b1 b2 b3 b4 b5 b6 G1 G2 G3 G4 G5 Fig2Tannergraphforencodingandtransmittingoftherstvebatches 0 1 KFirstsamplethedistribution whichreturnsadegreediwithprobability diThenuniformlyatrandomc ID: 328647
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alsotaketheadvantagesofnetworkcoding.BATScodesarefullycompatiblewithlinearnetworkcodingbyemployinganewdesignfreedomcalledbatch.AbatchisasetofMpacketsgeneratedbyusingasamesubsetofinputpackets.TheencodingcomplexityofaBATScodeisO(TKM)andthecorrespondingdecodingcomplexityisO(KM2+TKM).Moreover,whenapplyingBATScodes,anintermediatenodeusesO(TM)timetorecodeapacketandbuffersO(M)packets.Asanend-to-endcodingschemeworkingatthesourceanddestinationnodes,BATScodesaresuitableforalargerangeofnetworksaslongastheend-to-endoperationonthepacketsofabatchisalineartransformation,whichcanbedifferentfordifferentbatches.BATScodesarerobustagainstdynamicalnetworktopologyandpacketlosssincetheend-to-endoperationremainslinear.Moreover,BATScodesworkwithrandomlinearnetworkcodingwithsmallniteelds.Mostexistingworksonrandomlinearnetworkcodingrequiresalargeeldsizetoguaranteeafullrankforthetransfermatrix.ForBATScodes,however,thetransfermatricesofthebatchesareallowedtohavearbitraryrankdeciency.Thelineartransformationonbatchescanbemodelledbyalinearoperatorchannel(LOC),achannelmodelstudiedforlinearnetworkcoding[13].WeverifytheoreticallyforcertaincasesanddemonstratenumericallyforthegeneralcasesthatBATScodesachieveratesveryclosetothecapacityofmemorylessLOCs.ThebatchsizeMdeterminesthetradeoffbetweenthecomplexityandthemaximumachievablerate.WhenM=1,BATScodesdegeneratetoLTcodes,whichhavethelowestcomplexitybutwithoutthebenetofnetworkcoding.WhenM=K,BATScodeshasthesamecomplexityofrandomlinearnetworkcoding,andatthesametimethepotentialofnetworkcodingcanbefullyrealized.II.BATCHEDSPARSE(BATS)CODESConsiderencodingKinputpackets,eachofwhichhasTsymbolsinaniteeldFwithsizeq.Apacketisdenotedbyacolumnvector.Inthefollowingdiscussion,weequateasetofpacketstothematrixformedbyjuxtaposingthepacketsinthisset.Forexample,thesetoftheinputpacketsisdenotedbythematrixB=b1;b2;;bK;wherebiistheithinputpackets.Whentreatingasaset,wealsowritebi2B,B0B,etc.Weuserk(A)todenotethematrixrankofA.A.EncodingofBatchesAbatchisasetofMcodedpacketsgeneratedfromasubsetoftheseinputpackets.Fori=1;2;:::,theithbatchXiisgeneratedusingasubsetBiBoftheinputpacketsasXi=BiGi;whereGiiscalledthegeneratormatrixoftheithbatch.WecallthepacketsinBithecontributorsoftheithbatch.TheformationofBidependsonadegreedistribution = b1 b2 b3 b4 b5 b6 G1 G2 G3 G4 G5 Fig.2.Tannergraphforencodingandtransmittingoftherstvebatches.( 0; 1;; K):Firstsamplethedistribution whichreturnsadegreediwithprobability di;ThenuniformlyatrandomchoosediinputpacketsformingBi.Thedesignof isdiscussedlaterinSectionIII.ThedimensionofGiisdiM.TherearetwooptionsfordesigningGi.i)Giarepre-designed.ii)Giaregeneratedonthey.Inthispaper,weanalyzeBATScodeswithrandomgeneratormatrices,i.e.,allthecomponentsofGiareindependentlychosen,uniformlyatrandombytheencoder.Randomgenerationmatrixisnotonlygoodforanalysis,butalsoimplementable.E.g.,Gi,i=1;2;,canbegeneratedbyapseudorandomgeneratorandcanberecoveredinthedestinationsbythesamepseudorandomgenerator.TheencodingofBATScodescanbedescribedbyTannergraphs.ATannergraphhasKvariablenodes,wherethevariablenodeicorrespondstotheithinputpacketbi,andnchecknodes,wherethechecknodejcorrespondstothejthbatchXj.Checknodejisconnectedtovariablenodeiifbiisacontributorofbatchj.Fig.2illustratesanexampleofTannergraphforencoding.B.TransmissionofBatchesTotransmitabatch,thesourcenodetransmitsthepacketsinthebatch.Nofeedbackisrequiredtostopthetransmissionofeachbatch.BATScodesareratelesscodes,i.e.,thenumberofbatchesthatcanbetransmittedisnotxed.Whenapplyinglinearnetworkcoding,anintermediatenodeencodesthereceivedpacketsofabatchintonewpacketsusinglinearcombinationsandtransmitthesenewpacketsonitsoutgoinglinks.Thesenewpacketsareconsideredtobeinthesamebatch.Theruleisthatthepacketsindifferentbatchesarenotmixedinsidethenetwork.BATScodesarerobustagainstdynamicalnetworktopologyandpacketlosssincetheend-to-endoperationremainslinear.ToapplyBATScodes,wefurtherneedtoconsiderhowtoschedulethetransmissionofbatchesinthesourcenodeandtheintermediatenodesandhowtomanagethebuffersattheintermediatenodes.Thedesignofthesenetworkoperationsvariesfordifferentapplications.Forexample,fortheletrans-missioninadirectedacyclicnetwork,whentheintermediatenetworknodesdonotrequirethele,sequentialschedulingofbatchesatthesourcenodeandtheintermediatenodescanminimizethebufferrequirementattheintermediatenodes.Incontrast,fortheledistributioninapeer-to-peernetwork,sinceallnetworknodesrequestthele,randomschedulingofbatchescanreducetheprotocoloverhead. andforachecknodewithdegreed,theprobabilitythatithasrankrishd;r=Prfrk(GdH)=rg,whereGdisadMrandommatrixwithuniformi.i.dcomponents.ThegeneratormatrixofabatchwithdegreedisjustaninstanceofGd.hd;rcanbecomputedusingonlytherankdistributionofH(see[15]fortheexpression).Forconvenience,wealsocallthepair(d;r)thedegreeofachecknode.Let d;r= dhd;rbetheprobabilitythatachecknodehasdegree(d;r).AdecodinggraphwithKvariablenodesandnchecknodesisdenotedbyBATS(K;n;f d;rg).ThedesigncodingrateoftheBATScodeis=K=n.Weusetheresultofdensityevolutiontoshowtheasymp-toticdecodingperformanceofasequenceofdecodinggraphBATS(K;n;f d;rg)withconstant.WeapplyWormald'stheorem[16]toapproximatethedensityevolutionbydiffer-entialequations.Thedetailsoftheanalysisareomittedandcanbefoundin[15].AssumethatthemaximumDsuchthat DisnonzeroisnotrelatedtoK.Let (x)=MXr=1hr;rDXd=r+1d dIdr;r(x)+MXr=1hr;rr r;wherehr;r=1q1 1qr1hr+1;randIa;b(x)=a+b1Xj=aa+b1jxj(1x)a+b1jiscalledregularizedincompletebetafunction.Dene~1()=(1=C0) E[ ]( (=C0)+ln(1=C0));whereC0==E[ ].WeobtainthefollowingsufcientconditionofthedegreedistributionsuchthattheBPdecodingofBATScodessucceedswithhighprobabilitywhenKissufcientlylarge.Theorem1:ConsiderasequenceofdecodinggraphBATS(K;n;f d;rg)withconstant.Forany0,consideradegreedistributionwith~1()for2[0;C0(1)].ThereexistconstantK0,candc0suchthatwhenKK0,withprobabilityatleast1cn7=24exp(c0n1=8),thedecodingterminateswithatmostKinputpacketserased.Theorem1enablesustoconsiderthefollowingoptimizationproblemtondanasymptoticallyoptimaldegreedistributionthatmaximizesthecodingrate:max(2)s.t. (x)+ln(1x)0;0x1 d0;d=1;;DXd d=1:TheonlychannelinformationrequiredintheoptimizationproblemistherankdistributionofH.WecanfurthershowthatusingD-278;dM=e1doesnotgivebetteroptimalvalueintheaboveoptimizationproblem.ThuswesetD=dM=e1,whichcompliesourassumptionthatDisnotrelatedtoK.IV.ACHIEVABLERATESThecodingrateofaBATScodesisgivenbytheaveragenumberofpacketsthatcanbetransmittedusingonebatch.Theratecanalsobenormalizedbythebatchsize.A.AsymptoticallyAchievableRatesTheBPdecodingalgorithm,ifsucceeds,recoversatleast(1)Kpackets.Thus,themaximumachievablerateofBATScodesisatleast^(1),where^istheoptimalvalueoftheoptimizationproblem(2).Intermsofpacketsperuse,thecapacityofaLOCwiththetransfermatrixHisE[rk(H)][13].AschannelcodesforLOCs,themaximumachievablerateofBATScodesisupperboundedbythecapacityofLOCs.So^(1)E[rk(H)].ThemaximumachievablerateofBATScodesislowerboundedbythefollowingtheorem(provedin[15]).Theorem2:Let^betheoptimalvalueoftheoptimizationin(2).Then^maxr=1;2;;Mrhr;r:Eventhoughthelowerboundgivenbythetheoremislooseingeneral,itshowsthatBATScodesachieveratesarbitrarilyclosetothecapacityforthefollowingspecialcase.WecallanLOCwithtransfermatrixHfull-rankifh1=h2==hM1=0,wherehi=Prfrk(H)=ig.Forafull-rankLOC,^MhM;M!MhM=E[rk(H)]whentheeldsizeq!1.Sincecanbetakenarbitrarilysmall,BATScodesachieveratesarbitrarilyclosetothecapacityoffull-rankLOCsoversufcientlylargeniteelds.Toseetheachievableratesforthegeneralcases,wenu-mericallysolvetheoptimizationproblem(2)bytakingdiscretevaluesforx.Let~betheoptimalvalueofthisrelaxedversionof(2).SetM=5,q=16and=0:01.Arankdistributionfh0;h1;:::;hMgisgeneratedasfollows:First,h0=0andfori-278;1,hiisindependentlyanduniformlychosenbetweenzeroandone;Then,normalizetherankdistributionsuchthatPihi=1.Wecompute~for24345rankdistributionsindependentlygeneratedandcompare(1)~withPMr=1rhrbycomputing=(PMr=1rhr(1)~)=PMr=1rhr.Theresultsshowthatformorethan99%rankdistributions,issmallerthan0:05,andthelargestis0:1145.ThismeansthatBATScodesachieveratesveryclosetothecapacityevenforLOCsoversmallniteelds.Whenusinglargerelds,thegapbetweenthemaximumachievablerateandthecapacitybecomessmaller.E.g.,afterchangingtheeldsizetoq=64,formorethan99%rankdistributions,issmallerthan0:026,andthelargestreducesto0:0876.B.FiniteLengthPerformanceWeusethenetworkinFig.1toillustratetheperformanceofBATScodes.ThesourcenodesappliesBATScodeencoding.Ineachtimeslot,ssendsapackettoa.Assumetransmissionisinstantaneousandnodeareceivesthepacket,ifnoterased,atthesametimeslot.Nomatterwhetherparticularpacketsarereceivedornot,nodeatransmitsateachtimeslotalinearcombinationofthepacketsithasreceivedsofar.AfterMtime