On the representation of codes in Forney graphs

OntheRepresentationofCodesinForneyGraphsRalfKoetterCoordinatedSieneLaboratoryUniversityofIllinoisatUrbana-ChampaignUrbana,IL61801,U.S.A.koettersl.uiu.eduAugust17,2000AbstratWeinvestigatetherepresentationofodesingraphialmodels.Inpartiular,weusethenotionofatrellisformationonaForneygraphtovisualizethestrutureofaodeonagivengraph.WefousonthequestionofwhetheratrellisformationontainsmergeablevertiesandwhetherthedesriptionofaodeintermsofloalbehaviorsontheForneygraphanbemadesmaller.NeessaryandsuÆientonditionsformergeabilityaregivenleadingtoapolynomialtimealgorithmthatdeidesifagiventrellisformationontainsmergeableverties.OneofourmaintoolsisadualitytheorembyForney,forwhihwegiveashortproofintheontextofbinaryodes.Keywords:trellis,oding,graphialrepresentation,ForneygraphWorksupportedunderNSFgrantCAREER:CCR-9984515.1.IntrodutionCodesongraphsareaeldofodingtheorythatisnewlyemerginginthewakeoftheenormoussuessofiterativedeodingalgorithms.Itisbynowwellunderstoodthatmessagepassingalgorithms,likethesum-produt(beliefpropagation)ormin-sumalgorithm,haveanaturaldesriptionintermsofagraphthatrepresentstheodeinonsideration.Thethemeofodesongraphsisreurrentintheodingtheoryliterature.Gallager’slowdensityparityhek(LDPC)odes[1℄weretheearliestappearaneofiterativelydeodableodesandmuhofmodernodingtheoryouldhavebeenderivedintheearly1960s.OthernotableontributionstotheareaofodesongraphsaretheworkofTanner[2℄,Wibergetal.[3,4℄,andanumberofpapersexploringtheinterplaybetweeniterativedeodingandothergraph-basedalgorithms[5,6,7,8,9℄.TrellisrepresentationsofodeswereintroduedbyForney[10℄inordertoexplaintheworkingsoftheViterbialgorithm.Howeveritsoonbeamelearthattherepresentationofodesintrellis-likestruturesisadisiplinethatprovidesmuhinsightintothestrutureoftheunderlyingodeitself.Questionsaboutminimalrealizationsandanonialtrellisrepresentationsarebynowwellunderstoodintheontextofonventionaltrellises(wewilldenethistermshortly).AnexellentoverviewofthistopiisgivenbyVardy[11℄.Connetionsbetweensystemtheoryandtrellisesforlinearsystems,orequivalentlylinearodes,areexploredbyForneyandTrott[12℄.Theombinationoftheareasofodesongraphsandtrellisrepresentationofodes(orlinearsystems)isanaturaluniationbetweenthetwoelds,undertakenforthersttimebyWiberg[4℄.Thetraditionalproblemsonerningminimalityandothersystem-theoretinotionsareonsiderablymorediÆultinthisontext.Itisbynowunderstood[13,14℄thatstatespaerealizationsthataregeneralizedtoloopytime-axes(atime-orderingthatisnotlinearlyevolvingbutobeysmoregeneralrelationships)usuallydonotexhibitaminimalrealization.Indeed,itisnontrivialtoevendenethemeaningofminimalinthisontext(f.[13,14℄).Ingeneral,thedierentrealizationsofthesamelinearsystemformapartiallyorderedset.Nevertheless,mostonstrutiontehniquesforonventionaltrellises,liketheprodutonstrutionofKshishangandSorokine[15℄anbeappliedtotheonstrutionofstatespaerealizationsonloopytimeaxes.Inpartiular,fortheimportantaseoftail-bitingtrellises,KoetterandVardyouldexhibitapolynomialtimealgorithmthatharater-izestheminimal(forpratiallyallinterestingdenitionsofminimality)linearstatespaerealizationsofagivenlinearsystem[16℄.However,progressformoreomplexgraphstru-tureshasremainedslow.Inpartiular,noteventhesimplestquestionsaboutgeneralizedstatespaerealizations,likethemergeabilityofverties,haveasatisfatoryanswer.In1999,Forney[17℄derivedthenotionof\normalgraphsand\normalfatorgraphs,whihintroduesaruialnormalizationtotheproblemofodesongraphs.Thenor-malizationonsistsofrestritingthedegreeofthestatespaenodes.AknowledgingtheimportaneofthisnormalizationwewillrefertothesenormalizedgraphsasForneygraphs.Itanbeshown[17℄thatanyfatorgraphrepresentingalinearsystemhasan1equivalentForneygraph.Moreover,ForneyderivesatheoremthatrelatesthestatespaesizeofaForneygraphdesribingalinearsystemandthestatespaesizeofitsdual.InthispaperwereviewaspeializedformofForney’sbasiresult[17℄andaddressthesimplestformofminimalityofagivensystem.Inpartiular,wegiveapolynomialtimealgorithmtodeideifatrellisformation(denedinSetion2.1)onagivenForneygraphontainsapairofmergeableverties.ThissurprisinglydiÆultquestionintheontextofgeneralstatespaerealizationsisruialforanyfurtherinvestigationofminimalityofForneygraphs.InSetion2ofthepaperwereviewthebasinotionsoffatorgraphsandForneygraphs.TheonnetionbetweentrellisrepresentationsofodesandForneygraphsisdesribedinSetion2.1,whileSetion3givesasimpliedformofForney’sdualitytheorem.Setion4givesthemainresultsonerningthemergeabilityofvertiesintrellisformation.ThealgorithmforndingatrellisformationthatdoesnotontainmergeablevertiesisgiveninSetion5.2.ForneyGraphsAfatorgraphG=(V;E)withvertexsetVandedgesetEisabipartitegraphonsistingofasetofstate/symbolnodesVS,asetoffuntionnodesVf,i.e.V=VS[Vf,andasetofedgesEVSVf.Weonsideredgesassetsofpairsofvertiesandhenefv;ug2Eimpliesfu;vg2E.Wewillusetheterms\nodeand\vertexinterhangeably.Toeahvertexu2VSweassoiateanalphabetAu,i.e.asetofsymbols.Intheontextofthispaperwewillassumethatallalphabetsarenite.TheongurationspaeforthefatorgraphGisdenedastheCartesianprodutA=u2VSAu.Therest