On Achieving Optimal EndtoEnd Throughput in Data Networks Theoretical and Empirical Studies - PDF document

OnAchievingOptimalEnd-to-EndThroughputinDataNetworks:TheoreticalandEmpiricalStudiesZongpengLi,BaochunLi,DanJiang,LapChiLauWiththeconstraintsofnetworktopologiesandlinkcapacities,achievingtheoptimalend-to-endthroughputindatanetworkshasbeenknownasafundamentalbutcomputationallyhardproblem.Inthispaper,weseekefcientsolutionstotheproblemofachievingoptimalthroughputindatanetworks,withsingleormultipleuni-cast,multicastandbroadcastsessions.Towardsthisobjective,werstinvestigateupperandlowerboundsofsuchoptimalthroughputinthecaseofasinglemulticastsession,andshowthatitiscompu-tationallyhardtocomputethesebounds.Wethenshowthesurpris-ingresultthat,facilitatedbytherecentadvancesofnetworkcoding,computingthestrategiestoachievetheoptimalend-to-endthrough-putcanbeperformedinpolynomialtime,usingthealgorithmwepropose.Inaddition,weextendourresultstothecasesofmulti-plesessionsofunicast,multicast,broadcastandgroupcommunica-tion,aswellasthemodelofoverlaynetworks.Inallthesecases,wealsoshowthattheoptimalachievablethroughputisindependentfromtheselectionofdatasourceswithineachsession.Finally,sup-portedbyempiricalstudies,wepresentthesurprisingobservationthatinmosttopologies,applyingnetworkcodingmaynotimprovetheachievableoptimalthroughput;rather,itfacilitatesthedesignofsignicantlymoreefcientalgorithmstoachievesuchoptimality.I.INTRODUCTIONInthemostgeneralcase,adatanetworkconsistsofasetofendhostsandswitchesinterconnectedviaundirected(orduplex)communicationlinks.Indatanetworkswithknowntopologiesandbandwidthcapacityboundsforeachundirectedlink,afun-damentalproblemistocomputeandachievethemaximumend-to-endthroughputforoneormultipleactivecommunicationses-sions.Dependingontheobjectivesofapplications,acommuni-cationsessionmaybeintheformofunicast(onetoone),multi-cast(onetomany),broadcast(onetoall),orgroupcommunica-tion(manytomany).ThesolutionstothisproblemmayleadtofundamentalandnewinsightswithrespecttooptimalroutingandtrafÞcengineering.Forexample,therecentparadigmofselÞshrouting[1]allowsendhoststochooseroutesthemselvesusingsourceroutingstrategies.Findingtheoptimalstrategytodissem-inatedatatomultipledestinationswithmaximumthroughputisofnaturalinterestsinsuchaparadigm,especiallywhenwewishtooptimallyexploitexistingnetworkcapacitiestodisseminatelargevolumesofdata.Nevertheless,computingthestrategiestoachieveoptimalthroughputindatanetworkshasbeen,unfortunately,veryhardwithrespecttoitscomputationalcomplexity.Asanexampletosupportthisclaim,considerthecaseofoptimizingonemulticastZongpengLi,BaochunLiandDanJiangarewiththeDepartmentofElec-tricalandComputerEngineering,UniversityofToronto.Theiremailad-dressesarearcane,bli,djiang@eecg.toronto.edu.LapChiLauiswiththeDepartmentofComputerScience,UniversityofToronto.Hisemailaddressischi@cs.toronto.edusessionfromasourcetoasetofdestinations.Itcanbeshownthat,traditionally,suchaproblemisequivalenttotheproblemofsteinertreepacking,whichseekstoÞndthemaximumnumberofpairwiseedge-disjointsteinertrees,ineachofwhichthemul-ticastgroupremainsconnected.Anintuitiveexplanationtosuchequivalenceisthat,eachunitthroughputcorrespondstoaunitinformationßowbeingtransmittedalongatreethatconnectsev-erynodeinthegroup.ThemaximumnumberoftreeswecanÞndcorrespondstotheoptimalthroughputforthesession.ThesteinertreepackingproblemhasbeenshowntobeNP-completeandAPX-hard[2],[3],[4].Brute-forcesolutionstothesteinertreepackingproblemareobviouslynotrealisticallyfeasible,nottomentionmoregeneralcaseswheremultiplesessionsco-existinthenetwork.Inthispaper,weseektobringfundamentallynewinsightsandefÞcientsolutionstotheproblemofoptimizingend-to-endthroughputindatanetworks.WeÞrstillustratethepowerofworkcoding[5],[6]withrespecttoachievingoptimalthrough-put.Intheparadigmofnetworkcoding,informationßowsindatanetworksmaynotonlybestoredandforwarded,butalsobeencodedanddecodedinanynodesinthenetwork.Weprovethat,ononeend,theachievableoptimalthroughputwithnetworkcodingislowerboundedbythatachievedwithoutnetworkcod-ing,whichcorrespondstotheproblemofsteinertreepacking.Ontheotherend,itisalsoupperboundedbythesteinerstrengthofthenetwork.Weshowthat,unfortunately,thecomputationofbothboundsareNP-completeproblems,whichseemstosuggestthattheproblemofachievingoptimalthroughputwithnetworkcodingisalsocomputationallyhardtosolve.Surprisingly,asoneofourmostimportantcontributions,weareabletoprovethefollowing:Givenasinglemulticastsessionandundirecteddatanetworkswithper-linkcapacitybounds,acompletesolutiontotheproblemofachievingoptimalthrough-putcanbecomputedinpolynomialtime,usinganewalgorithmwepropose.Wethenshowthatthisconclusioncanbereadilyex-tendedtomultipleconcurrentsessions,aswellastoothertypesofcommunication,includingunicast,broadcastandgroupcom-munication.EvenwhenthegeneralformofdatanetworksismodiÞedtoreßectrealisticcharacteristicsofoverlaynetworks(whereonlyendhostsattheedgemaybeabletoreplicate,en-codeanddecodedata),thesameconclusionstillholds.Thesolu-tionstotheproblemsincludenotonlyoptimalroutingstrategiestotransmitdatainthenetwork,butalsohowdatamaybeen-codedanddecodedastheyarerelayedtowardsthedestinations.Thoughthereexistpreviousresultsonnetworkcodedthroughputdirectednetworks,tothebestofourknowledge,thispaperistheÞrstworkthatsystematicallystudiestheeffectsofnetworkcodingwithrespecttooptimizingthroughputinundirected networks.BeyondtheseefÞcientsolutionsthatcomputeachievableopti-malthroughput,additionalempiricalobservationsandtheoreticalcontributionsthatwepresentinthispaperareasfollows.First,fromagraphtheoreticperspective,weprovethattheadvantageofnetworkcodingÑtheratioofoptimalthroughputwithnet-workcodingoverthatwithoutcodingÑisboundedbyacon-stantfactorof.Thisisincontrasttopreviousresults[7]indirectednetworks,whichshowthatsucharatiomaybearbitrar-ilyhigh.Second,weprovethat,theachievableoptimalthrough-putisnotaffectedbytheselectionofsourcesineachcommu-nicationsession.Third,evenmoresurprisingly,supportedbyempiricalstudiesusingovertopologies,weshowthattheachievedoptimalthroughputwithcodingismostlyidenticaltothatachievedwithoutcoding.Thisimpliesthatthefundamen-talbeneÞtofnetworkcodingishigheroptimalthroughput,buttofacilitatesigniÞcantlymoreefÞcientcomputationofstrate-giestoachievesuchoptimalthroughput.Finally,ouralgorithmstoefÞcientlycomputeoptimalthroughputarealsoinstrumentaltowardsÞndingefÞcientapproximationalgorithmsforgraphthe-oreticproblemssuchassteinertreepackingminimumsteinertreeTheremainderofthispaperisorganizedasfollows.WeÞrstdiscussrelatedworkinSec.II.InSec.III,wepresentourmaintheoremsandalgorithmwithrespecttoachievingoptimalend-to-endthroughputindatanetworkswithasinglemulticastsession.InSec.IV,weextendourresultstothecasesofmultiplesessionsofunicast,multicast,broadcast,andgroupcommunication.Wealsoconsiderthemodelofoverlaynetworks,whereonlyasubsetofnodesarecapableofreplicationandcoding.Basedonthealgorithmswepropose,wepresentbothempiricalandtheoreticalinsightswithrespecttoachievableoptimalthroughputinSec.V.Finally,weconcludethepaperinSec.VI.II.RELATEDTheopenproblemofachievingoptimalend-to-endthrough-putwithefÞcientalgorithmshasnotbeendiscussedindepthinexistingliterature.Thereexist,however,similarproblemsthathavebeenextensivelystudied.TowardsthedirectionofQual-ityofService(QoS)routing,theobjectiveistoÞndend-to-endpathsormulticasttreesthatsatisfyspeciÞcbandwidthordelayconstraints,andthereforeprovidingthedesiredQoSguarantees[8].Withrespecttoend-to-endthroughput,Þndinggoodtopolo-giesthatsatisfybandwidthrequirementsisobviouslydifferentfromÑandarguablyeasierthanÑÞndingThereexistsanextensivebodyofresearchintheareaofmul-ticastroutinginwide-areaIPnetworks(e.g.,[9]).TheadvantageofIP-basedmulticastisbroughtbydatapacketreplicationonmulticast-capableswitches,improvingbandwidthefÞciencyandthroughputcomparedtoall(naive)unicastbetweenthesourceandthemulticastreceivers.However,sinceitisbasedontheconstructionofasingletree,theend-to-endthroughputisnotoptimalcomparedtowhatisachievablebyatopologybeyondaAsIPmulticastisnotreadilydeployed,algorithmspromot-ingapplication-layeroverlaymulticasthaverecentlybeenpro-posedasremedialsolutions,focusingontheissueofconstruct-ingandmaintainingamulticasttreeusingonlyendhosts[10],[11].Thoughasinglemulticasttreemaynotleadtooptimizedthroughput,recentstudies(e.g.,SplitStream[12],CoopNet[13],DigitalFountain[14]andBullet[15])haveproposedtoutilizeeithermultiplemulticasttrees(forest)oratopologicaldeliverstripeddatafromthesource,usingeithermultiplede-scriptioncodingorsourceerasurecodestosplitcontenttobemulticast.Theseproposalshaveindeedimprovedend-to-endthroughputbeyondthatofasingletree,buttherehavebeennodiscussionsonwhethertheoptimalthroughputmaybeachieved,orhowclosetheproposedalgorithmsapproachoptimality.Inthispaper,westudysuchachievableoptimality,whileconsid-eringthemostgeneralcasewherethedatasourcetransmitsastreamofbytes,andisnotassumedtoperformanysourceorerrorcorrectioncoding.Therehavebeenstudiesonachievingoptimalitywithrespecttocomputingobliviousroutingstrategiesindatanetworks.Theobjectivesaretomaximizethroughputforasource-destinationpair,andtominimizecongestiononthenetwork.Mostno-tably,usinglinearprogrammingtechniques,polynomialtimegorithms(withapolynomialnumberofvariablesandconstraintsintheLPformulation)canbeconstructedtocomputestrate-giesforobliviousroutingforanynetwork,directedorundirected[16],[17].Thoughwealsoemploylinearoptimiza-tiontoolsandstudyundirectednetworks,ourproblemdomainismoregeneral:whileoptimalobliviousroutingfocusesonorigin-destinationpairsofsessions(possiblyexploitingpathdi-versity),wefocusonavarietyofcommunicationsessions,in-cludingunicast,multicast,broadcastandgroupcommunication.Weseekfundamentalinsightsonhowoptimalaroutingstrategymaybecome,andwhatisthemaximumachievablethroughputinacommunicationsession. S R (a) Maximum throughput without network coding is 1.875.If one multicast tree is used, the achieved throughput is 1. S R (b) Maximum throughput withnetwork coding is 2. Fig.1.Theadvantageofnetworkcodingwithrespecttoimprovingtheend-to-endmulticastthroughputfromThetheoryofnetworkßows[18]studiesthetransmissionofcommoditiesofthesametype(unicommodityßows)throughanetworkwithknowncapacities.Themaximumßowratebe-tweenthesourceandthedestinationisknowntobeequaltotheminimumcutbetweenthem,whichmaybecomputedwithefÞ-cientalgorithms.Whencommoditiestobetransmittedareofdif-ferenttypes(multicommodityßows),computingthemaximumßowratecanbeformulatedasalinearoptimizationproblem,andthensolvedusinggenerallinearprogramsolvers[18],[19].Inbothunicommodityandmulticommodityßows,commoditiesmayonlybeforwardedatintermediatenodes,comparabletoallunicastindatanetworks.Theconceptofnetworkcodingextendsthecapabilitiesofnetworknodesinacommunicationsession:frombasicdataforwarding(asinallunicast)anddatareplication(asinIPoroverlaymulticast),tocodinginGaloisÞelds.Fig.1 illustratesaclassicexampleofhownetworkcodingassiststoim-proveend-to-endthroughput.Asreceivesboth(encodedoverGF(2)),itisabletodecodeandretrieveboth.Ifthelinkcapacitiesare,itisapparentthatthemaximumachievablethroughputwithnetworkcodingis.Withoutcod-ing,itcanbecomputedthattheoptimalthroughputis.Ifonlyonemulticasttreeisused(asinIPmulticast),theachievedthroughputisTherecentbreakthroughtheoreminnetworkcodingshowsthat,foramulticastsessionindirectednetworks,ifaratecanbeachievedfromthesendertoeachofthemulticastreceiversinde-pendently,itcanalsobeachievedfortheentiremulticastsession(refertoindependentproofsofAhlswedeetal.[5]andKoetteretal.[6]).Inaddition,Lietal.[20]showthatlinearcodesÞcetoachievesuchaproperty.AlllinearcodingoperationsaredeÞnedaslinearcombinationsoverGaloisÞeldswithÞxedel-ementlengths,thusthesizeofthedatadoesnotincreaseafterbeingencoded.Thepowerofnetworkcodingcomesfromthedifferencebetweeninformationßowsandtraditionalityßows:informationmaynotonlybereplicated,butalsobecodedandforwardedwithoutincreasesinsize,whichareimpos-siblewithcommodities.Asaconsequence,thetheoryofnetworkßowsmayonlybeusedtostudyunicastcommunicationsessionsindatanetworks.ThispaperseekstodesignefÞcientsolutionsfortheproblemofachievingoptimalthroughputingeneral,andnetworkcodingisoneoftheavenuesleadingtosuchanobjec-tive.III.AHROUGHPUTINATAETWORKSULTICASTWebeginourtheoreticalstudyfromthecaseofasinglemul-ticastsession.Weconsiderthemostgeneralformofdatanet-works,representedbyasimplegraphV,Erectededgesbetweennetworknodes.Eachedgerepresentsacommunicationlink,andtheedgecapacitiesarespeciÞedbyadenotesthesetofpositivera-tionalnumbers),representingtheavailablebandwidthcapacitiesofcommunicationlinks.Throughoutthispaper,wefocusonthefractionalmodelofdatarouting,wherethecapacityofeachlinkmaybesharedfractionallyinbothdirections,andinformationßowsmaybesplitandmergedatarbitrarilyÞnescales.Nowconsiderasinglemulticastcommunicationsession.Wetospecifythesetofnodesinvolvedinthemulticastcommunication,withbeingthedatasource.Ingraphicalillustrationsthroughoutthispaper(e.g.,Fig.1),nodesinareshownasblack,andnodesinareshownaswhite.Linksarelabeledwiththeircapacities,andallunlabeledlinkshaveacapacityofA.Optimalthroughputwithoutcoding:steinertreepackingToachieveoptimalthroughputinundirecteddatanetworksingeneral,anaturaldirectionistoÞrstconsidertheproblemofachievingoptimalthroughputwithoutcoding,butwithdatarepli-cationonnetworknodes(asinIPmulticast).Thisproblemisequivalenttothegraphtheoreticproblemofsteinertreepacking[2],[21].Asteinertreeisasub-treeofthenetworkthatconnectsev-erymulticastnode.Inthesteinertreepackingproblem,weseektodecomposethenetworkintoweightedsteinertrees,suchthatthetotaloftreeweightsismaximized,referredtoasthetreepackingnumber.Alinkmayappearinasetofdifferentsteinertrees,;butitisrequiredthatistheweightoftreei.e.,thetotalweightsoftreesusingacommonedgeshouldnotexceedthecapacityof.Withoutnetworkcoding,asolutiontosteinertreepackingleadstoaspe-ciÞcstrategytoachieveoptimalmulticastthroughput,sincewecantransmitadatastreamofthroughputalongeachsteiner,andtheresultingthroughputispreciselythetotalweightsofalltrees.Toillustratehowsteinertreepackingcorrespondstoachiev-ableoptimalthroughputwithoutcoding,weshowanexampleinFig.2(a).Intheillustration,eachlettercorrespondstoadis-tinctsteinertreethatconnectsthemulticastgroup,consistingof.Ninesuchsteinertreesexistintheshownpackingscheme(from),wherethetreecorrespondingtoishighlighted.Sinceeachlinkwithunitcapacityneedstoaccom-steinertrees,theachievablethroughputoneachtreeis,.Thisleadstoamulticastthroughputof,whichisoptimalwithoutcoding. aaabbba+ba+ba+bm1m0m2m3 (a) steiner tree packing and multicast without coding.(b) multicast with network coding. Fig.2.Theachievableoptimalthroughputiswithoutcoding,andUnfortunately,steinertreepackinghasbeenshowntobeNP-completeandAPX-hard[2],[4],andthebestknownpolynomialtimealgorithmhasanapproximationratioofaround[3].ItisnotsufÞcientlyaccuratetobeusedinpractice.Onthebrighterside,withthesameexample,wealsoshowthattheachievableoptimalthroughputwithnetworkcodingis(Fig.2(b)),whichishigherthanthatachievedwithoutcoding.B.Optimalthroughputwithcoding:boundsandcomplexitiesSincenetworkcodingmayleadtohigheroptimalthroughput,weturnourfocusontheproblemofachievingoptimalthrough-putwithcoding,andconsiderthesteinertreepackingnumberasanaturallowerbound.Withcoding,ourÞrstattempttowardsasolutionistoapplythebreakthroughtheoremofnetworkcod-ingindirectednetworks,whichrevealsthatifaratecanbeachievedforeachreceiverinthegroupindependently,itcanalsobeachievedfortheentiremulticastsession.ItimpliesthatwemaycomputetheoptimalthroughputachievableinthesessionbyÞndingtheminimumofthemaximumßowratesfromthesourcetoeachofthereceivers,wherethemaximumßowratesmaybeefÞcientlycomputedbyanyofthepolynomialtimemaximumßowalgorithms. 4 m 0m 0m 0m 1 m 1 m 1 m 2m 2 m 2 0.50.511 abac partition.(b) Optimal throughput withnetwork coding is 1.5.network coding is also 1.5. Fig.3.Acounter-exampleontheeffectsofnetworkcodinginundirectednet-works.Theindependentlyachievablethroughputisforboth,butthemulticastthroughputisonlywithorwithoutnetworkcoding.Unfortunately,thistheoremisnotvalidinundirectednet-works.Fig.3showsacounter-example.Intheexample,ifthesendstoeitherindependently,theachiev-ablethroughputis.However,itisimpossibletoachieveathroughputoftobothsimultaneouslyinamulticastsession.Tounderstandthisinfeasibility,considera,showninFig.3(a),thatdividesthenetworkintothreecomponents.Inordertoachieveamulticastthroughputof,itisnecessarytohaveacapacityoftoconcurrentlyßowintoandthecomponent,respectively.Sincetheinter-componentcapacityis(threeunit-capacitylinksbeingcut),wehave.Theoptimalthroughputisthereforeupperbounded,evenwithnetworkcoding.Inthisexample,ourinformalintuitioncorrespondstothecon-ceptofsteinerstrength.Formally,inanundirectednetworkweconsiderpartitionsofthenetworkwherethereexistsatleastonesourceorreceivernodeineachcomponentofthepartition.bethesetofallsuchpartitions.ThesteinerstrengthofisdeÞnedas,whereisthetotalinter-componentcapacityonthesetoflinksbeingcut,andisthenumberofcomponentsinthepartition.Inourexample,thesteinerstrengthisOurdiscussionssofarhaveeffectivelyledtoTheorem1Ñinthecaseofasinglemulticastsession,theachievableoptimalthroughputisinbetweenthesteinertreepackingnumberandthesteinerstrengthinundirecteddatanetworks.Theorem1.Foranundirecteddatanetworkwithasinglemulticastsession,V,E,wehaveisthesteinertreepackingnumber,istheachievableoptimalthroughput,andisthesteinerstrength.Proof:First,optimalthroughputobtainedwithoutcodingisal-waysfeasiblewhencodingissupportedÑonejustignoresnet-workcodingandappliesthesameroutingstrategyasinthecasewithoutcoding.Thus,throughputwithcodingislowerboundedbythroughputwithoutcoding,and.Second,asarguedinthepreviousexample,thepartitionconditionisneces-saryforacertainmulticastthroughputtobefeasible.Thereforemulticastthroughputisupperboundedbysteinerstrength,andThoughtheinequalityincannotbereplacedbyequalityingeneral(anobviouscounter-exampleisshowninFig.2),onenaturallywondersifistrue,i.e.,iftheoptimalthroughputisequivalenttoitsupperboundÑthesteinerstrengthofthenetwork.Ifthisconjectureholdsingeneral,itwouldbeanicemax-mincharacterizationthatessentiallyren-dersboththeoptimalthroughputproblemandthesteinerstrengthproblemCo-NP.GiventhattheoptimalthroughputproblemisapparentlyNP,ifitisalsoCo-NP,itusuallyimpliestheexistenceofanefÞcientsolution.Ifwefurtherstudyourexample,wecanshowthatsteinerstrengthmaybepreciselyachievedbyusingthedirectednetworkshowninFig.3(b),obtainedbyappropriatelyorientingtheundi-rectedlinks.Usingthemaintheoremofnetworkcodingindi-rectednetworks,theoptimalmulticastthroughputistoboth.Inaddition,ifwecomputetheachievableoptimalthroughputwithoutcodingusingsteinertreepacking,sincewecanoptimallypacktrees(annotatedwithletters,withthetreehighlighted)withaweightofineach,theresultisalsoThefavorablepropertiesofthisexampleare,infact,duetoitsbroadcastnature:allnodesinthenetworkareinthemulticastgroup.Itisknownthatinthespecialcaseofasinglebroadcastsessionwherethesourcetransmitstoallnodes,steinertreepack-ingdegradestospanningtreepacking,andsteinerstrengthisre-namedtonetworkstrength.Thegoodnewsisthat,thespanningtreepackingnumberisalwaysequaltothenetworkstrengthinthecaseofbroadcast,independentlyprovedbyTutte[22]andNash-Williams[23].ItcanbederivedfromTheorem1that,inthebroadcastcase.However,wehaveinvestigatedthecomplexityofthesteinerstrengthproblem,andÞndittobeÑunfortunatelyÑNP-complete,asprovedinTheorem2.Theorem2.ThesteinerstrengthproblemisNP-complete.Proof:Wepresentabriefoutlineoftheproof.Wecanre-duceanotherwell-knownNP-completeproblem,maxcut[24],tothesteinerstrengthprobleminpolynomialtime.ThereductionworksinessentiallythesamewayastheonegivenbyDahlhausetal.,intheirNP-completenessproofforthemultiterminalcutproblem[25].Weobservethatintheinstancegraphconstructedintheirproof,theoptimalmultiterminalcutalwaysleadstotheratio,andisthereforealwaystheoptimalpartitionthatcorrespondstothesteinerstrengthvalueofthenet-work.Sincethemaxcutintheoriginalgraphcorrespondstotheoptimalmultiterminalcut,italsocorrespondstotheoptimalpar-titionforthesteinerstrength.Theremainingstepsoftheproofcanbefoundin[25]andareomittedinthispaper.Itimmediatelybecomesunlikelyforoptimalthroughputtobeequaltosteinerstrengthingeneral,sincethisimpliesthatthesteinerstrengthproblemisbothCo-NPandNP-complete,andsuchproblemsarenotbelievedtoexist.Still,itisexceed-inglyhardtoÞndaspeciÞccounter-examplewheretheoptimalthroughputisnotequaltothesteinerstrength.Fig.4showsoneofthecounter-examplesthatwehavefound,inwhichcasethesteinerstrengthis.Wewilllatershowthattheachievableoptimalthroughputis.Withthiscounter-example,wecanconcludethatneitheroftheinequalitiesinTheorem1canbere-placedbyequalityingeneral.Thoughbothsteinertreepackingandsteinerstrengtharenatu-ralresearchdirectionstowardsoptimalthroughputandoffertightThereasonisthat,theexistenceofapartitionwithserveasashortcertiÞcatefortheclaimthatnotransmissionstrategymayachieveahighermulticastthroughputthan,andviceversa. 5 444 4444488m0m1m2 Fig.4.Optimalthroughputandsteinerstrength:acounter-example.Thesource.Theoptimalpartitionisshown,cuttingthenetworktothreecomponents.Thesteinerstrengthofthenetworkisboundstotheproblem,theyarecomputationallyintractable.Itseemstosuggestthatcomputingtheoptimalthroughputanditscorrespondingtransmissionstrategiesmayalsobecomputation-allyintractableingeneral.OursearchforefÞcientsolutionstoachieveoptimalthroughputcontinues.C.EfÞcientsolutionsforthroughputoptimization:ThecFlowLinearProgramContrarytothepreviouspessimisticviews,wepresentthesurprisingresultthatefÞcientsolutionsdoexistforcomputingoptimalthroughputinundirectednetworks,includingthecorre-spondingroutingandcodingstrategiesthatachievesuchthrough-put.TodesignsuchefÞcientsolutions,weÞrstformulatetheproblemasalinearoptimizationproblem,inwhichboththenum-berofvariablesandthenumberofconstraintsareboundedby.Wethenshowthattheresultofsuchoptimizationexactlygivesthemaximumachievablethroughput,aswellasthecorrespondingroutingstrategy. ve a1 a2 orientation Fig.5.Orientingeachundirectedlinkinthenetworkintotwodirectedones,suchthatWebeginbypresentingtheorientationconstraintsofthelin-earprogramthatcomputesoptimalthroughput.AsillustratedinFig.5,anofanetworkisastrategytoreplaceeachundirectedlinkwithtwodirectedlinks,suchthat.Af-tertheorientation,theundirectedlinksetbecomesadirectedlinkset,withthenumberoflinksinthesetdoubled.Sincethecapacityofadirectedlinkmustbenon-negative,wealsohave.Forexample,Fig.3(b)isanorientationoftheundirectednetworkinFig.3(a).Weproceedtoconsiderßowsfromthesourcetothemulticastreceivers.Totakeadvantageofthepowerofnetworkcodingtoresolvecompetitionforlinkcapacities,weintroducetheconceptconceptualßows).WedeÞneconceptualßowsasnet-workßowsthatco-existinthenetworkcontendingforlinkcapacities.Incomparison,ifweviewinformationßowsascommodityßows(wheredatacanonlybeforwarded,andcannotbecodedorreplicated),theydocompetefortheca-pacityofasharedlink,similartothecaseofallunicastbetweenthesourceandeachofthereceivers.Fig.6illustratesthefun-damentaldifferencebetweenconceptualandcommodityßowswithrespecttoachievableoptimalthroughput,inarandomnet-workofnodesgeneratedbyBRITE[26],withthesizeofthemulticastgroupincreasing.Therapiddecreaseofoptimalses-sionthroughputisduetocompetitionforsharedlinkcapacitiesamongdifferentcommodityßows,especiallyatthesource. 50100 2201015Conceptual flows Commodity flows Achievable throughput (Kbps)Number of nodes in the multicast group (|M|) Fig.6.Achievableoptimalthroughput:acomparisonbetweenßows,asthenumberofnodesinthemulticastsessionincreases.Ourlinearprogramtocomputetheoptimalthroughputwithcoding,showninTableI,isreferredtoastheLPsinceitisbasedonconceptualßows.IntheLP,isthesourceandarethemulticastreceivers.aretheconcep-tualßowstoeachofthereceivers.EachspeciÞesaßowrateforeachdirectedlinkdenotesthetotalin-ßowrateatanode,similarfor.Finally,theisthetargetßowrateofoptimization.Inadditiontotheorientationconstraints,theLPalsoincludesthenetworkßowconstraintsforeachconceptualßow,andtheequalrateconstraints.ThenetworkßowconstraintsarespeciÞedinacompactformforallconceptualßows,whichre-quiresthefollowing:(1)Flowratesmustbenon-negativeandupperboundedbylinkcapacities;(2)ßowconservation,whichrequiresthattheincomingßowrateintheconceptualßowequalstooutgoingßowrateinatarelaynodefor;and(3)theincomingßowrateatthesourceandtheoutgoingßowratesatthereceiverareallzero,foreach.Theequalrateconstraintsrequirethattheßowratesofconceptualßowsareidentical,withbeingtheuniformßowrate.Withtheselinearconstraints,thetargetßowrateisthenmaximized.TABLEI f Subjectto: Orientationconstraints:Independentnetworkßowconstraintsforeachconceptualßow:w:..k],aDfi(a)C(a)i[1..k],aDfiin(v)=fiout(v)i[1..k],vVŠ{m0,mi}fiin(m0)=00..k]fiout(mi)=0i[1..k]Equalrateconstraints: Wearenowreadytoformallypresentoneofourmaincontri-butionsofthispaper.WeshowthattheLPprovidesanefÞcientalgorithmtocomputetheachievableoptimalthrough-put,aswellastheroutingstrategy.Theorem3.Foranundirecteddatanetworkwithasinglemulticastsession,V,E,themaximumend-to-endthrough-itscorrespondingoptimalroutingstrategycanbecomputedinpolynomialtimeusingtheLP,inwhichboththenumberofvariablesandthenumberofconstraintsarepoly-nomial,andontheorderof.Theconceptualßowsconstitutetheoptimalroutingstrategy.Proof:Theorientationconstraintsreßectcompleteßexibilityinorientingtheundirectednetwork,withoutbeingtoorestrictiveortoorelaxed.ForeachÞxedorientation,conceptualßowsarebeingmaximizedwithindependentandstandardnetworkßowconstraints,aswellastheextraconstraintthatconceptualßowratesareequaltoeachother.Therefore,theresultofthemaxi-mizationisthemaximumpossibleßowratethatcanbeindepen-dentlyachievedfromthesourcetoallreceivers,overallpossibleorientationsofthenetwork:=max[minßowratedenotesallpossibleorientationsofthenetwork,andisthesetofmulticastreceivers.Therecentbreak-throughinnetworkcoding[5],[6]showsthat,foraÞxedorien-tationofthenetwork,aratecanbeachievedfortheentiremul-ticastsessionifandonlyifitcanbeachievedforeachmulticastreceiverindependently.Thisimpliesthat,themaximumthrough-putineachorientationequalstotheminimumofthemaximumreceiverßowrate.TheLPessentiallymaxi-mizesthismin-maxßowoverallpossiblenetworkorientations,andobtainsthemax-min-maxßowthatispreciselythemaximummulticastthroughputintheoriginalundirectednetwork.Further,thesourcemaytransmitinformationtoeachreceiveringtotheconceptualßow.Shouldmorethanoneconceptualßowsutilizecapacityonthesamelink,theconßictcanalwaysberesolved,providedthatnetworkcodingisappliedappropriately[5],[6].LPcontainsorientationvariablesvirtualßowvariables,andonetargetßowratevariable.Therefore,thetotalnumberofvariablesis+1)whichisontheorderof.Inaddition,theorientationconstraints,workßowconstraints,aswellasequalrateconstraints.Thetotalnumberofconstraintsis,therefore,1)+3,whichisalsoontheorderofTheoptimalroutingstrategycomputedbyLPspeci-Þestherateofdatastreamsbeingtransmittedalongeachlink.Basedontheroutingstrategy,weneedtoperformtheadditionalstepofcodeassignmenttocomputethestrategy,beforedatastreamsmaybetransmitted.Thecodingstrategyincludesonetransformationmatrixforeachnode,whichspeciÞeshowincomingdatastreamsarelinearlycodedintooutgoingstreams.GiventheroutingstrategyfromtheLP,thereexistpoly-nomialtimealgorithmstoperformsuchcodeassignments.Forcompletenessofthispaper,weincludeamoredetaileddiscus-sionintheAppendix.TheexistenceofpolynomialtimecodeassignmentalgorithmsleadstothefollowingcorollaryofTheo-rem3:Corollary.Thecompletesolutionthatachievesoptimalthrough-putinundirecteddatanetworkswithasinglemulticastsessioncanbecomputedinpolynomialtime,includingboththeroutingandcodingstrategies.AsanexampleofapplyingtheLPonactualnetworks,Fig.7showstheoptimalorientationtoachievetheoptimalthroughputofinthepreviousnetworkinFig.4. 444 4444488 2.51.5 2.51.5 0.50.5 0.50.5m0m1m2 Fig.7.TheoptimalnetworkorientationcomputedbytheLP,inwhichthemaximumßowandthemaximumßowarebothInordertoevaluatetheadvantageofnetworkcodingwithre-specttoimprovingachievableoptimalthroughput,wehaveim-plementedboththeLPandabrute-forcealgorithmtocom-putethesteinertreepackingnumber.Thesteinertreepackingalgorithmenumeratesallsteinertreesinthenetwork,assignsaßowvariabletoeachtree,andthenmaximizesthesummationofalltreeßows,subjecttotheconstraintsthatthetotalweight(throughput)oftreesusingeachlinkshouldnotexceeditscapac-ity.WehaveevaluatedboththeLPandthesteinertreepack-ingalgorithmusingourpreviousexamples,aswellasasetofuniformbipartitenetworks,whicharebelievedtobegoodcandi-datestoshowthepowerofcodingonimprovingthroughput[7],[27].Auniformbipartitenetworkn,kconsistsofthedatasourceandtwolayers:onewithrelaynodesandtheotherwithreceivers.Eachrelaynodeisconnectedtothesender,andeachreceiverisconnectedtoadifferentgroupofrelaynodes,andalllinkshaveacapacityof.Forinstance,isshowninFig.8,theexampleinFig.2is,andtheclassicexampleofnetworkcodinginFig.1isisomorphicto Fig.8.TheuniformbipartitenetworkTableIIsummarizestheresultsofourempiricalstudies,fromwhichwehavederivedthefollowingobservations.First,theLPismuchmorescalableandefÞcientthansteinertreepacking,whichfailstocomputeasolutionforanetworkassmall ,withonlynodesandlinks,butalmostliondifferentsteinertrees.Inseparateexperiments,theLPisabletocomputetheoptimalthroughputfornetworkshav-ingthousandsofnodes.Second,theresultssupportTheorem1,andshowthattheoptimalthroughputwithcodingisalwayslowerboundedbythatwithoutcoding;however,networkcodingonlyintroducesaslightadvantage,withthenohigherthan,eveninuniformbipartitenetworkswherecodingisparticularlybeneÞcial.TABLEIIOMPUTINGOPTIMALTHROUGHPUTSTEINERTREEPACKING Network |V| |M| |E|
(N) (N)
(N) (N) oftrees Fig.1 7 3 9 2 1.875 1.067 17 C 4 9 2 1.8 1.111 26 Fig.7 8 3 16 13.5 13.5 1.0 298 C 5 16 3 2.667 1.125 1,113 C 7 16 2 1.778 1.125 1,128 C 6 25 4 3.571 1.12 75,524 C 11 25 2 1.786 1.12 119,104 C 11 35 3 Ð Ð 49,956,624 IV.AHROUGHPUTINATAETWORKSOurefÞcientsolution,theLP,canbeextendedtosolvetheoptimalthroughputproblemincasesbeyondasinglemulti-castsession.WepresentourcontributionstoextendtheLPtounicast,broadcastandgroupcommunicationsessions,aswellastothecaseofmultiplecommunicationsessionsandtothemodelofoverlaynetworks.A.Thecasesofunicast,broadcastandgroupcommunicationSinceunicastandbroadcastcanbeviewedasspecialcasesofmulticastwheretwonodesandallnodesareinthemulticastgroup,respectively,oursolutioninthesinglemulticastcasecanbereadilyappliedtoasingleunicastorbroadcastsessionwithoutmodiÞcations.Inthecaseofaunicastsession,theLPes-sentiallysolvesalinearprogramforasinglenetworkßow.Inthecaseofabroadcastsession,theLPcomputestheoptimalbroadcastthroughput,whichhasbeenshowntobethesameasboththespanningtreepackingnumberandthenetworkstrength.Traditionally,thesethreeequalquantitieshavebeencomputedfromeithertheperspectiveofnetworkstrengthorspanningtreepacking.Cunningham[28]Þrstgaveacombinatorialalgorithmthatcomputesthenetworkstrength,whichwaslaterimprovedbyBarahona[29].Bothalgorithmsarebasedonmatroidtheory,andarehighlysophisticated.ThoughthespanningtreepackingproblemhasanLPformulation,thenumberofvariablesisex-ponential.Itisthereforenecessarytoworkonitsdualprogram,wheretheminimumspanningtreealgorithmscanserveastheseparationoracle.Incomparison,theLPismuchsim-plertoimplementyetefÞcienttorun,withapolynomialnumberofconstraintsandvariables,buildinguponmaturetheoriesandalgorithmsoflinearoptimization. S SSff1221 Fig.9.Transforminggroupcommunicationintomulticasttransmission.Groupcommunicationreferstomany-to-manycommunica-tionsessionswheremultiplesourcesmulticastindependentdatatothesamegroupofreceivers,thesetofsendersandthesetofreceiversmayormaynotoverlap.Previouswork[6]hasshownthatamany-to-manysessioncanbeeasilytransformedintoamulticastsession,byaddingasource,whichisatradi-tionaltechniqueinnetworkßows.AsillustratedinFig.9,wecanaddanadditionalsourcetothenetwork,andconnectittoeachofthesourcesinthegroupcommunicationsession,withlinksofunboundedcapacity.Wemaythenapplythetomaximizethemulticastthroughputfromtoallthereceivers.Additionalconstraintscanbeappliedtoßowratesonthenewlyaddedlinksbetweenthesupersourceandtheoriginalsourcesinthesession,governingfairnessamongtheoriginalsources.TheoutcomefromtheLPistheoptimalthroughputanditscorrespondingroutingstrategyfortheoriginalgroupcommuni-cationsession.B.ThecaseofmultiplesessionsInitsmostgeneralform,theoptimalthroughputproblemal-lowsmultiplecommunicationsessionsofdifferenttypestoco-existinthesamenetwork.SincemulticastisrepresentativeÑinthatunicast,broadcastandgroupcommunicationcanallbetrans-formedintomulticastÑitissufÞcienttoconsidertheoptimalthroughputprobleminthecaseofmultiplemulticastsessions.Toachieveoptimalthroughputwithmultiplesessions,weneedtoconsidertheproblemofinter-sessionfairness.ThedeÞnitionoffairnessisusuallyapplicationdependent;however,aslongasitcanbeexpressedusinglinearconstraints,wecaneasilyin-cludethemintheLPformulation.Withrespecttonetworkcod-inginmultiplesessions,itistheoreticallypossibletoapplynet-workcodingonmultipleincomingstreamsofdifferentsessions.However,weargueagainstthispossibility,andusecodingbybyi.e.,networkcodingisonlyappliedtoincom-ingstreamsofthesamesession.Thisargumentismainlysup-portedbythecomputationalintractabilityoftheoptimalthrough-putproblemifinter-sessioncodingisallowed.Inaddition,ourempiricalexperiencesshowthatallowinginter-sessioncodingcanhardlyimproveoptimalthroughput,anditisnotpracticaltocodedatastreamsfromdifferentapplications.LPgiveninTableIIIisdesignedtosolvetheopti-malthroughputproblemwithmultiplemulticastsessions,whereweuseweightedproportionalfairnessasthefairnessmodel.ItItisknownthatÞndingsufÞcientandnecessaryconditionsforthefeasibilityofmultiplesessionsinthiscaseisequivalenttoÞndingapointinanalgebraicvariety,whichisNP-hard[6]. istheresultofextendingtheLPtoitsmulticommodityvariant.Weassumethereexistatotalofmulticastsessions,numberedas.Eachsessionhasasource,anumberofreceivers,asetofconceptualßowsaswellasaweightindicatingtheimportanceofthesession.Thescalaristhecommonrateforconceptualßowswithin,thescalaristhecommonweightedthroughputforallthemulticastsessions,andthetargetoftheLPistoLPreplacesthestandardnetworkßowconstraintsintheLPwithasetofmulticommoditySinceßowsofdifferentsessionscontendforlinkcapacity,thesummationoftheper-sessionßowratesshouldnotexceedlinkcapacities.Sinceßowswithinthesamesessiondonotcompeteforlinkcapacity,theeffectiveßowratewithinasessiononlink)=maxmax..ki]fij(a).Thefunctionisnotlinear,sothisconstraintisrelaxedtoto...ki].ThevalidityofthisrelaxationwillbeshownintheproofofThe-orem4.ItiseasytocheckthatthetotalnumberofvariablesandthetotalnumberofconstraintsintheLParebothontheorder,whereisthenumberofsessionsandthesizeofthelargestmulticastgroup.TABLEIII f Subjectto: Orientationconstraints:constraints:..s],j[1..ki],aDfij(a)fi(a)i[1..s],j[1..ki],aD
si=1fi(a)C(a)aDfijin(v)=fijout(v)i[1..s],j[1..ki]vVŠ{mi0,mij}fijin(mi0)=00..s],j[1..ki]fijout(mij)=00..s],j[1..ki]Equalrateconstraints:constraints:..s],j[1..ki]Fairnessconstraints: Theorem4.Inthecaseofmultiplemulticastsessions,theopti-malend-to-endthroughputanditscorrespondingoptimalroutingstrategyinundirecteddatanetworkscanbecorrectlycomputedbytheLP,withcodingbysuperposition.Proof:ThecorrectnessoftheLPbuildsuponthecor-rectnessoftheLP,whichwehaveproved.Weonlyneedtoverifythattherelaxationof)=maxmax...ki]fij(a)tofi(a)fij(a),j[1...ki]doesnotaffecttheoverallre-sultofoptimization.Notethatvariablesappearonlyoncelaterinthelinearprogram,wherethetotaleffectiveßowratesarerequiredtobeupperboundedbylinkcapacities.Therefore,ifthereisafeasiblesolutionpointforthelinearprograminwhich=maxmax...ki]fij(a),changingthevaluesof)=maxmax..ki]fij(a)mustresultinanotherfeasiblesolutionpointwiththesameoptimizationresult,sincethenewvaluesstillsatisfyalltheconstraints.Therefore,therelax-ationwillnotleadtoahigheroveralloptimalvalue.C.ThecaseofoverlaynetworksSinceneithernetworkcodingnordatareplication(forIPmul-ticast)arewidelysupportedinthecurrent-generationnetworkelementsinthecore,weconsiderthecaseofoverlaynetworkswhereonlytheendhostshavethefullcapabilitiestoforward,replicateandcodedatastreams,andthecorenetworkelements(henceforthreferredtoasrouters)mayonlyforwarddatapacketsasis.Wenotethatthecaseofoverlaynetworksisactuallymoregeneralthantheclassicalmodelofundirecteddatanetworkswehaveusedsofar,whichhintsthattheoptimalthroughputproblemmaybecomehardertosolve.V,Ebeanoverlaynetworkwithamulti-castsession.Themulticastgroupisasubsetoftheendhosts.Ifi.e.,allendhostsareinthemulticastgroup,Gargetal.[30]hasshownthattheoptimalmulticastthroughputcanbeefÞcientlycomputedinthiscase,byworkingonthedualpro-gramofanaturalLPformulation.Ithasalsobeenshownin[30]that,inthegeneralcasetheoptimalthroughputproblemwithoutnetworkcodingistheoverlaysteinertreepackingproblem,andisstillAPX-hard.Withthesupportofnetworkcoding,however,weareabletoextendtheLPtoitsoverlayvariant,referredtoastheLP,tosolvetheoptimalthroughputprobleminthemodelofoverlaynetworks.TheLPtakesahierarchicalviewofthemulticasttransmission,withanandanoverlaylevel.Theunderlaylevelcorrespondstothephysicalnetworktopology,andhasmulticommodityßowsconnectingeachpairofendhosts,onlyviaroutersasintermediatenodes.Theoverlaylevelisconceptual,andcontainsendhostsfullyconnectedasacompletegraph.Thelinkhasacapacityequaltotheunderlayßowrate.WethenapplytheLPintheoverlayleveltomaximizetheend-to-endthroughput,whereeachnodeiscapableofreplicationandIntheLPshowninTableIV,weincludethreegroupsofconstraints.First,theorientationconstraintsareidenticaltothoseincludedintheLP.Second,thestandardmulticom-modityßowconstraintsarespeciÞedfortheunderlayßowsbe-tweenendhostsandonlyviarouters.Third,weintroducethemappingconstraintsthatmaptheunderlayßowratetotheoverlaylinkcapacity(referredtoas),andthenapplytheoriginalconstraintsintheLPattheoverlaylevel.ThetargetoftheLPistomaximizethroughputintheover-laylevel.ItiseasytoderivethatthenumberofvariablesandthenumberofconstraintsintheLParebothboundedby TABLEIV f Subjectto: Orientationconstraints:Underlaymulticommodityßowconstraints:i,jj..h],aD
gij(a)C(a)i,jj..h],aDgijin(v)=gijout(v)i,jj..h],vVŠHgijin(v)=0i,jj..h],vHŠ{mj}gijout(v)=0i,jj..h],vHŠ{mi}Overlayi,jj..h]0fi(a)i[1..k],aD={aij|1i,jj..k]fiin(v)=fiout(v)i[1..k],vHŠMfiin(m0)=0 Theorem5.Inthecaseofasinglemulticastsessioninthemodelofoverlaynetworks,theoptimalend-to-endthroughputanditscorrespondingoptimalroutingstrategycanbecorrectlycomputedbytheLP.Proof:Sincerelaynodesintheoverlaynetworkcannotreplicateorencodedata,adatastreamthatistransmittedbetweentwoendhostswithoutpassingathirdendhostremainsunchangedthroughoutthetransmissionanduponarrival.Therefore,itisvalidtomodelthesedirecttransmissionsbetweenendhostsasmulticommodityßows.ThevalidityoftheintheoverlaylayermaybederivedfromthecorrectnessoftheLP,whichwehaveprovedinTheorem3.Similartotheextensionfrom,onemayextendLPintoitsmulticommodityvarianttoaccommodatemultiplesessionsinoverlaynetworks.MorespeciÞcally,oneneedstoreplacetheoverlayconstraintswiththeoverlayconstraintsinthethirdgroupofconstraintsoftheLP.TheresultinglinearprogramhasbothitsnumberofvariablesandnumberofconstraintsboundedbyThisisusuallynotworsethanthoseofthesingle-sessionLP,sinceinmostcases.V.AHROUGHPUTMPIRICALANDDuetothelackofefÞcientalgorithms,previousstudiesontheproblemofimprovingsessionthroughputarelargelybasedonexperimentalorintuitiveinsights.WearguethattheavailabilityoftheLPshassigniÞcantlychangedthelandscape,andhasmadeitcomputationallyfeasibletostudytheexactbeneÞtsofvariousproposalstoachievehigherthrough-put,includingasinglemulticasttreewithdatareplication,multi-plemulticasttrees,andnetworkcoding.Basedonbothempiricalandtheoreticalstudies,weseektoansweranumberofimpor-tantquestionsandtopresentfundamentalinsightsonachievingoptimalthroughput.Whileourtheoreticalinsightsaresupportedbyproofs,ourem-piricalstudiesarebasedontheimplementationofallthreeLPsthatwehaveproposed.Incomparisonstudies,wehavealsoimplementedalgorithmstocomputetheoptimalthroughputwithmultiplemulticasttreesbutwithoutcoding(i.e.,thesteinertreepackingnumber),theoptimalthroughputwithasinglemulti-casttree,aswellastheoptimalthroughputwithallunicastfromthesourcetoallreceivers.TopologiesusedinoursimulationsaregeneratedbytheBRITEtopologygenerator[26],withsizesrangingfromnodes,bothwithandwithoutpower-lawproperties,withheavy-tailedorconstantlinkcapacities.WeuseasourLPsolverofchoice.Howadvantageousisnetworkcodingwithrespecttoimprovingoptimalthroughput?Theratioofachievableoptimalthroughputwithcodingoverthatwithoutcoding(i.e.,)isreferredtoastheingadvantage.Recallthatwehaveinvestigatedthecodingad-vantageinTableI,wherewetestseveraluniformbipartitenet-works,andareunabletoexperimentallyÞndcaseswherenet-workcodingmayimproveoptimalthroughputbyafactorhigher.Wearenaturallyledtothequestion:Whatistheupperboundofthecodingadvantage?Theoretically,previouswork[7]arguesthatthereexistmulti-castnetworkswherethecodingadvantagegrowsproportionallylog(,andisthusnotÞnitelyboundedingeneral.Thisresultisobtainedindirectedacyclicnetworkswiththeintegralroutingmodel.Weshowthat,inundirectednetworkswiththefractionalmodel,thecodingadvantageisboundedbyaconstantfactorofTheorem6.Formulticasttransmissionsinundirecteddatanet-works,thecodingadvantageisboundedbyaconstantfactorofProof:Weusetorepresenttheminimumofthepair-wiseconnectivityamongnodesinthemulticastgroup.Weproceedtoprovethatalwaysholdsbyshowing Sincemulticastthroughputisalwaysboundedbyedgecon-nectivitybetweenthesourceandamulticastreceiver,obviouslyholds.Intheremainderofthisproof,weshow intwosteps:(1)from,weobtainanetworkcontainingabroadcasttransmissionwithoutrelaynodes,such holdsonlyif holds;(2)weprove holdsforInstep(1),werepeatedlyapplyoneofthefollowingtwooper-ationsonthenetwork:(a)applyacompletesplittingatanon-cutrelaynode,preservingpairwiseedgeconnectivitiesamongmul-ticastnodesin;or(b)addarelaynodethatisacutnodeintothemulticastgroupAllthealgorithmswehaveimplemented,aswellasallnetworktopologiesusedinthispaper,arepubliclyavailable.Thelocationisomittedtoaccommodatedouble-blindedreviews. Asplitoffoperationatanodereferstothereplacementofsome2-hopunitpathbyadirectunitedgebetween,asillustratedinFig.10.Acompletesplittingatmeanstorepeatedlyapplysplitoffoperationsatisisolated. vz uvz Fig.10.AsplitoffatnodeMaderÕsUndirectedSplittingTheorem[31]guaranteesthatoperation(a)isalwayspossible,giventhatnon-cutrelaynodesexist:z,Ebeanundirectedgraphsothatthereisnonode-cutincidenttoandthedegreeiseven.Thenthereexistsacompletesplittingatpreservingthelocaledge-connectivitiesofallpairsofnodesin.Notethattheevennodedegreerequirementisirrelevantinthefractionalmodel.Therefore,aslongastherearerelaynodesinthenetwork,wecansuccessfullyapplyeither(a)or(b)todecreasethenumberofrelaynodesbyone.Eventuallyweobtainanetworktainingnodesinthecommunicationgrouponly.Furthermore,operation(a)doesnotincrease,andoperation(b)doesnotaffecteither.Therefore,if afterapplyingeitheroperation,itholdsbeforeapplyingtheoper-ationaswell.Wenowprovestep(2), .ByNash-WilliamsÕWeakGraphOrientationTheorem[31],agraphhasan-edgeconnectedorientationifandonlyifitis-edgeconnectedThereforethereexistsanorientationofwherethedirectedconnectivityisatleast .Withinsuchanorientation,themax-ßowfromthesourcetoeachreceiverisatleast ,andthereforethebroadcastthroughputachievedisatleast Recallthatforbroadcasttransmissions,optimalthroughputwithcodingequalstothatwithoutcoding.Wethushave Havingprovedatheoreticalboundof,Wehavefurtherstud-iedthecodingadvantageusingoverrandomlynetworks,ratherthanuniformbipartitenetworksusedinTableI.Thenetworkswetestareofsmallsizes(withuptoduetothepoorscalabilityofthesteinertreepackingalgorithmtocomputeoptimalthroughputwithoutcoding.Wehaveobservedthecodingadvantageremainsinallrandomtopolo-i.e.,networkcodingisnotbeneÞcialtoimproveoptimalthroughput.Webelievetheobservationholdsforlargernetworksaswell,wherethetopologiesandlinkcapacitiesremainrandom.ThisimpliesthatthefundamentalbeneÞtofnetworkcodingishigheroptimalthroughput,buttofacilitatesigniÞcantlymoreefÞcientcomputationandimplementationofstrategiestoachievesuchoptimalthroughput.Ourtheoreticalstudyhasshownaboundof,whilethehigh-ratiowehaveobservedinourempiricalstud-iesisonly.Ifthehighestratioingeneralisindeednearend,itwouldleadtobetterapproximationalgorithmsfortwotheoreticalproblems:steinertreepackingsteinertree,whichcanbeapproximatedtoexactlythesameratio[2],[30].Todate,thebestknownapproximationratiofortheseproblemsisaroundaroundHowadvantageousisstandardmulticastcomparedtounicastandoverlaymulticast?LPisinstrumentaltopreciselycomputetheachiev-ableoptimalthroughputwithonemulticastcommunicationses-sion,eitherwithnetworkcodingorwithmultiplemulticasttrees,sincetheoutcomesfromthetwoarehardlydifferent.Ineithercase,datareplicationneedtobesupportedonallnetworknodes,includingcorenetworkelements.Ithasbeencommonknowl-edgethat,whencomparedtounicastfromthesourcetoallre-ceivers,standardmulticastbringsbetterbandwidthefÞciencyandhigherend-to-endsessionthroughput.However,eveninthecaseofunicast,pathdiversityneedstobeexploitedtoachieveopti-malthroughput,equivalenttothemaximumunicommodityßowproblem.Itisnotimmediatelyclearhowadvantageousstandardmulticastis.Overlaymulticastbalancesthetradeoffbetweenthepractical-ityofstandardmulticastandunicast.Itreferstothecasewhereonlythemembersofthemulticastgroupmayreplicateorcodedata,whereasallothernodesmayonlyforwarddata.TheoptimalthroughputachievedbyoverlaymulticastisefÞcientlycomputedbytheLP.Weperformaquantitativestudythatcomparestheoptimalthroughputachievedwithstandardmulticast,overlaymulticastandunicast.Thestudyisperformedinrandomnetworksofdif-ferentsizes,thelargestofwhichhasnodesandoverlinks.Therearemembersinthemulticastgrouprespec-tively,intwodifferentsetsoftests.Multicastnodesarerandomlyselectedfromallnetworknodes,withdifferentmulticastgroupsbeingasdisjointaspossible.Foreachnetworksize,multipletestsareperformedwithdifferentnetworktopologiesanddiffer-entchoicesofthemulticastgroup,theresultsarethenaveraged. 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25 30 35 Number of nodes in the networkOptimal throughput (Kbps)(a) Size of multicast group = 3 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25 30 35 Number of nodes in the networkOptimal throughput (Kbps)(b) Size of multicast group = 10 Overlay multicast Fig.11.Achievableoptimalthroughputusingstandardmulticast,overlaymul-ticast,andallunicastfromthesendertoallreceivers.AswemayobservefromFig.11,thereexistsobviousdif-ferencesbetweenstandardmulticastthroughputandallunicastthroughput,andthedifferencesaremoresigniÞcantinFig.11(b),wherethescaleofthemulticasttransmissionislarger.Sur-prisingly,theÞgurealsosuggeststhat,theoptimalthroughputachievedbyoverlaymulticastisalmostidenticaltothatachievedbystandardmulticast,whereallnetworknodesareabletorepli-cateorcodedata.Onaverage,theoptimalthroughputofover-laymulticastisoverofstandardmulticast.Thisobserva-tionshowsthat,fromtheperspectiveofmaximumachievablethroughput,whiletheremayexistcontrivednetworktopologiesthatshowmoresigniÞcantadvantagesofstandardmulticastoveroverlaymulticast,littledifferenceremainsoncelargescaleprac-ticalnetworktopologiesareconsidered. 11 10 20 30 40 50 60 70 2030405060708090100200300400500 Overlay multicast Fig.12.BandwidthefÞciencyofstandardmulticastandoverlaymulticast.Fig.12showsthatthenear-optimalthroughputofoverlaymul-ticastcomeswithaprice,intheformofaslightlylowerwidthefÞciencythanthatofstandardmulticast.Inthiscompari-bandwidthefÞciencyiscomputedastheachievablesessionthroughputdividedbythetotalbandwidthconsumptiononalllinks,andthenscaledbythenumberofreceiversinthemulticastgroup.ThisdeÞnitionimpliesthatanend-to-endtransmissionstrategythatconsumesbandwidthonalargesetoflinksisnotfavorableforthesamesetofreceivers.WemayobservefrombothFig.11andFig.12that,forrandomnetworktopologies,therenaturallyexistresidualcapacitiesafterapplyingtheopti-maltransmissionstrategyofstandardmulticast.Suchresidualcapacitiesmaybeexploitedbyoverlaymulticasttoimproveitsoptimalthroughput,andcontributetotheinsigniÞcantdifferencebetweentheoptimalthroughputachievedbystandardandover-laymulticast.Incaseswherelittleresidualcapacityexistsinthestandardmulticaststrategy,throughputofoverlaymulticastshouldbeno-ticeablylower.ThisisconÞrmedbythenetworkinFig.7,pre-viouslypresentedinSec.III.Inthisexample,thetransmissionstrategyofstandardmulticastutilizesalllinkcapacitiestoinordertoachievetheoptimalthroughputof.Themax-imumthroughputforoverlaymulticastinthistopologyisatalowervalueof,asexpected.Howsensitiveisoptimalthroughputtonodejoins?Whennewnodesjointhemulticastsession,howmayachiev-ableoptimalthroughputbeaffected?Intuitively,ifarelaynodejoinsthemulticastgroupandbecomesanewreceiver,theachiev-ablesessionthroughputshoulddecrease,duetothefollowingtwocauses:(1)alargernumberofreceiversmayleadtomoreintensecompetitionforbandwidth;and(2)anewnodewithlowcapacitymaybecomeabottleneckandlimitthethroughputfortheentiresession.Oursimulationresultsshowthat,thesecondcausehasamuchmoresigniÞcantimpactthantheÞrstone.Fig.13(a)showsvariationsofoptimalthroughputasthenum-berofnodesinthemulticastgroupincreasesfromthreeto,andthento(effectivelyabroadcastsession),forvariousnetworksizes.Inthisexperiment,networktopolo-giesaregeneratedwithtwoedgespernodewithoutpower-lawrelationships,withheavy-tailedbandwidthdistributionbetween10and50Kbpsonthelinks.Aswecanobserve,whenthesizeofthemulticastgroupincreasesfromthreeto,theef-fectsonachievablethroughputisrathersigniÞcant.However,furtherexpandingthemulticastgrouptotheentirenetworkleadstoamuchsmallerdecrease.Bothcausesthatwehavediscussedcontributetotheinitialdecreaseofthroughput,whilethesecondcause(i.e.,theeffectsofabottlenecknode)playsalessimpor-tantroleinthesubsequentdecreaseÑwhenthemulticastgroupcontainshalfofthenodesinthenetwork,itisverylikelyforthegrouptohavealreadycontainedanodewithlowcapacity. 25 30 35 40 45 50 55 60 65 0 5 10 15 20 25 30 35 40 45 (a) Heavytailed link capacity 25 30 35 40 45 50 55 60 65 0 5 10 15 20 25 30 (b) Constant link capacity |M|=|V|/2 Fig.13.VariationsofoptimalthroughputduetonewnodesjoiningthemulticastWefurtherperformedthesametestsonpower-lawnetworktopologieswithKbpsconstantlinkbandwidth,andthere-sultsareshowninFig.13(b).Inthepower-lawtopologies,mostnodeshavesmalldegreesoftwoorthree,whileasmallnum-berofnodeshavehighdegrees.Therefore,theinitialmulti-castgroupusuallycontainsanodewithasmalldegreealready,whichalsohasalowcapacity,sincethelinkbandwidthiscon-stant.Inthiscase,onlyinter-receiverbandwidthcompetitionre-mainsasamajorconcern.However,aswecanobserveintheÞgure,inmostcasestheoptimalmulticastthroughputremainsroughlyconstant,evenafterallthenodeshavejoinedthemulti-castsession.Thiscounter-intuitiveobservationshowsthat,newreceiversmaysharebandwidthwithexistingreceiverswell,anddonotsigniÞcantlyaffecttheachievablethroughput,aslongastheircapacitiesarenottoolow.SpikesinFig.13(b)correspondtotheoccasionalcaseswherenodesintheinitialmulticastgroupallhaverelativelyhighcapacities.BothresultsinFig.13(a)and13(b)haveledtothesameobservationthat,whennewnodesjoinamulticastsession,thedecreasedoptimalthroughputismainlyduetobottleneckreceiverswithlowercapacities.Howsensitiveisoptimalthroughputtotheadditionofnewses-Whennewsessionsareaddedtothenetwork,howdotheyaf-fectachievableoptimalthroughput?TheLP,presentedinSec.IV,makesitfeasibletocarryoutourempiricalstud-ies.Fig.14showsthevariationofoptimalthroughputasnewcommunicationsessionsarecreated.Threetypesofthroughputareshown:(1)previousoptimal,whichrepresentstheoptimalweightedsessionthroughputbeforethenewsessionisadded;(2)incremental,whichistheweightedthroughputforthenewses-sionusingresiduallinkcapacitiesonly,orjustthepreviousop-timalthroughputiftheachievablethroughputofthenewsessionishigher;and(3)re-optimized,whichisthere-computedoptimalsessionthroughputafterthenewsessionisadded.Fourgroupsofsimulationsareperformed,withtwo,three,four,andÞveexist-ingsessions,respectively,beforethenewsessionisestablished.EachmulticastgrouphasasizeÞve,andnodesindifferentmulti-castgroupsarechosentobeasdisjointaspossible.Eachsession isassignedanequalweight. 5 10 15 20 25 121416182050100200300500 5 10 15 20 25 1216182050100200300500Optimal throughput (Kbps) 5 10 15 20 25 1216182050100200300500 5 10 15 20 25 1216182050100200300500Number of nodes in the networkOptimal throughput (Kbps) incremental Fig.14.Throughputvariationsasanewsessioniscreated.ResultsinFig.14showthat,theadditionofanextrasessiondoesnotdramaticallyaffecttheachievableoptimalthroughput,especiallywhenthenetworksizeislargeincomparisontothenumberofnodesinvolvedinthetransmissions.However,iftheexistingsessionsremaintransmittingaccordingtotheoptimaltransmissionstrategycomputedbeforethenewsessionjoins,andonlyresidualcapacitiescanbeutilizedtoservethenewsessionincrementalthroughputcase),thentheresultingthroughputisnotsatisfactoryunlessthenumberofsessionsisverysmall).Ingeneral,thismayleadtoverylow,evenzero,through-putforthenewsession.Thereforeitisnecessarytoperformre-optimizationbeforeanewsessionstartstotransmit.Howsensitiveisoptimalthroughputtofairnessconstraints?Inordertoinvestigatehowinter-sessionfairnessrequirementsaffecttheoptimalthroughput,weestablishthreeone-to-twomul-ticastsessionsinnetworksofvarioussizesbetween10and350,andcomputedtheirtotaloptimalthroughputwiththefollowingfairnessconstraints,respectively:(a)nofairnessrequirement,whichleadstothemaximumvaluepossibleforthetotalthrough-put;(b)absolutefairness,inwhicheachsessionisrequiredtohaveexactlythesamethroughput;(c)weightedproportionalfair-ness,wherethethroughputofeachsessionisproportionaltotheassociatedweightofthatsession;and(d)max-minfairness,inwhichnosessionthroughputcanbeincreasedwithoutdecreas-inganotheralreadysmallersessionthroughput.AsaÞrstsmall-scaleexperimenttogainsomeinsights,Fig.15showsthetotalthroughputofthreesessionsinanetworkwithtwentynodes,usingtheLP.Multicastgroupsarechosentobeasdisjointaspossible.Thetotalweightofthreesessions.Aswecansee,theweightdistributionhasasigniÞcantimpactontheachievabletotalthroughput.Whenthethreeweightsareheavilyunbalanced,thesessionwiththesmall-estweightcannotrealizeitsthroughputpotential,andconse-quentlyleadstoasmallvalueoftotalthroughput.TheachievablethroughputwithabsolutefairnessatKbps.TheglobaloptimalthroughputKbpsisachievedat)=(0,whichturnsouttobeidenticaltothethroughputwithmax-minfairnessinthiscase. 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 Fig.15.Totalthroughputofthreemulticastsessions,asinter-sessionfairnessrequirementschange.FurtherresultsinTableVshowthattheexcellentperformanceofmax-minfairnessintheaboveexampleisnotacoincidence.Aswemayobserve,whenthenetworksizeisrelativelylargeandaboveinthetable),max-minfairnessalwaysleadstooptimalthroughput.Whenthenetworksizeissmall(thetable),theinter-sessioncompetitionforbandwidthbecomesmoreintense.Thethroughputwithmax-minfairnessmaybeinferiortotheoptimalthroughputinthiscase,butthedifferenceisusuallysmall.TABLEVOTALACHIEVABLETHROUGHPUTWITHMAXMINFAIRNESSVSGLOBALOPTIMALTHROUGHPUT networksize 10 20 50 100 150 250 350 max-min(Kbps) 120.0 136.7 173.3 160.0 146.7 146.7 183.3 optimal(Kbps) 126.1 140.0 173.3 160.0 146.7 146.7 183.3 DoesoptimalthroughputleadtolowbandwidthefÞciency?InordertoÞndoutwhetherachievingoptimalthroughputsac-riÞcesbandwidthefÞciency,wehaveconductedperformancecomparisonsbetweenoptimalthroughputmulticastandsingletreemulticast.Inthelattercase,wecomputethewideststeinertree,whichhasthehighestthroughputfromallpossiblemulticasttrees.ThethroughputofatreeisdeÞnedasthelowestcapacityofitslinks.WechoosethetreewiththehighestthroughputratherthantheonethatismostbandwidthefÞcient,sincethelatterisequivalenttotheminimumsteinertreeproblem,whichishardtocomputeortoapproximate.EvenwhenwecanÞndsuchaband-widthefÞcienttree,itmayhaveanexceedinglylowthroughput,whichisnotpracticalfordatatransmissions.InFig.16,wecomparebothachievablethroughputandband-widthefÞciencybetweenthetwoapproaches.Wetestedtwogroupsofnetworks,onewithvariablelinkcapacityconformingtotheheavy-taileddistribution,theotherwithconstantlinkca-pacity.Forthevariablelinkcapacitycase,optimalthroughputishigherthanthewideststeinertreethroughputbyafactorofoveronaverage,showingtheadvantageofusingtheoptimaltrans-missionstrategycomputedwiththeLP,beyondasingle 13 10 20 30 40 1050100200300400500 20 40 60 80 1050100200300400500 10 20 30 40 1050100200300400500 20 40 60 80 1050100200300400500 Number of nodes in the network Widest tree Fig.16.AchievablethroughputandbandwidthefÞciency:acomparisonbe-tweentheoptimalthroughputmulticast(LP)andthewideststeinertree.multicasttree.Interestingly,thebandwidthefÞciencyofoptimalthroughputmulticastalsooutperformsthatofthewideststeinertreemulticast.Thewideststeinertreeinsiststouselinkswiththehighestbandwidthpossible,andthereforemayresultinratherlongtreebranches,especiallywhenthenetworksizeislarge.Fortheconstantlinkcapacitycase,thedifferencebetweentheoptimalandwideststeinertreethroughputbecomesevenlarger.Everytreeinthiscasehasthesamethroughput,thereforetheÒwidestÓselectioncriterionbecomesirrelevant.However,thedifferenceinbandwidthefÞciencydecreases,sinceitisnolongernecessarytoincludelongtreebranchestoachievethemaximumtreethroughput.Istheachievableoptimalthroughputdependentontheselec-tionofdatasourcesinacommunicationsession?Itisratherstraightforwardtoseethatoptimalthroughputwith-outcodingissourceindependent.Forexample,inthecaseofmulticastwithoutcoding,eachunitthroughputisachievedalongaunit-weightsteinertree.Thesamethroughputcanbeachievedregardlessofwhichnodeinthemulticastgroupisselectedastherootofthetrees.Itismuchlessobvious,though,whetherthesourceindependentpropertystillholdswhennetworkcodingisconsidered.Oursimulationresultsshowthat,whenthedatasourcewithinamulticastgroupswitchesfromonenodetoan-other,thesameoptimalthroughputisalwaysachievedusingouralgorithms.Wefurtherprovideatheoreticalproofforthisclaim.Theorem7.Foranundirecteddatanetworkwithoneormorecommunicationsessions,theachievableoptimalthroughputiscompletelydeterminedbythetopologyofthenetwork,thelinkcapacities,thesetofnodesineachcommunicationsession,aswellasthedeÞnitionofinter-sessionfairness.Itisirrelevanttheselectionofthedatasourcewithineachsession.Proof:Assumenodesarethesourceandoneofthereceiversinsession,respectively.Considermovingthedatasourceofsessionfromnodetonode,andexchangingthesource/receiverrolesbetween.Letbeanoptimaltransmissionstrategybeforethemovement.Letbetheßowinsessionbeforethemovement.Weshowthatthetransmissionstrategyobtainedbyreversingthedirectionsofinformationßowsinonly,achievesthesamethroughputforeachsessionasin v f1f1f2f2f3f3f4f4 Fig.17.Flowsreversingatanintermediatenode,eachreceiverisreceivingindependentinformationßowsatthesamerateasitisin.Weonlyneedtoverifythatisatransmissionscheme,i.e.,theinformationßowsoneachlinkcanbegeneratedfromitsprecedingßows.Consideranodeinthenetwork.Ifdoesnotpass,thenthevalidityatisguaranteedbythevalidityat.Otherwise,asshowninFig.17,letbethesetofincomingandoutgoingßowsatthatbelongto,respectively,andletthesetofincomingandoutgoingßowsatthatdonotbelong,respectively.Thevalidityofimpliesthat.Let,andlet,wherearecodematricesofscalerespectively.,sinceotherwisecontainsredundantßowsthatarenotnecessarytobeincludedin.Therefore,andlinearlyspansFinally,linearlyspansandtherefore.Hence,wehaveveriÞedthevalidityof aaabbbba+ba+ba+b aaabbbba+ba+ba+bABAB aabbba+ba+ba+b AB (a) Source independence: (b) Source independence: the case of multiple sessions Fig.18.Theachievableoptimalthroughputisindependentfromtheselectionofdatasourcesineachsession:twoexamples.Fig.18showstwoexamplesoftheßowreversingprocedurediscussedintheproofofTheorem7.TheÞguresontheleftshowtheoptimaltransmissionstrategieswhenthesourceisat.Theonesontherightshowthenewtransmissionstrategieswiththeßowreversed.ItmaybeveriÞedthatthenewtransmissionstrategiesarevalidandoptimal,giventhatthesourceismovedVI.CThemainproblemwehavestudiedinthispaperistocomputeandachieveoptimaltransmissionthroughputfordatacommuni-cationsofvarioustypes.WehavebeenpleasantlysurprisedathowresultsfromnetworkcodingareabletofacilitatethedesignofefÞcientsolutionstothisfundamentalproblemthatwaspre-viouslyviewedasveryhard.Wealsoshowthecounter-intuitive conclusionthat,themostsigniÞcantbeneÞtofnetworkcodingisnottoachievehigheroptimalthroughput,buttomakeitfeasi-bletoachievesuchoptimalityinpolynomialtime.WeshowthatsuchefÞcientalgorithmsmaybedesignedformultiplecommu-nicationsessionsofavarietyoftypes,andforthemorerealisticmodelofoverlaynetworks.Thereexistanumberofavenuesforfurtherresearchexploit-ingthefullpotentialofnetworkcodingindatanetworks.Forexample,itmaybefeasibletoseamlesslyintegratenetworkcod-ingwithsourceerasurecodes(e.g.,DigitalFountain[14])toim-provefaultresilienceofdatadissemination.Wemayalsoextendthetheoreticalfoundationsinthispapertowirelessnetworks,ad-dressingtheiruniquecharacteristicssuchasspatialcontentionofcapacities.Nevertheless,weÞrmlybelievethatitisnecessarytostartwiththeoreticalstudiesbasedonthemostgeneralformofundirectednetworkswithboundedlinkcapacitiesandarbi-trarysourcedatastreams,clearingthepathsleadingtoresearchinmorespeciÞcsituations.OLYNOMIALetal.[20]proposedtheÞrstcodeassignmentalgorithm,whichperformsanexponentialnumberofvectorindependencetests.Sandersetal.[7]observedthat,exploitingßowinforma-tionintheroutingstrategydramaticallysimpliÞesthetask,andgaveapolynomialtimealgorithm,referredtoasNIF.TheNIFalgorithmproceedsfromthesourcetothereceiversalonglinksinvolvedinthetransmission,intopologicalorder.Ateachstep,codesareassignedforoutgoinglinksofanode,guaranteeingthataftertheassignment,ßowsarrivingateachreceiverarelinearlyindependent.TheÞniteÞeldrequiredis,forTheNIFalgorithmworksinharmonywith,sincetheLPgeneratesthenecessaryßowinformationforNIFasin-put.However,thereareafewadditionalnoteworthyissues.First,thefractionaloutputfromneedstobetransformedintoanintegralone,duetothediscretenatureofcodeassignment.Anatomicßowratecanbeselected,suchthateachßowintherout-ingstrategyhasanintegralrate.Round-offoperationsmaybeappliedifnecessary,toavoidgeneratingalargenumberofsmallßows.Second,theNIFalgorithmassumesadirectedacyclicnet-workasinput,sothatnodescanbenaturallyprocessedintopo-logicalorder.Generallyspeaking,anoptimalroutingstrategyforanundirectednetworkmaycontaindirectedcycles,asinthecaseoftheoutput.Inthiscase,nodesshouldbeprocessedalongconceptualßowssuchthatßowsaresynchronizedatwheretheyareencoded.Third,theroutingstrategyfromthemaycontain,intheformofredundantßowscirculatingaroundcycleswithresidualcapacities.Thisproblemisactuallyinheritedfromregularnetworkßowlinearprograms.Toremovetheredundantßows,onemayconsidersolvingtheLPwithmultipletargetfunctions,withtheprimarygoalofmaximizingtheeffectiveßowrate,andthesecondarygoalofminimizingthetotalbandwidthconsumption.Asimplersolutionthatworkswellinpracticeistojustmaximizetheweighteddifferencebe-tweenthesetwotargetfunctions,withamuchsmallerweightassociatedwiththesecondone.[1]L.Qiu,Y.R.Yang,Y.Zhang,andS.Shenker,ÒOnSelÞshRoutinginInternet-LikeEnvironments,ÓinProc.ofACMSIGCOMM,2003.[2]K.Jain,M.Mahdian,andM.R.Salavatipour,ÒPackingSteinerTrees,ÓinProceedingsofthe10thAnnualACM-SIAMSymposiumonDiscreteAlgo-rithms(SODA),2003.[3]G.RobinsandA.Zelikovsky,ÒImprovedSteinerTreeApproximationinGraphs,ÓinProceedingsofthe7thAnnualACM-SIAMSymposiumonDis-creteAlgorithms(SODA),2000.[4]M.Thimm,ÒOnTheApproximabilityOfTheSteinerTreeProblem,ÓMathematicalFoundationsofComputerScience2001,SpringerLNCS2136,678-689,2001.[5]R.Ahlswede,N.Cai,S.R.Li,andR.W.Yeung,ÒNetworkInformationFlow,ÓIEEETransactionsonInformationTheory,vol.46,no.4,pp.1204Ð1216,July2000.[6]R.KoetterandM.Medard,ÒAnAlgebraicApproachtoNetworkCoding,ÓIEEE/ACMTransactionsonNetworking,vol.11,no.5,pp.782Ð795,Octo-ber2003.[7]P.Sanders,S.Egner,andL.Tolhuizen,ÒPolynomialTimeAlgorithmforNetworkInformationFlow,ÓinProceedingsofthe15thACMSymposiumonParallelisminAlgorithmsandArchitectures,2003.[8]Z.WangandJ.Crowcroft,ÒQualityofServiceRoutingforSupportingMultimediaApplications,ÓIEEEJournalonSelectedAreasinCommuni-,vol.14,no.7,pp.1228Ð1234,September1996.[9]A.J.Ballardie,P.F.Francis,andJ.Crowcroft,ÒCoreBasedTrees,ÓAugust[10]Y.Chu,S.G.Rao,S.Seshan,andH.Zhang,ÒACaseforEndSystemMulticast,ÓIEEEJournalonSelectedAreasinCommunications,pp.1456Ð1471,October2002.[11]S.Banerjee,B.Bhattacharjee,andC.Kommareddy,ÒScalableApplicationLayerMulticast,ÓinProc.ofACMSIGCOMM,August2002.[12]M.Castro,P.Druschel,A.-M.Kermarrec,A.Nandi,A.Rowstron,andA.Singh,ÒSplitStream:High-BandwidthMulticastinCooperativeEn-vironments,ÓinProc.ofthe19thACMSymposiumonOperatingSystemsPrinciples(SOSP),October2003.[13]V.Padmanabhan,H.Wang,P.Chou,andK.Sripanidkulchai,ÒDistribut-ingStreamingMediaContentUsingCooperativeNetworking,ÓinProc.ofNOSSDAV2002,May2002.[14]J.ByersandJ.Considine,ÒInformedContentDeliveryAcrossAdaptiveOverlayNetworks,ÓinProc.ofACMSIGCOMM,August2002.[15]D.Kostic,A.Rodriguez,J.Albrecht,andA.Vahdat,ÒBullet:HighBand-widthDataDisseminationUsinganOverlayMesh,ÓinProc.ofthe19thACMSymposiumonOperatingSystemsPrinciples(SOSP2003),2003.[16]Y.Azar,E.Cohen,A.Fiat,H.Kaplan,andH.Racke,ÒOptimalObliviousRoutinginPolynomialTime,ÓinProc.ofthe35thACMSymposiumontheTheoryofComputing(STOC),2003.[17]D.ApplegateandE.Cohen,ÒMakingIntra-DomainRouti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