Coding for a network coded fountain - PDF document

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 dId�r;r(x)+MXr=1hr;rr r;wherehr;r=1�q�1 1�q�r�1hr+1;randIa;b(x)=a+b�1Xj=aa+b�1jxj(1�x)a+b�1�jiscalledregularizedincompletebetafunction.Dene~1()=(1�=C0) E[ ]( (=C0)+ln(1�=C0));whereC0==E[ ].WeobtainthefollowingsufcientconditionofthedegreedistributionsuchthattheBPdecodingofBATScodessucceedswithhighprobabilitywhenKissufcientlylarge.Theorem1:ConsiderasequenceofdecodinggraphBATS(K;n;f d;rg)withconstant.Forany�0,consideradegreedistributionwith~1()for2[0;C0(1�)].ThereexistconstantK0,candc0suchthatwhenKK0,withprobabilityatleast1�cn7=24exp(�c0n1=8),thedecodingterminateswithatmostKinputpacketserased.Theorem1enablesustoconsiderthefollowingoptimizationproblemtondanasymptoticallyoptimaldegreedistributionthatmaximizesthecodingrate:max(2)s.t. (x)+ln(1�x)0;0x1� d0;d=1;


;DXd d=1:TheonlychannelinformationrequiredintheoptimizationproblemistherankdistributionofH.WecanfurthershowthatusingD&#x-278;dM=e�1doesnotgivebetteroptimalvalueintheaboveoptimizationproblem.ThuswesetD=dM=e�1,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=


=hM�1=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&#x-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