amesh Departmen of Computer Science Engg Indian Institute of ec hnology Hauz Khas New Delhi 110016 India Abstract In this paper presen algorithms for en umeration of spanning trees in undi rected graphs with and without eigh ts The algorithms use se ID: 83847
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paper,bebetoperationsnodesbebeNVrespectivbertroductionimportanortanWeightedspanningtreeenumerationinorderwould ndapplicationinasubroutineofagenerate-and-testprocedurepoinbebet appearedproceedings TR75],selectivegen-erationandtesting[Ch68],andedgeexchanging[Ga77].Ofthese,Gaboo]seemstobeNVNVbebebetbeNVoperations,paper,betnodedescribesprocedurenodenodeberepeatingberepeatedlyprocedure.Repetitionnodebetassociatednodedescribemethodologybepaper.nodesboundedberNVoperations.bettersGaboo]whichresultsinatotaltimecomplexityofO(NEpapererweusesimilartechniquestoenumerateallspanningtreesofadirectedgraphinO(NVuponbestboundNEE].Section2describesdescribesproofs bebecorrespondingberbecorrespondingberootroot.nodeassociatedypenodenodeINOUTINOUTbeCYCLEberespectrespectassociatednode.root.benodebeassociatedbeOUT:;berespectlabeledcorrespondsINbecauseINcorrespondsnodenodeINOUTININNfe1;e2;:::;;ffg,for1jkOUTOUT ININOUTOUTTffgSBj=SA fejg[ffg,for1jkE(GBj)=E(GA) fejgwithedgese1;e2;:::;CYCLECYCLE:;operationCYCLECYCLECYCLEINOUTrootbothCYCLErespectallProof:proofberootnoderootedrootrooted:Bprocessrepeatednodes:Brootedcorresponds:;corresponding :;benodecorresponding:;aboberepeatedrespectnodenodecorrespondsdoesnodesnodeassociatednodebothnodesnode.importannodebeenbe bothnodeperformed,proportionalberbercorrespondingelaborate.repeatedoccursspeeddescribenoderootnodepost-orderberupperroot)poinrespectpoinupperpoindescribebecomesendpoinbebecause MaincorrespondingrootrespectMainbedescribednoderooted:;respect:;beingOUTmodi ed,bepoin:;INbeINprocedurerootednodecorrespondingbeINprocedureprocedurebeforeocation.CHANGESCHANGESCHANGESMainmodi edmodi cation :;occurCHANGESCHANGESCHANGESCHANGESCHANGESCHANGESf;aboForloop;describesubproceduresperformancesupportsoperationsdescribeddescribesubproceduresoperationsperformeduponabooperationbedescribedcorresponding ModifyModifyoperationsperformeduponsubprocedures.spenoperationsbenodesnodeclubbedOperationspoinpoinproportionalberperformingbecomeself-loop;allowscorrespondingbebounds.appear.correspondingININbe 1'proportionalberpoinlocationOperationsbebemodi edproportionalbebeproportionalberdescribeoperationberespect:;proportionaldoesbothoperationbothbepoinassociatedbepoinassociatedpoinportionproportionalj jpoinbebound.proportionalpoin:pperpoinberpoinbeenpoinpoinpoinprocedurepoinstoppedber becauseaboprocedurepoinbepoinbepositionspoinpoinaboprocedure.dependinguponpoinpoinpositionssupposepoinpoinbelocatedendpoinpoinbesupposepoinpoinpoinpoinbelocatedpoinsupposepoinpoinbepoinpoinpoinbelocatedpoinpoinbelocatedpoinbeproportionalberallowsProof:nodeCHANGESoppositeoppositef;CHANGESrootedbeenberCHANGESCHANGES Proof.operationsperformednodespenoperationsnodesoperationsperformedperformedproportionalberlookoperationsperformedsubproceduresproportionalberspenbenodenodeproportionalbercorrespondingclubbedself-loopsrespect:;proportionalberuponpoinself-loopuponself-loopspossiblepoinspenbenodenodeproportionalberMainllMainProofnode updatedINOUTupdatesoperationsbefore.CHANGESbetbeingpreprocessingMainbeforeberootnodeMainocationsspenocationocation.ocationcorrespondnodeocationnodesMainMainProofnodebenoderootbernodesrespectivberboundedbecausedescribebeforerootnoderespectpropertbeINcorrespondingbepoinrootpoinroot.bounddescribebelo beprocess.occurdescribebeloroot,respectivendpoinroot.Supposenodebeenberedpostordernodeupperpoinpostorderberrespectrootproceeding:;be:; nodesdescribeoperationsproofbepoinpoinbeen:;INbe:;become:;poinpoin:;beenbeforebepoinpoinbeallallProof:beupperpoinbepoinrespectcomponencomponenbecomponennode:;beingbe:;happenbecomesbeberbepoin:;:;procedureabopoin:;:;abo before:;bebe:;:;all:;describeoperationstoo.poinpoinnode,INproportionalberoccursallowsbeupperpoinbepoinimportanindependennodebeingpostorderroot +1Proof:supposeupperpoinpostorderportionmodi edpostorderberbefore,componenModi cationsbepoinbetcorrespondingappearingnodepoinpoinpoinbeself-looploopperbeproportionalberallPseudo-codepseudo-codeprocedurecomponencomponenprocedures perform:;respectivbeforecomponen:;componenendpoin:;bottomLASTlocalLASTLAST;;f;ININUpdateINOUTOUTUpdateOUTf;ININOUTOUTININ inlistinlististe;EndFor;ININinlistOUTOUTaboforloopopx1] ADJ-LIST[x1] f;/*PreparesforlastrecursionassociatedLASTLASTTx1];OUTOUTProcedurefatheraboupperendpoinADJLISTTxi];/*EdgesinlarenowloopsaboaboProcedureLAST;LASTLASTLASTLASTLASTProcedureADJLISTTx1]isemptyandx1hasnosoninTthenLASTLAST LASTLASTtoo.too.Proof.oper-performednodespenoperationsnodesoperationsperformedperformedproportionalberlookoperationsperformedsubproceduresocationsprocedurespenproceduresproportionalbermajoroperationsprocedurepoinendpoinoperationpoinproportionalberINbelongspoinbenodepoinoperationendpoinproportionalberuponuponupperpoinself-loopaboprocess.uponspenmodifyingpoinbenodeendpoin allProof:processMaincomponenbepossiblecomponencomponenbecomponenbenodenodesProof:bebepoinnodesboundrootassociatedMSTrootbeforeupdatedcorrespondingrepeatednodes rootrootdescribenodesboundnodecorrespondingbeMSTRepeatnodecorrespondingMSTMSTProof:proofrootnodenodebe:;be:;:;:;:;MSTProof:associatedroot.becauseassociatednodeINnodebeassociatedroot,becauseassociatednodeOUTnode.bound ProofbenodesprocessingnodesProcessingbound.Proof:benodesberednodesbeNVpapermethodologypaperdescribesbepossiblepoinoinJ.P.Char,Generationoftrees,2treesandstorageofmasterforests,IEEETrans.CircuitTheory,vol.CT-15,pp.128-138,1968.[Ga77]H.N.Gabo H.N.GabooAnnJames,Astudyofalgorithmstoenumerateallstablematchingsandspanningtrees,M.Techthesis,Dept.ofMathematics,Dec1989,IITITy81]R.Jayakumar,Analysisandstudyofaspanningtreeenumerationalgorithm,MSthesis,Dept.ofComp.Sc.,IITreportedortedR.Jayaraman,Thulasiraman,M.N.S.Swamy,ComplexityofComputationofaspanningtreeenumerationalgorithm,IEEETrans.CircuitsandSystems,vol.CAS-31,pp.853-860,1984.[KJ89]N.C.KalraandS.S.Jamuar,MicroprocessorcessorS.KapoororW.Mayeda,GraphTheory,JohnWiley,pp.252-364,NY1972.[Mi65]G.J.Minty,Asimplealgorithmforlistingalltreesofagraph,IEEETrans.CircuitTheory,vol.CT-12,pp.120,1965.[Ra90]H.Ramesh,Analgorithmforenumeratingallspanningtreesofanundirectedweightedgraphinincreasingorderofweight,manuscript,alsopresentedatAllIndiaStudentSeminar-Tryst-90,IIT