THE throughput of information transmission within a data

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1 IEEE TRANACTION ON INFORMATION THEORY A Constnt Bound on Throughput Improvement of Multicst Network Coding in Undirected Networks Zongpeng Li, Memer, IEEE Bochun Li, enior Memer, IEEE Lp Chi Lu Deprtment of Computer cience Dept. of Electricl & Computer Engineering Dept. of Computer cience & Engineering University of Clgry University of Toronto Chinese University of Hong Kong zongpeng@uclgry.c li@eecg.toronto.edu chi@cse.cuhk.edu.hk Astrct Recent reserch in network coding shows tht, joint considertion of oth coding nd routing strtegies my led to higher informtion trnsmission rtes thn routing only. A fundmentl question in the field of network coding is: how lrge cn the throughput improvement due to network coding e? In this pper, we prove tht in undirected networks, the rtio of chievle multicst throughput with network coding to tht without network coding is ounded y constnt rtio of 2, i.e., network coding cn t most doule the throughput. This result holds for ny undirected network topology, ny link cpcity configurtion, ny multicst group size, nd ny source informtion rte. This constnt ound 2 represents the tightest ound tht hs een proved so fr in generl undirected settings, nd is to e contrsted with the unounded potentil of network coding in improving multicst throughput in directed networks. Index Terms Network Coding, Multicst, Undirected Networks, Routing, Complexity I. INTRODUCTION THE throughput of informtion trnsmission within dt network is constrined y the network topology nd link cpcities. Trditionl techniques in improving trnsmission throughput focus on strtegiclly routing informtion flows long high ndwidth or multiple pths from the source to the destintions. Recently, it is shown tht such routing strtegies lone my not e sufficient. Rther, it is necessry to consider encoding/decoding dt on nodes in the network, in order to chieve the optiml throughput [], [2]. ince these coding opertions re not restricted to source or destintion nodes, they re referred to s network coding. A clssic exmple tht illustrtes the power of network coding is shown in Fig., where ech link hs unit cpcity. With network coding, the chievle throughput is two. Without coding, the chievle throughput is only one, if integrl routing is required, i.e., if ll link flow rtes re integers (0 or in this exmple). imilr to source ersure codes, encoding nd decoding opertions in network coding re lso defined over finite fields, which hve fixed length representtion of symols. Therefore, informtion flows do not increse in size fter eing encoded. The introduction of network coding hs essentilly expnded the ville strtegies to chieve optiml trnsmission throughput: rther thn only relying on routing strtegies, n optiml trnsmission strtegy to chieve the mximum throughput includes oth routing scheme nd corresponding coding scheme. Optiml throughput chieved Mnuscript received Ferury 2007; revised Ferury 2008; ccepted August T Fig.. A clssic exmple from []: network coding helps chieve multicst throughput of 2, in topology where ech link hs unit cpcity. with coding is lwys lower ounded y tht chieved without coding. An importnt direction in network coding reserch is to understnd nd quntify the potentil of network coding in improving end-to-end throughput of informtion trnsmission, in generl multi-hop communiction networks [3], [], [5], [6], [7], [8], [9], [0], []. In this pper, we consider undirected networks with idirectionl links. We compre the chievle throughput with coding oth routing scheme nd corresponding coding scheme. Optiml throughput chieved with coding is lwys lower ounded y tht chieved without coding. An importnt direction in network coding reserch is to understnd nd quntify the potentil of network coding in improving end-to-end throughput of informtion trnsmission, in generl multi-hop communiction networks [3], [], [5], [6], [7], [8], [9], [0], []. In this pper, we consider undirected networks with idirectionl links. We compre the chievle throughput with coding to other prmeters tht hve een previously defined to reflect communiction network s connectivity or cpcity. uch prmeters include the pcking numer (which is lso the chievle throughput without coding), strength, nd connectivity. We consider three types of communiction sessions: unicst (one-to-one), rodcst (oneto-ll) nd multicst (one-to-mny). We exmine the reltive order mong the ove four quntities, from which we derive upper ounds for the coding dvntge, i.e., the rtio of throughput improvement due to network coding. In contrst to previous work, which shows the coding dvntge is not finitely ounded in directed networks [5], [9], [2], we show tht the coding dvntge is lwys ounded y constnt fctor of two in undirected networks. Our proof holds for either frctionl routing, where informtion flows cn e split +

2 IEEE TRANACTION ON INFORMATION THEORY 2 nd merged t ritrrily fine scles, or hlf-integer routing, where ech informtion flow eing trnsmitted hs either n integer or hlf-integer rte. We extend the result to n Internet-like idirected network model, where the ound for the coding dvntge depends on how imlnced link cpcities cn e on two opposite links etween the sme pir of nodes. We lso provide rief discussions on the integrl flow model nd the hypergrph model. In ddition, we prove tht the chievle throughput, nd therefore the coding dvntge, re independent of the loction of the informtion source within the communiction group, which is unique property tht is only vlid in undirected settings. Finlly, we show tht in mny cses, including in oth directed nd undirected networks, with oth integrl nd frctionl routing, optiml throughput with network coding is much esier to compute thn optiml throughput without coding. In prticulr, the smll ounds on the coding dvntge my led to nice pproximtion lgorithms. The reminder of the pper is orgnized s follows. We introduce relted work on network coding in ec. II, prove ounds on the coding dvntge for unicst, rodcst, nd multicst in ec. III, extend the result to group communictions in ec. IV, nd to other network models in ec. V. We then discuss the coding dvntge nd complexity issues in ec. VI nd finlly conclude the pper in ec. VII. II. PREVIOU REEARCH Ahlswede et l. [] initited the study of network coding. They show exmples tht demonstrte the enefit ofnetwork coding, in terms of throughput improvement. They lso prove the fundmentl result tht, for multicst trnsmission in directed network, if rte x cn e chieved for ech receiver independently, it cn lso e chieved for the entire session. Koetter et l. [2] lso derived this result for directed cyclic networks within n lgeric frmework. They further extend the discussion to multiple trnsmissions, nd exmined the enefit of network coding in terms of roust networking. Li et l. [3] show tht liner codes suffice in chieving optiml throughput for multicst trnsmission. The ound on the necessry se field size is first given y Koetter et l. [2]. They show tht for multicst session with throughput r nd numer of receivers k, there exists solution sed on finite field GF (2 m ),forsomem log 2 (kr +). This ound is then improved y Ho et l. to m log 2 (r +) []. Li et l. [3] lso proposed the first code ssignment lgorithm, which performs n exponentil numer of vector independence tests. Jggi et l. [9] oserved tht, exploiting flow informtion in the routing strtegy drmticlly simplifies the tsk, nd designed polynomil time code ssignment lgorithm ccordingly. They lso show tht, in directed networks with integrl routing, the coding dvntge my increse proportionlly s Ω(log V ), nd therefore my e ritrrily high. Agrwl nd Chrikr [] show tht the potentil of network coding to improve throughput is equivlent to the integrlity gp of the idirected cut relxtion of the minimum teiner tree prolem. This result is relted to the ound 2 on coding dvntge proved in this pper in the following two spects. First, it provides n lterntive proof method for the ound 2 in the frctionl flow cse, since the integrlity gp of the idirected cut relxtion is ounded y the gp of the undirected relxtion of the minimum teiner tree prolem, which is known to e ounded y 2 [5], [6]. econd, it shows tht the dvntge of network coding in decresing multicst cost is lso ounded y 2 (ssuming the liner link cost model in the clssic min-cost flow literture [7]), since the minimum teiner tree IP nd its idirected cut relxtion yield the optiml cost without nd with network coding, respectively. The proof of ound 2 in this pper is comintoril in nture nd does not rely on liner progrmming techniques; it works for not only frctionl routing ut lso hlf-integer routing. Chekuri et l. studied the coding dvntge in directed networks, without the symmetricl throughput requirement on multicst receivers [5]. They demonstrte clsses of networks where the coding dvntge is ounded y 2, including networks with two unit sources, or networks with two receivers. They lso construct network pttern where the coding dvntge grows t rte Θ( V ). In this pper, we focus on undirected networks with symmetricl multicst throughput. The ound we prove holds for ny multicst throughput nd ny multicst group size. Cnnons et l. [8] nd Dougherty et l. [9] lso performed comprison studies etween network cpcity with network coding (coding cpcity) nd network cpcity with routing only (routing cpcity). In prticulr, they show tht the network cpcity is independent of the coding lphet, nd tht while routing cpcity of network is lwys chievle, coding cpcity is not. Empiricl comprisons of multicst throughput with nd without network coding cn e found in [3] nd [20]. Both comprisons suggest tht for rndom network topologies, the coding dvntge is mrginl t est. Other dvntges re suggested insted, such s ese of mngement, roustness nd ese for lgorithm design. The coding dvntge in the cse of multiple unicst sessions hs lso een exmined in recent literture. While it is known tht the coding dvntge is lrger thn for either directed networks or integrl flows, it ws conjectured y Hrvey et l. [2] nd y us [22] tht network coding does not mke difference in undirected networks with frctionl flows. This conjecture remins unsettled except for some specil network cses [6], [7], [8]. Erly network coding reserch often focus on directed cyclic networks, where the lck of cycles fcilittes the definition nd computtion of the glol coding vectors t ech network node. For converting n undirected network into directed one, our pproch in this pper is the sme s tht used in clssic network flow reserch nd tht dopted y Hrvey et l. [6] nd Krmer et l. [23]: ech undirected link uv with cpcity C(uv) is viewed s pir of directed links uv nd vu, whose cpcities C( uv) nd C( vu) cn e flexily chosen s long s C( uv)+c( vu) C(uv). For extending the definition of network coding from directed cyclic networks to generl directed networks, we refer the reders to the technique

3 IEEE TRANACTION ON INFORMATION THEORY 3 of time-prmeterized lyered grphs y Ahlswede et l. [] nd y Yeung [2]. III. FINITE BOUND ON CODING ADVANTAGE IN UNDIRECTED NETWORK Network coding introduces new dimension to the informtion trnsmission prolem. Trditionlly, only the routing dimension is considered in trnsmission strtegy; with network coding, trnsmission strtegy includes oth the routing scheme nd the coding scheme. Considering oth dimensions together is necessry to chieve the mximum informtion trnsmission rte. We use simple grph G =(V,E) to represent the topology of network, nd use function C : E Z + to denote link cpcities. However, when we study frctionl routing models, hving integer vlues in C is neither importnt nor necessry, nd we ssume C cn tke rtionl vlues too. The communiction group is M = {, T,...,T k } V, with eing the sender of the unicst, rodcst, or multicst session, y defult. In our grphicl illustrtions, terminl nodes (nodes in the communiction group) re lck, nd rely nodes (nodes not in the communiction group) re white. We focus on frctionl routing in this section, nd will extend our discussion to integrl routing lter in the pper. For integrl routing, ll link cpcities nd flow rtes hve integer vlues. For frctionl routing, we ssume link cpcities my e shred frctionlly in oth directions, nd flows cn e split nd merged t ritrrily fine scles. We use χ(n) to denote the mximum throughput of network N contining single trnsmission session. We compre χ(n) with other prmeters tht re defined to chrcterize the connectivity or cpcity of communiction network, including the pcking numer, strength nd connectivity. We study nd compre the four prmeters for unicst, rodcst, nd multicst trnsmissions, respectively. Bsed on results otined from such comprison studies, we derive ound on the coding dvntge in ech cse. We refer the reders to our previous work [20], [25] for precise liner progrmming formultions of χ(n). Pcking refers to the procedure of finding pirwise edgedisjoint su-trees of G, in ech of which the communiction group remins connected. The pcking numer of communiction network N is denoted s π(n), nd is equl to the mximum throughput without coding. The reson is tht, ech tree cn e used to trnsmit one unit informtion flow from the sender to ll receivers, therefore the pcking numer gives the mximum numer of unit informtion flows tht cn e trnsmitted. Formlly, let T denote the set of ll possile trees connecting the communiction group, then the pcking numer cn e defined using the following liner progrm: P t T f(t) Mximize uject to: j P uv t f(t) C(uv) uv f(t) 0 E t T In the LP ove, f(t) is vrile representing the mount of informtion flow one ships long tree t. One cn either require f(t) to tke integer vlues only (ssuming ll inputs in C(uv) re integers), leding to integrl pcking, or llow f(t) to tke rtionl vlues, leding to frctionl pcking. trength is kind of prtition connectivity of the network [26], nd is denoted s η(n). LetP denote ll possile wys of prtitioning N into numer of disconnected components, such tht ech component contins t lest one terminl node. Then: η(n) =min p P E c p Here E c is the set of inter-component links, nd E c = uv E c C(uv); p is the numer of components tht the network is seprted into, in p. Connectivity refers to the minimum edge connectivity etween pir of nodes in the communiction group, nd is denoted s λ(n). It is lso the minimum size of cut tht seprtes the communiction group. Fig. 2 illustrtes the concept of these four prmeters using n exmple multicst network. T T efgh cdi defh defgh cdi Pcking trength cdi fghi cegi cefgh T T + Throughput + Connectivity Fig. 2. The four prmeters of communiction network N: pcking numer π(n), throughput χ(n), strength η(n) nd connectivity λ(n). In this prticulr network with unit cpcity t ech undirected link: π(n) =.8, nine trees (ech leled with letter etween nd i ) ech of rte 0.2 cn e pcked; χ(n) =2, two unit informtion flows nd cn e delivered to ll receivers simultneously; η(n) =2, in the illustrted optiml prtition, E c p = 6 = 2; λ(n) = 2, ech pir of terminl nodes is 2-edgeconnected. + A. Unicst In n undirected network with unicst session N = {(G(V,E),C:E Z +,M = {, T } V }, the pcking numer π(n) ecomes the numer of edge-disjoint -T pths. The throughput χ(n) is the mximum informtion rte tht

4 IEEE TRANACTION ON INFORMATION THEORY cn e chieved in the T trnsmission. The strength η(n) is now minimized over ll cuts seprting nd T no vlid prtition with more thn two components exists, since ech vlid component contins t lest one terminl node nd there re two terminl nodes only in the network. The connectivity ecomes the edge-connectivity etween nd T, i.e., the minimum mount of edge cpcity one needs to remove from the network, in order to seprte nd T into disconnected components. Bsed on previous results, we cn show tht the four quntities turn out to e ll equl for unicst trnsmission: Theorem. For unicst trnsmission in n undirected network, N, π(n) =χ(n) =η(n) =λ(n). Proof: Due to the fct tht η(n) cn e minimized over simple cuts (cuts which seprtes the network into exctly two components) only, it ecomes identicl to λ(n) y definition; oth re equl to the size of the min-cut etween nd T. Furthermore, oserve tht uncoded throughput is lwys ounded y coded throughput network coding llows insted of requires coding, nd ny vlid trnsmission scheme with routing only is still vlid in the prdigm of network coding. Therefore π(n) χ(n). Next, the T informtion rte is ounded y the -T min-cut, i.e., χ(n) η(n). In order to finish the proof, it is sufficient to show π(n) =λ(n), which is implied y Menger s Theorem [27]: Let u, v e two vertices of grph G. The mximum numer of pirwise edgedisjoint u-v pths equls to the minimum numer of edges whose removl seprtes u from v in G. It follows from Theorem tht network coding is not necessry in order to chieve the mximum throughput for unicst session: Corollry. The coding dvntge for unicst session is lwys. B. Brodcst Let N = {G(V,E),C : E Z +,M = V = {, T,...,T k }} e n undirected network contining rodcst session, with eing the rodcst sender, nd ll other nodes in the network eing receivers. The pcking numer π(n) ecomes the spnning tree pcking numer, i.e., the mximum numer of pir-wise edge-disjoint spnning trees tht cn e identified in the network. A spnning tree is tree tht connects every node in the network. The throughput χ(n) is the mximum informtion rte from to every other node in the network, simultneously. The strength η(n) is still s defined; just note tht for rodcst network, every prtition is vlid, since ech component of prtition lwys contins some node from the communiction group. Connectivity λ(n) ecomes the size of the minimum simple cut of the network. The fct tht ll nodes in the network request the sme informtion leds to the following nice property for rodcst trnsmissions: Theorem 2. For rodcst trnsmission in n undirected network, N, λ(n) π(n) =χ(n) =η(n) λ(n) 2 Proof: We first show tht π(n) =χ(n) =η(n). Tutte-Nsh- Willims Theorem chrcterizes the reltionship etween integrl spnning tree pcking nd network strength [27], [28], [29]: A grph G hs x pirwise edge-disjoint spnning trees if nd only if, for every vertex prtition, there re t lest (p )x edges with endpoints in different components, where p is the numer of components in the prtition. Tutte-Nsh-Willims Theorem shows tht, for the integrl spnning tree pcking prolem, π(n) = η(n). In the frctionl flow model, one cn pply the technique of scling edge cpcities up, nd then scling the solution down ccordingly, to derive π(n) = η(n) from the integrl pcking result. Furthermore, since the spnning tree pcking numer π(n) is equl to the uncoded throughput, it cn not exceed the coded throughput χ(n), i.e. π(n) χ(n). Next, we oserve tht, if the network is prtitioned into p components, ech component not contining the source needs totl incoming edge cpcity x in order to chieve throughput x; therefore (p )x inter-component edge cpcity is required in totl. This leds to χ(n) η(n). Comining the ove results, we hve π(n) =χ(n) =η(n). We next show tht 2λ(N) π(n) nd η(n) λ(n). By definition, η(n) λ(n) holds since λ(n) cn e viewed s specil cse of η(n), where only prtitions contining two components re considered. We now prove tht 2 λ(n) χ(n), using Nsh-Willims Wek Grph Orienttion Theorem [30]: grph G hs n x-edge-connected orienttion if nd only if it is 2x-edge-connected. The wek orienttion theorem implies tht, in the frctionl model, rodcst network N lwys hs 2 λ(n)-edge-connected orienttion, i.e., n orienttion where the mx-flow etween ech pir of nodes is t lest 2λ(N). Comined with the result in directed networks, trnsmission rte tht cn e independently chieved for ech receiver cn e chieved for the entire session, this implies 2λ(N) χ(n). From Theorem 2, we cn see tht network coding hs no potentil in improving rodcst throughput either: Corollry 2. The coding dvntge for rodcst session is lwys. C. Multicst Multicst is more generl form of communiction thn oth unicst nd rodcst. A unicst session cn e viewed s specil cse of multicst, where exctly two nodes in V re in the multicst group M. A rodcst session cn lso e viewed s specil cse of multicst, where ll nodes in V re in the multicst group M. In generl, the multicst group M cn e ny suset of V tht hs size two or lrger, nd the pcking prolem ecomes teiner tree pcking. A teiner tree is sutree in the network connecting ll the terminl nodes, ech rely node my or my not pper in the tree. Theorem 3. For multicst trnsmission in n undirected network, N={G(V,E),C : E Z +,M = {, T,...,T k }

5 IEEE TRANACTION ON INFORMATION THEORY 5 V }, λ(n) π(n) χ(n) η(n) λ(n). 2 Proof: The fct tht uncoded throughput is ounded y coded throughput gin leds to π(n) χ(n). Furthermore, the prtition connectivity condition is necessry for certin throughput to e fesile in multicst networks s well, therefore χ(n) η(n). Next, η(n) λ(n) still holds due to the sme rgument s in the rodcst cse λ(n) cn e considered s specil cse of η(n) where only prtitions contining two components re llowed. We now hve π(n) χ(n) η(n) λ(n), nd need only to focus on the vlidity of 2λ(N) π(n) in the rest of the proof, which is more complex in the cse of multicst. We uild the proof of 2λ(N) π(n) in the multicst cse upon the sme result proven in the rodcst cse. More specificlly, we trnsform the multicst network into rodcst network, y eliminting the existence of rely nodes, while gurnteeing tht 2λ(N) π(n) holds fter the trnsformtion only if it holds efore it. Before descriing the trnsformtion in more detil, we first introduce Mder s Undirected plitting Theorem [30]: Let G(V + z,e) e n undirected grph so tht (V,E) is connected nd the degree d(z) is even. Then there exists complete splitting t z preserving the edge-connectivity etween ll pirs of nodes in V. A split-off opertion t node z refers to the replcement of 2-hop pth u-z-v y direct edge etween u nd v, s illustrted in Fig. 3. A complete splitting t z is the procedure of repetedly pplying split-off opertions t z until z is isolted. u Fig. 3. A split-off t node z. z v The Undirected plitting Theorem sys tht, if grph hs n even-degree non-cut node, then there exists split-off opertion t tht node, fter which the pirwise connectivities mong the other nodes remin unchnged; nd y repetedly pplying such split-off opertions t this node, one cn eventully isolte it from the rest of the grph, without ffecting the edge-connectivity of ny pir of nodes in it. Now, consider repetedly pplying one of the following two opertions on multicst network: () pply complete splitting t non-cut rely node, preserving pirwise edge connectivities mong terminl nodes in M; or(2)ddrely node tht is M-cut node into the multicst group M, i.e., chnge its role from rely node to receiver node. Here M-cut node is one whose removl seprtes the multicst group into more thn one disconnected components. Fig. illustrtes these two opertions with concrete exmple. In order to meet the even node degree requirement in the Undirected plitting Theorem, we first doule ech link u z v B A () T B () Fig.. Trnsforming multicst network into rodcst network, where the vlidity of λ(n) π(n) cn e trced ck. () The originl multicst 2 network, with unit cpcity on ech link. () The network fter pplying opertion (2), moving the M-cut node A into the multicst group. A ecomes receiver. (c) The network fter pplying opertion (). A split-off ws done t rely node B. A rodcst network is otined. cpcity in the input network, then we scle the solution down y fctor of 2 t the end. Note tht, ssuming the input network hs integer link cpcities, then ech node hs n even degree fter douling link cpcities. Furthermore, split-off opertion does not ffect the prity of the degree of ny node in the network. Therefore the Undirected plitting Theorem gurntees tht s long s there re rely nodes tht re not cut nodes, opertion () is possile. Furthermore, opertion T () does not increse π(n). Therefore, if 2λ(N) π(n) holds fter pplying opertion (), it holds efore pplying opertion () s well. Opertion (2), pplied to M-cut nodes, does not ffect either π(n) or λ(n). o, gin we cn clim tht for opertion (2), if 2 λ(n) π(n) holds fter pplying the opertion, it holds efore pplying the opertion s well. As long s there re rely nodes in the multicst network, t lest one of the two opertions cn e pplied. If oth opertions re possile, opertion () tkes priority. ince ech opertion reduces the numer of rely nodes y one, eventully we otin rodcst network with terminl nodes only. By Theorem 2, 2λ(N) π(n) holds. Finlly, note tht we otined n integrl trnsmission strtegy fter douling ech link cpcity. Therefore, fter we scle the solution ck y fctor of 2, the trnsmission strtegy is hlf-integrl. Corollry 3. For multicst trnsmission in n undirected network, the coding dvntge is upper-ounded y constnt fctor of two, s long s hlf-integer routing is llowed. Proof: By Theorem 3, 2λ(N) π(n) nd χ(n) λ(n) s long s hlf integer routing is llowed. Therefore we conclude 2 χ(n) π(n), i.e., the coding dvntge χ(n)/π(n) 2. Fig. 5 shows the optiml throughput without coding of the multicst session given in Fig., ssuming hlf-integrl routing nd ritrrily frctionl routing respectively, with the network eing undirected. Links leled with the sme letter or numer form teiner tree. For exmple, the tree leled with is highlighted in old edges. In (), ech tree hs cpcity 0.5; in (), trees leled with letter ech hs cpcity 0.25, nd trees leled with numer hve cpcity As result, uncoded throughput chieved is.5 in () nd.875 in () respectively, y trnsmitting flow long ech teiner tree, with the flow rte equl to the tree cpcity. ince optiml throughput with coding is 2, the corresponding coding dvntges re.333 nd.067, respectively. (c) T

6 IEEE TRANACTION ON INFORMATION THEORY 6 c c T c c () de2 ce2 cde cef df23 df df23 cf3 ce3 T Fig. 5. Throughput without network coding, for the exmple shown in Fig.. () With hlf-integer routing, optiml throughput is.5. Ech flow is of rte 0.5. () With ritrry frctionl routing, optiml throughput is.875. Ech flow leled with letter nd numer hs rte 0.25 nd 0.25, respectively. We point out tht none of the inequlities in Theorem 3 cn e replced with equlity in generl multicst networks. In prticulr, the network in Fig. is n exmple where π(n) <χ(n), showing tht pcking numer is not lwys equl to multicst throughput; the network in Fig. 6 is n exmple where χ(n) <η(n), showing tht strength is not lwys equl to multicst throughput either. The vlue of χ(n) ws computed using the liner optimiztion pproch in [25]; the vlue of η(n) ws computed y optimizing over ll vlid network prtitions. In the sme network, π(n) =3.5 cn e otined y enumerting ll different multicst trees nd then solving the tree-pcking LP, nd λ(n) =6cn e otined y solving two mx-flow LPs. Therefore, we lso hve 2λ(N) < π(n) nd η(n) < λ(n) here. The construction of the network topology ws inspired y the gdget used in the proof y Dhlhous et l. [3] tht shows the multiterminl cut prolem is NP-Complete. T 8 8 Fig. 6. A multicst network where pcking numer π(n) =3.5, mximum throughput χ(n) =3.5, strength η(n) =nd connectivity λ(n) = 6. Unleled links ech hs cpcity. This exmple shows tht multicst throughput is in generl not equivlent to strength in undirected networks. IV. OURCE INDEPENDENCE AND CODING ADVANTAGE FOR GROUP COMMUNICATION In this section, we show tht the chievle throughput for multicst trnsmission does not depend on which node in the multicst group cts s the sender. In other words, if we move the informtion source from one node in the multicst group onto nother, the optiml coding throughput remins unchnged. First, note tht such property does not hold in directed networks, where the connectivity etween two nodes () cn e ritrrily different in the two directions. econd, it is rther ovious tht this property holds for multicst without coding. The uncoded multicst prolem is equivlent to the teiner tree pcking prolem, nd the pcking numer is defined upon the network topology nd the terminl set, regrdless of which node in the terminl set is the sender. However, with network coding considered, it is less ovious whether the source independence property still holds. In Theorem, we provide n ffirmtive nswer, sed on which we then extend the ound 2 on coding dvntge for multicst to the cse of group communiction. Theorem. The optiml throughput of multicst trnsmission in n undirected network is completely determined y the network topology, the link cpcities, nd the multicst group; it is not dependent on the selection of the sender within the multicst group. A B C Fig. 7. Two scenrios of reversing the A B flow. Thicker links re eing reversed. Proof: The proof we present elow is sed on the following fct: directed multicst trnsmission is fesile if nd only if it stisfies ll the simple cut conditions [2], i.e., ifevery simple cut etween the source nd ny receiver hs size no less thn the multicst rte. uppose we exchnge the sender nd receiver roles etween two terminl nodes A nd B, nd the optiml throughput efore the exchnge is f. Then there must exist network flow of rte f from the sender to every receiver (including, in prticulr, from A to B). Consider reversing the A B network flow, which hs rte f. We show tht fter the reversl, simple cut conditions re still stisfied. Let C e nother multicst node. Consider cut tht seprtes B nd C. There re two cses, either A is in the sme prtition s B, ora is in the sme prtition s C, s shown in Fig. 7. In the first cse, we hve net flow of rte f trversing the cut from the AB component to the C component efore the reversl, nd n equl mount of flow in oth directions will e reversed; therefore fter the reversl, we still hve the sme mount of flow going from the AB component towrds the C component. In the second cse, similrly, the totl flow going from the AC component towrds the B component is f efore the reversl, nd ll flows crossing the cut will e reversed. Therefore, fter the flow reversl, we hve flows of totl rte f going from the B component towrds the AC component. Our proof lso shows tht, fter the informtion source is moved, the sme multicst throughput cn e chieved with exctly the sme ndwidth consumption on ech link. Therefore, we cn derive the following corollry: Corollry. A multicst rte is fesile if nd only if it is A C B

7 IEEE TRANACTION ON INFORMATION THEORY 7 fesile with the informtion source seprted into independent su-sources nd redistriuted mong the multicst group. Fig. 8 shows n exmple contining the sme network s in Fig., with the two unit informtion sources t the top multicst node moved onto the two ottom multicst nodes respectively. Ech informtion source cn still e trnsmitted to ll three multicst nodes, fter the movement. + P 3 + P P 2 Fig. 8. Optiml trnsmission strtegy fter splitting nd moving the informtion source, for the network shown in Fig.. Here P is the source for nd P 2 is the source for. Corollry. is relevnt to video conferencing, where ech prticipnt multicsts his/her locl udio/video dt to every other prticipnt, nd receives udio/video dt from them s well. By Corollry., video conferencing session is fesile with certin sending throughput requirement t ech prticipnt, if nd only if the multicst trnsmission otined y congregting ll throughput requirement t one of the prticipnts is fesile. ince oth throughput with nd without network coding re source independent in undirected networks, we cn extend the ound of 2 proved in Theorem 3 to the cse of group communiction: Corollry. The coding dvntge for group communiction session in n undirected network is upper-ounded y constnt fctor of 2. As finl note of this section, we point out tht the flow reversing technique used in the proof of Theorem is not lwys pplicle in generl networks. For exmple, Dougherty nd Zeger [32] demonstrted specific (directed) network N 7 consisting of multiple unicst sessions, such tht N 7 hs network coding solution while its reverse network does not. A reverse network is defined s the network otined from reversing every link direction nd exchnging the sender/receiver role of every unicst session. V. CODING ADVANTAGE IN RELATED NETWORK MODEL A. Internet-like Bidirectionl Networks Given the fct tht the coding dvntge is finitely ounded in undirected networks ut not in directed ones, it is nturl to sk which model is closer to rel-world networks, nd whether the coding dvntge is ounded in such networks. A rel-world computer network, such s the current genertion Internet, is usully idirectionl ut not undirected. If u nd v re two neighoring routers in the Internet, the mount of ndwidth ville from u to v nd tht from v to u re oth fixed nd independent of ech other. At certin + moment, if the u v link is congested nd the v u link is idling, it is not fesile to orrow ndwidth from the v u direction to the u v direction, due to the lck of dynmic ndwidth lloction module. Therefore, the Internet resemles n undirected network in tht communiction is idirectionl, nd resemles directed network in tht ech link is directed with fixed mount of ndwidth. The Internet cn e etter modeled with lnced directed network. In lnced directed network, ech link hs fixed direction. However, pir of neighoring nodes u nd v re lwys mutully rechle through direct link, nd the rtio etween c( uv) nd c( vu) is upper-ounded y constnt rtio α. In cses where α =, we hve n solutely lnced directed network. This is rther close to the relity in the Internet ckone, lthough lst-hop connections to the Internet exhiit higher level of symmetry in upstrem/downstrem cpcities. Bsed on the constnt ound developed in the previous section, we now show tht the coding dvntge in such n α-lnced network is lso finitely ounded. Theorem 5. For multicst session in n α-lnced idirectionl network, the coding dvntge is upper-ounded y 2(α +). Proof: We first define few nottions. Let N :α e the α- lnced network; let N e n undirected network with the sme topology s N :α,wherec(uv) in N is equl to the smller one of c( uv) nd c( vu) in N :α ;letn α+ e the undirected network otined y multiplying every link cpcity in N with α +.Thenwehve: π(n :α ) π(n ) α + π(n α+) α +2 χ(n α+) 2(α +) χ(n :α) In the derivtions ove, the third inequlity is n ppliction of Theorem 3; the other inequlities re sed on definitions. From Theorem 5, we cn see tht the more lnced directed network is, smller ound on the coding dvntge cn e climed. In the cse of n solutely lnced network, the ound is. In ritrry directed networks, α my pproch, nd correspondingly finite ound on the coding dvntge does not exist. B. Integrl Flows nd Hypergrph Models We now riefly discuss the cses of integrl flow rtes nd hypergrph network models. The motivtion of hving integrl flows lies ehind the fct tht in pcket-switched networks, ech pcket constitutes n tomic dt unit nd dt flow rtes should e discrete. We showed in [33] tht in the model of undirected networks with integrl flows, χ(n) 26π(N), i.e., the coding dvntge is ounded y constnt fctor of 26. A hypergrph is similr to grph except tht ech edge my connect more thn two vertices. It nturlly models dt trnsmission in wireless networks equipped with omnidirectionl ntenne, where ech trnsmission is herd y ll users within certin effective reception rnge. Király nd

8 IEEE TRANACTION ON INFORMATION THEORY 8 Lu [3] recently showed tht in the hypergrph model, if communiction group is 2x-hyperedge-connected, then there is n orienttion within which it is x-hyperedge-connected. This demonstrtes tht the multicst throughput cn chieve t lest hlf hyperedge connectivity. Furthermore, the hypertree pcking numer is upper-ounded y the hyperedge connectivity. In conclusion, the coding dvntge is ounded y fctor of 2 in hypergrph models. C. Non-Uniform Demnd Networks nd Averge Throughput Internet clients often exhiit significnt heterogeneity in their network connections. Dil-up, cle modem, ADL nd cmpus networks ech my provide different connection cpcity. Consequently, the mx-flows etween the multicst sender nd different receivers my e drsticlly different. For rel-world multicst pplictions such s medi streming, common technique for ccommodting heterogeneous clients in the sme session is lyered coding [35]. With lyered coding, the source medi is encoded into se lyer nd severl enhncement lyers. Depending on the receiving cpcity, receiver my choose to receive nd decode suset of ll the lyers only, nd ccept suoptiml plyck qulity. uch rel-world heterogeneity is nturlly modeled y the non-uniform demnd network model [36], [37]. A non-uniform demnd network is very similr to multicst network, with one node cting s the informtion source nd set of nodes cting s receivers. However, the requirement tht ll receivers receive informtion t the sme rte is relxed. Different users my receive informtion t rte tilored ccording to her own cpcity. The gol is to mximize the totl throughput of ll the receivers, or equivlent, the verge throughput mong ll the receivers. Unfortuntely, non-uniform multicst is computtionlly much hrder prolem thn multicst. Cssuto nd Bruck [37] proved tht determining whether rte vector χ,...,χ k (χ i denotes the desired throughput for receiver T i ) is fesile is NP-hrd; consequently, mximizing the verge throughput k i χ i is lso NP-hrd. A prcticl heuristic to circumvent the complexity of nonuniform multicst is to use successive lyering [35]. The ide is to first compute the multicst rte tht cn e chieved for ll receivers, nd the corresponding multicst flow; then remove the utilized network cpcities from N, remove the ottleneck receivers who cn not chieve higher rtes, nd compute the mximum multicst rte in the residul network. These two steps re repeted until there is no more receivers in the residul network. By Theorem 3, within ech lyer of norml multicst, we cn still chieve t lest hlf of the coding throughput. Therefore the upper ound of 2 for the coding dvntge holds if the verge throughput with network coding is computed using this prcticl solution. Chrcterizing the coding dvntge using the true mximum verge throughput with network coding is theoreticlly more interesting direction. A similr frmework s in the proofs of Theorem 2 nd Theorem 3 cn e dopted, i.e., utilize the fct tht coding throughput is upper-ounded y the connectivity, nd then ound the rtio etween throughput without coding nd connectivity. Bsed on prtil teiner tree pcking result y Bng-Jensen et l. [38], Chekuri et l. [36] proved tht without network coding, rte vector χ,...,χ k cn e chieved, such tht for ech T i, χ is no less thn hlf of the mx-flow from to T i. This implies tht the upperound of 2 for the coding dvntge remins vlid, for verge throughput of non-uniform multicst. VI. CODING ADVANTAGE AND COMPLEXITY By estlishing constnt ounds on the coding dvntge, so fr in the pper we hve een rguing in the direction tht the enefit of network coding in terms of improving throughput is limited. Empiricl studies on the coding dvntge revel similr picture. For multicst in undirected networks, the lrgest coding dvntge vlue oserved is only slightly lrger thn one. In [25], we showed smll network where the vlue is 9 8 ; in [], the uthors there pointed out network pttern in which the coding dvntge pproches 8 7 s the network size grows towrds infinity. It ws lso oserved in [25] tht the coding dvntge vlue is lmost lwys in rndomly constructed network topologies. On the other hnd, however, network coding my drmticlly reduce the complexity of mny optimiztion prolems tht rise in informtion dissemintion. In some cses, the underlying network flow structure of coded multicst trnsmission leds to efficient liner optimiztion lgorithms; in some other cses, the smll ound on coding dvntge leds to nice polynomil-time pproximtion lgorithms. We provide some of such exmples in the rest of this section. A. Informtion Exchnge In n informtion exchnge session, two nodes A nd B need to trnsmit informtion to ech other, with throughput requirement f AB nd f BA respectively. This form of communiction rises in scenrios such s: two sensor nodes exchnge sensed dt with ech other [39], [0], two receivers in n synchronous file downloding session reconcile received dt with ech other [], or two online messging pplictions strem multimedi dt to ech other concurrently. An informtion exchnge session cn lso e viewed s two simultneous unicst sessions etween pir of nodes, in opposite directions. If network coding is ignored, then even prolem s simple s determining the fesiility of single informtion exchnge session is NP-hrd, in directed networks with integrl routing. One my derive this NP-hrdness result from the proof given y Fortune et l. [2] tht shows the edge-disjoint pth prolem is NP-hrd for two opposite commodities. On the other hnd, when network coding is tken into considertion, the informtion exchnge prolem ecomes nicely trctle. As shown in Fig. 9, we cn trnsform the coded informtion exchnge prolem into coded multicst prolem, which requires just two mx-flow computtions []. In the trnsformtion, we dd n extr source node to e the multicst sender, then ssume the two unicst nodes re multicst receivers. Connect the sender with A nd B with n edge of cpcity f AB nd f BA respectively. Then we cn verify tht the originl

9 IEEE TRANACTION ON INFORMATION THEORY 9 informtion exchnge session is fesile if nd only if the resulting multicst session cn chieve throughput f AB + f BA with coding. The ltter requirement is equivlent to hve oth the A mx-flow nd the B mx-flow to e t lest f AB + f BA []. A f AB Fig. 9. Trnsforming the coded informtion exchnge prolem to the coded multicst prolem. B. Multicst with Frctionl Routing We now switch ck to the undirected network model, nd consider gin multicst with frctionl routing. Without network coding, the optiml multicst throughput prolem is equivlent to the frctionl teiner tree pcking prolem, which is NP-Complete nd APX-hrd [3]. However, once network coding is supported, the optiml throughput cn then e computed efficiently, due to the nice network flow structure underlying the coded informtion dissemintion prolem. We pointed out in [] tht optiml multicst with network coding in the frctionl flow model cn e solved using the liner progrmming pproch. Vrious efficient nd distriuted lgorithms hve een designed therefter within the liner optimiztion frmework, for chieving mximum multicst throughput or minimum multicst cost [5], [6], [25], [7], [8], [9]. To conclude, in ll the ove three exmples we hve shown, the optiml trnsmission throughput prolem is much more trctle with network coding considered. In the first nd the third exmple, the prolem is NP-Complete without network coding, nd is P with network coding. In the second exmple, the prolem is NP-hrd either with or without network coding, ut with network coding the optiml solution is much esier to pproximte using polynomil-time lgorithms. C. Multicst with Integrl Routing Continuing from the previous susection, we now further restrict our solutions to integrl flows only. Without network coding, the chievle multicst throughput equls to the (integrl) teiner tree pcking numer. It hs een shown tht the teiner tree pcking prolem is NP-Complete [3], [50]; it is even worse in the integrl cse: there did not exist ny known polynomil time lgorithm tht cn pproximte the prolem to ny constnt rtio, until the result 26π(N) λ(n) [33] ws recently proven in the integrl flow model. On the other hnd, y tking network coding into considertion, we re led to 2-pproximtion for the optiml multicst throughput. We show this clim y exmining the reltion etween connectivity nd throughput in the integrl f BA B model. We hve shown tht 2λ(N) π(n) χ(n) holds in the frctionl model; more ccurtely, it holds s long s hlf-integer flows re llowed. In the integrl model, it is not known whether 2λ(N) π(n) still holds or not. In fct, it is well known open prolem in grph theory. The est result known so fr is the forementioned rtio 26. On the other hnd, we show tht throughput with network coding cn still chieve hlf connectivity in the integrl routing model. Theorem 5. For multicst trnsmission in n undirected network, N, 2λ(N) χ(n) holds under the integrl routing requirement. Proof: Our proof is sed on Nsh-Willims trong Grph Orienttion Theorem [30]: every undirected grph G(V,E) hs n orienttion G = (V,D) for which λ G (u, v) 2 λ G(u, v), for ll u, v V. This theorem gurntees the existence of well-lnced orienttion of n undirected network, within which the connectivity from ny node u to ny other node v is t lest hlf of the connectivity etween u nd v in the originl undirected network. From the theorem ove, we know tht if λ(n) =x, then there is n integrl orienttion of the network, such tht the directed connectivity mong the multicst group M is t lest 2x. This implies tht the integrl mx-flow from to ech receiver T i is t lest 2x. Then y the fesiility condition of directed multicst [], [2], there is n integrl routing scheme to chieve χ(n) 2 x. Corollry 5. The optiml multicst throughput prolem in undirected networks with integrl routing cn e pproximted within fctor of two in polynomil time. Proof: The clim χ(n) λ(n) in Theorem 3 still holds in the integrl cse. Comined with Theorem 5, we hve 2λ(N) χ(n) λ(n). Therefore computing λ(n) gives 2-pproximtion for χ(n). Note tht λ(n) is oviously computle in polynomil time in the worst cse, one cn compute the mx-flow etween ech pir of multicst nodes, nd tke the minimum vlue mong them. It is lso possile to find the detiled trnsmission strtegy tht chieves the pproximted throughput vlue, since polynomil time lgorithms exist for oth the orienttion [30] nd code ssignment [9], [2]. To conclude, in ll the ove three exmples we hve shown, the optiml trnsmission throughput prolem is much more trctle with network coding considered. In the first nd the third exmple, the prolem is NP-Complete without network coding, nd is P with network coding. In the second exmple, the prolem is NP-hrd either with or without network coding, ut with network coding the optiml solution is much esier to pproximte using polynomil-time lgorithms. VII. CONCLUDING REMARK In this pper, we hve compred the coded trnsmission throughput with the pcking numer, strength, nd connectivity, in n undirected network with unicst, rodcst, nd multicst trnsmission, respectively. Our results led to smll constnt ounds on the coding dvntge in these cses: the coding dvntge is lwys for unicst or rodcst, nd

10 IEEE TRANACTION ON INFORMATION THEORY 0 is t most 2 for multicst. The ound 2 for multicst is then extended to the cse of group communiction, y showing tht chievle throughput is source independent in undirected networks, either with or without network coding. We lso mke the oservtion tht pplying network coding mkes it possile to design efficient lgorithms tht compute nd chieve the optiml trnsmission throughput. Finite ounds on the coding dvntge discussed throughout this pper re summrized in Tle I. The following questions on the coding dvntge re still open. First, for multiple unicst sessions tht concurrently coexist in the sme network, is the coding dvntge lwys, ssuming undirected networks with frctionl routing [2], [22]? econd, with ritrry frctionl routing, cn the ound of two for coding dvntge e further tightened? Third, is the ound of two still vlid if we replce the hlf-integer routing requirement with integrl routing? 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