423,837,292112.16 6,4,4 4,698,8424,139,211112.17 1,690,2293,404,49972.47 6,4,5 29,345,81673,522,8731909.09 8,052,12623,457,604678.49 6,8,4 59,1582,469,21135.9 22,7561,082,82718.05 8,4,4 35,626,71448,525,8271303.08 12,246,48047,273,9271171.17 ParSBDS ParSBDSincNGs ParSBDSWNC=GWIC ft ft ft 8,918281,72021.62 8,918281,55316.68 8,654275,23618.0114.43 6625,8246.22 6625,8104.96 6626,73254.09 297,819307,14812.91 297,819306,3768.32 289,822334,2639.737.80 508,5852,042,20076.83 508,5852,036,42551.68 490,6872,095,83659.546.83 95,3804,143,524185.03 95,3804,129,503126.99 90,7674,163,933147.23116.14 1,943,6081,890,047102.97 1,943,6041,888,67962.08 1,859,8001,961,61071.8453.99 9,472,05625,509,6151,260.44 9,472,02125,493,887806.56 9,034,35325,186,758920.08696.83 24,3551,033,52932.45 24,3551,031,16118.92 24,1921,090,15422.6917.39 14,541,82621,963,9881,450.38 14,541,82221,958,213750.56 13,877,57422,137,598889.59637.30 LReSBDSGAC incNGs WNC=GWIC ft ft ft 235,07415.01 7,698234,806 237,30619.63 20,8205.6 5620,806 21,8016.36 253,7297.34 235,866252,866 262,4838.78 1,614,08844.47 392,2211,608,523 1,645,38953.79 3,210,014106.3 72,1503,197,525 3,260,502139.17 1,568,77659.95 1,608,5361,566,251 1,650,21071.88 20,494,554747.13 7,631,83320,474,513 21,024,022904.68 793,25816.28 18,933790,810 806,89720.56 17,464,022768.35 11,582,46717,453,197 18,047,630998.21 DoublelexxFleneretal.,2002,andalsoLexLeadertobreakamuchlargersubsetofsymmetries.WethenreporttheresultsoftwodynamicmethodsParSBDSandLReSBDS.Eachdy-namicmethodwouldbeimplementedwiththefourpropaga-tors:GAConeachnogood(),thelteringalgorithmofincNGsgivenbyLeeandZhu[2014b](incNGs),WNConeachnogood(WNC)andGWIConeachincNGsconstraintGWICWealsodidextraexperimentstondthebestsubsetofma-trixsymmetriesforeachmethod.ParSBDSisgivenanytworowsorcolumnsbeingpermutableandtheCartesianproductsofthesetwosubsets.LReSBDSandLexLeadercanbreaktheentirerowsymmetriesandcolumnsymmetriesbyonlypost-ingadjacentrowsorcolumnsbeingpermutablewiththein-putordervariableheuristicheuristicLeeandZhu,2014a.WethusonlypostadjacentrowsorcolumnsbeingpermutableandtheCartesianproductsofanytworowsorcolumnsbeingper-mutabletoLReSBDSandLexLeader.LDSBandReSBDSarediscardedinthecomparisonsinceLReSBDSissubstan-tiallymoreefcientcientLeeandZhu,2014a;2014bthanthesetwomethods.AllexperimentsareconductedusingGecodeSolver4.2.0onXeonE56202.4GHzprocessorswith7GB.Duetothemanycolumns,eachtableissplitintothreerows.Therstcolumnalwaysgivestheinstanceparameters.Inaddition,denotesthenumberofsolutions,thenumberoffailuresanddenotestheruntimes.SinceWNCandGWIChavethesamepruningpower,weshowtheirso-lutionsandfailurestogetherandusetodenotetheruntimeofWNCandGWICrespectively.Thesearchtimeoutlimitis1hour.Anentrywiththesymbolthatmemoryisexhausted.ThebestresultsarehighlightedinUnlessotherwisespecied,searchisdefaultedtoinputvariableorderandminimumvalueorder.Table2:CoverArrayProblem(allsolutions) t;k;g;b DoubleLex LexLeader ft ft 2,4,4,16 3,456661,72623.67 123,4398.19 3,4,2,13 29,738202,7232.86 11,04776,235 3,4,2,14 107,224496,2467.29 38,007185,786 3,4,2,15 348,8571,149,97417.91 120,832431,794 3,4,2,16 1,039,6412,548,94142.21 357,662965,531 3,4,2,17 2,870,7345,433,94394.55 991,7002,085,752 3,4,2,18 7,413,39411,181,194210.61 2,590,0004,362,860130.88 3,4,2,19 18,043,63022,265,801443.24 6,404,2818,851,675257.07 3,4,3,27 2464,7775.66 26,19317.69 ParSBDS ParSBDSincNGs ParSBDSWNC=GWIC ft ft ft 432130,81114.18 432130,7078.41 432128,62911.216.37 16,08595,3437.01 16,08595,3433.86 15,36193,8963.842.20 54,702229,58817.53 54,702229,5889.34 52,655226,7159.815.45 170,263526,76641.98 170,263526,76622.04 165,093521,43124.2113.08 491,1351,162,22597.74 491,1351,162,22550.45 479,2001,152,82358.0131.25 1,325,2542,477,180222.36 1,325,2542,477,180113.12 1,299,6572,461,304135.7671.07 3,370,1565,114,350495.8 3,370,1565,114,350246.88 3,318,5335,088,491314.99158.25 8,127,24910,247,8301,077.68 8,127,24910,247,830525.8 8,028,43610,206,998-347.16 26,6908.19 26,6906.45 26,6545.993.56 LReSBDSGAC incNGs WNC=GWIC ft ft ft 120,8158.54 424120,758 121,35310.38 11,04776,235 11,04776,235 11,04776,235 38,007185,786 38,007185,786 38,007185,786 120,832431,794 120,832431,794 120,832431,794 357,662965,531 357,662965,531 357,662965,531 991,7002,085,752 991,7002,085,752 991,7002,085,752 2,590,0004,362,860 2,590,0004,362,860 2,590,0004,362,860702.6136.78 6,404,2818,851,675 6,404,2818,851,675 6,404,2818,851,675-300.97 825,962 825,962 25,9857.19 ErrorCorrectingCode-LeeDistance(ECCLD)EachECCLDinstanceisparameterizedbyn;c;b.WeusethesamemodelbyLeeandZhu(2014a).Table1showstheresultsforECCLD.ParSBDSWNCis1.35timesfasterthanParSBDSonaverage.WNCdoesnotperformwellsincethenumberofextranogoodsaddedbyLReSBDStopruneextracomposi-tionsymmetriesisbigandthesymmetriesarebrokenlate.ParSBDSGWICandLReSBDSGWICrun1.13and1.15timesfasterthanParSBDSincNGsandLReSBDSincNGsonaver-agerespectively.Theimprovementisnotthatmuchduetothesmallnumberofgivensymmetries.WhenwecomparethenumberoffailuresforParSBDSandLReSBDS,WNCandGWICincreaseonlyslightlythesearchtreesize.Thisshowsourweakerconsistencieslosefewpruningopportuni-ties.LReSBDSGWICperformsthebestandruns1.50and2.67timesfasterthanLexLeaderandDoubleLexonaveragerespectively.ParSBDSGWICperformsslightlyslowerthanGWICduetoitsbiggersearchtreesizeandmoresymmetriestohandle.Thus,lazinesscansaveustimecom-paringwithothersymmetrybreakingmethods.CoverArrayProblem(CA)CAinstancesareparameterizedbyt;k;g;bWeusethesamemodelbyLeeandZhu(2014b).Table2showstheresultsforCA.ParSBDSWNCis1.61timesfasterthanParSBDSonaverage.WNCstilldoesnotperformwell.ParSBDSGWICandLReSBDSGWICrun1.62and1.75timesfasterthanParSBDSincNGsandLReSBDSincNGsonaveragerespec-tively.ForLReSBDS,WNCandGWIConlyslightlyincreasethesearchtreesize.LReSBDSGWICperformsthebestandruns1.67and1.82timesfasterthanLexLeaderandDou- 344 Table3:BIBDwithMaximumValueOrdering(allsolutions) v;k; DoubleLex LexLeader ft ft 7,3,5 33,304191,223 5,97941,978 7,3,6 250,8781,814,42521.06 33,824292,634 7,3,7 1,460,33213,149,270154.79 203,2962,069,840 7,3,8 6,941,12476,463,115886.95 --- 8,4,6 2,058,52314,156,697157.75 596,3993,873,360118.12 ParSBDS ParSBDSincNGs ParSBDSWNC=GWIC ft ft ft 12,93683,57834.95 12,93683,5787.25 7,91654,6082.394.84 93,713717,959377.91 93,713717,95944.37 41,388353,23218.0720.50 476,7524,486,5873,349.82 476,7524,486,587270.15 226,1762,292,110137.22114.92 305,3123,583,1923,600.00 --- 1,134,25313,599,864-694.02 932,0226,450,1512,366.52 932,0226,450,183281.22 925,5046,483,468351.32177.09 LReSBDSGAC incNGs WNC=GWIC ft ft ft 5,97941,978 5,97941,978 5,97941,9782.894.20 33,824292,634 33,824292,634 33,824292,634 --- 203,2962,069,840 203,2962,069,840 --- 1,075,69412,921,639 1,075,69412,921,639 596,3993,873,339 596,3993,873,339 3,956,200287.32129.06 bleLexonaveragerespectively.Thisagainshowstheadvan-tageofourlazyincNGspropagator.BalancedIncompleteBlockDesign(BIBD)ABIBDinstancecanbedeterminedbyitsparametersv;k;.WeusethesamemodelbyLeeandZhu(2014b).ThevalueorderingismaximumvalueorderingandDou-bleLexordersrowsandcolumnsdecreasingly.Table3showstheresultsforBIBD.ParSBDSWNCandLReSBDSWNCrun16.67and4.07timesfasterthanParSBDSandLReSBDSonaveragerespectively.Onereasonfortheimprovementisthereductionofoverhead.TheotherreasonforthegoodefciencyofParSBDSWNCthatitprunessymmetrieslateandcanbreakmuchmorecom-positionsymmetries.ParSBDSGWICandLReSBDSGWICrun1.90and1.34timesfasterthanParSBDSincNGsincNGsonaveragerespectively.ThesearchtreesizebyenforcingWNCandGWICstilldoesnotincreasetoomuchmorethanGAC.LReSBDSGWICperformsthebestandruns7.90and1.12timesfasterthanLexLeaderandDou-bleLexonaveragerespectively.Fromtheabove,wecancon-cludethatlazinesscansavemuchtimeandmemorycompar-ingwithothersymmetrybreakingmethods.ConclusionandFutureWorkOurcontributionsarefourfold.First,weproposeWNCfornogoodsandgivealazypropagatortosymmetrybreakingno-goodsaddedbySBDS-basedmethods.WegivealsothespaceandtimecomplexitiesoftheWNClazypropagator.Second,weproposeGWICforincNGsglobalconstraintandalsogivealazypropagatoranditsspaceandtimecomplexities.Third,weprovethatGWIConaconjunctionisequivalenttoWNContheindividualnogoods.Fourth,weuseexperimentstoshowthelazymethods'pruninglossissmallwhilethegaininefciencyisworthwhile.TheimprovementfromtheoriginalincNGstothelazyver-sionisnotasgoodasthatfromGAConnogoodstoWNConnogoods.Thatisbecausetheglobalconstraintversionmustbeincrementalinnaturetocaterfortheadditionofnewno-goods,andistriggeredeverytimewhennewnogoodsaregenerated,whichisoften.Soitisnotaslazyaswehopeittobe.ThispointstonewresearchopportunitiestoinvestigatehowbesttoincorporatelazinessintoincNGsconstraints.Otherpossibilitiesincludeinvestigatingsymmetricno-goodscollectedfromrestartsrestartsLecoutreandTabary,2011anddetectingsymmetrydynamically.ReferencesencesCrawfordetal.,1996Crawford,J.;Ginsberg,M.;Luks,E.;andRoy,A.1996.Symmetrybreakingpredicatesforsearchproblems.In,148–159.148–159.Fahle,Schamberger,andSellmann,2001Fahle,T.;Scham-berger,S.;andSellmann,M.2001.Symmetrybreaking.In,93–107.93–107.Fleneretal.,2002Flener,P.;Frisch,A.;Hnich,B.;Kiziltan,Z.;Miguel,I.;Pearson,J.;andWalsh,T.2002.Breakingrowandcolumnsymmetriesinmatrixmodels.InInFrischetal.,2002Frisch,A.;Hnich,B.;Kiziltan,Z.;Miguel,I.;andWalsh,T.2002.Globalconstraintsforlexi-cographicorderings.In,93–108.93–108.GentandSmith,2000Gent,I.,andSmith,B.2000.Sym-metrybreakinginconstraintprogramming.InInGentetal.,2003Gent,I.;Harvey,W.;Kelsey,T.;andLin-ton,S.2003.GenericSBDDusingcomputationalgrouptheory.In,333–347.333–347.Gent,Harvey,andKelsey,2002Gent,I.;Harvey,W.;andKelsey,T.2002.Groupsandconstraints:Symmetrybreak-ingduringsearch.In,415–430.415–430.Gent,Jefferson,andMiguel,2006Gent,I.;Jefferson,C.;andMiguel,I.2006.Watchedliteralsforconstraintpropa-gationinMinion.In,182–197.182–197.LawandLee,2006Law,Y.,andLee,J.2006.Symmetrybreakingconstraintsforvaluesymmetriesinconstraintsat-isfaction.ConstraintsaintsLecoutreandTabary,2011Lecoutre,C.,andTabary,S.2011.Symmetry-reinforcednogoodrecordingfromrestarts.In,13–27.13–27.LeeandZhu,2014aLee,J.,andZhu,Z.2014a.BoostingSBDSforpartialsymmetrybreakinginconstraintprogram-ming.In,2695–2702.2695–2702.LeeandZhu,2014bLee,J.,andZhu,Z.2014b.Anincreasing-nogoodsglobalconstraintforsymmetrybreak-ingduringsearch.In,465–480.465–480.Mackworth,1977Mackworth,A.1977.Consistencyinnet-worksofrelations.ArticialintelligenceenceMearsetal.,2013Mears,C.;delaBanda,M.G.;Demoen,B.;andWallace,M.2013.LightweightdynamicsymmetryConstraintsaintsMoskewiczetal.,2001Moskewicz,M.;Madigan,C.;Zhao,Y.;Zhang,L.;andMalik,S.2001.Chaff:Engineeringanefcientsatsolver.InDAC'01,530–535.530–535.Rossi,VanBeek,andWalsh,2006Rossi,F.;VanBeek,P.;andWalsh,T.2006.HandbookofconstraintprogrammingElsevier. 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