Network Coding for Unicast: Exploiting Multicast Subgraphs to Achieve Capacity in 2-User Networks

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1 Netwok Codng fo Uncast: xplotng Multcast Subgaphs to Acheve Capacty n -Use Netwoks Jennfe Pce Taa Javd Depatment of lectcal Compute ngneeng Unvesty of Calfona, San Dego La Jolla, CA 9093 jenn-pce@ucsd.edu, taa@ece.ucsd.edu Abstact In ths pape, we addess optmal netwok codng schemes fo two-use uncast netwoks. Usng pevous esults on the elatonshp between gaph stuctue nfomatontheoetc capacty, we chaacteze the capacty egon fo a class of two-use uncast netwoks by fndng a combnaton netwok codng/outng scheme to acheve an oute bound on capacty. Ths scheme has the emakable popety of decomposng the dffcult uncast netwok codng poblem nto thee sub-poblems: a multcast netwok codng sub-poblem, two mult-commodty flow sub-poblems. Usng nsghts fom the ate assgnment scheme, we constuct a dstbuted ate contol algothm based on optmzaton dual theoy that acheves capacty of the netwok. Fomulatng the uncast netwok codng poblem n tems of optmzaton theoy also allows us to mplement a tanspot-type mechansm that povdes systemwde pefomance objectves. I. INTRODUCTION Snce the semnal wok n [], netwok codng has eceved a geat deal of attenton as a way to help bdge the gap between mult-commodty flow capacty nfomaton theoetc capacty of netwoks [], []. Although the multcast netwok codng poblem s well undestood, the poblem becomes moe dffcult when addessng uncast. In such netwoks, the poblem of how to acheve capacty of the netwok emans a lagely open one. In addton, the poblem of fndng optmal netwok codng schemes typcally assumes that desed communcaton ates, as well as capacty of lnks n the netwok, ae fxed known. In ealstc communcaton netwoks, howeve, not only should the ate allocaton account fo system-wde QoS equements, but t should be able to adapt to fluctuatons n use dem be mplemented n a dstbuted manne at the end-uses. In othe wods, t s desable to have a dstbuted ate-contol scheme smla to exstng ate contol mechansms fo tadtonal IP netwoks (see e.g. [3], [4], but that ncopoates the use of netwok codng to acheve nfomaton-theoetc capacty of the netwok. The man contbutons of ou pape ae as follows: Chaacteze the nfomaton-theoetc capacty egon fo a class of two-use uncast netwoks by fndng ate assgnments that acheve the oute bound on capacty. Implement an adaptve dstbuted ate contol scheme that acheves the nfomaton-theoetc capacty of the netwok whle povdng system-wde pefomance objectves. The authos n [] have shown that t s possble to elate the nfomaton theoetc capacty of a netwok to ts undelyng stuctue. Usng these esults, we gve a pocedue fo fndng an oute bound on the nfomaton theoetc capacty of a uncast netwok. We then show that unde cetan techncal assumptons on the netwok stuctue, t s possble to acheve ths oute bound wth lnea XOR netwok codes by usng a combnaton of netwok codng tadtonal outng. Intutvely, ths ate assgnment pocedue conssts of thee components:. Use tadtonal outng to assgn ates along paths fom a souce to ts own destnaton on whch netwok codng cannot occu.. Solve the multcast netwok codng poblem on the emanng gaph. 3. Use tadtonal outng to assgn ates along paths fom a souce to ts own destnaton on the emanng gaph. Ths esult s qute emakable. Fo the gven class of netwoks, achevng capacty fo the dffcult uncast netwok codng poblem can be decomposed nto thee subpoblems: a multcast netwok codng poblem, two mult-commodty flow poblems! In addton, by usng nsghts fom the ate assgnment pocedue, we show that s s possble to fomulate the uncast netwok codng ate assgnment as an optmzaton poblem n whch netwok-coded flows non-netwok coded flows ae teated as sepaate commodtes that must satsfy cetan constants. Although ths appoach s shown to be optmal only fo the two-use case, we conjectue that the multpleuse extenson of ths algothm also acheves capacty of the netwok unde assumptons smla to the ones gven n ths pape. Thee s, of couse, an extensve ch lteatue on netwok codng fo both multcast []-[9] uncast [0]- [4] netwoks. Netwok codng fo multcast s well studed, n tems of both theoetcal achevablty []-[6] pactcal

2 Fg.. The xtended Buttefly Netwok Fg.. Step : Mult-Commodty Flow Sub-Poblem mplementaton [7]-[9]. In tems of motvaton ou wok s most smla to that of [8], n whch the authos povde an adaptve ate contol mechansm fo mult-cast netwoks that can be mplemented at the end-uses. The authos combne netwok codng wth a utlty maxmzaton famewok to develop dual-based adaptve ate contol mechansms. The bggest dffeence between [8] ou wok s that [8] only consdes codng fo mult-cast netwoks, whle we develop a scheme that employs a combnaton of netwok codng tadtonal outng to acheve capacty n two-use uncast netwoks. The othe man dffeence s n the mplementaton of the dual-based algothms. Whle the authos n [8] use a back-pessue appoach combned wth sesson schedulng, ou appoach dstngushes between coded non-coded flows by vewng them as sepaate commodtes. As we wll see, ths allows us to utlze smple mult-path outng technques smla to [5]-[7]. Netwok codng fo uncast s less well undestood. Reseach n ths aea typcally takes on one of two foms: theoetcal esults examnng the fundamental dffeences between multcast uncast [], [0],[8],[9], [?] the constucton of suboptmal codng schemes []-[4]. Ou wok s most smla to the latte, n that we constuct a codng scheme fo uncast netwoks. The man dffeence between these ou wok s that []-[4] focus on pactcal but sub-optmal netwok codng fo uncast, whle we pesent a scheme that acheves capacty fo a gven class of netwoks whle also consdeng adaptve ate contol. The eme of the pape s oganzed as follows. Secton II pesents a smple example of an extended buttefly netwok. Ths netwok s used to llustate the dffcultes n uncast netwok codng, to povde ntuton fo how the capacty achevablty esults ae obtaned. Secton III pesents ou model notaton. Secton IV gves the pocedue to oute bound capacty of the netwok, whle Secton V shows that ths oute bound s achevable. Secton VI gves the dstbuted ate contol algothm, Secton VII gves conclusons aeas fo futue wok. II. A SIMPL XAMPL: TH XTNDD BUTTRFLY NTWORK In ths secton, we povde va an example ntuton fo how to fnd an oute bound on the capacty of uncast netwoks, how to acheve the oute bound. Consde the smple extended buttefly netwok shown n Fgue (a, whch conssts of two uncast sessons: one fom X to X, one fom Y to Y, wth lnk capactes as ndcated on the gaph. Consde cuts D D as ndcated on Fgue (b. As we wll see n late sectons, the noton of nfomatonal domnance [] can be used to pove that cuts D D each povde an oute bound fo capacty -.e. H(X + H(Y H(D H(X + H(Y H(D. The noton of nfomatonal domnance can be ntutvely descbed as follows: gven complete knowledge about the symbols on lnks n cut D (o D, one can fully detemne the symbols at X, X, Y, Y. Now, consde achevng ths oute bound by decomposng the uncast netwok codng poblem nto thee subpoblems. Fst, we constuct a subgaph consstng of the paths fom each souce to ts own destnaton ove whch netwok codng cannot occu -.e. the paths fom X to X that do not shae any lnks wth the paths fom Y to Y, vce-vesa. Solve the mult-commodty flow poblem ove ths subgaph, shown n Fgue (a. Subtact these ates fom the capacty of the ognal netwok, gvng us the netwok shown n Fgue (b. The second step s to take the netwok fom Fgue (b solve the mult-cast netwok codng poblem, as shown n Fgue 3(a. Agan, subtact these ates fom the capacty of the netwok fom Fgue (b, gvng us the netwok shown n Fgue 3(b. The last step s to take the netwok fom Fgue 3(b solve the mult-commodty flow poblem. Fo example, we could assgn ates as shown n Fgue 4 wth use X ecevng all of the avalable ate. The total ate acheved by ths soluton s R x +R y = ( ( =.3. Ths s exactly the capacty of the mnmum cut (D fom Fgue (b. In the eme of ths pape, we pesent smla capacty esults ate assgnment schemes fo a class of genealzed -use uncast netwoks. In lookng at these genealzed net-

3 Fg. 3. Step : Mult-Cast Netwok Codng Sub-Poblem Fg. 4. Step 3: Mult-Commodty Flow Sub-Poblem woks, a fa amount of notaton techncal machney ae needed; the basc ntuton pocedues, howeve, mmc those pesented above. III. NOTATION AND PRLIMINARY RSULTS We use the followng notaton. Consde a dected uncast netwok G = (V,, whee N = {, } s the set of soucedestnaton pas. Followng the conventon ntoduced n [] ( wthout loss of genealty, we assume that each souce has a sngle outgong lnk S( wth nfnte capacty, each destnaton has a sngle ncomng lnk T ( wth nfnte capacty. Let ρ = {S(, S(} ϕ = {T (, T (}. Let L = {,..., L} be the set of lnks (edges, each wth capacty c l, let K = {,..., K} be the set of all possble paths fom any souce S( to any destnaton T (j. We denote{ by p k the set of lnks tavesed by path k, set f l pk ψ kl = 0 else We now ntoduce the followng defntons that wll ad us n ou descpton of the netwok capacty codng scheme. Defnton : A cut A s a set of lnks. We defne the capacty of cut A to be C(A = l A c l. Defnton : The set of paths P s a mnmal ntesectng path goup on gaph G f only f k P, ˆk K\P such that l L\ρ\ϕ fo whch ψ kl ψˆkl =, one of the followng thee condtons s met: a. P s a sngleton b. k, k P ethe thee exsts a lnk l such that ψ klψ kl =, o c. thee exsts some sequence of paths a,..., a M such that thee exsts a coespondng sequence of lnks l,..., L M+ fo whch ψ kl ψ al =, ψ aml m+ ψ am+l m+ = m =,..., M, ψ am l M+ ψ kl M+ = In othe wods, a mnmal ntesectng path goup s a set of paths such that no path n mnmal ntesectng path goup P shaes a lnk wth a path whch s n K\ρ\ϕ but not n P, f P contans moe than one path, then evey pa of paths n P ethe shaes a lnk, o s connected by a sequence of paths n whch adjacent paths shae at least one lnk. Futhemoe, let P = {l : l p k fo some k P } be the set of lnks tavesed by the paths contaned n mnmal ntesectng path goup P. Defnton 3: A mnmal ntesectng path goup P s called shaed f {S(, S(, T (, T (} P. Othewse, t s called dedcated. Futhemoe, let P S be the set of shaed mnmal ntesectng path goups n a gven gaph, let P D be the set of dedcated mnmal ntesectng path goups n a gven gaph. Fnally, we ntoduce a lemma egadng the pattonng of paths n K. Lemma : Fo any netwok G = (V,, thee exsts a unque patton of paths n K nto mnmal ntesectng path goups. Poof: By the vey defnton, mnmal ntesectng path goups ae equvalence classes ove sets of paths (the elaton defned by s n the same mnmal ntesectng path goup as s eflexve, symmetc, tanstve. We note that the equvalence classes defned by such a elaton unquely patton a set (see e.g. [0], page 3, we ae done. IV. OUTR BOUND ON CAPACITY OF -USR NTWORKS In ths secton, we pesent a pocedue to fnd cuts n the netwok that gve an oute bound on the netwok capacty fo netwok codng. In late sectons, we wll see that unde cetan techncal assumptons, ths oute bound s achevable by a combnaton netwok codng/outng scheme, hence descbes the capacty egon of those netwoks. Pocedue P Step P- Consde gaph G. Fnd the smallest cut A that sepaates S( fom T ( on G. Step P- Consde gaph G. Fnd the smallest cut A that sepaates S( fom T ( on G. Step P-3 Ceate subgaph G = (V,, whee = P m P D ( P m. StepP-3. Fnd the smallest cut A 3 that sepaates S( fom {T (, T (} S( fom T ( on G. StepP-3. Fnd the smallest cut A 3 that sepaates S( fom {T (, T (} S( fom T ( on G.

4 Fg. 5. oute Bound on Netwok Capacty: Veson I Fg. 6. oute Bound on Netwok Capacty: Veson II Step P-4 Ceate subgaph G = (V,, whee = P ( m P S P m. StepP-4. Fnd the smallest cut A 4 that sepaates S( fom {T (, T (} S( fom T ( on G \A 3. StepP-4. Fnd the smallest cut A 4 that sepaates S( fom {T (, T (} S( fom T ( on G\A 3. Step P-5 Set { A = A 3 A 4 f C(A 3 A 4 < C(A 3 A 4 A 3 A 4 else Theoem : Let R R be the total nfomaton ate conveyed fom S( to T ( fom S( to T (, espectvely. We denote by the set of ates R, R that satsfy the followng condtons: R C(A ( R C(A ( R + R C(A (3 whee A, A, A ae as defned n Pocedue P. Then the capacty egon of gaph G s a subset of. Due to space consdeatons, we have omtted the full poof of Theoem. The dea s to use max-flow mn-cut condtons (see e.g. [], Chapte 3 to show ( (. Then, we use the noton of nfomatonal domnance fom [] to show (3. The complete poof of Theoem can be found n Chapte 5 of []. Fgues 5 6 show the two possble shapes the egon gven by can take. Fgue 5 shows the case when the lne R +R = C(A does not ntesect the lnes R = C(A R = C(A. Ths shape esults, fo example, f use has a set of paths along whch t can only send nfomaton fom S( to T (j, whch have much lage capactes than the paths along whch use can send nfomaton fom S( to T (. Fgue 6 shows the case when the lne R +R = C(A ntesects the lnes R = C(A R = C(A. If takes on ths shape, we see that thee may be a mult-use gan that can be acheved wth netwok codng. We wll see n the next secton that whch of these two shapes descbes the egon plays an mpotant ole n descbng a netwok codng scheme to acheve capacty of the netwok. V. ACHIVABILITY RSULTS FOR A CLASS OF NTWORKS In ths secton, we ntoduce a combnaton netwok codng/outng scheme that acheves capacty fo cetan types of netwoks. Ths scheme uses lnea XOR codes, only employs nte-sesson netwok codng on pevously uncoded nfomaton. A. Pelmnaes In ths secton, we estct ou attenton to a subset of netwoks whch satsfy cetan popetes, as defned by the assumptons below, povde achevablty esults unde these assumptons. These assumptons ae deved fom the ntuton ganed by consdeng the smple extended buttefly netwok fom Secton II. Techncal Assumpton : Let P = {P,..., P M } be the set of mnmal ntesectng path goups fo netwok G. Then m =,..., M whee P m >, k, k P m such that p k p k = Pm. Techncal Assumpton eques that the set of lnks tavesed by the paths n any gven mnmal ntesectng path goup ethe belong to a sngle path, o can be tavesed by exactly two paths. Techncal Assumpton : Shaed mnmal ntesectng path goups do not contan undected cycles. Unde the above assumptons, usng the dea of mnmal ntesectng path goups, we can ceate an altenate enumeaton of paths n the netwok as follows. Pocedue P Let P = {P,..., P M } be the set of mnmal ntesectng path goups fo netwok G, let K D, KS, KNC, K =, be ntally empty sets of paths. Fo each mnmal ntesectng path goup m =,..., M n P, do the followng: Step P- When P m =, thee s a sngle path k P m. If path k taveses fom S( to T ( (fom S( to T (, add t to K D (to K D. If path k taveses fom S( to T ( (fom S( to T (, add t to K S (to K S.

5 Step P- When P m > P m s a dedcated mnmal ntesectng path goup, Techncal Assumpton guaantees that thee exsts at least one pa of paths k k such that p k p k = Pm. If k taveses fom S( to T ( (fom S( to T (, add t to K D (to K D. If path k taveses fom S( to T ( (fom S( to T (, add t to K S (to K S. Do the same fo path k. Step P-3 When P m > P m s a shaed mnmal ntesectng path goup, fom Techncal Assumpton Techncal Assumpton we know that thee exsts exactly one pa of paths k k such that path k goes fom S( to T(, path k goes fom S( to T(, p k p k = P. Add k to K NC k to K NC. Smlaly, thee s exactly one pa of paths ˆk ˆk such that path ˆk goes fom S( to T ( ˆk goes fom S( to T (. Add ˆk to K ˆk to K. Step P-4 Set K = {K D K S K NC K } K = {K D K S K NC K } F = {p k : k (K D K D K S K S } F = {p k : k (K NC K NC }. We now ntoduce the followng lemma, whch delneates mpotant popetes of ths enumeaton of lnks. We note that t s exactly these popetes that Techncal Assumptons ae desgned to ensue. Lemma : Let K D, KS, KNC, K, F, F be as constucted above. Then these sets satsfy the followng popetes: F F = (4 F F = ρ ϕ (5 ( p D k ( p D k ρ ϕ (6 ( p S k ( p S k ρ ϕ (7 k K NC exactly one k K NC such that (p k p k (ρ ϕ (8 Poof: Pocedue P s a constuctve poof of ths theoem fo netwoks satsfyng Techncal Assumpton -. B. Rate Assgnment Pocedue We now tun to the constucton of ou desed netwok codng/outng scheme to acheve capacty. Notce that not only does the enumeaton of paths gven by Pocedue P captue all of the elevant nfomaton about netwok G, but t has a natual stuctue n tems of ou desed combnaton netwok codng/outng scheme. In patcula, the sets K D contan paths along whch uses tansmt dedcated nfomaton to the own destnaton, along whch netwok codng does not occu. The sets K S contan paths along whch uses can at best tansmt sde nfomaton to othe destnatons If thee exsts moe than one such pa, abtaly choose any sngle pa. fo use n decodng. The sets K NC contan paths along whch uses tansmt nfomaton to the own destnaton, but along whch netwok codng may occu. Fnally, the sets K contan addtonal paths along whch sde nfomaton can be tansmtted fo use n decodng. The pocedue fo fndng the ate assgnments that acheve capacty consst of seveal steps, but has thee man components: use tadtonal outng to assgn ates along dedcated paths (a mult-commodty flow poblem, solve a mult-cast netwok codng poblem, then use tadtonal outng to assgn leftove ates (a mult-commodty flow poblem. Agan, the beauty of ths pocedue s that the dffcult poblem of achevng capacty n uncast netwok codng poblems s decomposed nto thee sepaate subpoblems. In addton, the ate assgnment pocedue gven below beaks the soluton to the mult-cast netwok codng poblem nto seveal sub-steps. Rate Assgnment Pocedue RA Step RA-: Solve a Mult-Commodty Flow Poblem Consde the subgaph H = (V, F. Fnd the mnmum cut B D that sepaates S( fom T (, the mnmum cut B D that sepaates S( fom T (. Assgn ates k along evey path k K D such that k ψ kl C l l L D D k = C(B D =, (9 Let D = D k be the total dedcated ate use eceves. Notce that (6 ensues t s possble to assgn dedcated ates n a way that satsfes (9. Step RA-: Solve a Mult-Cast Netwok Codng Poblem In a netwok employng lnea XOR codes, netwok coded nfomaton can only be decoded f the amount of sde nfomaton avalable at the destnaton s equal to the amount of nfomaton that has been netwok coded. Hence, the total amount of nfomaton that can be netwok coded s uppe bounded by the amount of sde nfomaton that can be tansmtted. On the othe h, the avalablty of sde nfomaton s not useful unless thee s enough capacty n the netwok to pefom netwok codng. Befoe we can assgn netwok codng ates, we need to know whch of these s the lmtng facto. Thus, we beak the soluton to the mult-cast netwok codng poblem nto the followng thee steps. St ep RA-.: Assgn Sde Infomaton Rates Agan, consde the subgaph H = (V, F. Fnd the mnmum cut B S that sepaates S( fom {T (, T (}, the mnmum cut B S Although the ate assgnments along ndvdual paths may not be unque, the total ate D s. Ths s tue of the ate assgnments thoughout the ente pocedue

6 that sepaates S( fom {T (, T (}. Assgn ates k along evey path k K S such that N k ψ kl + k ψ kl C l = D D k + S S k = C(B S (0 Let S = S k be the total sde nfomaton ate use eceves. Notce that (7 ensues t s possble to assgn sde ates n a way that satsfes (0. St ep RA-.: Detemne the Amount of Netwok Codng Befoe detemnng a patcula codng scheme, we must fst devse how much netwok codng the netwok can sustan. To do so, consde the subgaph H = (V, F. Fnd the followng mnmum cuts: B T : sepaates S( fom T ( B L : sepaates S( fom T ( S( fom T ( B : sepaates S( fom {T (, T (} S( fom T ( Along evey path k K NC quanttes q k such that = NC K = = NC K K, assgn q k ψ kl C l l L q k = C(B T ( q k = C(B L ( q k = C(B (3 Let T = q NC k, L = q NC k, = q k. Now epeat ths pocess but statng wth use to obtan T, L, fom cuts B T, B L, B. The total ates T T ae the amount of nfomaton each use can send to ts own destnaton, whle the total ates S + S + ae the amount of nfomaton each use can send to the othe use s destnaton. Thus, we set the total amount of nfomaton netwok coded by each use as NC = mn( S +, S +, T, T. St ep RA-.3: Assgn Netwok Coded Rates In the pevous step we detemned the amount of netwok codng that the netwok can sustan. Hee, we assgn the actual ates along paths on whch netwok codng occus. To do so, ecall fom ( that fo each path k K NC, thee exsts exactly one path k K NC such that k k shae at least one lnk. Fo each such pa, assgn ates k = k such that max kψ kl, C l l L k = kψ kl k = NC (4 Note that, due to XOR codng at the ntemedate nodes, ( t s the physcal flow, gven by max NC k ψ kl, NC k ψ kl athe than the nfomaton flow, gven by NC k ψ kl + NC k ψ kl that must satsfy the lnk capacty constants. Snce NC mn ( T, T, t s always possble to constuct ates that satsfy (4. Step RA-3: Solve a Mult-Commodty Flow Poblem Statng wth use, assgn ates k along evey path k K NC such that k ψ kl + + max C l l L kψ kl, kψ kl ( k + k = C(B T (5 Now consde use. Assgn ates k along evey path k K NC such that k ψ kl = NC + max C l l L kψ kl, kψ kl

7 = ( k + k = C(B L (6 Fnally, assgn ates k along evey path k K such that k ψ kl = NC K + max C l l L kψ kl, kψ kl k = (7 Let R = NC k be the total outng ate use eceves, R = NC k be the total outng ate use eceves. Note that the constucton of ensues that t s always possble to assgn ates that satsfy (7. Theoem : Let (R A, R A be the ate assgnment gven by Pocedue RA. Then R A = C(A R A = C(A f C(A C(A + C(A R A = C(A R A = C(A C(A f C(A < C(A + C(A whee A A ae the cuts defned n the pocedue to oute bound capacty (.e. Pocedue P. Agan, we have omtted the poof due to space consdeatons. The basc dea of the poof follows fom the fact that the subgaphs used to constuct the ate assgnments ae the same subgaphs used to oute bound the capacty. The poof of shows that the ate assgnments ae elated to the capacty of the cuts constucted on these subgaphs, can be found n full n []. C. Achevng the Oute Bound on Capacty The esult of Pocedue RA s a sngle, feasble ate assgnment (R A, R A. Ultmately, howeve, ou goal s to show that the oute bound gven by the egon fom Secton IV s achevable, makng t the capacty egon. Futhemoe, ecall that the egon can take on one of two shapes, as shown n Fgues 5 6. If C(A C(A + C(A, then (R A, R A concdes wth the cone pont RI, meanng the ente egon s achevable. When C(A < C(A + C(A, (R A, R A concdes wth the cone pont RII,, hence we stll need to show that the othe cone pont s achevable n ode to establsh the achevablty of (by appopate tme shang. Notce that n Step RA-3 of the ate assgnment pocedue, use eceves pefeence n the assgnment of outng ates. Altenately, we could have gven use pefeence n the assgnment of outng ates, whch gves the followng pocedue. Rate Assgnment Pocedue RB Steps RB- & RB-: As n Pocedue RA. Step RB-3: As n Pocedue RA, but statng wth use. As t so happens, ths s exactly what we need to acheve the second cone pont when C(A < C(A +C(A. When C(A > C(A + C(A, Pocedues RA RB acheve the same pont. We fomalze ths noton n the followng coollay. Coollay : Any ate vecto (R, R s acheved by an appopate tme shang of the RA RB pocedues. In othe wods, egon s the capacty egon of the netwok. VI. DISTRIBUTD TRANSPORT-LAYR RAT CONTROL The ate assgnment pocedue descbed n Secton V not only eques knowledge about netwok-wde paametes (such as capactes of lnks n the netwok, but t also eques centalzed ate assgnments. It most netwoks, t s desable to be able to mplement ate assgnments locally at the souces n a dstbuted manne. In addton, we may want to suppot system-wde pefomance objectves - fo nstance, mantanng popotonal faness among uses - whle allowng fo adaptaton to fluctuatng dems. It tuns out that the path enumeaton gven n Pocedue P, combned wth nsghts ganed fom the ate assgnment pocedues, allows us to fomulate the uncast netwok codng poblem as the followng optmzaton poblem. Poblem P max, subject to ( = = U k ψ kl S D + max k + k + k = ( k + k kψ kl C l l L k =, The key equement hee s that the path enumeaton gven by Pocedue P s fxed known. Ths s smla to the appoach used n [8], whee authos assume ate contol occus ove fxed mult-cast subgaphs. Although Poblem P s a convex optmzaton poblem, the objectve functon s not stctly concave n the pmal vaables. Although ths means we cannot use tadtonal dualbased algothms to mplemented dstbuted ate assgnments, we can use methods based on poxmal technques, followng the appoach ecently used n multpath flow contol poblems (see e.g. [5]-[7]. In ode to do so, we constuct an augmented veson of Poblem P by ntoducng auxlay vaables coespondng to each of the pmal vaables.

8 Poblem P max,,y,y subject to ( = log k + = D ( k y k + max k ψ kl k + S K K k = ( k + k ( k y k kψ kl C l l L k =, It s clea that the soluton to Poblem P concdes wth the soluton to Poblem P. In ode to solve Poblem P, we use stad poxmal algothms fom [3]. Algothm A At the t th teaton:. Fx y = y(t y = y (t. Solve Poblem P wth espect to.. Set y(t + = (t y (t + = (t. In othe wods, we can thnk of the auxlay vaables y y as cuent estmates of the ate assgned to each path. Algothm A stll eques the soluton to a global optmzaton poblem, but ths can now be done usng stad dual technques. The esult s the followng set of algothms. Lnk Algothm ach lnk poduces a egulatoy sgnal (Lagange multple λ l that ndcates the level of congeston at that lnk (accountng fo netwok codng when applcable. Ths sgnal evolves accodng to the dffeence equaton: λ l = β[ kψ kl Il C l ] + (8 = k ψ kl + whee β s a constant ( = kψ kl + NC k ψ klil s the total physcal taffc flow on lnk l. Souce Vtual Buffe Algothm ach souce poduces two ntenal coodnaton sgnals (Lagange multples µ + µ that ndcate the dffeence between the total sde nfomaton netwok-coded nfomaton ates. These sgnals evolve accodng to the dffeence equatons: µ + = α[ k + k k] + (9 µ = α[ S k S k + k ncs k ] + (0 whee α s a constant, S k + k s the total ate at whch souce s sendng sde nfomaton, NC k s the ate at whch souce s sendng nfomaton flagged as netwok codng. These sgnals ae used to geneate the aggegate coodnaton sgnal µ = µ + µ. Souce Rate Assgnment Algothm I ach souce adjusts ts dedcated, outng, netwok codng ates accodng to:, = ag max, D KNC D KNC log k + D ( k y k k q k + (µ + µ NC k K NC kq k ( k + k ( k y k whee y y ae fxed values fo duaton of a sngle teaton of Algothm A. Souce Rate Assgnment Algothms II ach souce adjusts ts sde exta ates accodng to: k = ag max ( ( k y k k q k µ k ( k whee y y ae fxed values fo duaton of a sngle teaton of Algothm A. Fnally, we pesent a theoem egadng the convegence of these algothms. Theoem 3: Fo any fxed value of y y gven an appopate choce of step-szes, the algothm descbed by (8-( wll convege to the optmal soluton to Poblem P. Futhemoe, f y y ae updated accodng to Algothm A at a slow enough tme-scale, the algothm wll convege to the optmal soluton to Poblem P. The poof of Theoem 3 s a staghtfowad applcaton of known esults fo poxmal gadent pojecton algothms (see [3] fo elevant esults. Due to space lmtatons, the eade s efeed to [] fo a moe detaled explanaton of the algothm desgn convegence popetes. VII. CONCLUSIONS In ths pape, we have shown that unde cetan assumptons, t s possble to decompose the uncast netwok codng poblem fo two uses nto mult-commodty flow mult-cast netwok codng subpoblems. Futhemoe, we have shown that such a decomposton actually acheves capacty of the netwok. Usng nsght fom the achevablty esults, we fomulate the optmal two-use uncast netwok codng poblem as an optmzaton poblem. Ths allows us to develop dstbuted ate contol algothms that acheve system pefomance objectves n a decentalzed manne.

9 Thee ae many nteestng mpotant extensons to ths wok, we dscuss a few of them befly. Fst, n ths pape we have pesented esults fo uncast netwoks wth two uses. The oute bound on netwok capacty can, wth slght modfcatons, be used to bound the capacty of uncast netwoks wth an abtay numbe of uses. The achevablty esult, howeve, eques moe cae. Ths s due, n lage pat, to the fact that netwok codng can occu ove dffeent sets (possbly of moe than two uses. In such a case, t s not clea how to assgn the sde netwok codng ates to solve the mult-cast subgaph poblem. We conjectue, howeve, that a smla pocedue n whch the uncast netwok codng poblem s decomposed nto seveal mult-cast netwok codng mult-commodty flow sub-poblems wll acheve capacty of the netwok (unde assumptons smla to those pesented n hee. In addton, the poblem of netwok codng fo uncast netwoks pesents some nteestng ssues elated to the ncentves of netwok codng. Coopeaton among uses s nheently equed to maxmze ate assgnments when netwok codng s used. In the moe tadtonal mult-cast settng, uses do not lose anythng by sendng sde nfomaton snce the nfomaton needs to be boadcast to all uses n the netwok. In a uncast netwok, howeve, a use gans nothng by sendng sde nfomaton unless the othe uses send sde nfomaton as well. The queston of what a use wll do when faced wth the choce between sendng nfomaton to ts own destnaton, sendng nfomaton fo puposes of decodng at othe uses, s an nteestng one. ACKNOWLDGMNT The authos would lke to thank Pof T. Ho at Caltech, R. Appuswamy Pof K. Zege at UCSD fo the thoughtful dscussons. Ths wok was suppoted n pat by the NSF CARR No. CNS , the ARO-MURI No. W9NF , the AFOSR No. FA , CWC at UCSD. [9] T. Ho, M. Medad, R. Koette, D. Kage, M. ffos, J. Sh, B. Leong, A om lnea netwok codng appoach to multcast, I Tansactons on Infomaton Theoy, vol. 5, no. 0, pp , Oct 004. [0] Z. L B. L, Netwok codng: The case of multple uncast sessons, n Poceedngs of the Foty-Second Annual Alleton Confeence on Communcatons, Contol Computng, Sep 004. [] C. Wang N. Shoff, Beyond the buttefly - a gaph-theoetc chaactezaton of the feasblty of netwok codng wth two smple uncast sessons, n Poceedngs of the I Intenatonal Symposum on Infomaton Theoy (ISIT, June 007. [] T. Ho, Y. Chang, K. Han, On constucte netwok codng fo multple uncasts, n Poceedngs of the Foty-Fouth Annual Alleton Confeence on Contol, Communcaton Computng, Sep 006. [3] A. ylmaz D. Lun, Contol fo nte-sesson netwok codng, n Poceedngs of the Wokshop on Netwok Codng, Theoy Applcatons (NetCod, Jan 007. [4] D. Taskov, N. Ratnaka, D. Lun, R. Koette, M. Medad, Netwok codng fo multple uncasts: An appoach based on lnea optmzaton, n Poceedngs of the I Intenatonal Symposum on Infomaton Theoy (ISIT, July 006. [5] X. Ln N. Shoff, Utlty maxmzaton fo communcaton netwoks wth multpath outng, I/ACM Tansactons on Automatc Contol, vol. 5, no. 5, pp , May 006. [6] H. Han, S. Shakkotta, C. Hollot, R. Skant, D. Towsley, Multpath tcp: A jont congeston contol outng scheme to explot path dvesty n the ntenet, I/ACM Tansactons on Netwokng, vol. 4, no. 6, Dec 006. [7] W. Wang, M. Palanswam, S. Low, Optmal flow contol outng n mult-path netwoks, Pefomance valuaton, 003. [8] R. Doughety, C. Felng, K. Zege, Insuffcency of lnea codng n netwok nfomaton flow, I Tansactons on Infomaton Theoy, vol. 5, no. 8, pp , Aug 005. [9] A. Lehman. Lehman, Complexty classfcaton of netwok nfomaton flow poblems, n Poceedngs of the Foty-Fst Alleton Confeence on Contol, Communcaton Computng, Oct 003. [0] G. Foll, Real Analyss: Moden Technques The Applcatons. John Wley & Sons, Inc., 999. [] R. Rockafella, Netwok Flows Monotopc Optmzaton. Athena Scentfc, 998. [] J. Pce, Resouce Allocaton n Mult-Use Communcaton Systems. Ph.D. Thess, 007. [3] D. Betsekas J. Tstskls, Paallel Dstbuted Computaton. Pentce-Hall Inc., 989. RFRNCS [] R. Ahlswede, N. Ca, S. L, R. Yeung, Netwok nfomaton flow, I Tansactons on Infomaton Theoy, vol. 46, no. 4, pp. 04 6, Jul 000. [] N. Havey, R. Klenbeg, A. Lehman, On the capacty of nfomaton netwoks, I Tansactons on Infomaton Theoy, vol. 5, no. 6, pp , June 006. [3] F. Kelly, Mathematcal modellng of the Intenet, n Mathematcs Unlmted - 00 Beyond, B. ngqust W. Schmd, ds. Spnge-Velaq, 00, pp [4] S. H. Low D.. Lapsley, Optmzaton flow contol, I: basc algothm convegence, I/ACM Tansactons on Netwokng, vol. 7, no. 6, pp , Dec 999. [5] R. Koette M. Medad, An algebac appoach to netwok codng, I/ACM Tansactons on Netwokng, vol., no. 5, pp , Oct 003. [6] S. L, R. Yeung, N. Ca, Lnea netwok codng, I Tansactons on Infomaton Theoy, vol. 49, no., pp , Feb 003. [7] T. Ho H. Vswanathan, Dynamc algothms fo multcast wth nta-sesson netwok codng, Submtted to I Tansactons on Infomaton Theoy. [8] L. Chen, T. Ho, S. Low, M. Chang, J. Doyle, Optmzaton based ate contol fo multcast wth netwok codng, n Poceedngs of I Infocom, May 007.