2024年5月14日发(作者：步慧捷)

ASimpleMin-CutAlgorithm

MECHTHILDSTOER

TeleverketsForskningsinstitutt,Kjeller,Norway

AND

FRANKWAGNER

FreieUniversita¨tBerlin,Berlin-Dahlem,Germany

entanalgorithmforfindingtheminimumcutofanundirectededge-weighted

shortandcompactdescription,iseasytoimplement,and

timematchesthatofthefastestalgorithm

rasttonearlyallapproachessofar,the

yspeaking,thealgorithmconsistsofabout͉V͉nearly

identicalphaseseachofwhichisamaximumadjacencysearch.

CategoriesandSubjectDescriptors:G.L.2[DiscreteMathematics]:GraphTheory—graphalgorithms

GeneralTerms:Algorithms

AdditionalKeyWordsandPhrases:Min-Cut

uction

Graphconnectivityisoneoftheclassicalsubjectsingraphtheory,andhasmany

practicalapplications,forexample,inchipandcircuitdesign,reliabilityof

communicationnetworks,transportationplanning,g

theminimumcutofanundirectededge-weightedgraphisafundamental

ely,itconsistsinfindinganontrivialpartitionofthe

graphsvertexsetVintotwopartssuchthatthecutweight,thesumoftheweights

oftheedgesconnectingthetwoparts,isminimum.

ApreliminaryversionofthispaperappearedinProceedingsofthe2ndAnnualEuropeanSymposium

eNotesinComputerScience,vol.855,1994,pp.141–147.

ThisworkwassupportedbytheESPRITBRAProjectALCOMII.

Authors’addresses:,TeleverketsForskningsinstitutt,Postboks83,2007Kjeller,Norway;

e-mail:@.;,Institutfu¨rInformatik,FachbereichMathematikund

Informatik,FreieUniversita¨tBerlin,Takustraße9,Berlin-Dahlem,Germany;e-mail:wagner@-

.

Permissiontomakedigital/hardcopyofpartorallofthisworkforpersonalorclassroomuseis

grantedwithoutfeeprovidedthatthecopiesarenotmadeordistributedforprofitorcommercial

advantage,thecopyrightnotice,thetitleofthepublication,anditsdateappear,andnoticeisgiven

thatcopyingisbypermissionoftheAssociationforComputingMachinery(ACM),

otherwise,torepublish,topostonservers,ortoredistributetolists,requirespriorspecificpermission

and/orafee.

᭧1997ACM0004-5411/97/0700-0585$03.50

JournaloftheACM,Vol.44,No.4,July1997,pp.585–591.

586

Theusualapproachtosolvethisproblemistouseitscloserelationshiptothe

ousMax-Flow-Min-Cut-TheorembyFordand

Fulkerson[1956]showedthedualityofthemaximumflowandtheso-called

,sandtaretwoverticesthatarethesourceandthesink

intheflowproblemandhavetobeseparatedbythecut,thatis,theyhavetolie

ecentlyallcutalgorithmswere

gaminimumcutwithout

specifiedverticestobeseparatedcanbedonebyfindingminimums-t-cutsfora

fixedvertexsandall͉V͉Ϫ1possiblechoicesoftʦVگ{s}andthenselecting

thelightestone.

RecentlyHaoandOrlin[1992]showedhowtousethemaximumflow

algorithmbyGoldbergandTarjan[1988]inordertosolvetheminimumcut

problemintimeᏻ(͉VʈE͉log(͉V͉

2

/͉E͉),whichisnearlyasfastasthefastest

maximumflowalgorithmssofar[Alon1990;Ahujaetal.1989;Cheriyanetal.

1990].

NagamochiandIbaraki[1992a]publishedthefirstdeterministicminimumcut

algorithmthatisnotbasedonaflowalgorithm,hastheslightlybetterrunning

timeofᏻ(͉VʈE͉ϩ͉V͉

2

log͉V͉),nweighted

case,theyuseafast-searchtechniquetodecomposeagraph’sedgesetEinto

subsetsE

1

,...,E

␭

suchthattheunionofthefirstkE

i

’sisak-edge-connected

spanningsubgraphofthegivengraphandhasatmostk͉V͉mulate

orkisoneofasmallnumberof

paperstreatingquestionsofgraphconnectivitybynon-flow-basedmethods

[NishizekiandPoljak1989;NagamochiandIbaraki1992a;Matula1992].Karger

andStein[1993]suggestarandomizedalgorithmthatwithhighprobabilityfinds

aminimumcutintimeᏻ(͉V͉

2

log͉V͉).

Inthiscontext,wepresentinthispaperaremarkablysimpledeterministic

minimumcutalgorithmwiththefastestrunningtimesofar,establishedin

NagamochiandIbaraki[1992b].Wereducethecomplexityofthealgorithmof

NagamochiandIbarakibyavoidingtheunnecessarysimulateddecompositionof

ablesustogiveacomparablystraightforwardproofof

correctnessavoiding,forexample,thedistinctionbetweentheunweighted,

integer-,rational-,andreal-weightedcase.

ThisalgorithmwasfoundindependentlybyFrank[1994].

Queyranne[1995]generalizesoursimpleapproachtotheminimizationof

submodularfunctions.

ThealgorithmdescribedinthispaperwasimplementedbyKurtMehlhorn

fromtheMax-Planck-Institut,Saarbru¨ckenandispartofthealgorithmslibrary

LEDA[MehlhornandNa¨her1995].

orithm

Throughoutthepaper,wedealwithanordinaryundirectedgraphGwithvertex

dgeehasnonnegativerealweightw(e).

Thesimplekeyobservationisthat,ifweknowhowtofindtwoverticessandt,

andtheweightofaminimums-t-cut,wearenearlydone:

T

HEOREM

/{s,t}bethe

inimumcutofGcanbeobtainedby

takingthesmallerofaminimums-t-cutofGandaminimumcutofG/{s,t}.

2024年5月14日发(作者：步慧捷)

ASimpleMin-CutAlgorithm

MECHTHILDSTOER

TeleverketsForskningsinstitutt,Kjeller,Norway

AND

FRANKWAGNER

FreieUniversita¨tBerlin,Berlin-Dahlem,Germany

entanalgorithmforfindingtheminimumcutofanundirectededge-weighted

shortandcompactdescription,iseasytoimplement,and

timematchesthatofthefastestalgorithm

rasttonearlyallapproachessofar,the

yspeaking,thealgorithmconsistsofabout͉V͉nearly

identicalphaseseachofwhichisamaximumadjacencysearch.

CategoriesandSubjectDescriptors:G.L.2[DiscreteMathematics]:GraphTheory—graphalgorithms

GeneralTerms:Algorithms

AdditionalKeyWordsandPhrases:Min-Cut

uction

Graphconnectivityisoneoftheclassicalsubjectsingraphtheory,andhasmany

practicalapplications,forexample,inchipandcircuitdesign,reliabilityof

communicationnetworks,transportationplanning,g

theminimumcutofanundirectededge-weightedgraphisafundamental

ely,itconsistsinfindinganontrivialpartitionofthe

graphsvertexsetVintotwopartssuchthatthecutweight,thesumoftheweights

oftheedgesconnectingthetwoparts,isminimum.

ApreliminaryversionofthispaperappearedinProceedingsofthe2ndAnnualEuropeanSymposium

eNotesinComputerScience,vol.855,1994,pp.141–147.

ThisworkwassupportedbytheESPRITBRAProjectALCOMII.

Authors’addresses:,TeleverketsForskningsinstitutt,Postboks83,2007Kjeller,Norway;

e-mail:@.;,Institutfu¨rInformatik,FachbereichMathematikund

Informatik,FreieUniversita¨tBerlin,Takustraße9,Berlin-Dahlem,Germany;e-mail:wagner@-

.

Permissiontomakedigital/hardcopyofpartorallofthisworkforpersonalorclassroomuseis

grantedwithoutfeeprovidedthatthecopiesarenotmadeordistributedforprofitorcommercial

advantage,thecopyrightnotice,thetitleofthepublication,anditsdateappear,andnoticeisgiven

thatcopyingisbypermissionoftheAssociationforComputingMachinery(ACM),

otherwise,torepublish,topostonservers,ortoredistributetolists,requirespriorspecificpermission

and/orafee.

᭧1997ACM0004-5411/97/0700-0585$03.50

JournaloftheACM,Vol.44,No.4,July1997,pp.585–591.

586

Theusualapproachtosolvethisproblemistouseitscloserelationshiptothe

ousMax-Flow-Min-Cut-TheorembyFordand

Fulkerson[1956]showedthedualityofthemaximumflowandtheso-called

,sandtaretwoverticesthatarethesourceandthesink

intheflowproblemandhavetobeseparatedbythecut,thatis,theyhavetolie

ecentlyallcutalgorithmswere

gaminimumcutwithout

specifiedverticestobeseparatedcanbedonebyfindingminimums-t-cutsfora

fixedvertexsandall͉V͉Ϫ1possiblechoicesoftʦVگ{s}andthenselecting

thelightestone.

RecentlyHaoandOrlin[1992]showedhowtousethemaximumflow

algorithmbyGoldbergandTarjan[1988]inordertosolvetheminimumcut

problemintimeᏻ(͉VʈE͉log(͉V͉

2

/͉E͉),whichisnearlyasfastasthefastest

maximumflowalgorithmssofar[Alon1990;Ahujaetal.1989;Cheriyanetal.

1990].

NagamochiandIbaraki[1992a]publishedthefirstdeterministicminimumcut

algorithmthatisnotbasedonaflowalgorithm,hastheslightlybetterrunning

timeofᏻ(͉VʈE͉ϩ͉V͉

2

log͉V͉),nweighted

case,theyuseafast-searchtechniquetodecomposeagraph’sedgesetEinto

subsetsE

1

,...,E

␭

suchthattheunionofthefirstkE

i

’sisak-edge-connected

spanningsubgraphofthegivengraphandhasatmostk͉V͉mulate

orkisoneofasmallnumberof

paperstreatingquestionsofgraphconnectivitybynon-flow-basedmethods

[NishizekiandPoljak1989;NagamochiandIbaraki1992a;Matula1992].Karger

andStein[1993]suggestarandomizedalgorithmthatwithhighprobabilityfinds

aminimumcutintimeᏻ(͉V͉

2

log͉V͉).

Inthiscontext,wepresentinthispaperaremarkablysimpledeterministic

minimumcutalgorithmwiththefastestrunningtimesofar,establishedin

NagamochiandIbaraki[1992b].Wereducethecomplexityofthealgorithmof

NagamochiandIbarakibyavoidingtheunnecessarysimulateddecompositionof

ablesustogiveacomparablystraightforwardproofof

correctnessavoiding,forexample,thedistinctionbetweentheunweighted,

integer-,rational-,andreal-weightedcase.

ThisalgorithmwasfoundindependentlybyFrank[1994].

Queyranne[1995]generalizesoursimpleapproachtotheminimizationof

submodularfunctions.

ThealgorithmdescribedinthispaperwasimplementedbyKurtMehlhorn

fromtheMax-Planck-Institut,Saarbru¨ckenandispartofthealgorithmslibrary

LEDA[MehlhornandNa¨her1995].

orithm

Throughoutthepaper,wedealwithanordinaryundirectedgraphGwithvertex

dgeehasnonnegativerealweightw(e).

Thesimplekeyobservationisthat,ifweknowhowtofindtwoverticessandt,

andtheweightofaminimums-t-cut,wearenearlydone:

T

HEOREM

/{s,t}bethe

inimumcutofGcanbeobtainedby

takingthesmallerofaminimums-t-cutofGandaminimumcutofG/{s,t}.