On Achieving Optimal Throughput with Network Coding


 Ariel Rogers
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1 On Achieving Optiml Throughput with Network Coding Zongpeng Li, Bochun Li, Dn Jing, Lp Chi Lu Astrct With the constrints of network topologies nd link cpcities, chieving the optiml endtoend throughput in dt networks hs een known s fundmentl ut computtionlly hrd prolem. In this pper, we seek efficient solutions to the prolem of chieving optiml throughput in dt networks, with single or multiple unicst, multicst nd rodcst sessions. Although previous pproches led to solving NPcomplete prolems, we show the surprising result tht, fcilitted y the recent dvnces of network coding, computing the strtegies to chieve the optiml endtoend throughput cn e performed in polynomil time. This result holds for one or more communiction sessions, s well s in the overly network model. Supported y empiricl studies, we present the surprising oservtion tht in most topologies, pplying network coding my not improve the chievle optiml throughput; rther, it fcilittes the design of significntly more efficient lgorithms to chieve such optimlity. Index terms: Grph theory, Informtion theory, Mthemticl progrmming/optimiztion, Simultions. I. INTRODUCTION In its most generl form, dt network consists of set of end hosts nd switches interconnected vi undirected (or duplex) communiction links. In dt networks with known topologies nd ndwidth cpcity ounds for ech undirected link, fundmentl prolem is to compute nd chieve the mximum endtoend throughput for one or multiple ctive communiction sessions. Depending on the ojectives of pplictions, communiction session my e in the form of unicst (onetoone), multicst (onetomny), rodcst (onetoll), or group communiction (mnytomny). The solutions to this prolem my led to fundmentl nd new insights with respect to optiml routing nd trffic engineering. For exmple, the recent prdigm of selfish routing [1] llows end hosts to choose routes themselves using source routing strtegies. Finding the optiml strtegy to disseminte dt to multiple destintions with mximum throughput is of nturl interests in such prdigm, especilly when we wish to optimlly exploit existing network cpcities to disseminte lrge volumes of dt. The focus on the undirected network model is supported y the following justifictions. First, s pst reserch in network flow theory [2] nd informtion theory [3] suggests, the undirected network model hs its own rhythm, nd results otined there my e drsticlly different from those otined in the directed network model. In fct, the undirected model is more generl nd fundmentl in tht, solution constructed for undirected networks cn usully e pplied to solve the sme prolem in directed networks, ut not vice vers. This is prticulrly true for our prolem nd solution in this pper. Second, undirected communiction links provide the complete flexiility in cpcity lloction, nd consequently leds to higher trnsmission rtes tht etter represent the optiml informtion flow rte. Finlly, in specil network scenrios such s wireless d hoc networks, communiction links re nturlly undirected, in the sense tht dt trnsmission long oth directions of the wireless link shre the ville spectrum. In this pper, we seek to ring fundmentlly new insights nd efficient solutions to the prolem of optimizing endtoend throughput in undirected dt networks. We first illustrte the power of network coding [4], [] with respect to chieving optiml throughput. In the prdigm of network coding, informtion flows in dt networks my not only e stored nd forwrded, ut lso e encoded nd decoded in ny nodes in the network. We show tht, lthough previous directions of computing optiml multicst throughput involve solving NPcomplete prolems, the mximum multicst throughput nd the corresponding optiml multicst strtegy cn indeed e computed efficiently in polynomil time, with the unique encodle property of informtion flows considered. We then show tht this conclusion cn e extended to multiple concurrent sessions, s well s to other types of communiction, including unicst, rodcst nd group communiction. Even when the generl form of dt networks is modified to reflect relistic chrcteristics of overly networks (where only end hosts t the edge my e le to replicte, encode nd decode dt), the sme conclusion still holds. The solutions to the prolems include not only optiml routing strtegies to trnsmit dt in the network, ut lso how dt my e encoded nd decoded s they re relyed towrds the destintions. Though there exist previous results on network coded throughput in directed networks, to the est of our knowledge, this pper is the first work tht systemticlly studies the effects of network coding with respect to optimizing throughput in undirected dt networks. The vilility of efficient solutions mkes it finlly possile to study vrious spects of properties of the chievle throughput, in relisticlly sized networks. We present empiricl studies sed on simultion results over thousnds of test scenrios using our lgorithms. We compre the optiml multicst throughput with nd without network coding, nd show tht noticele throughput gins cn only e experienced in contrived network topologies; for rndom nd irregulr network topologies it is lmost lwys zero. This grees with out previous theoreticl results on the upper ound of the dvntge of network coding in undirected networks [3]: rther thn incresing throughput, the dvntge of network coding
2 is indeed to fcilitte significntly more efficient computtion of the strtegies to chieve optiml throughput of informtion flows. Our empiricl studies lso show tht overly multicst, which hs recently ttrcted extensive reserch efforts, my pproch optiml throughput quite well. The reminder of this pper is orgnized s follows. We first discuss relted work in Sec. II. In Sec. III, we present our min theorems nd lgorithm with respect to chieving optiml endtoend throughput with single multicst session. In Sec. IV, we extend our results to the cses of multiple sessions of unicst, multicst, rodcst, nd group communiction. We lso consider the model of overly networks, where only suset of nodes re cple of repliction nd coding. We then present empiricl studies in Sec. V, nd conclude the pper in Sec. VI. II. RELATED WORK The open prolem of chieving optiml endtoend throughput with efficient lgorithms hs not een discussed in depth in existing literture. There exist, however, similr prolems tht hve een extensively studied. Towrds the direction of Qulity of Service (QoS) routing, the ojective is to find endtoend pths or multicst trees tht stisfy specific ndwidth or dely constrints, nd therefore providing the desired QoS gurntees [6]. With respect to endtoend throughput, finding good topologies tht stisfy ndwidth requirements is oviously different from nd rguly esier thn finding optiml ones. There exists n extensive ody of reserch in the re of multicst routing in widere IP networks (e.g., [7]). The dvntge of IPsed multicst is rought y dt pcket repliction on multicstcple switches, improving ndwidth efficiency nd throughput compred to ll (nive) unicst etween the source nd the multicst receivers. However, since it is sed on the construction of single tree, the endtoend throughput is not optiml compred to wht is chievle y topology eyond tree. As IP multicst is not redily deployed, lgorithms promoting pplictionlyer overly multicst hve recently een proposed s remedil solutions, focusing on the issue of constructing nd mintining multicst tree using only end hosts [8], [9]. Though single multicst tree my not led to optimized throughput, recent studies (e.g., SplitStrem [1], CoopNet [11], Digitl Fountin [12] nd Bullet [13]) hve proposed to utilize either multiple multicst trees (forest) or topologicl mesh to deliver striped dt from the source, using either multiple description coding or source ersure codes to split content to e multicst. These proposls hve indeed improved endtoend throughput eyond tht of single tree, ut there hve een no discussions on whether the optiml throughput my e chieved, or how close the proposed lgorithms pproch optimlity. In this pper, we study such chievle optimlity, while considering the most generl cse where the dt source trnsmits strem of ytes, nd is not ssumed to perform ny source or error correction coding. There hve een studies on chieving optimlity with respect to computing olivious routing strtegies in dt networks. The ojectives re to mximize throughput for sourcedestintion pir, nd to minimize congestion on the network. Most notly, using liner progrmming techniques, polynomil time lgorithms (with polynomil numer of vriles nd constrints in the LP formultion) cn e constructed to compute strtegies for optiml olivious routing for ny network, directed or undirected [14]. Though we lso employ liner optimiztion tools nd study undirected networks, our prolem domin is more generl: while optiml olivious routing focuses on origindestintion pirs of unicst sessions (possily exploiting pth diversity), we focus on vriety of communiction sessions, including unicst, multicst, rodcst nd group communiction. We seek fundmentl insights on how optiml routing strtegy my ecome, nd wht is the mximum chievle throughput in communiction session. The theory of network flows studies the trnsmission of commodities of the sme type (unicommodity flows) through cpcitied network. The mximum flow rte etween the source nd the destintion which my e computed with vrious efficient comintoril lgorithms [2]. When commodities to e trnsmitted re of different types (multicommodity flows), computing the mximum flow rte cn e solved s liner optimiztion prolem. In oth unicommodity nd multicommodity flows, commodities my only e forwrded t intermedite nodes, comprle to ll unicst in dt networks. The concept of network coding extends the cpilities of network nodes in communiction session: from sic dt forwrding (s in ll unicst) nd dt repliction (s in IP or overly multicst), to coding in Glois fields. Fig. 1 illustrtes clssic exmple of how network coding ssists to improve endtoend throughput. As R 1 receives oth nd + (encoded over GF(2)), it is le to decode nd retrieve oth nd. If the link cpcities re 1, the mximum chievle throughput with network coding is 2. Without coding, it cn e computed tht the optiml throughput is 1.87 [3]. If only one multicst tree is used (s in IP multicst), the chieved throughput is 1. S R1 R 2 () Mximum throughput with one multicst tree is 1(1.87 with multiple trees). + S + R1 R 2 () Mximum throughput with network coding is 2. Fig. 1. The dvntge of network coding with respect to improving the endtoend multicst throughput from S to R 1 nd R 2. The recent rekthrough theorem in network coding shows tht, for multicst session in directed networks, if rte x cn e chieved from the sender to ech of the multicst receivers independently, it cn lso e chieved for the entire multicst session (refer to independent proofs of Ahlswede +
3 et l. [4] nd Koetter et l. []). In ddition, Li et l. [1] show tht liner codes suffice to chieve such property. All liner coding opertions re defined s liner comintions over Glois fields with fixed element lengths, thus the size of the dt does not increse fter eing encoded. III. ACHIEVING OPTIMAL THROUGHPUT IN UNDIRECTED DATA NETWORKS: THE SINGLE MULTICAST CASE We egin our study from the cse of single multicst session. We consider the most generl form of dt networks, represented y simple grph G = (V,E) with undirected edges etween network nodes. Ech edge represents communiction link, nd the edge cpcities re specified y function C : E Q + (where Q + denotes the set of positive rtionl numers), representing the ville ndwidth cpcities of communiction links. Throughout this pper, we focus on the frctionl model of dt routing, where the cpcity of ech link my e shred frctionlly in oth directions, nd informtion flows my e split nd merged t ritrrily fine scles. We use M = {m,m 1,...,m k } V to specify the set of nodes in the multicst group, with m eing the sender. In grphicl illustrtions throughout this pper, nodes in M re shown s lck, nd nodes in V M re shown s white. Links re leled with their cpcities, nd ll unleled links hve cpcity of 1. A. Steiner tree pcking nd Steiner strength To compute the optiml throughput of multicst sessions, Steiner tree pcking [16], [17] nd Steiner strength hve een the stteofthert. Unfortuntely, oth re NPhrd solutions. Steiner tree pcking. Consider the cse of informtion flows in one multicst session from source to set of destintions. It cn e theoreticlly shown tht, if coding is not considered, chieving optiml throughput vi multiple multicst trees is equivlent to the prolem of Steiner tree pcking, which seeks to find the mximum numer of pirwise edgedisjoint Steiner trees, in ech of which the multicst group remins connected. An intuitive explntion to such equivlence is tht, ech unit throughput corresponds to unit informtion flow eing trnsmitted long tree tht connects every node in the group. The mximum numer of trees we cn find corresponds to the optiml throughput for the session. Fig. 2() shows such n exmple. In the figure, ech letter corresponds to distinct Steiner tree, nd nine such Steiner trees ( to i) exist in the shown pcking scheme, where the tree corresponding to is highlighted. Since ech link with unit cpcity needs to ccommodtes Steiner trees, the chievle throughput on ech tree is, therefore,.2. This leds to multicst throughput of 1.8, which is optiml without coding. Unfortuntely, Steiner tree pcking hs een shown to e NPcomplete [17], [18], nd the est known polynomil time lgorithm hs n pproximtion rtio of round 1. [18]. With the sme exmple, we cn lso show tht the chievle optiml throughput with network coding is 2 (Fig. 2()), which is higher thn tht chieved without coding. Consequently, defh m 3 efgh defgh cegi m 1 cdi m fghi cdi cefgh () steiner tree pcking nd multicst without coding. cdi m 2 m3 m 1 + m + + m 2 () multicst with network coding. Fig. 2. The chievle optiml throughput is 1.8 without coding, nd 2 with coding. even if Steiner tree pcking is computtionlly fesile, it my not lwys yield the ctul optiml multicst throughput. Steiner strength. In n undirected cpcitied network N, we consider prtitions of the network where there exists t lest one source or receiver node in ech component of the prtition. Let P e the set of ll such prtitions. The Steiner strength of N is defined s min p P E c /( p 1), where E c is the totl intercomponent link cpcity on the set of links E c eing cut, nd p is the numer of components in the prtition p. It is nturl extension of network strength [19] defined for rodcst network. It is known from our previous work tht network strength is equivlent to the chievle optiml throughput in rodcst sessions [3]. Therefore, it is nturl direction to compute optiml multicst throughput y computing the Steiner strength. Unfortuntely, the Steiner strength prolem turns out to e NPcomplete s well. The fct tht computing Steiner strength is NPcomplete lso rules out the possiility tht Steiner strength nd optiml multicst throughput re lwys equl. In fct, we find tht Steiner strength is either equl to or higher thn the chievle optiml throughput 1. B. Efficient solutions for throughput optimiztion: the cflow Liner Progrm Contrry to the previous pessimistic views, we present the surprising result tht efficient solutions do exist for computing optiml throughput in undirected networks. We first formulte the prolem s liner network optimiztion prolem, in which oth the numer of vriles nd the numer of constrints re ounded y O( M E ). We then show tht the result of such optimiztion exctly gives the mximum chievle throughput, s well s the corresponding routing strtegy. We lso discuss possile solutions to the liner progrm. We egin y presenting the orienttion constrints of the liner progrm tht computes optiml throughput. An orienttion of network N is strtegy to replce ech undirected link e = uv with two directed links 1 = uv nd 2 = vu, such tht C(e) = C( 1 ) + C( 2 ). After the orienttion, the 1 Oserving spce constrints, we exclude the proofs of this result nd the NPcompleteness of Steiner strength. Interested reders re referred to our technicl report [], which lso includes more detiled explntions nd n exmple in which the Steiner strength is higher thn the optiml throughput.
4 set of undirected links E ecomes set of directed links A, with the numer of links in the set douled. We proceed to consider flows from the source to the multicst receivers. To tke dvntge of the power of network coding to resolve competition for link cpcities, we introduce the concept of conceptul flows (cflow). We define conceptul flows s network flows tht coexist in the network without contending for link cpcities. Our liner progrm to compute the optiml throughput, shown in Tle I, is referred to s the cflow LP since it is sed on conceptul flows. In the LP, f 1...f k re the conceptul flows from sender m to ech of the receivers. Ech flow vector f i specifies flow rte f i () for ech directed link A. f i in (v) denotes the totl incoming fi flow rte t node v, similr for f i out(v). Finlly, the sclr χ is the trget flow rte of optimiztion. In ddition to the orienttion constrints, the cflow LP lso includes the network flow constrints for ech conceptul flow, nd the equl rte constrints. The network flow constrints re specified in compct form for ll conceptul flows, which requires (1) flow rtes must e upper ounded y link cpcities; (2) flow conservtion, i.e., the incoming flow rte in the conceptul flow f i equls to outgoing flow rte in f i t rely node for f i ; nd (3) the incoming flow rte t the source nd the outgoing flow rtes t the receiver re ll zero, for ech f i. The equl rte constrints require tht the flow rtes of conceptul flows re identicl, with χ eing the uniform flow rte. With these liner constrints, the trget flow rte χ is then mximized. TABLE I THE cflow LP Mximize: χ Suject to: Orienttion j constrints: C() A C( 1) + C( 2) = C(e) e E Independent 8 network flow constrints for ech conceptul flow: f i () i [1..k], A >< f i () C() i [1..k], A fin(v) i = fout(v) i i [1..k], v V {m, m i} f >: in(m i ) = i [1..k] fout(m i i) = i [1..k] Equl rte constrints: χ = fin(m i i) i [1..k] We re now redy to present one of our min contriutions of this pper, y showing tht the cflow LP provides n efficient lgorithm to compute the chievle optiml throughput, s well s the routing strtegy. Theorem 1. For n undirected dt network with single multicst session, N = {G(V,E),C : E Q +,M = {m,m 1,...,m k } V }, the mximum endtoend throughput χ(n) nd its corresponding optiml routing strtegy cn e computed in polynomil time using the cflow LP, in which oth the numer of vriles nd the numer of constrints re polynomil, nd on the order of O( M E ). The conceptul flows f 1...f k constitute the optiml routing strtegy. Proof: The orienttion constrints reflect complete flexiility in orienting the undirected network N, without eing too restrictive or too relxed. For ech fixed orienttion, conceptul flows re eing mximized with independent nd stndrd network flow constrints, s well s the extr constrint tht conceptul flow rtes re equl to ech other. Therefore, the result of the mximiztion is the mximum possile flow rte tht cn e independently chieved from the source to ll receivers, over ll possile orienttions of the network: χ = mx [ min (mximum m m i flow rte)], o O m i M {m } where O denotes ll possile orienttions of the network, nd M {m } is the set of multicst receivers. Recll the recent rekthrough in network coding [4], [] shows tht, for fixed orienttion of the network, rte x cn e chieved for the entire multicst session if nd only if it cn e chieved for ech multicst receiver independently. This implies tht, the mximum throughput in ech orienttion equls to the minimum of the mximum source to receiver flow rte. The cflow LP essentilly mximizes this minmx flow over ll possile network orienttions, nd otins the mxminmx flow tht is precisely the mximum multicst throughput in the originl undirected network. Further, the source my trnsmit informtion to ech receiver m i ccording to the conceptul flow f i. Should more thn one conceptul flows utilize cpcity on the sme link, the conflict cn lwys e resolved, provided tht network coding is pplied ppropritely [4], []. The cflow LP contins 2 E orienttion vriles C(), 2 M E virtul flow vriles f i (), nd one trget flow rte vrile χ. Therefore, the totl numer of vriles is 2( M + 1) E + 1, which is on the order of O( M E ). In ddition, the cflow LP contins 3 E orienttion constrints, (4 E + V )( M 1) network flow constrints, s well s M 1 equl rte constrints. The totl numer of constrints is, therefore, (4 E + V + 1)( M 1) + 3 E, which is lso on the order of O( M E ). The optiml routing strtegy computed y cflow LP specifies the rte of dt strems eing trnsmitted long ech link. Bsed on the routing strtegy, we need to perform the dditionl step of code ssignment to compute the coding strtegy, efore dt strems my e trnsmitted. The coding strtegy includes one trnsformtion mtrix for ech node, which specifies how incoming dt strems re linerly coded into outgoing strems. Given the routing strtegy from the cflow LP, there exist polynomil time lgorithms to perform such code ssignments [21]. Therefore, we hve the following corollry of Theorem 1: Corollry 1. The complete solution tht chieves optiml throughput in undirected dt networks with single multicst session cn e computed in polynomil time, including oth the routing nd coding strtegies. In order to evlute the dvntge of network coding with
5 respect to improving chievle optiml throughput, we hve implemented oth the cflow LP nd ruteforce lgorithm to compute the Steiner tree pcking numer. The Steiner tree pcking lgorithm enumertes ll steiner trees in the network, ssigns flow vrile to ech tree, nd then mximizes the summtion of ll tree flows, suject to the constrints tht the totl weight (throughput) of trees using ech link should not exceed its cpcity. We hve evluted oth the cflow LP nd Steiner tree pcking (denoted s π(n)) using our previous exmple in Fig. 1, s well s set of uniform iprtite networks, which re elieved to e good cndidtes to show the power of coding on improving throughput [21], [22]. A uniform iprtite network C(n,k) consists of the dt source nd two lyers: one with n rely nodes nd the other with ( n k) receivers. Ech rely node is connected to the sender, nd ech receiver is connected to different group of k rely nodes, nd ll links hve cpcity of 1. For instnce, the network in Fig. 2 is C(3,2), nd the clssic exmple of network coding in Fig. 1 is isomorphic to C(3,2). Tle II summrizes the results of our empiricl studies, from which we hve derived the following oservtions. First, the cflow LP is much more sclle nd efficient thn Steiner tree pcking, which fils to compute solution for network s smll s C(,3), with only 16 nodes nd 3 links, ut lmost million different Steiner trees. In seprte experiments, the cflow LP is le to compute the optiml throughput for networks hving thousnds of nodes. Second, optiml throughput with coding is lwys lower ounded y tht without coding; however, network coding only introduces slight dvntge, with the χ(n)/π(n) rtio no higher thn Third, coded trnsmission my led to more integrl flow rtes nd throughput thn uncoded trnsmission. TABLE II COMPUTING OPTIMAL THROUGHPUT: cflow LP VS. STEINER TREE PACKING χ(n) π(n) Network V M E χ(n) π(n) # of trees Fig C(3, 2) C(4, 3) ,113 C(4, 2) ,128 C(, 4) ,24 C(, 2) ,14 C(, 3) ,96,624 throughput. IV. ACHIEVING OPTIMAL THROUGHPUT IN UNDIRECTED DATA NETWORKS: MORE GENERAL CASES Our efficient solution, the cflow LP, cn e extended to solve the optiml throughput prolem in cses eyond single multicst session. We now present its extensions (1) to unicst, rodcst nd group communiction sessions, (2) to the cse of multiple communiction sessions, nd (3) to the model of overly networks. A. The cses of unicst, rodcst nd group communiction sessions Since unicst nd rodcst cn e viewed s specil cses of multicst, where two nodes nd ll nodes re in the multicst group, respectively, our solution in the single multicst cse cn e redily pplied to single unicst or rodcst session without modifictions. In the cse of unicst session, the cflow LP essentilly solves liner progrm for single network flow. In the cse of rodcst session, the cflow LP computes the optiml rodcst throughput, which hs een shown y our previous work to e the sme s oth the spnning tree pcking numer nd the network strength [3]. Trditionlly, these three equl quntities hve een computed from either the perspective of network strength or spnning tree pcking. Cunninghm [19] first gve comintoril lgorithm tht computes the network strength, which ws lter improved y Brhon [24]. Both lgorithms re sed on mtroid theory, nd re highly sophisticted. Though the spnning tree pcking prolem hs n LP formultion, the numer of vriles is exponentil. It is therefore necessry to work on its dul progrm, where the minimum spnning tree lgorithms cn serve s the seprtion orcle. In comprison, the cflow LP provides n efficient lterntive, with polynomil numer of constrints nd vriles, nd with oth generl LP solvers nd customtilored distriuted sugrdient solutions [23] ville. f1 S1 S f2 S 2 As finl note, we point out tht eyond pplying generl liner progrmming solutions such s the simplex method it is lso possile to design customtilored lgorithms for the cflow LP, to tke dvntge of its underlying network flow structure. In n ccompnying pper [23], we pply Lgrngin relxtion on the dul progrm of the cflow LP, nd design distriuted sugrdient solution. The lgorithm itertively refines n existing orienttion of the originl network, until n optiml one is reched. At this point, M mxflow computtions re invoked to find the optiml multicst Fig. 3. Trnsforming group communiction into multicst trnsmission. Group communiction refers to mnytomny communiction sessions where multiple sources multicst independent dt to the sme group of receivers, the set of senders nd the set of receivers my or my not overlp. Previous work [] hs shown tht mnytomny session cn e esily trnsformed into multicst session, y dding super source, which is trditionl technique in network flows. As illustrted in Fig. 3,
6 we cn dd n dditionl source S to the network, nd connect it to ech of the sources in the group communiction session, with links of unounded cpcity. We my then pply the cflow LP to mximize the multicst throughput from S to ll the receivers. Additionl constrints cn e pplied to flow rtes on the newly dded links etween the super source nd the originl sources in the session, governing firness mong the originl sources. The outcome from the cflow LP is the optiml throughput nd its corresponding routing strtegy for the originl group communiction session. B. The cse of multiple sessions In its most generl form, the optiml throughput prolem llows multiple communiction sessions of different types to coexist in the sme network. Since multicst is representtive in tht unicst, rodcst nd group communiction cn ll e trnsformed into multicst it is sufficient to consider the optiml throughput prolem in the cse of multiple multicst sessions. To chieve optiml throughput with multiple sessions, we need to consider the prolem of intersession firness. The definition of firness is usully ppliction dependent; however, s long s it cn e expressed using liner constrints, we cn esily include them in the LP formultion. With respect to network coding in multiple sessions, it is theoreticlly possile to pply network coding on multiple incoming strems of different sessions. However, we rgue ginst this possiility, nd use coding y superposition [4], i.e., network coding is pplied only to incoming strems of the sme session. This rgument is minly supported y the computtionl intrctility of the optiml throughput prolem if intersession coding is llowed 2. In ddition, our empiricl experiences show tht llowing intersession coding cn hrdly improve optiml throughput, nd it is not prcticl to code dt strems from different pplictions either. The mflow LP given in Tle III is designed to solve the optiml throughput prolem with multiple multicst sessions, where we use weighted proportionl firness s the firness model. It is the result of extending the cflow LP to its multicommodity vrint. We ssume there exist totl of s multicst sessions, numered s 1...s. Ech session i hs source m i, numer of receivers m i1...m iki, set of conceptul flows f i1...f i k i, s well s weight w i indicting the importnce of the session. The sclr χ i is the common rte for conceptul flows within session i, the sclr χ is the common weighted throughput for ll the multicst sessions, nd the trget of the mflow LP is to mximize χ. The mflow LP replces the stndrd network flow constrints in the cflow LP with set of multicommodity cflow constrints. Since flows of different sessions contend for link cpcity, the summtion of the persession flow rtes should not exceed link cpcities. Since flows within the sme session do not compete for link cpcity, the effective flow rte within 2 It is known tht finding sufficient nd necessry conditions for the fesiility of multiple sessions in this cse is equivlent to finding point in n lgeric vriety, which is NPhrd []. session i on link is f i () = mx j [1..ki] f ij (). The mx function is not liner, so this constrint is relxed to f i () f ij (), j [1...k i ]. TABLE III THE mflow LP Mximize: χ Suject to: Orienttion j constrints: C() A C( 1) + C( 2) = C(e) e E Multicommodity 8 cflow constrints: f i j () i [1..s], j [1..k i], A f i j () f i () i [1..s], j [1..k i], >< P A s i=1 fi () C() A f i j in (v) = fi j out(v) i [1..s], j [1..k i] v V {m i, m ij } f >: j in (mi ) = i [1..s], j [1..k i] f i j out(m ij ) = i [1..s], j [1..k i] Equl rte constrints: χ i = f i j in (mi ) j Firness constrints: i [1..s], j [1..ki] χ = χ i /w i i [1..s] Theorem 2. In the cse of multiple multicst sessions with coding y superposition, the optiml endtoend throughput nd its corresponding optiml routing strtegy in undirected dt networks cn e computed in polynomil time, y the mflow LP. Proof: The correctness of the mflow LP uilds upon the correctness of the cflow LP, which is proved in Theorem 1, plus the fct tht for coding y superposition, dt trnsmission from different sessions constitute totlly different commodities when competing for link cpcity. Furthermore, it is esy to check tht oth the numer of vriles nd the numer of constrints in the mflow LP re on the order of O(s M E ), where s is the numer of sessions. C. The cse of overly networks Since neither network coding nor dt repliction (for IP multicst) re widely supported in the currentgenertion network elements in the core, we consider the cse of overly networks where only the end hosts hve the full cpilities to forwrd, replicte nd code dt strems, nd the core network elements (henceforth referred to s routers) my only forwrd dt pckets s is. We note tht the cse of overly networks is ctully more generl thn the clssicl model of undirected dt networks we hve used so fr, which hints tht the optiml throughput prolem my ecome hrder to solve. Let N = {G(V,E),C : Q +,M = {m,...,m k },H = M {m k+1,...m h } V } e n overly network with multicst session. The multicst group M is suset of the end hosts H. If M = H, i.e., ll end hosts re in the multicst group, Grg et l. [2] hs shown tht the optiml multicst throughput cn e efficiently computed in this cse,
7 y working on the dul progrm of nturl LP formultion. It hs lso een shown in [2] tht, in the generl cse the optiml throughput prolem without network coding is the overly Steiner tree pcking prolem, nd is still NPcomplete. With the support of network coding, however, we re le to extend the cflow LP to its overly vrint, referred to s the oflow LP, to solve the optiml throughput prolem in the model of overly networks. The oflow LP tkes hierrchicl view of the multicst trnsmission, with n underly nd n overly level. The underly level corresponds to the physicl network topology, nd hs multicommodity flows g ij connecting ech pir of end hosts m i nd m j, vi only routers s intermedite nodes. The overly level is conceptul, nd contins end hosts fully connected s complete grph. The link ij from m i to m j hs cpcity equl to the underly flow rte g ij. We then pply the cflow LP in the overly level to mximize the endtoend throughput, where ech node is cple of repliction nd coding. In the oflow LP shown in Tle IV, we include three groups of constrints. First, the orienttion constrints re identicl to those included in the cflow LP. Second, the stndrd multicommodity flow constrints re specified for the underly flows etween end hosts nd vi routers only. Third, we introduce the mpping constrints tht mp the underly g ij flow rte to the overly link cpcity (referred to s C ( ij )), nd then pply the originl constrints in the cflow LP t the overly level. The trget of the oflow LP is to mximize throughput in the overly level. TABLE IV THE oflow LP Mximize: χ Suject to: Orienttion j constrints: C() A C( 1) + C( 2) = C(e) e E Underly 8 multicommodity flow constrints: P g ij () i, j [1..h], A >< g ij () C() i, j [1..h], A g ij in (v) = gij out(v) i, j [1..h], v V H g >: in (v) = i, j [1..h], v H {mj} gout(v) ij = i, j [1..h], v H {m i} Overly 8 cflow constrints: C ( ij) = gout(m ij i) i, j [1..h] f i ( ) i [1..k], >< A = { ij 1 i, j h} f i ( ) C ( ) A, i [1..k] fin(v) i = fout(v) i i [1..k], v H M fin(m i ) = i [1..k] f >: out(m i i) = i [1..k] χ = fin(m i i) i [1..k] Theorem 3. In the cse of single multicst session in the model of overly networks, the optiml endtoend throughput nd its corresponding optiml routing strtegy cn e computed in polynomil time, using the oflow LP. Proof: Since rely nodes in the overly network cn not replicte or encode dt, dt strem tht is trnsmitted etween two end hosts without pssing third end host remins unchnged throughout the trnsmission nd upon rrivl. Therefore, it is vlid to model these direct trnsmissions etween end hosts s multicommodity flows. The vlidity of the cflow constrints in the overly lyer my e derived from the correctness of the cflow LP, which we hve proved in Theorem 1. Furthermore, inspection on the vriles nd constrints in the oflow LP revels tht, the numer of oth re on the order of O( H 2 E ). Similr to the extension from cflow to mflow, one my extend the oflow LP into its multicommodity vrint to ccommodte multiple sessions in overly networks. More specificlly, one needs to replce the overly cflow constrints with the overly mflow constrints in the third group of constrints of the oflow LP. The resulting liner progrm hs oth its numer of vriles nd numer of constrints ounded y O(( H 2 + s M ) E ). This is usully not worse thn those of the singlesession oflow LP, since H 2 domintes s M in most cses. V. EMPIRICAL STUDIES Due to the lck of efficient lgorithms, previous studies on the prolem of improving session throughput re lrgely sed on experimentl or intuitive insights. We rgue tht the vilility of the cflow, mflow nd oflow LPs hs significntly chnged the lndscpe, nd hs mde it computtionlly fesile to study the exct enefits of vrious proposls to chieve higher throughput, including single multicst tree with dt repliction, multiple multicst trees, nd network coding. Our empiricl studies re sed on the implementtion of ll three LPs tht we hve proposed. In comprison studies, we hve lso implemented lgorithms to compute the optiml throughput with multiple multicst trees ut without coding, the optiml throughput with widest multicst tree, s well s the optiml throughput with ll unicst from the source to ll receivers. Topologies used in our simultions re generted y the BRITE topology genertor [26], with sizes rnging from 1 to nodes, oth with nd without powerlw properties, with hevytiled or constnt link cpcities. How dvntgeous is network coding with respect to improving optiml throughput? The rtio of chievle optiml throughput with coding over tht without coding is referred to s the coding dvntge. Recll tht we hve investigted the coding dvntge in Tle I, nd re unle to experimentlly find cses where network coding my improve optiml throughput y fctor higher thn We re nturlly led to the question: Wht is the upper ound of the coding dvntge? Previous work [21] shows tht in directed cyclic networks with integrl routing requirement, there exist multicst networks where the coding dvntge grows proportionlly s log( V ), nd is thus not finitely ounded. However, we found the sitution is drsticlly different in undirected networks. In
8 [3], we use undirected splitting nd grph orienttion to prove tht, for multicst trnsmissions in undirected networks, the coding dvntge is ounded y constnt fctor of 2. Given the ound 1.12 otined for contrived networks, nd the ound 2 proven in theory, we further studied the coding dvntge in over one thousnd rndomly generted topologies. Our oservtion is tht, for ll the rndom topologies we tested, the coding dvntge lwys remins 1., i.e., network coding does not introduce ny improvement in chievle throughput. This implies tht the fundmentl enefit of network coding is not higher optiml throughput, ut to fcilitte significntly more efficient computtion nd implementtion of strtegies to chieve such optiml throughput. How dvntgeous is stndrd multicst compred to unicst nd overly multicst? The cflow LP is instrumentl to precisely compute the chievle optiml throughput with one multicst communiction session, either with network coding or with multiple multicst trees, since the outcomes from the two re hrdly different. In either cse, dt repliction need to e supported on ll network nodes, including core network elements. It hs een common knowledge tht, when compred to unicst from the source to ll receivers, stndrd multicst rings etter ndwidth efficiency nd higher endtoend session throughput. However, even in the cse of unicst, pth diversity needs to e exploited to chieve optiml throughput, equivlent to the mximum unicommodity flow prolem. It is not immeditely cler how dvntgeous stndrd multicst is. Overly multicst lnces the trdeoff etween the prcticlity of stndrd multicst nd unicst. It refers to the cse where only the memers of the multicst group my replicte or code dt, wheres ll other nodes my only forwrd dt. The optiml throughput chieved y overly multicst is efficiently computed y the oflow LP. We perform quntittive study tht compres the optiml throughput chieved with stndrd multicst, overly multicst nd unicst. The study is performed in rndom networks with up to nodes nd over 1 links. There re 3 nd 1 memers in the multicst group respectively, in two different sets of tests. Multicst nodes re rndomly selected, with different multicst groups eing s disjoint s possile. For ech network size, multiple tests re performed with different network topologies nd different choices of the multicst group, the results re then verged. As we my oserve from Fig. 4, there exists ovious differences etween stndrd multicst throughput nd ll unicst throughput, nd the differences re more significnt in Fig. 4(), where the scle of the multicst trnsmission is lrger. This is due to the fct tht with lrge numer of receivers, the numer of unicst flows increses in the ll unicst pproch, nd links incident to the sender ecome ottlenecks for the trnsmission. Surprisingly, the figure lso suggests tht, the optiml throughput chieved y overly multicst is lmost identicl to tht chieved y stndrd multicst, where ll network nodes re le to replicte or code Optiml throughput (Kps) Optiml throughput (Kps) () Size of multicst group = Numer of nodes in the network () Size of multicst group = 1 Stndrd multicst Overly multicst All unicst Numer of nodes in the network Fig. 4. Achievle optiml throughput using stndrd multicst, overly multicst, nd ll unicst from the sender to ll receivers. dt. On verge, the optiml throughput of overly multicst is over 9% of stndrd multicst. This oservtion shows tht, from the perspective of mximum chievle throughput, while there my exist contrived network topologies tht show more significnt dvntges of stndrd multicst over overly multicst, little difference remins once lrge scle prcticl network topologies re considered. In summry, the ll unicst pproch does not scle, while overly multicst my closely pproch optiml throughput without requiring core routers to e modified. How sensitive is optiml throughput to node joins? When new nodes join the multicst session, how my chievle optiml throughput e ffected? Intuitively, if rely node joins the multicst group nd ecomes new receiver, the chievle session throughput should decrese, due to the following two cuses: (1) lrger numer of receivers my led to more intense competition for ndwidth; nd (2) new node with low cpcity my ecome ottleneck nd limit the throughput for the entire session. Our simultion results show tht, the second cuse hs much more significnt impct thn the first one. Fig. () shows vritions of optiml throughput s the numer of nodes in the multicst group increses from three to V /2, nd then to V (effectively rodcst session), for vrious network sizes V. In this experiment, network topologies re generted with two edges per node without powerlw reltionships, with hevytiled ndwidth distriution etween 1 nd Kps on the links. As we cn oserve, when the size of the multicst group increses from three to V /2, the effects on chievle throughput is rther significnt. However, further expnding the multicst group to the entire network leds to much smller decrese. Both cuses tht we hve discussed contriute to the initil decrese
9 of throughput, while the second cuse (i.e., the effects of ottleneck node) plys less importnt role in the susequent decrese when the multicst group contins hlf of the nodes in the network, it is very likely for the group to hve lredy contined node with low cpcity. Optiml throughput (Kps) Optiml throughput (Kps) () Hevy tiled link cpcity Numer of nodes in the network () Constnt link cpcity M =3 M = V /2 M = V Numer of nodes in the network Fig.. Vritions of optiml throughput due to new nodes joining the multicst session. We further performed the sme tests on powerlw network topologies with 1 Kps constnt link ndwidth, nd the results re shown in Fig. (). In the powerlw topologies, most nodes hve smll degrees of two or three, while smll numer of nodes hve high degrees. Therefore, the initil multicst group usully contins node with smll degree lredy, which lso hs low cpcity, since the link ndwidth is constnt. In this cse, only interreceiver ndwidth competition remins s mjor concern. However, s we cn oserve in the figure, in most cses the optiml multicst throughput remins roughly constnt, even fter ll the nodes hve joined the multicst session. This counterintuitive oservtion shows tht, new receivers my shre ndwidth with existing receivers well, nd do not significntly ffect the chievle throughput, s long s their cpcities re not too low. Spikes in Fig. () correspond to the occsionl cses where nodes in the initil multicst group ll hve reltively high cpcities. Both results in Fig. () nd () hve led to the sme oservtion tht, when new nodes join multicst session, the decresed optiml throughput is minly due to ottleneck receivers with lower cpcities. How sensitive is optiml throughput to the ddition of new sessions? When new sessions re dded to the network, how do they ffect chievle optiml throughput? The mflow LP, presented in Sec. IV, mkes it fesile to crry out our empiricl studies. Fig. 6 shows the vrition of optiml throughput s new communiction sessions re creted. Three types of throughput re shown: (1) previous optiml, which represents the optiml weighted session throughput efore the new session is dded; (2) incrementl, which is the weighted throughput for the new session using residul link cpcities only, or just the previous optiml throughput if the chievle throughput of the new session is higher; nd (3) reoptimized, which is the recomputed optiml session throughput fter the new session is dded. Four groups of simultions re performed, with two, three, four, nd five existing sessions, respectively, efore the new session is estlished. Ech multicst group hs size five, nd nodes in different multicst groups re chosen to e s disjoint s possile. Ech session is ssigned n equl weight. Optiml throughput (Kps) Numer of sessions = Numer of sessions = Fig Optiml throughput (Kps) Numer of sessions = Numer of sessions = Numer of nodes in the network prev optiml incrementl re optimized Throughput vritions s new session is creted. Results in Fig. 6 show tht, the ddition of n extr session does not drmticlly ffect the chievle optiml throughput, especilly when the network size is lrge in comprison to the numer of nodes involved in the trnsmissions. However, if the existing sessions remin trnsmitting ccording to the optiml trnsmission strtegy computed efore the new session joins, nd only residul cpcities cn e utilized to serve the new session (the incrementl throughput cse), then the resulting throughput is not stisfctory unless the numer of sessions is very smll (s = 2). In generl, this my led to very low, even zero, throughput for the new session. Therefore it is necessry to perform reoptimiztion efore new session strts to trnsmit. How sensitive is optiml throughput to firness constrints? In order to investigte how intersession firness requirements ffect the optiml throughput, we estlish three onetotwo multicst sessions in networks of vrious sizes etween 1 nd 3, nd computed their totl optiml throughput with the following firness constrints, respectively: () no firness requirement, which leds to the mximum vlue possile for
10 the totl throughput; () solute firness, in which ech session is required to hve exctly the sme throughput; (c) weighted proportionl firness, where the throughput of ech session is proportionl to the ssocited weight of tht session; nd (d) mxmin firness, in which no session throughput cn e incresed without decresing nother lredy smller session throughput. As first smllscle experiment to gin some insights, Fig. 7 shows the totl throughput of three sessions in network with twenty nodes, using the mflow LP. Multicst groups re chosen to e s disjoint s possile. The totl weight of three sessions w 1 + w 2 + w 3 = 1. As we cn see, the weight distriution hs significnt impct on the chievle totl throughput. When the three weights re hevily unlnced, the session with the smllest weight cn not relize its throughput potentil, nd consequently leds to smll vlue of totl throughput. The chievle throughput with solute firness t w 1 = w 2 = w 3 =.333 is 91.8 Kps. The glol optiml throughput 17. Kps is chieved t (w 1,w 2,w 3 ) = (.287,.47,.36), which turns out to e identicl to the throughput with mxmin firness in this cse. In order to find out whether chieving optiml throughput scrifices ndwidth efficiency, we hve conducted performnce comprisons etween optiml throughput multicst nd single tree multicst. In the ltter cse, we compute the widest Steiner tree, which hs the highest throughput from ll possile multicst trees. The throughput of tree is the lowest cpcity of its links. We choose the tree with the highest throughput rther thn the one tht is most ndwidth efficient, since the ltter is equivlent to the minimum Steiner tree prolem, which is hrd to compute or to pproximte. Even when we cn find such ndwidth efficient tree, it my hve n exceedingly low throughput, which is not prcticl for dt trnsmissions. Optiml throughput (Kps) Hevy tiled link cpcity Constnt link cpcity Bndwidth efficiency (%) Hevy tiled link cpcity Constnt link cpcity cflow Widest tree Totl throughput of 3 sessions (Kps) W W Numer of nodes in the network Fig. 8. Achievle throughput nd ndwidth efficiency: comprison etween the optiml throughput multicst (cflow LP) nd the widest Steiner tree. Fig. 7. Totl throughput of three multicst sessions, s intersession firness requirements chnge. Further results in Tle V show tht the excellent performnce of mxmin firness in the ove exmple is not coincidence. As we my oserve, when the network size is reltively lrge ( nd ove in the tle), mxmin firness lwys leds to optiml throughput. When the network size is smll (1 nd in the tle), the intersession competition for ndwidth ecomes more intense. The throughput with mxmin firness my e inferior to the optiml throughput in this cse, ut the difference is usully smll. TABLE V TOTAL ACHIEVABLE THROUGHPUT WITH MAXMIN FAIRNESS VS. GLOBAL OPTIMAL THROUGHPUT network size mxmin (Kps) optiml (Kps) Does optiml throughput led to low ndwidth efficiency? In Fig. 8, we compre oth chievle throughput nd ndwidth efficiency etween the two pproches. Bndwidth efficiency is computed s the totl receiving rte t ll receivers divided y the totl ndwidth consumption. We tested two groups of networks, one with vrile link cpcity conforming to the hevytiled distriution, the other with constnt link cpcity. For the vrile link cpcity cse, optiml throughput is higher thn the widest Steiner tree throughput y fctor of over 2 on verge, showing the dvntge of using the optiml trnsmission strtegy computed with the cflow LP, eyond single multicst tree. Interestingly, the ndwidth efficiency of optiml throughput multicst lso outperforms tht of the widest Steiner tree multicst. The widest Steiner tree insists to use links with the highest ndwidth possile, nd therefore my result in rther long tree rnches, especilly when the network size is lrge. For the constnt link cpcity cse, the difference etween the optiml nd widest Steiner tree throughput ecomes even lrger. Every tree in this cse hs the sme throughput, therefore the widest selection criterion ecomes irrelevnt. However, the difference in ndwidth efficiency decreses, since it is no longer necessry to include long tree rnches to chieve the mximum tree throughput.
11 VI. CONCLUDING REMARKS The min prolem we hve studied in this pper is to compute nd chieve optiml throughput in dt networks, in the generl cse of undirected communiction links. We hve een plesntly surprised t how results from network coding re le to fcilitte the design of efficient solutions to this fundmentl prolem tht ws previously viewed s very hrd. We lso show the counterintuitive conclusion tht, the most significnt enefit of network coding is not to chieve higher optiml throughput, ut to mke it fesile to chieve such optimlity in polynomil time. We show tht such efficient lgorithms my e designed for multiple communiction sessions of vriety of types, nd for the more relistic model of overly networks. Simultion studies lso suggest tht, overly multicst techniques my pproch optiml multicst throughput quite well. REFERENCES [18] G. Roins nd A. Zelikovsky, Improved Steiner Tree Approximtion in Grphs, in Proceedings of the 7th Annul ACMSIAM Symposium on Discrete Algorithms (SODA),. [19] W. H. Cunninghm, Optiml Attck nd Reinforcement of Network, Journl of the ACM, vol. 32, pp , 198. [] Z. Li, B. Li, D. Jing, nd L. C. Lu, On Achieving Optiml Endtoend Throughput in Dt Networks: Theoreticl nd Empiricl Studies, Tech. Rep., ECE, University of Toronto, 4. [21] P. Snders, S. Egner, nd L. Tolhuizen, Polynomil Time Algorithm for Network Informtion Flow, in Proceedings of the 1th ACM Symposium on Prllelism in Algorithms nd Architectures, 3. [22] C. Gröpl, S. Hougrdy, T. Nierhoff, nd H. J. Prömel, Steiner Trees in Uniformly Qusiiprtite Grphs, Informtion Processing Letters, vol. 83, no. 4, pp. 19, 2. [23] Z. Li nd B. Li, Efficient Computtion of Mximum Multicst Rtes, in Proc. of IEEE INFOCOM,. [24] F. Brhon, Pcking Spnning Trees, Mthemtics of Opertions Reserch, vol., no. 1, pp , 199. [2] N. Grg, R. Khndekr, K. Kunl, nd V. Pndit, Bndwidth Mximiztion in Multicsting, in Proceedings of the 11th Europen Symposium on Algorithms (ESA), 3. [26] A. Medin, A. Lkhin, I. Mtt, nd J. Byers, BRITE: Boston University Representtive Internet Topology Genertor, [1] L. Qiu, Y. R. Yng, Y. Zhng, nd S. Shenker, On Selfish Routing in InternetLike Environments, in Proc. of ACM SIGCOMM, 3. [2] R. K. Ahuj, T. L. Mgnnti, nd J. B. Orlin, Network Flows: Theory, Algorithms, nd Applictions, Prentice Hll, Upper Sddle River, New Jersey, [3] Z. Li nd B. Li, Network Coding in Undirected Networks, in Proc. of the 38th Annul Conference on Informtion Sciences nd Systems (CISS), 4. [4] R. Ahlswede, N. Ci, S. R. Li, nd R. W. Yeung, Network Informtion Flow, IEEE Trnsctions on Informtion Theory, vol. 46, no. 4, pp , July. [] R. Koetter nd M. Medrd, An Algeric Approch to Network Coding, IEEE/ACM Trnsctions on Networking, vol. 11, no., pp , Octoer 3. [6] Z. Wng nd J. Crowcroft, Qulity of Service Routing for Supporting Multimedi Applictions, IEEE Journl on Selected Ares in Communictions, vol. 14, no. 7, pp , Septemer [7] A. J. Bllrdie, P. F. Frncis, nd J. Crowcroft, Core Bsed Trees, August [8] Y. Chu, S. G. Ro, S. Seshn, nd H. Zhng, A Cse for End System Multicst, IEEE Journl on Selected Ares in Communictions, pp , Octoer 2. [9] S. Bnerjee, B. Bhttchrjee, nd C. Kommreddy, Sclle Appliction Lyer Multicst, in Proc. of ACM SIGCOMM, August 2. [1] M. Cstro, P. Druschel, A.M. Kermrrec, A. Nndi, A. Rowstron, nd A. Singh, SplitStrem: HighBndwidth Multicst in Coopertive Environments, in Proc. of the 19th ACM Symposium on Operting Systems Principles (SOSP), Octoer 3. [11] V. Pdmnhn, H. Wng, P. Chou, nd K. Sripnidkulchi, Distriuting Streming Medi Content Using Coopertive Networking, in Proc. of NOSSDAV 2, My 2. [12] J. Byers nd J. Considine, Informed Content Delivery Across Adptive Overly Networks, in Proc. of ACM SIGCOMM, August 2. [13] D. Kostic, A. Rodriguez, J. Alrecht, nd A. Vhdt, Bullet: High Bndwidth Dt Dissemintion Using n Overly Mesh, in Proc. of the 19th ACM Symposium on Operting Systems Principles (SOSP 3), 3. [14] D. Applegte nd E. Cohen, Mking IntrDomin Routing Roust to Chnging nd Uncertin Trffic Demnds: Understnding Fundmentl Trdeoffs, in Proc. of ACM SIGCOMM, August 3, pp [1] S. Y. R. Li, R. W. Yeung, nd N. Ci, Liner Network Coding, IEEE Trnsctions on Informtion Theory, vol. 49, pp. 371, 3. [16] S. Chen, O. Günlük, nd B. Yener, The Multicst Pcking Prolem, IEEE/ACM Trnsctions on Networking, vol. 8, no. 3, pp ,. [17] K. Jin, M. Mhdin, nd M. R. Slvtipour, Pcking Steiner Trees, in Proceedings of the 1th Annul ACMSIAM Symposium on Discrete Algorithms (SODA), 3.
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