TRANSFORMATIONS ON REGULAR NON DOMINATED COTERIES AND THEIR APPLICATIONS - PDF document

TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIESANDTHEIRAPPLICATIONSKAZUHISAMAKINOyANDTIKOKAMEDAzSIAMJ.DISCRETEMATH.c\r2001SocietyforIndustrialandAppliedMathematicsVol.14,No.3,pp.381{407Abstract.AcoterieunderanunderlyingsetUisafamilyofsubsetsofUsuchthateverypairofsubsetshasatleastoneelementincommon,butneitherisasubsetoftheother.AcoterieCunderUissaidtobenondominated(ND)ifthereisnoothercoterieDunderUsuchthat,foreveryQ2C,thereexistsQ02DsatisfyingQ0Q.WeintroducetheoperationwhichtransformsaNDcoterietoanotherNDcoterie.Aregularcoterieisanaturalgeneralizationofavote-assignablecoterie.WeshowthatanyregularNDcoterieCcanbetransformedtoanyotherregularNDcoterieDbyjudiciouslyapplyingtheoperationtoCatmostjCj+jDj2times.Asanotherapplicationoftheoperation,wepresentanincrementallypolynomial-timealgorithmforgeneratingallregularNDcoteries.Wethenintroducetheconceptofag-regularfunctionalasageneralizationofavailability.WeshowhowtoconstructanoptimumcoterieCwithrespecttoag-regularfunctionalinO(n3jCj)time,wheren=jUj.Finally,wediscussthestructuresofoptimumcoterieswithrespecttoag-regularfunctional.Keywords.coterie,nondominatedness,regularcoterie,availability,mutualexclusion,positiveself-dualBooleanfunction,regularself-dualBooleanfunction,g-regularfunctionalAMSsubjectclassications.68M14,68M15,68P15,68Q25,68R05PII.S08954801003711101.Introduction.AcoterieCunderanunderlyingsetU=f1;2;:::;ngisafamilyofsubsets(calledquorums)ofUsatisfyingtheintersectionproperty(i.e.,foranypairS;R2C,S\R6=;holds)andminimality(i.e.,noquoruminCcontainsanyotherquoruminC)[18,23].Theconceptofacoteriehasapplicationsindiverseareas,suchasmutualexclusionindistributedsystems[13,18,23],datareplicationprotocols[14],nameservers[27],selectivedisseminationofinformation[39],anddistributedaccesscontrolandsignatures[30].Forexample,toachievemutualexclusioninadistributedsystem,lettheelementsinUrepresentthesitesinthedistributedsystem.AprocessisallowedtoenteracriticalsectiononlyifitcangetpermissionsfromallthemembersofaquorumQ2C,whereeachsiteisallowedtoissueatmostonepermissionatatime.Bytheintersectionproperty,itisguaranteedthatatmostoneprocesscanenterthecriticalsectionatanytime.AcoterieCunderUissaidtodominateanothercoterieD(6=C)underUif,foreachquorumQ2D,thereisaquorumQ02CsatisfyingQ0Q.Acoteriewhichisnotdominatedbyanyothercoterieissaidtobenondominated(ND)[18].NDcoteriesareimportantinpracticalapplications,sincetheyhavemaximal\eciency"insomesense[4,18,21].ReceivedbytheeditorsApril17,2000;acceptedforpublication(inrevisedform)June6,2001;publishedelectronicallyAugust29,2001.AnextendedabstractofthispaperappearsinProceedingsoftheNineteenthACMSymposiumonPrinciplesofDistributedComputing(PODC2000),Portland,OR,2000,pp.279{288.ThisworkwassupportedinpartbytheScienticGrantinAid,bytheMinistryofEducation,Science,Sports,andCultureofJapan,andinpartbytheNaturalSciencesandEngineeringResearchCouncilofCanada.http://www.siam.org/journals/sidma/14-3/37111.htmlyDivisionofSystemsScience,GraduateSchoolofEngineeringScience,OsakaUniversity,Toyon-aka,Osaka,560-8531,Japan(makino@sys.es.osaka-u.ac.jp)zSchoolofComputingScience,FacultyofAppliedSciences,SimonFraserUniversity,Burnaby,BritishColumbia,V5A1S6Canada(tiko@cs.sfu.ca)381 382KAZUHISAMAKINOANDTIKOKAMEDAGivenafamilyCofsubsetsofU,whichisnotnecessarilyacoterie,wedeneapositive(i.e.,monotone)BooleanfunctionfCsuchthatfC(x)=1iftheBooleanvectorx2f0;1gnisgreaterthanorequaltothecharacteristicvectorofsomesubset1inC,and0otherwise,wheren=jUj.Itwasshownin[20]thatCisacoterieifandonlyiffCisdual-minor,andCisNDifandonlyiffCisself-dual.(Seesection2.2.)Basedonthischaracterization,themethodsdevelopedinthericheldofBooleanfunctionscanbeexploitedtoderivevariouspropertiesofcoteriesandNDcoteries.AcoterieCissaidtobevote-assignableifthereexistavoteassignmentw:U7!R+andathresholdt2R+suchthatw(S)tifandonlyifSQforsomeQ2C[18,19,37],whereR+isthesetofnonnegativerealnumbersandw(S)=Pi2Sw(i).Itiseasytoseethatthereisaone-to-onecorrespondencebetweenvote-assignablecoteries(resp.,NDcoteries)Canddual-minor(resp.,self-dual)thresholdBooleanfunctionsfC.(ForthedenitionofathresholdBooleanfunction,seesection2.)Thevote-assignablecoteriesareimportantandhavebeenusedinmanypracticalproblems,sincetheycanbehandledeciently(see,e.g.,[18,19,37,38]).Weassumeinthispaperthatavoteassignmentwsatisesw(i)w(j)forallij,sinceweareinterestedincoterieswhicharenonequivalentunderpermutationonU.AcoterieCisequivalenttoacoterieC0underpermutationifCcanbetransformedintoC0bypermutingtheelementsofU.Forexample,C=ff1;2g;f1;3ggisequivalenttoC0=ff2;3g;f2;1ggunderpermutation.AcoterieCissaidtoberegularif,foreachQ2Candeverypair(i;j)2UUwithij,i62Qandj2Q,thereexistsQ02CsuchthatQ0(Qnfjg)[fig.2Bydenition(andthediscussioninsection2),avote-assignablecoterieCisalwaysregular,thoughingeneraltheconverseisnottrue.TheregularBooleanfunctionsweredenedasageneralizationofthethresholdfunctions[28].Itisknownthatmostregularcoteriesarevote-assignable[28];inparticular,allregularNDcoteriesunderUwithn=jUj9arevote-assignable.Amongtheimportantproblemsregardingcoteriesarethefollowing:(i)decidewhetheragivencoterieisND(equivalently,whetheragivenpositivedual-minorfunctionisself-dual);(ii)construct\optimal"NDcoteriesaccordingtoacertaincriterion,suchasavailabilityandload[29](equivalently,construct\optimal"positiveself-dualfunctions);and(iii)generateallNDcoteries(equivalently,allpositiveself-dualfunctions)sys-tematically.Unfortunately,thecomplexityofproblem(i)isstillunknown[8,16,22],althougharesultbyFredmanandKhachiyan[17]suggeststhatitisunlikelythattheproblemisNP-hard.[8,16]giveanumberofinterestingequivalentproblemswhichariseinvariouseldsofapplications.However,itisknownthatifwerestrictourselvestoregularcoteries,(i)ispolynomiallysolvable[6,32].Although(i)isaninterestingproblem,wedonotconsider(i)furtherinthispaper.Instead,wefocusonproblems(ii)and(iii).Asfor(ii),letusconsidertheavailabilityofacoterie,wheretheconceptofavailabilityhasbeenextensivelystudiedunderdierentnamesinreliabilitytheory(see,e.g.,[35]).Assumethateachelementcanbeineitheroftwostates,operationalorinoperational,andtakesonitsstaterandomlyandindependently,elementibeingoperational(resp.,inoperational)withprobabilitypi(resp.,1pi).Givenoperationalprobabilitiespi,i2U,whereweassumewithoutlossofgeneralitythat1p1p2


pn0,theavailabilityofacoterieCis1Theithcomponentofthecharacteristicvectoris1(0)ifi2Uis(not)containedinthesubset.2ThisdenitionwasmotivatedbythedenitionofregularBooleanfunctions.Seesection2.3. TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES383theprobabilitythatthesetofoperationalelementscontainsatleastonequoruminC.Availabilityisundoubtedlyanimportantconceptinpracticalapplications,andhenceitisnaturaltoconstructacoteriewiththemaximumavailability.Theavailabilityofcoterieshasbeenstudiedextensively[1,5,15,33,36,38].Itisknown[1,36]thattheelementsi2Uwithpi1=2canbeignored;i.e.,thereexistsamaximum-availabilitycoterieCsuchthatnoquoruminCcontainsi.(Inthecasewherepi1=2holdsforalli,C=ff1gghasthemaximumavailability[1,15,33].)Thus,weshallassumethat(1)p1p2


pn1=2:Itisalsoknownthatifeitherp1=1orp11=2,thenC=ff1gghasthemaximumavailability.If16=p1�1=2,ontheotherhand,itisdemonstratedin[36,38]thatthecoterieCmax,givenbelow,maximizesavailability.Firstdenetheweightfori2Ubyw(i)=log2(pi=(1pi))(1)introducethenotationw(S)=Pi2Sw(i)forSU.Now,Q2Cmaxif(a)w(Q)(=w(UnQ))=w(U)=2and12Q(1isanelementofU),or(b)QisaminimalsubsetofUsatisfyingw(Q)�w(U)=2,andQdoesnotcontainanyquorumoftype(a).SincethiscoterieCmaxisvote-assignable,AmirandWool[1],SpasojevicandBerman[36],andTongandKain[38]proposedalgorithmstocomputeavoteas-signmentwfromw,calledtie-breakingalgorithm,inordertoremovecase(a).Anexponentialalgorithmisproposedin[38]tondthe\optimal"tie-breakingrule,whileAmirandWool[1]andSpasojevicandBerman[36]presentpolynomial-timeapproxi-mationalgorithmsforit.ThemainproblemwiththeabovedenitionofCmaxisthattheremayexistasubsetSUsuchthatw(S)=w(UnS)(case(a)),becauseofwhichasimplevoteassignmentw(showingthatCmaxisvote-assignable)isnoteasilyobtainable,andthattheweightw(i)is,ingeneral,notarationalnumber;hencewecannotcomputew(S)=Pi2Sw(i)inpolynomialtime.Fortheabovereasons,nopolynomialalgorithmforconstructingmaximum-availabilitycoterieswasknown.Inthispaper,wepresentapolynomial-timealgorithmforit.Moreprecisely,wedenea\g-regular"functionalasageneralizationofavailability(seesection6)andthenshowthat,givenag-regularfunctional,wecancomputeacoterieCwhichmaximizesinO(n3jCj)time,wherejCjisthenumberofquorumsinC.Problem(iii)isknowntobeusefultosolve(ii)[9,18].Tosolve(ii),werstenumerateall(orsome)NDcoteriesecientlyandselectthebestoneunderacertaincriterion,whichisnoteasilycomputable.Thisprocedureisusefulwhennissmallorwhenwehaveenoughtimetocomputeit.Wefeelthat(iii)ismathematicallyinteresting,givingusaninsightintothestructureofNDcoteries(or,equivalently,self-dualBooleanfunctions).ThegenerationofallNDcoteriesinacertainsubclassofvote-assignableNDcoterieswasdiscussedin[28],whichisusedtogivealowerboundonthenumberofallvote-assignableNDcoteries.However,theprocedureisnotpolynomialandcomputesapropersubclassofvote-assignableNDcoteries.Garcia-MolinaandBarbara[18]proposedanalgorithmtogenerateallNDcoteriesinacertainsuperclassofregularNDcoteries.However,itisalsonotpolynomial.BiochandIbaraki[9]latercameupwithapolynomial-timealgorithmtogenerateallNDcoteries,andcompiledalistcontainingallNDcoteriesofuptosevenelements,whichareessentiallydierent 384KAZUHISAMAKINOANDTIKOKAMEDA(i.e.,nonequivalentunderpermutation).Weremarkherethattheiralgorithmisnotpolynomialifequivalentduplicatesaretobedeletedfromtheoutput.Infact,theycompiledalistofallNDcoteriesundersevenorfewerelementsbyrstrunningtheiralgorithmandthenselectingnonequivalentrepresentativesfromamongthem.Inthispaper,wepresentapolynomialalgorithmtogenerateallregularNDcoteries.SincenoregularNDcoterieCisequivalenttoanyotherregularNDcoterieC0(6=C)underpermutation(seeLemma2.2),ouralgorithmdoesnotoutputNDcoterieswhichareequivalentunderpermutation.AlthoughouralgorithmoutputsonlyregularNDcoteries,itispracticallyusefulbecauseallNDcoteriesundern=5orfewerelementsareallregular(ifweconsidertheirrepresentatives),andwhennisrelativelysmall,alargefractionofNDcoteriesareregular[28].Moreover,iftheobjectivefunctionof(ii)citedaboveisg-regular(e.g.,theavailabilityofacoterie),thenwecanrestrictourattentiontoregularcoteries.Afterdeningnecessaryterminologyinsection2(weuseBooleanterminology,whichissimplerthanthatofsettheory),wediscussinsection3twooperations,namedand,whichtransformthepositiveself-dualfunctionf(representingaNDcoterie)intoanotherpositiveself-dualfunction(representinganotherNDcoterie)bymakingaminimalchangeinthesetofminimaltruevectorsoff.Theoperationwasintroducedin[9],andwasimplicitlyintroducedin[18],whereitiscalledcoterietransformation.Section4showsthatanyregularself-dualfunctionf(representingaregularNDcoterie)canbetransformedintoanyotherregularself-dualfunctiong(represent-inganyotherregularNDcoterie)byjudiciouslyapplyingtheoperationtofatmostjminT(f)j+jminT(g)j2times.(ForthedenitionofminT(f),seesection2.)Insections5and6,weconsidertheproblemsofgeneratingallregularself-dualfunctionsandofcomputinganoptimalself-dualfunctionwithrespecttoag-regularfunctional(seethedenitionofg-regularityinsection6)asapplicationsoftheabovetransformation.Inadditiontothetheoryofcoteries,theconceptsofself-dualityandregularityplayimportantrolesindiverseareassuchascomputationallearningtheory(e.g.,iden-ticationofpositiveBooleanfunctions[8,10,24,25]),thresholdlogic[28],operationsresearch[6,11,31,32],cluttersinsettheory[7],minimaltransversalsinhypergraphs[16],andcoherentsystemsofreliabilitytheory[35].Theresultsofthispaperarerelevanttoalltheseproblems.2.Denitionsandbasicproperties.ABooleanfunction,orafunctioninshort,ofnvariablesisamappingf:f0;1gn!f0;1g,wherev2f0;1gniscalledaBooleanvector(avectorinshort).Iff(v)=1(resp.,0),thenviscalledatrue(resp.,false)vectoroff.Thesetofalltruevectors(resp.,falsevectors)offisdenotedbyT(f)(resp.,F(f)).Throughoutthispaper,theconstantfunctionswithT(f)=;(empty)andF(g)=;aredenotedbyf=?andg=�,respectively.Foranytwofunctionsfandg,wewritefgifT(f)T(g).Foravectorv=(v1;v2;:::;vn),wedeneON(v)=fjjvj=1gandOFF(v)=fjjvj=0g.Theargumentxoffunctionfisrepresentedasavectorx=(x1;x2;:::;xn),whereeachxiisaBooleanvariable.Avariablexiissaidtoberelevantifthereexisttwovectorsvandwsuchthatf(v)6=f(w),vi6=wi,andvj=wjforallj6=i;otherwise,itissaidtobeirrelevant.ThesetofallrelevantvariablesofafunctionfisdenotedbyVfV=fx1;x2;:::;xng.Aliteraliseitheravariablexioritscomplementxi,whicharereferredtoasapositiveornegativeliteral,respectively.Thecomplementofvectorx=(x1;x2;:::;xn)isdenedbyx=(x1;x2;:::;xn).A TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES385termtisaconjunction(Vi2P(t)xi)^(Vj2N(t)xj)ofliteralssuchthatP(t);N(t)f1;2;:::;ngandP(t)\N(t)=;.Forexample,t1=x1x4x5x6isaterm,whilet2=x2x4x2isnot.Inparticular,thetermtwithP(t)=N(t)=;represents�.Adisjunctivenormalform(DNF)isadisjunctionofdistinctterms.ItiseasytoseethatanyfunctionfcanberepresentedinDNFwhosevariablesetisVf.Wesometimesdonotdistinguishaformula(e.g.,DNF)fromthefunctionitrepresentsifnoconfusionarises.2.1.Positivefunctions.Forapairofvectorsv;w2f0;1gn,wewritevwifvjwjholdsforallj2V,andvwifvwandv6=w,wherewedene01.ForasetofvectorsSf0;1gn,minS(resp.,maxS)denotesthesetofallminimal(resp.,maximal)vectorsinSwithrespectto.Forexample,forafunctionf,minT(f)denotesthesetofallminimaltruevectorsoff,andmaxF(f)denotesthesetofallmaximalfalsevectorsoff.WesometimesuseminS(resp.,maxS)insteadofminS(resp.,maxS)ifnoconfusionarises.Afunctionfissaidtobepositiveormonotoneifvwalwaysimpliesf(v)f(w).Aprimeimplicantofafunctionfisaterm(i.e.,monomial)tsuchthattf,butt06fforanypropersubtermt0oft.Thereisaone-to-onecorrespondencebetweenminT(f)andthesetofallprimeimplicantsoffsuchthatavectorvcorrespondstothetermtvdenedbytv=xi1xi2


xikifvij=1;j=1;2;:::;k,andvi=0otherwise.Forexample,thevectorv=(1010)correspondstothetermtv=x1x3.Inparticular,ifv=(00


0),thentv=�.Notethattvtw(asfunctions)holdsifandonlyifvw.Wealsousethenotationtvtodenotethetermxj1xj2


xjl,wherefj1;j2;:::;jlg=f1;2;:::;ngnfi1;i2;:::;ikg.Fortheabovev=(1010),wehavetv=x2x4.ItisknownthatapositivefunctionfisuniquelydeterminedbyminT(f)(henceapositivefunctionfcanberepresentedbyastringoflengthnjminT(f)j)andthatfhastheuniqueminimaldisjunctivenormalform(MDNF),consistingofalltheprimeimplicantsoff,whereN(t)=;foreachprimeimplicantt.Inthispaper,wesometimesrepresenttheMDNFofapositivefunctionsuchasf=x1x2+x2x3+x3x1inasimpliedformf=12+23+31,usingonlythesubscriptsoftheliterals.ThesetofminimaltruevectorsofthisfunctionisminT(f)=f(110);(011);(101)giffisa3-variablefunction.CoteriescanbeconvenientlymodeledbyBooleanfunc-tionsbasedonthefactthatminT(f)canrepresentafamilyofsubsets,noneofwhichincludestheother.Forexample,theaboveminT(f)representsacoterieC=ff1;2g;f2;3g;f3;1gg,whileT(f)representsthefamilyofallsubsetsthatcontainamemberofC.2.2.Dual-comparablefunctions.Thedualofafunctionf,denotedfd,isdenedbyfd(x)=f(x);wherefandxdenotethecomplementoffandx,respectively.Asiswellknown,fdisobtainedfromfbyinterchanging+(OR)and
(AND),aswellastheconstants0and1.Recallthatforanytwofunctionsfandg,wewritefgifT(f)T(g),andfgiffgandf6=g.Wesaythatfiscoveredbygiffg.Itiseasytoseethat(f+g)d=fdgd;(fg)d=fd+gd,fgifandonlyiffdgd,andsoon.Afunctioniscalleddual-minorifffd,dual-majorifffd,andself-dualiff=fd.Itisknown[20]that1.fisdual-minorifandonlyifatmostoneofvandvbelongstoT(f)foranyv2f0;1gn; 386KAZUHISAMAKINOANDTIKOKAMEDA2.fisdual-majorifandonlyifatleastoneofvandvbelongstoT(f)foranyv2f0;1gn;and3.fisself-dualifandonlyifexactlyoneofvandvbelongstoT(f)foranyv2f0;1gn.Forexample,f=123isdual-minorsincefd=1+2+3satisesffd.Thedualoff=12+23+31isfd=(1+2)(2+3)(3+1)=12+23+31:Thisfunctionfisself-dualandiscalledthebasicmajorityfunction;itisknowntobetheonlypositiveself-dualfunctionofthreerelevantvariables.Thereisnopositiveself-dualfunctionofexactlytworelevantvariables.However,eachfunctionf=xiisapositiveself-dualfunctionofonevariable.Iffispositive,thenfdisalsopositive.Inthiscase,analternativedenitionoffdisgivenbytheconditionthatv2T(fd)ifandonlyifvisatransversalofminT(f);i.e.,itsatisesON(v)\ON(w)6=;forallw2minT(f).LetCSD(n):theclassofallpositiveself-dualfunctionsofnvariables,CDMA(n):theclassofallpositivedual-majorfunctionsofnvariables,CDMI(n):theclassofallpositivedual-minorfunctionsofnvariables.Notethatinthesedenitionsfunctionsmayhavesomeirrelevantvariables.2.3.Regular,2-monotonic,andthresholdfunctions.Apositivefunctionfissaidtoberegularif,foreveryv2f0;1gnandeverypair(i;j)withij,vi=0andvj=1,thefollowingconditionholds:f(v)f(v+e(i)e(j));(2)e(k)denotestheunitvectorwhichhasa1initskthpositionand0inallotherpositions.Inordertodeneanimportantpartialorderonf0;1gn,werstdenetheconceptoftheproleofavectorv2f0;1gnasfollows:profv(k)=Xjkvj;wherek=1;2;:::;n.Ifv;w2f0;1gn,wherev6=w,satisfyprofv(k)profw(k)forallk,thenwewritevw(orwv),andwesaythatwmajorizesv.Ifvworv=w,thenwewritevw(orwv).ItishelpfultovisualizetheproleasinFigure2.1.14567832kProfileFig.2.1.Theprolesprofv(k)(solidlines)andprofw(k)(dashedlines). TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES387InFigure2.1theprolesofv=(00101001)andw=(01100001)aredepictedinthesolidstaircaseanddashedstaircase,respectively.Thedashedstaircaseisnotvisiblewhereitoverlapswiththesolidstaircase.Itiseasytoseethatifvismajorizedbyw,thentheproleofwdoesnotgobelowtheproleofv.Itisclearfromtheabovedenitionthatvwifandonlyifvw,sinceprofv(k)=kprofv(k).Notethatvwimpliesvw,buttheconverseisnotalwaystrue.Afunctionfissaidtobeprole-monotoneifvwimpliesf(v)f(w).Thefollowinglemmaisprovedin[28].Lemma2.1(seeMuroga[28]).Afunctionfisregularifandonlyiffisprole-monotone.Fortwofunctionsfandg,wesaythatfisequivalenttogunderpermutationifpermutingvariablesoffproducesg.Lemma2.2.Twodierentregularfunctionsarenotequivalentunderpermuta-tion.Proof.Letfandgberegularfunctionssuchthatgcanbeobtainedfromfbyapermutation,whereweregardasthepermutationonindices;i.e.,wewrite(i)=jinsteadof(xi)=xj.Weclaimthatf=g,whichprovesthelemma.Letiandjbeindicessatisfyingijand(i)�(j).Notethatifthereexistnosuchindices,thenistheidentitypermutation,implyingf=g.Bytheregularityoff,wehavef(v)f(v+e(i)e(j))(3)everyv2f0;1gnwithvi=0andvj=1,andbytheregularityofg,wehaveg(w)g(w+e(j)e(i))foreveryw2f0;1gnwithw(j)=0andw(i)=1,thatis,f(v)f(v+e(i)e(j))(4)everyv2f0;1gnwithvi=0andvj=1.Bycombining(3)and(4),f(v)=f(v+e(i)e(j))holdsforeveryv2f0;1gnwithvi=0andvj=1.Thismeansthatfissymmetricinvariablesxiandxj.Namely,thefunctionf0obtainedfromfbythepermutation0(k)=8iifk=j;jifk=i;kotherwiseisidenticaltof(i.e.,f0=f).Sincecanbeobtainedbyaconcatenationofsuchpermutations0,itfollowsbyinductionthatf=g.ForasetofvectorsSf0;1gn,minS(resp.,maxS)denotesthesetofallminimal(resp.,maximal)vectorsinSwithrespectto.ForanysetofvectorsSf0;1gn,wehaveminSminS(=minS)andmaxSmaxS(=maxS),sincevwimpliesvw.Inparticular,wehaveminT(f)minT(f);i.e.,anyelementofminT(f)nminT(f)majorizesanelementofminT(f).ItfollowsfromLemma2.1thataregularfunctionfisuniquelydeterminedbyminT(f).Apositivefunctionfiscalled2-monotonicifthereexistsalinearorderingonVforwhichfisregular.The2-monotonicityandrelatedconceptshavebeenstud-iedinvariouscontextsineldssuchasthresholdlogic[6,12,28,32],gametheory 388KAZUHISAMAKINOANDTIKOKAMEDA[35],hypergraphtheory[11],andlearningtheory[10,24,25].The2-monotonicitywasoriginallyintroducedinconjunctionwiththresholdfunctions(e.g.,[28]),whereapositivefunctionfisathresholdfunctionifthereexistnnonnegativerealnumbers(weights)w1;w2;:::;wnandanonnegativerealnumber(threshold)tsuchthatf(x)=1ifPwixit;0ifPwixit:Asthisfsatises(2)bypermutingvariablessothatwi�wjimpliesij,athresholdfunctionisalways2-monotonic,althoughtheconverseisnottrue[28].Itisknown[18]thatthereare\n(22cn)self-dualfunctions,wherecdenotessomepositiveconstant,butonlyO(2n2)thresholdself-dualfunctions.Itisnotknownif2-monotonicself-dualfunctionsaresubstantiallymorethanthresholdself-dualfunc-tions.3.Theoperationsand.Letfbeapositivefunctionofnvariables.Throughoutthispaper,weassumethatfisnontrivialinthesensethatf6=?;�andn1.Givenavectorv2minT(f),theoperationvappliedtofremovesvfromT(f)andthenaddsvtoT(f)[9].Moreprecisely,whileaddingv,allthevectorslargerthanvarealsoaddedtoT(f).Therefore,T(v(f))=(T(f)nfvg)[T(v);(5)T(v)=fw2f0;1gnjwvg:Anequivalentdenitionisv(f)=fnv+tv+tvtdv;(6)fnvdenotesthefunctiondenedbyalltheprimeimplicantsoffexcepttv,andtdvdenotesthedualoftv.Wenotethatiftv=xi1xi2


xikandtv=xj1xj2


xjl,thentvtdv=xi1xi2


xik(xj1+xj2+


+xjl)representsallthevectorslargerthanv.AsseeninExample3.1,theexpression(6)isnotnecessarilyinMDNF,eveniffnvisrepresentedbyitsMDNF,becausesomeoftheprimeimplicantsintv+tvtdvmaycoverormaybecoveredbysomeprimeimplicantsinfnv.Letusnotethattheoperationisessentiallythesameasthecoterietransfor-mation(CT)in[18]exceptthatCTassumesthefollowingadditionalconditions:(i)jOFF(v)j2,and(ii)thereisatleastoneprimeimplicantintvtdvwhichisnotcoveredbyfnv.Inthissense,CTisaspecialcaseoftheoperation.Givenavectorv2minT(f)andavariablesetIwithVfIV,wedenetheoperation(v;I)by(v;I)(f)=fnv+tv[I]+tv[I]tdv[I];(7)v[I]denotetheprojectionofvonI;e.g.,ifv=(1100),I1=fx1;x2;x3g,andI2=fx2;x3g,thenv[I1]=(110)andv[I2]=(10).Bydenition,wehave(v;V)=v.Thisoperation(v;I)isimplicitlyusedin[18]. TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES389Example3.1.Considerapositivefunctionofn=7variables,f=12+13+145+234+235:Forthisfunction,wehaveVf=f1;2;3;4;5g.3Forv=(1100000)andw=(0111000),weshowbelowhowoperationsandareapplied.v(f)=13+145+234+235+34567+12(3+4+5+6+7)=124+125+126+127+13+145+234+235+34567;w(f)=12+13+145+235+1567+234(1+5+6+7)=12+13+1567+2346+2347+235;(v;Vf)(f)=13+145+234+235+345+12(3+4+5)=124+125+13+145+234+235+345;(w;Vf)(f)=12+13+145+235+15+234(1+5)=12+13+15+235:LetfbeafunctiononthevariablesetV=f1;2;:::;ng.ForavariablesetIV,theprojectionoffonI,denotedbyProjI(f),isthefunctiononIobtainedfromfbyxingxi=0forallxi2VnI,i.e.,ProjI(f)(x1;x2;:::;xjIj)=f(x1;x2;:::;xjIj;0;0;:::;0)ifI=fx1;x2;:::;xjIjg.ForavariablesetJV,theexpansionofftoJ,denotedbyExpJ(f),isthefunctiononJobtainedfromfbyaddingirrelevantvariablesxi2JnV.Bydenition,fanditsexpansioncanberepresentedbythesameDNF.ForIVf,wehave(v;I)(f)=ExpV(v[I](ProjI(f))):(8)ushaspropertiessimilartothoseof.See,forexample,Theorem3.2below.Now,foraspeciedclassC(n)ofpositivefunctionsofnvariables,wesaythat(resp.,)preservesC(n)ifv(f)2C(n)holdsforallf2C(n)andv2minT(f)(resp.,(v;I)(f)2C(n)holdsforallf2C(n),v2minT(f),andIVf).Theorem3.2.TheoperationsanddenedabovepreservetheclassesCSD(n),CDMA(n),andCDMI(n).Proof.Thistheoremisprovedforin[9].Consideranyfunctionf2CSD(n)andanysetIsatisfyingVfIV.Iff=fd,thenclearlyProjI(f)=ProjI(fd).WethushaveProjI(f)2CSD(jIj),andhencev(ProjI(f))2CSD(jIj)bytheabove-citedresultin[9].Itisclearthat,foranyg2CSD(jIj),wehaveExpV(g)2CSD(n).Thusby(5),(v;I)preservesCSD(n),similarlyforCDMA(n)andCDMI(n).Notethatiffisself-dual,thenv(f),v2minT(f),isspeciedsimplybyT(v(f))=(T(f)nfvg)[fvg;(9)byinterchangingvwithvinT(f).Thisfollowsfrom(5)andthefactthatv(f)2CSD(n),hencejT(v(f)j=jT(f)j=2n1.Toseetheeectof(v;I)onT(f),whereVfIV,denev[I]=fu2f0;1gnju[I]=v[I]g:3Wesometimesrepresentavariablesetasanindexset;e.g.,fx1;x2gisrepresentedasf1;2g. 390KAZUHISAMAKINOANDTIKOKAMEDAItiseasytoseethatT((v;I)(f))=(T(f)nv[I])[v[I]:(10)oseethedierencebetween(9)and(10),refertoExample3.1,whereI=Vf.Nowconsiderasequenceoftransformationsfromapositiveself-dualfunctionftoanotherpositiveself-dualfunctiong,f0(=f)!f1!f2!:::!fm1(=g);g0(=f)!g1!g2!:::!gm2(=g);wherefi+1=v(i)(fi),v(i)2minT(fi),gi+1=(w(i);Ii)(gi),w(i)2minT(gi),andIiVgi.Wecanseethatm1;m2jminT(f)nminT(g)jandm1jT(f)nT(g)j.Thelatterimpliesthatm1mightbeexponentialinnandjminT(f)j,whilem2mightbesmall.Inthenextsection,weconsidertheandoperationsforregularself-dualfunctions,andgiveatransformationalgorithmbetweenanytworegularself-dualfunctionsfandgofnvariables,whichsatisfym2jminT(f)j+jminT(g)j2:4.Transformationofregularself-dualfunctions.Thegoalofthissectionistopresentanecientalgorithm,TRANS-REG-SD,whichtransformsagivenregularself-dualfunctionftotheone-variableregularself-dualfunctiong=x1.Itappliesasequenceofoperationstof,generatingasequenceofregularself-dualfunctionsintheprocess.Aswewillshow,thisalgorithmcanbeusedtotransformagivenregularself-dualfunctionofnvariablestoanyotherregularself-dualfunctionofnvariables,someofwhichmaybeirrelevant.Weneedtoproveanumberoflemmastoachievethisgoal.Westartwiththefollowinglemma,whichshowsthatvpreservesprole-monotonicity(i.e.,regularity)ifvsatisesacertaincondition.(Wehavealreadyseenthatvpreservesself-duality.)4Recallthatv(f)isspeciedby(9),andthereforeintheproofweconcentrateonthevectorsvandv.Lemma4.1.Letfbearegularself-dualfunction,andletv2minT(f).v(f)isregularifandonlyifv2minT(f)andv6v.Proof.Bydenition,v(f)isregular(i.e.,prole-monotone)ifv(f)(u)v(f)(w)foranyuw:(11)usrstconsidertheonly-ifpart.Recallthatv(f)canbespeciedby(9).Thuswehavev(f)(v)=0andv(f)(v)=1,which,togetherwith(11),impliesv6v.Moreover,sincev2minT(f),ifv62minT(f),thenthereexistsavectoru2minT(f)majorizedbyv,i.e.,uv.(SeetheparagraphafterLemma2.1.)Nowwehavev(f)(v)=0andv(f)(u)=1,whichcontradicts(11)withw=v.Thusv2minT(f)holds.Wenowturntotheproofoftheifpartandshowthatv(f)isprole-monotone.Equation(11)clearlyholdsifu;w=2fv;vg,sincefisprole-monotone.(See(9).)Ifu=vin(11),thentheleft-handsidebecomesv(f)(v)=0by(9),and(11)holdsforanyw.Similarly,ifw=vin(11),thentheright-handsidebecomesv(f)(v)=1by(9),and(11)holdsforanyu.Nowassumethatu=v(w),inwhichcase4Aswecommentedbefore,theoperationisaspecialcaseoftheoperation. TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES391w6=vbytheconditioninthelemma.Thenwehavevw,andv2minT(f)impliesf(w)=0,hencef(w)=v(f)(w)=1.Thus(11)holds.Finally,assumethatw=v(u),inwhichcaseu6=vbytheconditioninthelemma.v2minT(f)impliesf(u)=0,hencev(f)(u)=0.Thus(11)againholds.Thefollowinglemmashowshowtochoosevtobeusedinv(f)toguaranteethatv(f)isregular.Lemma4.2.Letfbearegularself-dualfunctionofn(2)variables.Ifv2minT(f)andvn=1,thenv(f)isregular.Proof.ByLemma4.1wehaveonlytoshowv6v.Wehavef(ve(n))=0fromv2minT(f).Thus,theself-dualityoffimpliesf(v+e(n))=1,which,togetherwithv2minT(f),inturnimpliesv+e(n)6v.Itfollowsfromv+e(n)6vandvn=1thatv6v.Interestingly,theexistenceofavectorvsatisfyingtheconditioninLemma4.2isequivalenttherelevanceofxntof,asprovedinthefollowinglemma.Lemma4.3.Foraregularfunctionf,xnisrelevanttofifandonlyifthereexistsavectorv2minT(f)suchthatvn=1.Proof.Ifsuchavectorvexists,thenwehavef(v)=1andf(ve(n))=0(byv2minT(f)).Thusxnisrelevanttof.Conversely,ifxnisrelevant,thenthereexistsavectorw2minT(f)suchthatwn=1,since,otherwise,theMDNFoffdoesnotcontainvariablexn,andhencexnisirrelevant.Theproofiscompleteifweshoww2minT(f).Assumethatw62minT(f).Thenthereexistsu2minT(f)suchthatuw.Notethatw2minT(f)andu2minT(f)implyu6we(n).Otherwise,bytheprole-monotonicityoff,wewouldhavef(we(n))f(u)=1,acontradictiontow2minT(f).Fromu6we(n)anduw,itfollowsthatun=1(possibleonlyifn2),implyingtheonly-ifpart.Lemma4.2dealswiththecasewherexnisrelevanttof.Beforedealingwiththecasewherexnisirrelevanttof,werstprovethefollowingproposition.Proposition4.4.Letfbearegularfunction.Foranyi;j2Vsuchthatij,ifxjisrelevanttof,thensoisxi.Proof.Assumethatxjisrelevant,butxiisnot.Thentheremustbetwovectors,vandw,suchthatf(v)�f(w),wherevk=wkforallk(1kn)exceptk=j,vj=1,andwj=0.Nowdenetwovectors,v0=(v1;:::;vi1;0;vi+1;:::;vn)andw0=(w1;:::;wi1;1;wi+1;:::;wn).Wethushavev0w0.Sincexiisirrelevant,weshouldhavef(v0)=f(v)�f(w)=f(w0);acontradictiontotheprole-monotonicity(regularity)off.TheabovepropositionimpliesthatxiisrelevanttofifandonlyifVff1;2;:::;ig;inparticular,xnisrelevanttofifandonlyifVf=f1;2;:::;ng=V.Corollary4.5generalizesLemma4.2tothecasewherexnmaybeirrelevanttof.Corollary4.5.Letfbearegularself-dualfunctionsuchthatjVfj=i(2).Ifv2minT(f)andvi=1,then(v;Vf)(f)isregular.Proof.LetI=Vfin(8).Thenv(ProjVf(f))isaregularfunctiononVfbyLemma4.2.Thiscompletestheproof,sinceExpV()preservesregularity.WenowhavethetheoreticalfoundationforTRANS-REG-SD.ByLemma4.2andCorollary4.5,ifxnisrelevanttoagivenf,wecanusetransformationv(f),withsomev,togenerateanewregularself-dualfunctionandrepeatthisprocedureaslong 392KAZUHISAMAKINOANDTIKOKAMEDAasxnisrelevant.Oncexnbecomesirrelevanttothenewlygeneratedfunction,f0,weusetheoperationwithrespecttoVf0,andsoforth.Whatremainsisthediscussionofdataweneedtokeeptrackofinimplementingasequenceoftransformations.ItwillbeusedlaterincomputingthecomplexityofTRANS-REG-SD.Torepresentthesequenceofregularself-dualfunctionsff0gthatTRANS-REG-SDgenerates,werepresenteachsuchfunctionf0intermsofminT(f0)andminT(f0)(seeLemma4.8).ThefollowingpropositionandcorollarywillprepareusforLemma4.8.Foravectorv,letusintroducethefollowingnotation:T(v)=fwjwvgandT(v)=fwjwvg:Proposition4.6.u2T(v)isaminimalmemberwithrespecttoinT(v)ifandonlyifprofu(i)=profv(i)+1forsomei;1in,andprofu(k)=profv(k)forallk6=i.Proof.Forsimplicity,wepresentaninformal\pictureproof"usingFigure4.1.InFigure4.1(a)and(b),itisclearthatthevectorvwhoseproleisrepresentedbythesolidstaircaseismajorizedbythevectoruwhoseproleisrepresentedbythedashedstaircaseandthatuisaminimalvectorwithrespecttoinT(v).ThedashedstaircaseinFigure4.1(c)showsanonminimalvectorwwithprofw(n)=jON(w)j=profv(n)=jON(v)j.wisnonminimal,sinceitmajorizesanothervectoru2T(v),whoseprolesatisesprofu(3)=1andprofu(4)=2.Similarly,itiseasytoseethatanymemberofT(v)violatingtheconditionsofthispropositionmajorizesanothermemberofT(v).14567832k14567832k14567832k(a)(b)(c)Fig.4.1.A\pictureproof"ofProposition4.6.Formula(12)inthefollowingcorollaryfollowsimmediatelyfromtheabovepropo-sition.Formula(13)isdualto(12).Corollary4.7.minT(v)=8fv+e(j)e(j+1)jvj=0;vj+1=1;1jn1g[fv+e(n)gifvn=0;fv+e(j)e(j+1)jvj=0;vj+1=1;1jn1gifvn=1:(12) TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES393maxT(v)=8fve(j)+e(j+1)jvj=1;vj+1=0;1jn1g[fve(n)gifvn=1;fve(j)+e(j+1)jvj=1;vj+1=0;1jn1gifvn=0:(13)WenowshowtheeectofoperationvonminT(f)andminT(f).Lemma4.8.Letfbearegularself-dualfunctionofn(2)variables,andletv2minT(f)withvn=1.ThenwehaveminT(v(f))=minT(f)n(fvg[fv+e(j)jmaxOFF(v)jng)[fvg;(14)minT(v(f))=minT(f)n(fvg[minT(v))[fvg(15)[fu2minT(v)ju6zforallz2(minT(f)nfvg)[fvgg:Proof.By(9),weneedtoconsideronlythein\ruenceof(i)addingvtoT(f)and(ii)removingvfromT(f).Tofollowthisproof,itwillbehelpfultohavethefollowingexample:Forafunctionf=12+13+145+234+235ofvevariables,ifv=(10011),thenv=(01100),maxOFF(v)=2,v(f)=12+13+234+235+23+145(2+3)=12+13+23,minT(f)=f(10100);(10011);(01101)g,andminT(v(f))=f(01100)g.(14)(i).Bytheself-dualityoff,wehavev2maxF(f),hencev2minT(v(f)).(See,e.g.,theprimeimplicant23ofv(f)inExample3.1.)Letusnextconsidervectorsoftheformv+e(j),j2ON(v)(=OFF(v)),whichmayceasetobeaminimalmemberasaresultofoperation(i).NotethatthesevectorsbelongtoT(f),sincev2maxF(f).Weclaimthatv+e(j)2minT(f)ifandonlyifj�maxOFF(v).(Seethentheprimeimplicants234and235offinExample3.1.)Ifj�maxOFF(v),v+e(j)2minT(f)holds,sinceotherwisethereexistsavectorw2minT(f)suchthatwv+e(j).Sincew6=vandwv+e(j),wv+e(j)e(i)holdsforsomei2OFF(v)(=ON(v)).Sinceij,wehavewv.Sincef(v)=0bytheself-dualityoff,itfollowsfromLemma2.1thatf(w)=0holds,acontradictiontow2minT(f).Thisprovestheifpartofourclaim.Letj2ON(v)(=OFF(v))withjmaxOFF(v)andconsiderthevectorw=v+e(j)e(maxOFF(v)).Sincej2ON(v),wehavejmaxOFF(v).Thuswv,andhencevwholds.Iff(w)=0,thenf(w)=1holdsbytheself-dualityoff.Sincevw,wehavev62minT(f),acontradiction.Thusf(w)=1,implyingthatv+e(j)62minT(f)holdsinthiscase.(14)(ii).Sincev62minT(v(f)),theremaybeaj2OFF(v)suchthatu=v+e(j)belongstominT(v(f)).Sincef(v)=1,f(v+e(j)e(n))=1,i.e.,v+e(j)e(n)2T(f),followsfromtheregularityoff.Thusnovectoroftheformv+e(j)belongstominT(v(f)),sincethereobviouslyexistsaw2minT(f)satisfyingwv+e(j)e(n)v+e(j),acontradiction.(15)(i).Itiseasytoseethatv2minT(v(f))holds,sinceotherwisef(v)=0andthereexistsavectorwsuchthatwvandf(w)=1,acontradictiontotheregularityoff.Nowconsideranyvectorw2minT(f)majorizingv,i.e.,satisfyingwv.SuchawmustberemovedfromminT(f)toconstructminT(v(f)).Clearly,eachsuchwiscontainedinminT(v)(e.g.,w=(01110)2minT(f)correspondingtotheprimeimplicant234offinExample3.1).(15)(ii).Asnotedin(14)(ii),wehavev62minT(v(f)),afortiori,v62minT(v(f)).LetusconsiderthevectorsinminT(v),givenby(12),since,besidesv,onlytheymaybecontainedinminT(v(f))nminT(f).Notethatavectoru2minT(v)belongstominT(v(f)),providedthereisnovectorz2minT(f)nfvg[fvgsuchthatzu.ThisisbecauseminT(f)n(fvg[minT(v))[fvgminT(v(f))(see(15)(i)above)andallvectorsinminT(v)majorizingv. 394KAZUHISAMAKINOANDTIKOKAMEDAFromtheproofofLemma4.8(case(14)(i)),wecanseethatv+e(n)2minT(f).Sincevn=1impliesn�maxOFF(v),fv+e(j)jmaxOFF(v)jngisnonempty,and(14)impliesthefollowinglemma.Lemma4.9.Letfbearegularself-dualfunctionofn(2)variables,andletv2minT(f)withvn=1.ThenjminT(v(f))jjminT(f)j1;(16)minT(v(f))xn=1[fv;v+e(n)g=minT(f)xn=1;(17)eSxn=1denotesthesetfv2Sjvn=1g.Wearenowreadytodescribethetransformationalgorithm,TRANS-REG-SD.Ifwerepeatedlyapplythevoperation(withdierentv's,ofcourse)toaregularself-dualfunctionf,untilthereisnovectorv2minT(f)withvn=1,thenbyLemmas4.2,4.3,and4.9wehavearegularself-dualfunctionf0towhichxnisirrelevant.This,togetherwith(17),impliesthatjminT(f)njiseven.Notethatf0isnotunique;i.e.,itdependsonthesequenceofvectorsv2minT(f)withvn=1thatareusedinv.Forexample,considerafunctionf=12+13+145+234+235ofvevariables.Forvectorsv=(10011)andw=(01101),v(f)=12+13+23andw(f)=12+13+14+234,respectively.NowVf0=f1;2;:::;j1gholdsforsomej1n1.Ifj1=1,wehavef0=x1andwearedone.Ifj16=1,ontheotherhand,weapply(v;Vf0)operationstof0insteadof(v;Vf)(=v)untilthereisnovectorv2minT(f0)withvj1=1.Sinceallthelemmaspresentedinthissectionarestillvalidfor(v;Vf0)andvj1=1inplaceof(v;Vf)(=v)andvn=1,weobtainaregularself-dualfunctionf00whoserelevantvariablesetisVf00=f1;2;:::;j2gwithj2j1.Byrepeatingthisargument,wereachtheone-variableregularself-dualfunctionx1.Formally,thissequenceoftransformationscanbestatedasfollows.AlgorithmTRANS-REG-SDInput:minT(f),wherefisaregularself-dualfunction.Output:Regularself-dualfunctionsf0(=f),f1,f2;:::;fm(=x1).Step0:Leti=0andf=f0.Step1:Outputfi.Iffi=x1,thenhalt.Step2:fi+1=(v(i);Vfi)(fi),wherev(i)2minT(fi)andv(i)maxVfi=1.i:=i+1.ReturntoStep1.By(16),thenumbermintheoutputfromalgorithmTRANS-REG-SDsatisesmjminT(f)j1.Sinceeveryself-dualfunctionfsatisesv(v(f))=f(see(9)),wecantransformx1intoanyregularself-dualfunctiongbyrepeatedlyapplyingtheoperationtox1atmostjminT(g)j1times.Thuswehavethefollowingtheorem.Theorem4.10.Letfandgbeanytworegularself-dualfunctions.ThenfcanbetransformedintogbyrepeatedlyapplyingoperationstofatmostjminT(f)j+jminT(g)j2times.Inthesubsequentsections,weconsidertheproblemsofgeneratingallregularself-dualfunctionsandofcomputinganoptimumself-dualfunctionwithrespecttoa\g-regular"functional(forthedenitionofg-regularity,seesection6)asapplicationsofalgorithmTRANS-REG-SD.5.Generationofallregularself-dualfunctions.LetCR-SD(n)denotetheclassofallregularself-dualfunctionsofnvariables.WepresentinthissectionanalgorithmtogenerateallfunctionsinCR-SD(n)byapplyingtheoperation.Thealgo- TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES395rithmisincrementallypolynomial[22]inthesensethattheithfunctioni2CR-SD(n)isoutputinpolynomialtimeinnandPi1j=0jminT(j)jfori=1;2;:::;jCR-SDj.Tovisualizethealgorithm,werstdeneanundirectedgraphGn=(CR-SD(n);E),where(g;f)2E,ifthereexistsavectorv2minT(g)suchthat(v;I)(g)=fforsomeIVg.Example5.1.Figure5.1showsthegraphG5.5(Ignorethearrowsonsomeedgesfornow.)e@@@I6eee@@@666f0=1f1=12+13+23f2=12+13+14+234f3=12+13+14+15+2345f4=12+13+145+234+235f5=12+134+135+145+234+235+245f6=123+124+125+134+135+145+234+235+245+345Fig.5.1.ThegraphG5:theprimeimplicantscorrespondingtothevectorsinminT(fi)aresetinboldface.Theorem4.10impliesthatGnisconnected.Moreover,thecondition(g;f)2Eholdsifandonlyif(f;g)2E;i.e.,Gnisindeedundirected,asshownbythefollowingproposition.Foravectorv2f0;1gnandIV,letv[I]0denotethevectorudenedbyON(u)=ON(v)\I;i.e.,u[I]=v[I]andtheremainingcomponentsofu,ifany,areallsetto0's.Proposition5.2.Letf2CR-SD(n)andg=(w;I)(f)forw2minT(f)suchthatIVfandw[I]6w[I].Theng2CR-SD(n)andf=(u;I)(g),whereu=w[I]02minT(g).Proof.g2CR-SD(n)followsfromTheorem3.2andLemma4.1.From(10),wehaveT(g)=(T(f)nw[I])[w[I]:Letf0=(u;I)(g).ThenT(f0)=(T(g)nw[I])[w[I]:WethushaveT(f0)=T(f),hencef0=f.u=w[I]02minT(g)followsfromLemma4.1andthefactthatfisregular.5NotethatinthissectionthesubscriptsofthefunctionsffigarereversedfromthoseusedinTRANS-REG-SD;forexample,f0nowdenotesthefunctionx1. 396KAZUHISAMAKINOANDTIKOKAMEDAExample5.3.Forexample,considerthefunctionf=f2=12+13+14+234inFigure5.1,whereweassumen=5.WehaveminT(f2)=f(10010);(01110)g.Pickw=(10010)2minT(f2),whichsatisesw[I]6w[I]forI=Vf4.Itiseasytoseethatg=(w;I)(f2)=12+13+14(2+3+5)+235+234=12+13+145+234+235=f4.Foru=w=(01101)2minT(g),wehave(u;I)(g)=12+13+234+235(1+4)+14=f2.Notethatw[I]0=w[I]sinceI=V.Fortwodistinctvectorsu=(u1;:::;un)andv=(v1;:::;vn),wesaythatuislexicographicallysmallerthanv,writtenu
v,ifforsomek(1kn)ukvkandui=viforalli(1ik).Thusamongthevectorsinf0;1g3,forexample,wehave(000)
(001)
(010)
(011)
(100)
(101)
(110)
(111).Letf0=x1bethedesignatedfunctioninCR-SD(n)andconsidertheproblemoftransforminganarbitraryfunctiong2CR-SD(n)tof0byrepeatedlyapplyingtheoperationasinalgorithmTRANS-REG-SD.Notethatthetransformationpathfromagivengtof0isnotunique.Thus,tomakethepathunique,wechoosefortheoperationthelexicographicallysmallestvector~v2minT(g)suchthat~vmaxVg=1.Letbesuchanoperation,i.e.,(g)=(~v;Vg)(g):(18)thisway,wedeneadirectedspanningtreeofGn,RTn=(CR-SD(n);ART),suchthat(g;f)isadirectedarcinARTifandonlyif(g)=f.If()isappliedrecursivelytog,weeventuallyreachafunctionhsuchthatVhVg(see(17)).Forexample,inFigure5.1,wehaveVf2Vf4.Thus,RTnisanin-treerootedatf0=x1.InFigure5.1,ARTisindicatedbythethickarcs.OuralgorithmGEN-REG-SDpresentedlaterinthissection,whichgeneratesallregularself-dualfunctionsofnvariablesforagivenn,willtraverseRTnfromf0inadepth-rstmannerinthereversedirectionofthearcsinART,outputtingeachregularself-dualfunctionfwhenitrstvisitsf.Thistypeofenumerationiscalledreversesearchin[2,3]andhasbeenappliedtomanyotherenumerationproblemssuchastheextremepointsofaconvexpolyhedron,thearrangementsofhyperplanes,thetriangulationsofapolygon,matroidbases,andsoon.WhenRTnistraversedfromf0,foreacharc(g;f)2ART,theendnodef(nearertheroot)isvisitedbeforetheendnodeg(fartherfromtheroot).Unfortunately,whenwerstvisitnodefwecannotidentifytheincomingarcs(inART)towardsnodeffromamongtheedgesinE(ofGn).Inotherwords,knowingf,wecannotndgsuchthat(g;f)2ART.Notethat(18)computesfgiveng,nottheotherwayaround.InLemma5.5below,wendthe\inverse"of(18)inthesensethatuinProposition5.2coincideswith~vin(18).ThefollowinglemmaisessentiallyarestatementofLemma4.8inaslightlygen-eralizedform.Lemma5.4.Letf;g2CR-SD(n)satisfyf=(g)=(~v;Vg)(g),where~visthelexicographicallysmallestvectorinminT(g)xmaxVg=1.Withw=(~v)[Vg]0,wehaveminT(f)=minT(g)n(f~vg[fw+e(j)j;(19)max(OFF(~v)\Vg)jmaxVgg)[fwg;minT(g)=(minT(f)nfwg)[f~vg(20)[fw+e(j)jmax(ON(w)\Vg)jmaxVgg;minT(f)=minT(g)n(f~vg[minT(w;Vg))[fwg(21)[fu2minT(~v;Vg)ju6zforallz2(minT(g)nf~vg)[fwgg; TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES397minT(g)=minT(f)n(fwg[minT(~v;Vg))[f~vg(22)[fu2minT(w;Vg)ju6zforallz2(minT(f)nfwg)[f~vgg;whereT(v;I)=fujuv;ON(u)IgforavectorvandanindexsetIV.Proof.TheprooffollowsfromLemma4.8,whichisaspecialcaseofthislemma(Vg=V).Notethatby(20)wehavew+e(maxVg)2minT(g),implying~v
w+e(maxVg),hence~v1=0andw1=1,sincew1=~v1.InLemma5.4,conceptually,~vwasexplicitlychosenrst,andwwasspeciedintermsof~v.Thefollowinglemmawillenableustochoosewexplicitly,sothat~vischosenimplicitly.Notethatcondition(c)inLemma5.5involvesthelexicographicorderinminT(f),whichonecancomputegivenf,while~vin(18)isdenedintermsofthelexicographicallysmallestvectorinminT(g),whichonecancomputegiveng.Notealsothat~visuniqueforagivenregularself-dualfunctiong,butvectorwwhichsatisestheconditionsofLemma5.5isingeneralnotuniqueforagivenregularself-dualfunctionf.Thisre\rectsthefactthatanonrootnodeingraphRTnhasoneparentbutingeneralhasmorethanonechildnode.Lemma5.5.Letf2CR-SD(n)andg=(w;Vg)(f)forw2minT(f)6suchthatw[Vg]6w[Vg]andVgVf.Thenf=(g)(=(~v;Vg)(g))i.e.,(g;f)2ART(thearcsetofRTn),ifandonlyif(a)wmaxVg=0,(b)w1=1,and(c)w[Vg]0islexicographicallysmallerthananyvectorinminT(f)xmaxVg=1.Proof.Letusrstconsidertheonly-ifpart,assumingf=(~v;Vg)(g),where~visthelexicographicallysmallestvectorinminT(g)xmaxVg=1asinLemma5.4.Theng=(w;Vg)(f)impliesthatw=(~v)[Vg]0.Since~vmaxVg=1bydenition,wehavewmaxVg=0,hence(a)holds.(b)wasshownaboveinthecommentimmediatelyafterLemma5.4.Toprove(c),by(21),itsucestoshowthat~v
wandthat~vislexicographicallysmallerthananyvectorinthesetminT(~v;Vg),sinceanyvectoruinthersttermin(21)withumaxVg=1mustbelexicographicallylargerthan~vbydenitionof~v.Theformerfollowsimmediatelyfrom(b)and~v=w[Vg]0,andthelatterisobvious,sinceuvimpliesthatu
vforanyvectorsuandv.Toprovetheifpart,let~v=w[Vg]0.Wewanttoshowthat~vislexicographicallythesmallestinminT(f)xmaxVg=1.By(c)and(22),itsucestoshowthat~vislexicographicallysmallerthananyvectorinminT(w;Vg)(thelasttermin(22)).Thisisobvioussincew1=1and~v1=0.Example5.6.UsingLemma5.5,wecanconstructRTn=(CR-SD(n);ART).RT6=(CR-SD(6);ART)isshowninFigure5.2.Thefunctionnumbersintheguredenotethoseregularself-dualfunctionsofsixvariablesshowninthefollowingtable,whichwasderivedfromtheworkbyBiochandIbaraki[8].(Somefunctionnumbershavebeenchanged.)6Thismeansthat(g;f)2E(theedgesetofGn).gdependsonthechoiceofw. 398KAZUHISAMAKINOANDTIKOKAMEDA#1#3#5 #6 #7#2#4Graph RT for |V| = 6.Vf = {1,2,...,5}Vf = {1,2,...4}Vf = {1,2,3}Vf = {1}#9 #10 #13 #17 #11 #14 #18 #12 #16 #20#21Vf = {1,2,...,6}#15 #19Fig.5.2.RT6=(CR-SD(6);ART).#jVfjFunction1112312+13+23y3412+13+14+234y4512+13+14+15+2345y512+13+145+234+235y612+134+135+145+234+235+245y7123+124+125+134+135+145+234+235+245+345y8612+13+14+15+16+23456y912+13+14+156+2345+2346y1012+13+145+146+156+2345+2346+2356y1112+13+145+146+234+2356y1212+13+234+235+236y+14561312+134+135+136+145+146+156+2345+2346+2356+2456y1412+134+135+136+145+146+234+2356+2456y1512+134+135+136+145+234+235+2456y1612+134+135+136+234+235+236+1456+2456y17123+124+125+126+134+135+136+145+146+156+2345+2346+2356+2456+3456y18123+124+125+126+134+135+136+145+146+234+2356+2456+3456y19123+124+125+126+134+135+136+145+234+235+2456+3456y20123+124+125+126+134+135+136+234+235+236+1456+2456+3456y21123+124+125+126+134+135+145+234+235+245+3456yInthetable,theprimeimplicantscorrespondingtothevectorsinminT(f)areshowninboldface.Lettwdenotetheprimeimplicantcorrespondingtovectorw2minT(f);i.e.,twcontainsthevariablexiasafactorifandonlyifi2ON(w).Inotherwords,i2P(tw)ifandonlyifi2ON(w)andN(tw)=;.(ForthedenitionofP()andN(),seethebeginningofsection2.)indicatesaprimeimplicanttwsuchthatvectorwsatisestheconditionsofLemma5.5.Wheneachfunctioniscalledg,theprimeimplicanttvwithycorrespondstotheuniquevector~v.Ingeneral,eachvectorwwithwmaxVf=1impliesthatfhas(nmaxVf)childrenduetowinRTn,oneeachforjVgj=jVfj+1;:::;n.Similarly,eachvectorwwithwmaxVf=0impliesthatfhas(nmaxVf+1)childrenduetow,oneeachforjVgj=jVfj;jVfj+1;:::;n.Forexample,itisobservedinFigure5.2thattw=14infunction#3givesriseto64=2children,whiletw=15infunction#4givesrise TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES399tojust65=1child.Similarly,tw=145infunction#5givesrisetojustonechild,whiletw=13infunction#5givesriseto65+1=2children,andsoforth.NotethatifVg6=Vf(i.e.,VgVf),Lemma5.5impliesthatf=(g)ifandonlyifw1=1,sinceVgVfandw2minT(f)implyconditions(a)and(c)arevacuousinthiscase.Thus,foranindexsetIVf,anyvectorw2minT(f)thatsatisesw1=1andw[I]6w[I]producesg=(w;I)(f)suchthatf=(g).WenowdiscussthedatastructuresforminT(f)andminT(f).ThesetminT(f)isrepresentedbyabinarytree,denotedbyB(minT(f)),ofheightn,inwhichtheleftedge(resp.,rightedge)fromanodeatdepthj1(therootisatdepth0)representsthecasexj=1(resp.,xj=0).AleafnodetofB(S)atdepthnstoresthevectorv2S(f0;1gn),thecomponentsofwhichcorrespondtotheedgesofthepathfromtheroottot.Inordertohaveacompactrepresentation,theedgeswithnodescendantleavesareremovedfromB(S).Example5.7.Figure5.3showsB(minT(f))forminT(f)=fv(1)=(110000),v(2)=(101000),v(3)=(100110),v(4)=(011100);v(5)=(011010)g(i.e.,f=12+13+145+234+235).Withthisdatastructure,itiseasytoseethatwecanapplyoperationsMEMBER(i.e.,checkifv2S),INSERT(i.e.,updateS:=S[fvg),andDELETE(i.e.,updateS:=Snfvg)allinO(n)time.Moreover,sincetherightmost(resp.,leftmost)pathinB(S)representsthelexicographicallysmallest(resp.,largest)vectorinS,wecanoutputfromB(S)thelexicographicallysmallest/largestvectorinSinO(n)time.LetVi=f1;2;:::;iganddene:f0;1gn!Z+by(v)=min(fijv[Vi]6v[Vi];ON(v)ViVg[fn+1g):(23)denition,ifnoisatisestheconditionontheright-handside,(v)=n+1holds.Wecaneasilyseethatv[Vi]6v[Vi]holdsforalli(v),if(v)n+1.Basedonthis(v)andmaxON(v),wedecomposeminT(f)intomanysubsetsasfollows:minT(f)=Sjin+1j=1;2;:::;nminT(f)(i;j),whereminT(f)(i;j)=fv2minT(f)j(v)=i;maxON(v)=jg:Fortheaboveexamplefunctionf=12+13+145+234+235,wehaveminT(f)=fv(2)=(101000);v(3)=(100110);v(5)=(011010)g.Fromthis,wegetminT(f)(5;3)=fv(2)g,minT(f)(5;5)=fv(3);v(5)g,andminT(f)(i;j)=;,other-wise.OuralgorithmkeepsminT(f)asSjin+1j=1;2;:::;nB(minT(f)(i;j))andSn=1B(minT(f)j),whereminT(f)j=Sn+1i=jminT(f)(i;j).Westartdepth-rstsearchfromtherootf0.NotethatminT(f0)(1;1)=(10:::0)andVf0=f1g.Duringthedepth-rstsearch,whenwevisitnodef,werstsetupI:=Vfastheindexset.Intheorder(i;j)=(1;1),:::;(1;maxI1),(2;1),:::;(2;maxI1),:::;(maxI;1),:::;(maxI;maxI1),wethencheckifthelexicographicallylargestw(i;j)inminT(f)(i;j)satises(b)w(i;j)1=1and(c)thatthevectorv(i;j),denedbyON(v(i;j))=OFF(w(i;j))\I,islexicographicallysmallerthananyvectorinminT(f)maxI(i.e.,minT(f)xmaxI=1).Ifthereexistssuchavectorw(i;j),wemovetog=(w(i;j);I)(f).Sincew(i;j)satisesconditions(a),(b),and(c)ofLemma5.5,wehavef=(g).Moreover,ifw(i;j)doesnotsatisfy(b)(resp.,(c)),thennovectorinu2minT(f)(i;j)satises(b)(resp.,(c)).ThismeansthatwedonothavetocheckthevectorsinminT(f)(i;j)otherthanw(i;j).Thus,ifnow(i;j)(i=1;2;:::;maxI, 400KAZUHISAMAKINOANDTIKOKAMEDAeeeeeeeeeeeeeeeeeeeeeeeeeXXXXXXXXXPPPPPHHHHHHHH@@@@@@LLLLLLLLLLLLLLLLx1=1x1=0x2=1x2=0x2=1x3=0x3=1x3=0x3=1x4=0x4=0x4=1x4=1x4=0x5=0x5=0x5=1x5=0x5=1x6=0x6=0x6=0x6=0x6=0v(1)v(2)v(3)v(4)v(5)Fig.5.3.AbinarytreeB(minT(f)),wheref=12+13+145+234+235(afunctionofsixvariables).=1;2;:::;maxI1)satises(b)and(c),thenwecheckifmaxI=n.Ifso,wereturntoh=(f)fromf(iff=f0=x1,thenhalt);otherwise,weupdateI:=I[fmaxI+1gandagaincheckifw(i;j)satises(b)and(c).Whenthedepth-rstsearchreturnstoffromgbybacktracking(i.e.,f=(g)=(v(i;j);Vg)(g)),wesetI:=Vgandmoveontothevectorw0whichlexicographicallyfollowsw(i;j)inminT(f)(i;j).Ifthisw0satisesconditions(b)and(c)ofLemma5.5,thenwemovetog0=(w0;I)(f);otherwise,checkifw(i;j)satises(b)and(c),accordingtotheordering(i;j)=(i;j+1);:::;(i;maxI1),(i+1;1);:::;(i+1;maxI1),:::;(maxI;1);:::;(maxI;maxI1).ThisprocedurehastheadvantagethatthereisnoneedtomaintainthedataoftheentiresearchtreeRTnbutonlytheinformationaboutthecurrentfunctionissucient.Ouralgorithmcanbestatedformallyasfollows:AlgorithmGEN-REG-SDInput:Apositiveintegern.Output:Allregularself-dualfunctionsofnvariables.Step0:Letf:=f0,I:=Vf,w:=(11


1),andoutputf./*Notethat(11


1)isthespecialvector,indicatingthatnooperationisappliedtof.*/Step1:Ifw=(11


1),thengotoStep2./*Inthiscase,nooperationwasappliedtof.*/ElseifI=Vf,thengotoStep3./*Inthiscase,Vg=Vfholdsforg=(w;I)(f);i.e.,wehaveappliedtheoperationwithI=Vfpreviously.*/Else,gotoStep4./*Inthiscase,VgVfholdsforg=(w;I)(f);i.e.,wehaveappliedtheoperationwithIVfpreviously,andhencewealreadyappliedtheoperationforwithVi,whereVfViI.*/Step2:Trytondthelexicographicallylargestw0=w(i;j)inminT(f)(i;j)satisfy-ingthetwoconditionsofLemma5.5,(b)w(i;j)1=1and(c)thevectorv(i;j),denedbyON(v(i;j))=OFF(w(i;j))\I,islexicographicallysmallerthananyvectorinminT(f)maxI,intheorder(i;j)=(1;1);:::;(1;maxI1),(2;1);:::;(2;maxI1);:::;(maxI;1);:::;(maxI;maxI1)./*Recall TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES401I=Vf.FirstapplywiththeindexsetI=Vf.*/Thenconsiderthefollowingfourcases:(i)Thereisaw0satisfyingtheaboveconditions:Letf:=(w0;I)(f)andw:=(11


1),outputf,andreturntoStep1(downwardmove).(ii)Thereisnosuchw0andmaxIn:7Trytondthelexicographicallylargestw00=w(i;j)inminT(f)(i;j)satisfying(b)w(i;j)1=1,accordingtotheordering(i;j)=(1;1);:::;(1;maxVf),(2;1);:::;(2;maxVf);:::;(n;1);:::;(n;maxVf).Ifthereisnosuchw00,thengotoeithercase(iii)or(iv).Else,letw002minT(f)(i;j).ThenletI:=Vi(=f1;2;:::;ig)ifimaxI+2;otherwise,letI:=I[fmaxI+1g,f:=(w00;I)(f),andw:=(11


1),outputf,andreturntoStep1(downwardmove).(iii)Thereisnosuchw0orw00,andf=f0:Halt(allfunctionshavebeenoutput).(iv)Thereisnosuchw0orw00,andf6=f0:LetvbethelexicographicallysmallestvectorinminT(f)maxVf.Updatef,I,andwbylettingI:=Vf,f:=(f)=(v;Vf)(f),andON(w)=OFF(v)\I,respectively.ReturntoStep1(backtrack).Step3:Letw2minT(f)(i;j).Trytondthevectorw0satisfyingconditions(b)and(c)ofLemma5.5,accordingtotheordering(1)thevectorthatlexicographicallyfollowswinminT(f)(i;j),followedby(2)w(i;j),where(i;j)=(i;j+1);:::;(i;maxI1),(i+1;1);:::;(i+1;maxI1);:::;(maxI;1);:::;(maxI;maxI1)./*TrytoapplywiththecurrentIwithI=Vf.*/ThenconsiderthefourcasesofStep2.Step4:Letw2minT(f)(i;j).Trytondthevectorw0satisfying(b)w01=1,accordingtotheordering(1)thevectorthatlexicographicallyfollowswinminT(f)(i;j),followedby(2)w(i;j),where(i;j)=(i;j+1);:::;(i;maxVf),(i+1;1);:::;(i+1;maxVf);:::;(maxI;1);:::;(maxI;maxVf)./*TrytoapplywiththecurrentIwithIVf.*/ThenconsiderthefourcasesofStep2.Incase(ii),wecheckifthereexistasetI00Iandavectorw002minT(f)suchthatf=(g),whereg=(w00;I00)(f).SinceI00IVf,wejustcheckifw001=1,asnotedinthesecondparagraphafterExample5.6.Accordingtotheorderingsonw0andw00,algorithmGEN-REG-SDtraversesRTndepth-rst;i.e.,eacharcinRTnistraversedonlytwice,downwardandupward.ToanalyzethetimecomplexityofGEN-REG-SD,weneedtwomorelemmas.Lemma5.8.GiventhedatastructuresB(minT(f)),B(minT(f)(i;j))(j=1;2;:::;n,i=j;j+1;:::;n+1)andB(minT(f)j)(j=1;2;:::;n),eachiterationofSteps14inalgorithmGEN-REG-SDcomputeseitherw0orw00(orconcludesthatnosuchvectorexists)inO(n3)time.Proof.Sincewecancheckifagivenvectorusatises(b)and(c)ofLemma5.5inO(n)time,andsincethereare2n2candidatesforeitherw0orw00,eachiterationrequiresO(nn2)=O(n3)time.Lemma5.9.Letf;g2CR-SD(n)satisfyf=(g).Letf=(v;I)(g)andg=(w;I)(f).Giventhedatastructuresforg(i.e.,B(minT(g)),B(minT(g)(i;j))(i=1;2;:::;n+1,j=1;2;:::;n),andB(minT(g)j)(j=1;2;:::;n)),v,andI,7NotethatIVf,andhenceminT(f)j=;holdsforalljmaxI+1. 402KAZUHISAMAKINOANDTIKOKAMEDAwecancomputedatastructuresoffinO(n3)time.Similarly,givendatastructuresoff,w,andI,wecancomputedatastructuresofginO(n3)time.Proof.Weshallproveonlytherstassertionofthelemma,sincethesecondassertioncanbeprovedanalogously.Itdirectlyfollowsfrom(19)inLemma5.4thatwecancomputeB(minT(f))inO(n2)time.Asfortherest,itsucestoshowthat,givenavectoru,themembershipu2minT(f)canbecheckedinO(n2)time,sinceatmostnvectorsaredeletedfromoraddedtominT(g)toconstructminT(f)by(21)and(12).NotethatavectoruiscontainedinminT(f)ifandonlyiff(z)=0holdsforallvectorsz2maxT(u).From(13),wehaveatmostnsuchvectorsz.Moreover,wecancheckifagivenvectorzsatisesf(z)=1inO(n)timeifB(minT(f))isprepared[34].Thiscompletestheproof.ByLemmas5.8and5.9,eachiterationofStep1canbecarriedoutinO(n3)time,exceptfortheoutputtingoffunctionf.Sinceeacharc(g;f)2ARTistraversedtwice,algorithmGEN-REG-SDrequiresO(n3jCR-SD(n)j)time,plusthetimeforoutputtingallfunctionsinCR-SD(n),i.e.,O(nMsum)time,whereMsum=Xf2CR-SD(n)jminT(f)j:AlgorithmGEN-REG-SDclearlyrequiresO(nMmax)space,whereMmax=maxf2CR-SD(n)jminT(f)j:Thuswehavethefollowingtheorem.Theorem5.10.AlgorithmGEN-REG-SDgeneratesallfunctionsinCR-SD(n)andisincrementallypolynomial.ItrequiresO(n3jCR-SD(n)j+nMsum)timeandO(nMmax)space.Corollary5.11.AllfunctionsinCR-SD(n)canbescannedinO(n3jCR-SD(n)j)time.ByLemma2.2,theregularfunctionsareallrepresentativesofequivalenceclassesunderpermutation;i.e.,thereisnoregularfunctionfthatisequivalenttoanotherregularfunctiong(6=f)underpermutation.Therefore,ouralgorithmgeneratesthenonequivalentfunctions.Letusremarkthatthealgorithmsin[9,18]arenotpolyno-mialifwetrytooutputonlynonequivalentfunctions.Itisknownthatthepositiveself-dualfunctionsofupton=5variablesareallthresholdfunctions(andhenceregularifweconsidertherepresentatives),buttherearemanynonregularself-dualfunctionsforn6,evenifweconsidertherepresentatives(seeExample5.6).Moreover,itisknown[28]thatallregularpositiveself-dualfunctionsforn9arethresholdfunctions.6.Optimumself-dualfunctionforregularfunctional.Let'beapseudo-Booleanfunction;i.e.,'isamappingfromf0;1gntothesetofrealnum-bersR.Apseudo-Booleanfunction'issaidtobeg-regularif'(v)'(w)holdsforallpairsofvectorsvandwwithvw.ForaBooleanfunctionf,let(f)=Xv2T(f)'(v);(24)'isapseudo-Booleanfunction.Thefunctionalisalsosaidtobeg-regularif'isg-regular.Asanexampleofag-regularpseudo-Booleanfunctionalofinterest, TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES403wecitetheavailabilityofaBooleanfunction,denedasfollows.Eachelementi2Vhastheoperationalprobabilitypi(0pi1);i.e.,theithelementisoperationalwiththeprobabilitypi,whileitisfailedwiththeprobability1pi.Weassumethateachelementtakesonitsstateindependently.ThentheavailabilityA(f)ofaBooleanfunctionfisdenedbyA(f)=Xv2T(f)0Yi2ON(v)piYi2OFF(v)(1pi)1:(25)weinterpretT(f)asthesetofstatesinwhichthen-elementsystemdenedbytheBooleanfunctionfisworking,thenA(f)representstheprobabilitythatthesystemrepresentedbyfisworking.Theavailabilityhasbeenstudiedextensively[35],especiallyinthecasewherefrepresentsNDcoteries(i.e.,fispositiveself-dual)[1,5,15,33,36,38].Itisknown[1,36]thatanyelementiwithpi1=2canbeignored;i.e.,xiisirrelevantforallpositiveself-dualfunctionsfhavingthemaximumavailability.Theonlyexceptioniswhenalltheelementshavepi1=2.Inthiscase,itisknown[1]thatf=xjhasthemaximumavailability,wherejistheelementsuchthatpjpiforalli.Thus,wecanassumethatpi1=2holdsforalli.Moreover,weassumewithoutlossofgeneralitythatp1p2


pn1=2:Now,let'(v)=Qi2ON(v)piQi2OFF(v)(1pi).Thenwehave(f)=A(f).ItfollowsfromtheassumptionontheorderofprobabilitiesthatA(f)isg-regular.Inthissection,weconsiderthefunctionsfthatmaximizeg-regularfunctionalamongallself-dualfunctions.Lemma6.1.Givenag-regularfunction',letbeag-regularfunctionaldenedby(24).Thenthefollowingstatementsregardingfareequivalent.(i)(f)ismaximumamongallself-dualfunctions.(ii)Allvectorsv2T(f)satisfy'(v)'(v).(iii)Allvectorsv2minT(f)satisfy'(v)'(v).Proof.Letusprove(i)=)(ii)=)(iii)=)(i).Since(ii)=)(iii)isobvious,weshow(i)=)(ii)and(iii)=)(i).(i)=)(ii):Ifthereexistsavectorv2T(f)suchthat'(v)'(v),thenthefunctiongdenedbyT(g)=(T(f)nfvg)[fvgsatises(g)�(f).Sincegisself-dual,(f)isnotmaximumamongallself-dualfunctions.(iii)=)(i):Assumethat(f)isnotmaximumamongallself-dualfunctions.Thenthereexistsaself-dualfunctiongsuchthat(g)�(f).Since(g)�(f),somevectorv2T(f)nT(g)satises'(v)'(v).Forthisv,letwbeavectorinminT(f)suchthatwv.Bytheg-regularityof','(w)'(v)holds.Moreover,sincewvbywv,'(w)'(v)holds.Thus,wehave'(w)'(v)'(v)'(w):Thismeansthatthiswsatisesw2minT(f)and'(w)'(w).Theorem6.2.Let(f)beag-regularfunctionaldenedby(24).Thenthereexistsaregularself-dualfunctionfwhichmaximizes(f)amongallself-dualfunc-tions.Proof.Letfbearegularself-dualfunctionthatmaximizesamongallregularself-dualfunctions.Weclaimthatfinfactmaximizesamongallself-dualfunctions. 404KAZUHISAMAKINOANDTIKOKAMEDAIfnot,byLemma6.1,thereexistsavectorv2minT(f)suchthat'(v)'(v).Notethatv6vholds,sinceotherwise(i.e.,vv)'(v)'(v),acontradiction.Thus,itfollowsfromLemma4.1thatv(f)isregularandself-dual.Moreover,by(9),wehave(v(f))�(f),whichcontradictstheassumption.However,theremaybenonregularfunctionsfthatalsomaximize(f).Forexample,letp1=p2=1=2.Thenf1=x1;f2=x2;f3=x1,andf4=x2havethemaximumavailability,and,clearly,f2,f3,andf4arenotregular.(f3andf4arenotevenpositive.)Apseudo-Booleanfunction'issaidtobeproperlyg-regularif'(v)�'(w)holdsforallpairsofvectorsvandwwithvw.Thefunctionaldenedby(24)isalsosaidtobeproperlyg-regularif'isproperlyg-regular.ThenexttheoremisaweakuniquenessresultforTheorem6.2,showingthatanyoptimalcoterieisregularif(f)isproperlyg-regular.Theorem6.3.Let(f)beaproperlyg-regularfunctionaldenedby(24).Thenanyfunctionfthatmaximizes(f)amongallself-dualfunctionsisregular.Proof.Assumethatanonregularfunctionfmaximizes(f)amongallself-dualfunctions.Sincefisnonregular,thereexiststwovectorvandwwithvwsuchthatf(v)=0andf(w)=1.ByLemma6.1,wehave'(w)'(w).This,togetherwithvw(andwv),impliesthat'(v)�'(v),whichcontradictstheassumptionbyLemma6.1.Theabovetheoremdirectlyimpliesthefollowingcorollary.Corollary6.4.Letpi,i=1;2;:::;n,betheoperationalprobabilityoftheithelement.Ifp1�p2�


�pn�1=2,thenanyfunctionfthatmaximizestheavailabilityA(f)amongallself-dualfunctionsisregular.AlgorithmOPT-REG-SDInput:Amembershiporacleofg-regularfunction'.Output:Aregularself-dualfunctionfthatmaximizes(f)amongallself-dualfunctions.Step0:Leti:=1andf:=x1.Step1:While9v2minT(f)suchthatvi=0,v[Vi]6v[Vi]and'(v0)'(v0)forv0=v+Pn=i+1e(j),letf:=(v;Vi)(f),whereVi=f1;2;:::;ig.Step2:Ifi=n,outputfandhalt.Otherwise,i:=i+1andreturntoStep1.NotethatthesetminT(f)inthe\while"statementofStep1isupdatedasaresultofapplyingthetransformationtofinStep2.Example6.5.Letusconsidertheavailabilityofthe6-variablefunctions,whenp1=9=10,p2=6=7,p3=4=5,p4=7=10,andp5=3=5.Recallthat(f)=A(f)isgivenby(25).WeapplyalgorithmOPT-REG-SDtothis(f).Step0:Leti:=1andf:=x1(thus,minT(f)=f(10000)g).Letu=(10000).FirstiterationofSteps1-2:Sinceu1=1,skipStep1.Step2setsi:=2.SeconditerationofSteps1-2:Sinceu[V2]=(10)u[V2]=(01)fortheonlyvectoruinminT(f),skipStep1.Step2setsi:=3.ThirditerationofSteps1-2:Vectoru2minT(f)satisesu3=0,u[V3]=(100)6u[V3]=(011),but'(10011)=9=101=71=57=103=5=189=17500�'(01100)=1=106=74=53=102=5=144=17500.(Ifweweretoapply(u;V3)tof,wewouldhave((u;V3)(f)(f).)ThusskipStep1.Step2setsi:=4.FourthiterationofSteps1-2:Vectorusatisesu4=0,u[V4]=(1000)6u[V4]=(0111).Moreover,wehave'(10001)'(01110),since'(10001)=9=101=71=53=103=5=81=17500and'(01110)=1=106=74=57=10 TRANSFORMATIONSONREGULARNONDOMINATEDCOTERIES4052=5=336=17500.Thusfistransformedtof:=(u;V4)(f)=12+13+14+234:Forthisnewf,wehaveminT(f)=fu=(10010);v=(01110)g.Sinceu4=v4=1,weskipStep1.Step2setsi:=5.FifthiterationofSteps1-2:usatisesu5=0,u[V5]=(10010)6u[V5]=(01101).Moreover,wehave'(10010)'(01101),since'(10010)=9=101=71=57=102=5=126=17500and'(01101)=1=106=74=53=103=5=216=17500.Thusfistransformedtof:=(u;V5)(f)=12+13+14(2+3+5)+234+235=12+13+145+234+235:(26)aresult,wehaveminT(f)=fu=(10100);v=(10011);w=(01101)g.Sincev5=w5=1,weneedtoconsideronlyu=(10100).Sinceu5=0,u[V5]=(10100)6u[V5]=(01011),but'(10100)=216=17500�'(01011)=126=17500,skipStep1.Sincei=n=5,outputfunctionfgivenby(26)andhalt.Asmentionedintheintroduction,theweights(log29;log26;2;log2(7=3);log2(3=2))denedby(1)produceanoptimalvote-assignablecoterie,sincetie-breakingisun-necessaryhere.However,someweightsareirrational,andhencewecannotexactlycomputethesumoftheweightsw(S)=Pi2Sw(i).Letfi,i=1;2;:::;n,bethefunctionfaftertheithiterationofStep1ofalgorithmOPT-REG-SDhasbeencompleted.ThenclearlyVfiVi(=f1;2;:::;ig)holds.Moreover,wehavethefollowinglemma.Lemma6.6.Letfi,i=1;2;:::;n,beasdenedabove.Foreachi=1;2;:::;n,allvectorsv2minT(fi)withv[Vi]6v[Vi]satisfy'(v0)'(v0);(27)ev0=v+Pn=i+1e(j).Proof.Ifi=1,thenf1=x1,andthelemmaholdsinthiscase,sinceminT(f1)=fv=(100


0)gandv[V1]v[V1].Assumingitholdsfori=k,considerthecasewherei=k+1.Letusconsiderthevectorv2minT(fk+1)withv[Vk+1]6v[Vk+1].Ifvk+1=0,thenvsatises'(v0)'(v0),wherev0=v+Pn=k+2e(j),sinceotherwise,vwouldhavebeenremovedfromT(fk+1)inStep1byapplyingtheoperation(v;Vk+1)tofk+1.Thiscontradictsthedenitionoffk+1.Ifvk+1=1,ontheotherhand,therearetwopossibilities:(1)fk(v)=0and(2)fk(v)=1.Incase(1),sincev2T(fk+1)nT(fk),fwasupdatedbyf:=(v0;Vk+1)(f)inthe(k+1)stiterationofStep1.Therefore,wehave'(v0)'(v0).Incase(2),assuming'(v0)'(v0),wederiveacontradiction.VfkVkim-pliesfk(ve(k+1))=1.Sincev[Vk+1]6v[Vk+1],wehave(ve(k+1))[Vk]6(ve(k+1))[Vk].Notethat(ve(k+1))0=ve(k+1)+Pn=k+1e(j)=v0.Thus,ifve(k+1)2minT(fk),byassumption,fkwouldhavebeenupdatedbyfk:=(ve(k+1);Vk)(fk).Thiscontradictsthedenitionoffk.Now,sinceve(k+1)62minT(fk),thereexistsavectorw2minT(fk)suchthatwve(k+1).Wecaneasilyseethatthiswsatisesw[Vk]6w[Vk]and'(w0)'(w0),wherew0=w+Pn=k+1e(j).Thisimpliesthatfkwouldhavebeenupdatedbyfk:=(w;Vk)(fk),againacontradiction. 406KAZUHISAMAKINOANDTIKOKAMEDALemma6.7.Letfnbeasdenedabove.Thenfnmaximizesamongallself-dualfunctions.Proof.Lemma6.6asserts(wheni=n)thatallvectorsv2minT(fn)withv6vsatisfy'(v)'(v).Asforvectorsv2minT(fn)withvv,theg-regularityof'implies'(v)'(v).TogetherwithLemma6.1,thiscompletestheproof.Theorem6.8.AlgorithmOPT-REG-SDcorrectlyoutputsaregularself-dualfunctionfthatmaximizesamongallself-dualfunctionsinO(n3jminT(f)j)time.Proof.Sincethealgorithm'scorrectnessfollowsfromLemma6.7,weconsideronlyitstimecomplexity.LetusassumethatOPT-REG-SDgeneratesthefollowingsequenceoffunctions:fm(=x1)!fm1!:::!f0,wheref0istheoutputofalgorithmOPT-REG-SD.Thentheremustbeasequenceoftransformations,f0!f1!:::!fm,whichcanbegeneratedbyalgorithmTRANS-REG-SD.ThismeansthatmjminT(f0)j.AsinalgorithmGEN-REG-SD,weusebinarytreesBasthedatastructuresofminT(f)andminT(f)inthefollowingway:Foravectorv,dene(v)by(v)=min08imaxON(v)j'(v0)'(v0)forv0=v+nXj=i+1e(j)9[fn+1g1:Itisclearthat'(v0)'(v0)holdsforv0=v+Pn=i+1e(j)withi(v),if(v)n.AlgorithmOPT-REG-SDthenpreparesB(minT(f))andfB(minT(f)(i;j))ji=1;2;:::;n+1;j=1;2;:::;ng,whereminT(f)(i;j)=fv2minT(f)ji=maxf(v);(v)g;j=maxON(v)g:AsinthetimecomplexityanalysisofalgorithmGEN-REG-SD,wecanprovethateachiterationofStep1inalgorithmOPT-REG-SDcanbeexecutedinO(n3)time.ThusitrequiresO(n3jminT(f)j)timeintotal.Acknowledgment.Theauthorsthanktheanonymousrefereesfortheirhelpfulandconstructivecommentswhichimprovedthepresentationofthispaper.REFERENCES[1]Y.AmirandA.Wool,Optimalavailabilityquorumsystems:Theoryandpractice,Inform.Process.Lett.,65(1998),pp.223{228.[2]D.AvisandK.Fukuda,Apivotingalgorithmforconvexhullsandvertexenumerationofarrangementsandpolyhedra,DiscreteComput.Geom.,8(1992),pp.295{313.[3]D.AvisandK.Fukuda,Reversesearchforenumeration,DiscreteAppl.Math.,65(1996),pp.21{46.[4]D.BarbaraandH.Garcia-Molina,Thevulnerabilityofvoteassignment,ACMTrans.Com-puterSystems,4(1986),pp.187{213.[5]D.BarbaraandH.Garcia-Molina,Thereliabilityofvotingmechanisms,IEEETrans.Comput.,36(1987),pp.1197{1208.[6]P.BertolazziandA.Sassano,AnO(mn)timealgorithmforregularset-coveringproblems,Theoret.Comput.Sci.,54(1987),pp.237{247.[7]L.J.Billera,Onthecompositionanddecompositionofclutters,J.CombinatorialTheorySer.B,11(1971),pp.234{245.[8]J.C.BiochandT.Ibaraki,ComplexityofidenticationanddualizationofpositiveBooleanfunctions,Inform.Comput.,123(1995),pp.50{63.[9]J.C.BiochandT.Ibaraki,Generatingandapproximatingpositivenon-dominatedcoteries,IEEETrans.ParallelDistrib.Systems,6(1995),pp.905{914.[10]E.Boros,P.L.Hammer,T.Ibaraki,andK.Kawakami,Polynomial-timerecognitionof2-monotonicpositiveBooleanfunctionsgivenbyanoracle,SIAMJ.Comput.,26(1997),pp.93{109. 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