Network Coding for Multi-Resolution Multicast

Transcription

1 Network Coding for Multi-Reolution Multicat MinJi Kim, Daniel Lucani, Xiaomeng Shi, Fang Zhao, Muriel Médard Maachuett Intitute of Technology, Cambridge, MA 02139, USA {minjikim, dlucani, xhi, zhaof, Abtract Multi-reolution code enable multicat at different rate to different receiver, a etup that i often deirable for graphic or video treaming. We propoe a imple, ditributed, two-tage meage paing algorithm to generate network code for ingle-ource multicat of multi-reolution code. The goal of thi puhback algorithm i to maximize the total rate achieved by all receiver, while guaranteeing decodability of the bae layer at each receiver. By conducting puhback and code aignment tage, thi algorithm take advantage of inter-layer a well a intra-layer coding. Numerical imulation how that in term of total rate achieved, the puhback algorithm outperform routing and intra-layer coding cheme, even with field ize a mall a 2 10 (10 bit. In addition, the performance gap widen a the number of receiver and the number of node in the network increae. We alo oberve that naïve inter-layer coding cheme may perform wore than intra-layer cheme under certain network condition. I. INTRODUCTION Many real-time application, uch a teleconferencing, video treaming, and ditance learning, require multicat from a ingle ource to multiple receiver. In conventional multicat, all receiver receive at the ame rate. In practice, however, receiver can have widely different characteritic. It become deirable to erve each receiver at a rate commenurate with it own demand and capability. One approach to multirate multicat i to ue multi-decription code (MDC, dividing ource data into equally important tream uch that the decoding quality uing any ubet of the tream i acceptable, and better quality i obtained by more decription. A popular way to perform MDC i to combine layered coding with the unequal error protection of a priority encoding tranmiion (PET ytem [1]. Another approach for multirate multicat i to ue multi-reolution code (MRC, encoding data into a bae layer and one or more refinement layer [2], [3]. Receiver ubcribe to the layer cumulatively, with the layer incrementally combined at the receiver to provide progreive refinement. The decoding of a higher layer alway require the correct reception of all lower layer including the bae layer. In thi paper, we conider multirate multicat with linear network coding. Propoed in [4], network coding allow and Thi work wa upported in part by the National Science Foundation under grant No , CNS-06221, and CCR , and by ONR MURI Grant No. N , by ubcontract # / iued by BAE Sytem National Security Solution, Inc. In addition, thi work wa partially upported by the Defene Advanced Reearch Project Agency (DARPA and the Space and Naval Warfare Sytem Center (SPAWARSYSCEN, San Diego under Contract No. N C- 2020/00151, and by ubcontract # C iued by Stanford Univerity and upported by DARPA. encourage mixing of data at intermediate node. It ha been hown that for a ingle rate multicat, network coding achieve the minimum of the maximum flow to each receiver; thi limit i generally not achievable through traditional routing cheme. Kötter and Médard alo tudied multirate multicat, deriving neceary algebraic condition for the exitence of network coding olution for a given network and receiver requet [5]. For n-layer multicat, linear network code can atify requet from all the receiver if the n layer are to be multicated to all but one receiver. If more than one ubcribe to ubet of the layer, linear code ceae to be ufficient. Previou work on multirate multicat with network coding include [6], [7], [8], [9], [10], [11]. For the MDC approach, [6] and [7] modified PET at the ource to cater for network coded ytem. Recovery of ome layer i guaranteed before full rank linear combination of all layer are received, and thi i achieved at the cot of a lower code rate. Wu et al. tudied the problem of Rainbow Network Coding, which incorporate linear network coding into multi-decription coded multicat [8]. For the MRC approach, [9] tudied multireolution media treaming, and propoed a polynomial-time algorithm for multicat to heterogeneou receiver. Zhao et al. conidered multirate multicat in overlay network [10], organizing receiver into layered data ditribution mehe, and utilizing network coding in each meh. Xu et al. propoed the Layered Separated Network Coding Scheme to maximize the total number of layer received by all receiver [11]. Note that if no additional coding at the ource uch a modified PET i ued, the aggregate rate to all receiver i maximized by olving the linear network coding problem eparately for each layer [8], [9], [10], [11]. Specifically, for each layer, a ubgraph i elected for network coding by performing linear programming. In other word, only intralayer network coding i allowed. On the other hand, interlayer network coding, which allow coding acro layer, often achieve higher throughput. Incorporating inter-layer linear network coding into multirate multicat, however, i ignificantly more difficult, a intermediate node have to know the network topology and the demand of all downtream receiver before determining it network code. Reference [12] conider inter-layer network coding by performing integer-programming (IP flow optimization on multicat layer i defined a the combination of layer from 1uptoi. In addition to IP, which i NP-Hard, thi algorithm require everal computation of edge dijoint path, which i alo NP-Hard. It alo require centralized knowledge of the

2 Bae Layer X 1, X 2, X 3 Refinement layer Multicat Network r 2 r 3 r 4 X 1 X 1 X 1, X 2 X 1, X 2, X 3 Fig. 1. A network with a ource with multi-reolution code X 1, X 2,and X 3, and receiver,r 2,r 3,r 4. network topology. Such centralized algorithm are difficult to perform on general network. On the other hand, the algorithm we tudy ha polynomial complexity and i fully ditributed. In thi paper, we propoe a imple, ditributed, two-tage meage paing algorithm to generate network code for ingle ource multicat of multi-reolution code. Unlike previou work, thi algorithm allow both intra-layer and inter-layer network coding at all node. It guarantee decodability of the bae layer at all receiver. In term of total rate achieved, with field ize a mall a 2 10, it outperform routing a well a network coding cheme that allow intra but not interlayer coding. The performance gain increae a the number of receiver increae and a the network grow in ize, if appropriate criterion i ued. Otherwie, naïve inter-layer coding may lead to an inappropriate choice of network code, which can be wore than intra-layer network coding. The ret of thi paper i organized a follow. Section II preent the network model and the network coding problem of multicat of multi-reolution code. The puhback algorithm i propoed in Section III, and analyzed in Section IV. Simulation reult are preented in Section V, while dicuion on future work conclude the paper in Section VI. II. PROBLEM SETUP We conider the network coding problem for ingle-ource multicat of multi-reolution code, a illutrated in Figure 1. A ingle-ource multicat network i modeled by a directed acyclic graph G =(V,E, V being the et of node, and E the et of link. Each link i aumed to have unit capacity, while link with capacitie greater than 1 are modeled with multiple parallel link. The ubet R = {,r 2,...r n } V i the et of receiver which wih to receive information from the ource node V. The ource procee, X 1, X 2,..., X L, contitute a multi-reolution code, where X 1 i the bae layer and the ret are the refinement layer. It i important to note that layer X i without layer X 1, X 2,..., X i 1 i not ueful for any i. For implicity, we aume each layer i of unit rate. Thu, given a link e E, we can tranmit one layer (or equivalent coded data rate on e at a time. The min-cut between and a node v i denoted by mincut(v, and we aume that every node v know it mincut(v. Note that there are efficient algorithm, uch a Ford-Fulkeron algorithm, that can compute mincut(v. Our goal i to deign a imple and ditributed algorithm that provide a coding trategy to maximize the total rate achieved by all receiver with the reception of the bae layer guaranteed to all receiver. By Min-Cut Max-Flow bound, each receiver r i can receive at mot mincut(r i layer (X 1, P(v q(v v Receiver: q(r = mincut(r q(u 2 C(v r u 2 q(v C(u 3 =Ø: q(u 3 =0 u 3 P(v c(e 1,m 1 c(e 3,m 3 C(v e 1 e 2 c(e 2,m 2 Fig. 2. Puhback tage and code aignment tage at node v. v e 3 e 4 c(e 4,m 4 X 2,.., X mincut(ri. We preent the puhback algorithm, and compare it performance againt other exiting algorithm and againt the theoretical bound of Min-Cut Max-Flow. III. PUSHBACK ALGORITHM The puhback algorithm i a ditributed algorithm which allow both intra-layer and inter-layer linear network coding. It conit of two tage: puhback and code aignment. In the puhback tage, meage initiated by the receiver are puhed up to the ource, allowing uptream node to gather information on the demand of any receiver reachable from them. Meage are paed from node to their parent. Initially, each receiver r i R requet for layer X 1, X 2,..., X mincut(ri to it uptream node, i.e., the receiver r i requet to receive at a rate equal to it min-cut. An intermediate node v V compute a meage, which dep on the value of mincut(v and the requet from it children. Node v then puhe thi meage to it parent, indicating the layer which the parent node hould encode together. The code aignment tage i initiated by the ource once puhback tage i completed. Random linear network code [13] are generated in a top-down fahion according to the puhback meage. The ource generate code according to the meage from it children: encode the requeted layer and tranmit the encoded data to the correponding child. Intermediate node then encode/decode the packet according to the meage determined during the puhback tage. To decribe the algorithm formally, we introduce ome additional notation. For a node v, let P (v be it et of parent node, and C(v it children a hown in Figure 2. P (v and C(v are dijoint ince the graph i acyclic. Let Ev in = {(v 1,v 2 E v 2 = v} be the et of incoming link, and Ev out = {(v 1,v 2 E v 1 = v} the et of outgoing link. A. Puhback Stage A hown in Figure 2, we denote the meage received by node v from a child u C(v a q(u, and the et of meage received from it children a q(c(v = {q(u u C(v}. A meage q(u mean that u requet it parent to code acro layer 1 to at mot q(u. Once requet are received from all children, v compute it meage q(v and the ame q(v to all of it parent. The requet q(v i a function of q(c(v and mincut(v, i.e. q(v = f(q(c(v,mincut(v. A peudocode for the puhback tage at a node v V i hown in Algorithm 1. It i important to note that the choice of f( i a u

3 q( =2 q(v 1 =2 q(v 2 =1 v 1 v 2 r 2 r 3 mincut( =2 mincut(r 2 =3 mincut(r 3 =1 Fig. 3. An example of puhback tage with min-req criterion. q( =2 q(r 1 =2 q(v 1 =2 q(v 2 =2 v 1 v 2 q(r 1 =2 q(v 3 =1 q(v 3 =1 q(v 3 =2 v 3 q(v 3 =2 mincut(v 1 =1 mincut(v 2 =1 mincut(v 3 =2 q(r 3 =1 mincut(v 1 =1 mincut(v 2 =1 mincut(v 3 =2 r 2 r 3 mincut( =2 mincut(r 2 =3 mincut(r 3 =1 Fig. 4. An example of puhback tage with min-cut criterion. Highlighted are the meage different from that of min-req criterion (Figure 3. v 3 q(r 3 =1 key feature of the algorithm a it determine the performance. We preent two different verion of f( : min-req criterion and min-cut criterion, which we dicu next. 1 Min-req Criterion: The min-req criterion, a the name ugget, define q(v =f(q(c(v,mincut(v a follow: q(v = { 0 if q(u =0for all u C(v, q min otherwie, where q min = min q(u 0, u C(v q(u i the minimum nonzero q(u from u C(v. Thi criterion may eem very peimitic and naïve, a the intermediate node erve only the minimum requeted by their downtream receiver to enure the decodability of the bae layer. Nonethele, a we hall ee in Section V, the performance of thi criterion i quite good. An example of puhback with min-req i hown in Figure 3. Receiver, r 2, and r 3 requet their min-cut value 2, 3, and 1, repectively. The intermediate node v 1, v 2, and v 3 requet the minimum of all the requet received, which are 2, 1, and 1, repectively. if v i a receiver then q(v =mincut(v; if v i an intermediate node then if C(v = then q(v =0; if C(v then q(v =f(q(c(v,mincut(v; Algorithm 1: The puhback tage at node v. 2 Min-cut Criterion: The min-cut criterion define the function q(v =f(q(c(v,mincut(v a follow: { q min if mincut(v q min, q(v = mincut(v otherwie, where q min = min q(u 0, u C(v q(u. Note if a node v receive mincut(v number of linearly indepent packet coded acro layer 1 to mincut(v, it can decode layer X 1, X 2,..., X mincut(v and act a a econdary ource for thoe layer. Thu, if there i at leat one child u C(v that requet fewer than mincut(v layer, i.e. mincut(v >q min, node v et it requet q(v to mincut(v. However, if all children requet more than mincut(v layer, node v doe not have ufficient capacity to decode the layer requeted by it children. Thu, it et q(v =q min. An example of puhback with min-cut i hown in Figure 4. The network i identical to that of Figure 3. Again, the node, r 2, r 3, and v 1 requet 2, 3, 1, and 2, repectively. However, node v 2 requet 2, which i the minimum of all the requet it received, and node v 3 requet mincut(v 3 =2. B. Code Aignment Stage Thi tage i initiated by the ource after puhback i completed. A hown in Figure 2, c(e, m denote the random linear network code v tranmit to it child u C(v, where e =(v, u, and m mean that packet on e are coded acro layer 1 to m. Notem may not equal to q(u, which we dicu in more detail in Section IV. Algorithm 2 preent a peudocode for the code aignment tage at any node v V. Algorithm 2 conider ource, intermediate, and receiver node eparately. The ource alway exactly atifie any requet from it children, while the receiver decode a many conecutive layer a they can. For an intermediate node v connected to the network (P (v, v collect all the code c(e i,m i from it parent and determine m, the number of layer up to which v can decode. It i poible that v cannot decode any layer, leading to an m equal to zero. For m 0, v can act a a econdary ource for layer 1, 2,..., m by decoding thee layer. In the cae where q(u m, u C(v, v can atify u requet exactly by encoding jut the layer 1 to q(u. Ifq(u >m, v cannot decode the layer u requeted; thu, cannot atify u requet exactly. Therefore, v ue a bet effort approach and deliver a packet coded acro m max layer, where m max i the cloet to q(u node v can erve without violating u requet, i.e. q(u m max. The code aignment tage require that every node check it decodability to determine m. Thi proce involve Gau-Jordan elimination, which i computationally cheaper than matrix inverion required for decoding. Note that only a ubet of the node need to perform (partial decoding. Figure 5 and 6 how the code aignment tage for the example in Figure 3 and 4. The algorithm for code aignment tay the ame, whether we ue min-req or mincut criterion during the puhback tage, but the reulting code aignment are different. Although the throughput achieved in thee example are the ame, thi i uually not the cae.

4 if v i the ource then foreach edge e =(v, u Ev out do v tranmit c(e, q(u; if v i an intermediate node then if P (v = then v et c(e, 0 for all e Ev out ; if P (v then v receive code c(e i,m i, e i Ev in ; v determine m, which i the maximum m uch that X 1, X 2,...X m are decodable from c(e i,m i ; foreach child u C(v do Let e =(v, u; if q(u m then v decode layer X 1, X 2,..., X m ; v tranmit c(e, q(u; if q(u >m then Let m max = max mi q(u m i ; v tranmit c(e, m max ; if v i a receiver then v receive code c(e i,m i, e i Ev in ; v decode m layer, which i the maximum m uch that X 1, X 2,...X m are decodable from c(e i,m i ; Algorithm 2: The code aignment tage at node v. Generally the min-cut criterion achieve higher throughput than the min-req criterion. C. Complexity analyi Given a ingle-ource multicat network with a L- layer multi-reolution code, each puhback meage q( {1,...,L} i log 2 L -bit long. On the other hand, the code aignment meage c( from node v to u C(v i of length O(q(v log 2 (p-bit where p i the field ize. Note that log 2 p 10 i ufficient (ee Section V; therefore, c( i O(L-bit long. Thu, a node v tranmit a total of O( log 2 L P (v + L C(v -bit. Given L, the meage complexity i linear in the number of neighbor v ha. To compute q(v, node v perform O( C(v comparion. Since v tranmit the ame q(v to all it parent, v only need to compute q(v once. To compute c(v, node v firt perform a Gau-Jordan elimination to determine m (ee Section V for more detail. Note that given a k l matrix, Gau-Jordan elimination ha complexity O(kl 2. By Lemma 4.2, v receive a coding matrix of ize P (v q(v, which i at mot a matrix of ize P (v L. Thu, computing m require at mot O( P (v L 2 operation. In addition, v may need to decode m layer, where m min{ P (v,l}. Decoding involve c(e 1,2 e c(e 2,1 e 2 1 c(e 3,1 v e 1 v 3 2 c(e 4,1 e 5 e 6 e 7 e 8 c(e 8,1 e c(e 5,2 9 c(e 7,1 c(e 6,2 c(e 9,1 r 2 r 3 mincut( =2 mincut(r 2 =3 mincut(r 3 =1 Decode 2 layer Decode 2 layer Decode 1 layer Fig. 5. e 4 v 3 e10 mincut(v 1 =1 mincut(v 2 =1 mincut(v 3 =2 c(e 10,1 An example of code aignment tage with min-req criterion. c(e 1,2 e c(e 2,2 1 e 2 v c(e 3,2 1 v 2 e 3 mincut(v 1 =1 mincut(v 2 =1 mincut(v 3 =2 e 4 c(e 4,2 e v 3 decode layer 1 and 2 5 e 6 e 7 e 8 c(e 8,2 e 9 c(e 5,2 c(e 7,2 e 10 c(e c(e 6,2 10,1 c(e 9,2 r 2 r 3 mincut( =2 mincut(r 2 =3 mincut(r 3 =1 Decode 2 layer Decode 2 layer Decode 1 layer Fig. 6. An example of code aignment tage with min-cut criterion. High lighted are the meage different from the min-req criterion (Figure 5. a matrix inverion, with complexity O(min{ P (v,l} 3. Latly, re-encoding require a vector-matrix multiplication, which ha complexity O( P (v L. Note that v need to reencode a eparate meage c( for each u C(v. Thu, re-encoding require O( C(v P (v L operation. The total complexity at v i O( P (v L 2 +O(min{ P (v,l} 3 + O( C(v P (v L =O( P (v L 2 + C(v P (v L. The overall time required for each tage i linearly proportional to the maximum ditance from the ource to any of the ink, auming that each link incur unit delay. If we aume the network i tatic, thi time delay can be well amortized over multiple tranmiion. IV. ANALYSIS OF PUSHBACK ALGORITHM In general, not all receiver can achieve their min-cut through linear network coding. Nonethele, we want to guarantee that no receiver i denied ervice, i.e. although ome node may not receive up to the number of layer deired, all hould receive at leat laye. In thi ection, we prove that the puhback algorithm guarantee decodability of the bae layer, X 1, at all receiver. Two related lemma are preented to prove Theorem 4.3. Lemma 4.1: Aume mincut(v =n for a node v in G. In the puhback algorithm, if m i n for all c(e i,m i, e i Ev in, then v can decode at leat laye with high probability. In other word, if all received code at v are combination of at mot n layer, v can decode at leat laye. Proof: Recall that a code c(e i,m i repreent coding acro layer 1 to m i ; if the field ize i large, with high probability, the firt m i element of thi coding vector are non-zero, wherea the ret are zero.

5 n n c 1 c 2 c 3 n M All non-zero element are indepently and randomly elected. n M M 1 r 2 M 2 r 3 M 3 (a Fig. 7. Coding matrix M; each row repreent a code received, and column repreent the layer. The maximum number of non-zero column in M, c k, can be equal to n (a hown in (a, or le than n (a hown in (b. Since mincut(v =n, there exit n edge-dijoint path from the ource to v, for all link are aumed to have unit capacity. Therefore, v receive from it incoming link at leat n code, each of which can be repreented a a row coding vector of length n, ince m i <nfor all i. Pickthe n code correponding to the edge-dijoint path to obtain an n n coding matrix. For the quare coding matrix, ort it row according to the number of non-zero element per row, obtaining the tructure hown in Figure 7. We denote thi orted matrix by M, and the unique number of non-zero element in it row by c 1,c 2,..., c k, in acing order. Since the row of M are generated along edge-dijoint path uing random linear network coding, the non-zero element of M are indepently and randomly elected. Next, define upper-left corner ubmatrice M 1, M 2,..., M k a hown in Figure 8, where each ubmatrix M i i of ize r i c i. Specifically, the row of M with exactly c 1 non-zero element form a c 1 ubmatrix M 1 ; the row of M with exactly c 1 or c 2 non-zero element form the r 2 c 2 ubmatrix M 2. M k i of ize r k c k, where r k = n, and c k n. Note for any i, ifrank(m i =c i, node v can decode layer 1 to c i, i.e., the bae layer i decodable. In other word, laye i not decodable at node v only if rank(m i <c i for all i. With thee definition, we aume laye i not decodable at node v, and prove the lemma by contradiction. Specifically, we prove by induction that thi aumption implie r i <c i for all i, leading to the contradiction r k <c k. For the bae cae, firt conider M 1. If laye i not decodable, rank(m 1 <c 1. Recall that element in M 1 are indepently and randomly elected [13]; if c 1, with high probability, rank(m 1 =c 1. Therefore, the above aumption implie <c 1 and rank(m 1 =. Next conider M 2. Under the aumption that laye i not decodable, rank(m 2 <c 2. Since rank(m 1 = and M 2 include row of M 1, rank(m 2.Row +1, +2,..., r 2 are called the additional row introduced in M 2. If there are more than c 2 additional row, M 2 ha full rank, i.e. rank(m 2 =c 2, with high probability. Hence, the number of additional row in M 2 mut be le than c 2, implying r 2 <c 2. For the inductive tep, conider M i, 3 i k. Aume that r j <c j for all j<i. If laye i not decodable, rank(m i < c i. By imilar argument a above, rank(m i 1 =r i 1, and there mut be le than c i r i 1 additional row introduced in M i. Thu, r i < c i. By induction, the total number of row r k = n in M i trictly le than c k n, which i a contradiction. We therefore conclude that node v can decode the bae layer. In fact, v can decode at leat c 1 layer. (b Fig. 8. Upper-left corner ubmatrice M 1, M 2,andM 3. Lemma 4.2: In the puhback algorithm, for each link e = (v, v, aume that node v requet q(v =q to node v. Then, the code c(e, m on link e (from node v to v i coded acro at mot q layer, i.e. m q. Proof: Firt, define the notion of level. A node u i in level i if the longet path from to u i i, a hown in Figure 9. Since the graph i acyclic, each node ha a finite level number. We hall ue induction on the level to prove that thi lemma hold for both min-req and min-cut criteria. For the bae cae, if v i in level 1, it i directly connected to the ource, and receive a code acro exactly q layer on e from. For the inductive tep, aume that all node in level 1toi, 1 i<k, get packet coded acro layer 1 to at mot their requet. Aume v i in level i +1.Let v P (v ; therefore, v i in level j i. Letq min be the mallet non-zero requet at v, that i q min = min q(u 0,u C(v q(u. For the min-req criterion, v alway requet q(v = q min to it parent, and the code v receive are linear combination of at mot q min layer. Therefore, the code v to it children i coded acro at mot q min layer, where q = q(v q min. In other word, the code received by v i coded acro at mot q layer. For the min-cut criterion, if q min >mincut(v, node v requet q(v =q min to it parent. By the ame argument a that for the min-req criterion, v receive packet coded acro at mot q layer. If q min mincut(v, v requet q(v =mincut(v. According to the code aignment tage, if v cannot atify requet q exactly, it will out a linear combination of the layer it can decode. Since v i in level j i, v receive code acro layer 1 to at mot mincut(v. By Lemma 4.1, node v can decode at leat the bae layer. Thu, we conclude that node v i alway able to generate a code for node v uch that it i coded acro layer 1 to at mot q. Theorem 4.3: In the puhback algorithm, every receiver can decode at leat the bae layer. Proof: The receiver with min-cut n receive linear combination of at mot n layer by Lemma 4.2. From Lemma 4.1, the receiver, therefore, can decode at leat the bae layer. level 1 Fig. 9. level 3 Level 1 and level 3 node in a network.

6 V. SIMULATIONS To evaluate the effectivene of the puhback algorithm, we implemented it in Matlab, and compared the performance with both routing and intra-layered network coding cheme. Random network were generated, with a fixed number of receiver randomly elected from the vertex et. We conider two metric to evaluate the performance: % Happy Node = # of receiver that achieve min-cut, # of trial # of receiver all trial total rate achieved all trial % Rate Achieved =. all total min-cut trial The % Happy Node metric i the average percentage of receiver that achieved their min-cut, i.e. receiver that received the ervice they requeted. The % Rate Achieved metric give a meaure of what percentage of the total requeted rate (equal to um of min-cut wa delivered to the receiver over all trial. It i important to note that the optimal achievable rate (denoted OPT for multirate multicat are generally unknown for a network. The min-cut i the theoretical upper bound on OPT. Therefore, % Rate Achieved i a lower bound on the total rate achieved in term of OPT, i.e. total rate achieved all trial % Rate Achieved. OPT Recall that thi total rate achieved i what we int to maximize with the propoed algorithm. A an example, conider two poible cae where the (min-cut, achieved-rate pair are ([1, 1, 2], [1, 1, 1] and ([2, 2, 3], [2, 2, 2]. In both cae, the demand of a ingle receiver i mied by one layer, but the correponding fraction of rate achieved are 3/4 and 6/7 repectively. Uing only the % Happy Node metric would tell u that 1/3 of the receiver did not received all requeted layer. However, the %Rate Achieved metric provide a more accurate meaure of how unhappy the overall network i. A. Algorithm for comparion 1 Point-to-point Routing Algorithm: the point-to-point routing algorithm conider each multicat a a et of unicat. The ource node firt multicat the bae layer X 1 to all receiver. To determine the link ued for layer X 1, compute the hortet path to each of the receiver eparately. Given the hortet path to all receiver, then ue the union of the path to tranmit the bae layer. Note the hortet path to receiver r i may not be dijoint with the hortet path to receiver r j. After tranmitting layer X i 1, 2 i L, ource ue the remaining network capacity to tranmit the next refinement layer X i to a many receiver a poible. Firt, update the min-cut to all receiver and identifie receiver that can receive X i.letr = {r i1,r i2,...} be the et of receiver with updated min-cut greater than 1 and, therefore, can receive layer X i. The ource then compute the hortet path to receiver in R. The union of thee path i ued to tranmit the refinement layer. Thi proce i repeated until no receiver can be reached or the layer are exhauted. 2 Tree Routing Algorithm: the tree routing algorithm compute the minimal-cot tree connecting ource and all the receiver. We aume that each link i of unit cot. For the bae layer X 1, compute and tranmit on the tree connecting to all receiver. For each new refinement layer X i, compute a new tree to receiver with updated min-cut greater than zero. Thi proce i repeated until no receiver can be reached or the layer are exhauted. It i important to note that tree routing algorithm i an optimal routing algorithm it ue the fewet number of link to tranmit each layer. Unlike the point-to-point algorithm, thi algorithm may make routing deciion that i not optimal to any ingle receiver, i.e. the ource may ue a non-hortet path to communicate to a receiver, but it ue fewer link globally. However, thi optimality come with a cot: the problem of finding a tree i NP-complete. 3 Intra-layer Network Coding Algorithm: the intra-layer network coding algorithm ue linear coding on each layer eparately. It iteratively olve the linear programming problem for linear network coding for layer X i with receiver R i = {r R mincut(r 1}, where i =1and R 1 = R initially [14]. After olving the linear program for layer X i, the algorithm increment i, update the link capacitie, and perform the next round of linear programming. Reference [8] and [9] are example of thi intra-layer coding approach. B. Implementation of Puhback Algorithm The puhback algorithm wa implemented with two different meage paing chedule. 1 Sequential: for the puhback tage, each node in the network a requet to it parent after requet meage from all it children have been received. For the code aignment tage, each node a code to it children after receiving code from all it parent. Thi correpond to the algorithm explained in Section III. 2 Flooding: for the puhback tage, each node update it requet to it parent upon reception of a new meage from it children. For the code aignment tage, each node a new code to it children after receiving a new meage from any of it parent node. Thi allow an update mechanim that converge to the ame olution a equential meage paing, where the convergence time dep on the diameter of the graph. Another important iue i the procedure to check decodability at each node. In general, Gau-Jordan elimination on the coding matrix in a field of ize p i neceary to determine which layer are decodable at a node after the code are aigned. However, thi i not the cae for 2-layer multi-reolution code (L =2. We define pattern of coding coefficient for a node with δ incoming link a [a 1,a 2,..., a δ ], where a i repreent the number of layer combined in the i-th incoming link. If a node receive only the bae layer on all incoming link, i.e. the pattern of coding coefficient i [1, 1,..., 1], it can decode the bae layer. If at leat one of the incoming link contain a combination of two layer,

7 % Happy Node Fig PB min cut p = 2 m PB min req p = 2 m PB min cut p = PB min req p = m (p = 2 m Varying field ize p in a network with 5 receiver and 25 node. i.e. the pattern of coding coefficient i one of the following: [1,..., 1, 2], [1,..., 1, 2, 2],..., [2,..., 2], both layer can be decoded. Thu, for L =2, the pattern of coding coefficient indicate decodability. Note that uing the pattern of coding coefficient i equivalent to uing Gau-Jordan elimination with infinite field ize. In more general cae with L>2, the pattern of coding coefficient i no longer ufficient. For example, a node with 4 incoming link of unit rate can have a min-cut of at mot 4. Aume that thi node ha a min-cut of 3, and that thi node i aigned a coding-coefficient pattern of [1, 1, 3, 3]. If coding vector are linearly indepent, all layer are decodable. However, it i poible that the third and the fourth link, both combining three layer, are not from dijoint path, i.e. linearly depent combination. Then, Gau-Jordan elimination i neceary to check that only the firt layer i decodable. In ubequent ection, we preent imulation reult for 2 and 3-layer multi-reolution code. However, our algorithm i not limited to 2 and 3-layer; it can be applied to general n-layer multi-reolution code. C. Simulation reult for 2-layer multi-reolution code The imulation for 2-layer multi-reolution code were carried out for random directed acyclic network. We averaged 0 trial for each data point on the curve plotted in thi ection. The network were generated uch that the min-cut and the in-degree of all node were le than or equal to 2. A tated in Section V-B, the pattern of coding coefficient are ufficient to check decodability for 2-layer multi-reolution code, and it i equivalent to uing Gau-Jordan elimination with an infinite field ize. Figure 10 how the effect of field ize in a network with 25 node and 5 receiver by performing Gau-Jordan elimination at every node during the code aignment tage with varying field ize p. Italohow the average performance in term of % Happy Node when uing the pattern of coding coefficient to check decodability. In eence, we are comparing the performance of our ytem uing a field ize p to that of an infinite field ize. Oberve that even for moderately mall field ize, uch a p 2 8, the puhback algorithm perform cloe to that of the ytem operating in an infinite field. From Figure 10, we ee that the min-cut criterion perform coniderably better than the min-req criterion for large field % of Happy Node % Rate Achieved Layered PB min req flooding PB min cut flooding PB min req equential PB min cut equential PB min req p = 2 10 PB min cut p = No. of Receiver ( R Fig. 11. Layered PB min req flooding PB min cut flooding PB min req equential PB min cut equential PB min cut p = 2 10 PB min req p = No. of Receiver ( R Varying number of receiver in a network with 25 node. ize. However, for mall field ize (p 2 3, the min-req criterion i lightly better. Thi i becaue it forward the minimum of the requet received at any node. For L =2, there will be more node requeting only the bae layer when uing the min-req criterion than when uing the min-cut criterion. Thu, network uing the min-req criterion will have more link carrying only the bae layer, which help improve redundancy for the receiver. Thi allow everal path to carry the ame information, enuring the bae layer i decodable at the receiver. By comparion, the min-cut criterion trie to combine both layer on a many link a poible. When the field ize i large, both layer are decodable with high probability; when the field ize i mall, the probability of generating linearly depent code i high, conequently preventing decodability of both layer at a ubet of receiver. Figure 11 and 12 compare the performance of the variou cheme in term of the two metric % Happy Node and % Rate Achieved. The puhback algorithm i compared to the Point-to-point Routing Algorithm (, the Tree Routing Algorithm (, and the Intra-layer Network Coding Algorithm ( Layered. We implement two verion of puhback: flooding and equential meage paing approache. The flooding cheme with an infinite field ize are labeled PB min-req flooding and PB min-cut flooding for the min-req and min-cut criteria, repectively. The equential meage paing cheme with an infinite field ize are labeled PB min-req equential and PB min-cut equential. Finally, puhback uing a moderate field ize of p =2 10 are labeled PB min-req p =2 10 and PB min-cut p =2 10.

8 % Happy Node % Rate Achieved Layered PB min req flooding 86 PB min cut flooding PB min req equential 84 PB min cut equential 82 PB min req p = 2 10 PB min cut p = No. Node (n Layered PB min req flooding 86 PB min cut flooding PB min req equential 84 PB min cut equential 82 PB min cut p = 2 10 PB min req p = No. Node (n Fig. 12. Varying number of node in a network with 3 receiver. Figure 11 how the performance of the variou cheme when the number of receiver i increaed. PB min-cut ha the bet performance overall. Both flooding and equential meage paing approache behave imilarly. Furthermore, uing a moderate field ize of p =2 10 yield reult cloe to that of an infinite field ize (for both min-cut and min-req. The performance of the variou cheme follow a imilar tr for both metric % Happy Node and % Rate Achieved. In addition, Figure 11 illutrate that the gap between the mincut criterion and, and Layered increae with the number of receiver. Note that the gap between the mincut and the min-req criteria increae more lowly than the gap between the min-cut and the other cheme. Figure 12 compare the different cheme when the number of receiver i fixed, but the network grow in ize. PB min-cut outperform the intra-layer network coding and the routing cheme; it alo conitently achieve cloe to % for both % Happy Node and % Rate Achieved while the econd bet cheme ( Layered achieve at mot 96% and 97% for the two metric. Figure 12 alo how that the min-cut criterion i very robut to the ize of the network. In fact, the performance improve a more node are added. However, the min-req degrade with the number of node. Thi i becaue with min-req, the requet from receiver with min-cut equal to 1 limit the rate of other receiver. A the network become larger, thi flooding of bae layer requet ha a more ignificant effect on the throughput ince more reource are wated in delivering jut the bae layer. Thi indicate that the choice of network code can % Happy Node 65 Layered PB min req p=2 m 60 PB min cut p=2 m m (p= 2 m Fig. 13. Varying field ize in a network with 9 receiver 25 node. greatly impact the overall network performance, deping on it topology and demand. An inappropriate choice of network code can be detrimental, a hown by PB min-req ; however, an intelligent choice of network code can improve the performance ignificantly, a hown by PB min-cut. D. Simulation reult for 3-layer multi-reolution code Similarly to the 2-layer cae, for 3-layer multi-reolution code, we generated random network to evaluate the puhback algorithm. For each data point in the plot, we averaged 0 trial. The min-cut and the in-degree of all node were le than or equal to 3. Recall that with 3 layer, the pattern of coding coefficient are not ufficient for checking the decodability of incoming packet. Intead, Gau-Jordan elimination i neceary at every node during the code aignment tage. Figure 13 how the effect of field ize in a network of 25 node and 9 receiver. PB min-cut outperform routing and intra-layer coding cheme with a field ize of p =2 5. In term of % Happy Node, PB min-cut achieve roughly 92% when the field ize i large enough, while the intra-layer coding cheme only achieve about 82%. Figure 13 alo how that intra-layer coding cheme till outperform the routing cheme, even when optimal multicat routing i ued for each layer. Our puhback algorithm achieve coniderably higher gain by uing inter-layer in addition to intra-layer coding. A the number of receiver increae, more demand need to be atified imultaneouly. It i therefore expected that the overall performance of multicat cheme will degrade with the number of receiver. Thi can be oberved in Figure 14. However, the performance gap between PB min-cut and PB min-req i approximately contant, while the performance gap over other cheme increae. Thi mean that our algorithm i more robut to change in the number of receiver than the other cheme, an important property for ytem that aim to provide ervice to a large number of heterogeneou uer. Figure 15 how the performance of the different cheme a the network grow in ize. A more node are added, there are more dijoint path within the network for tree routing and intra-layer coding to ue. Hence the performance of thee cheme improve. The oppoite behavior occur for PB minreq, i.e. the % Happy Node decreae a the network ize increae. Thi reult i imilar to that of Figure 12 for 2-layer

9 % Happy Node % Rate Achieved Layered PB min req p=2 12 PB min cut p= No. of Receiver ( R Layered PB min req p=2 12 PB min cut p= No. of Receiver ( R Fig. 14. Varying number of receiver in a network with 25 node uing field ize of % Happy Node 65 Layered PB min req p=2 10 PB min cut p= No. of Node (n Fig. 15. Varying number of node in a network with 9 receiver uing field ize of cae. A the network ize increae, it become more likely that a mall requet by one receiver uppree higher requet by many other receiver. Hence, puhback with the min-req criterion quickly deteriorate in term of % Happy Node. VI. CONCLUSIONS AND FUTURE WORK A imple, ditributed meage paing algorithm, called the puhback algorithm, ha been propoed to generate network code for ingle ource multicat of multi-reolution code. With two tage, the puhback algorithm guarantee decodability of the bae layer at all receiver. In term of total rate achieved, thi algorithm outperform routing cheme a well a intra-layer coding cheme, even with mall field ize uch a The performance gain increae a the number of receiver increae and a the network grow in ize a hown by numerical imulation. Poible future work include the addition of a third complaint tage, in which receiver whoe requet have not been atified pa another et of requet to their parent, ignaling their deire for more. In generating new code, parent node mut take into account the new updated requet, while maintaining decodability at receiver which did not participate in the complaint tage. It i important to determine what the complaint meage hould be, and to ae the improvement that can be achieved with uch an additional tage. Another poible extenion i to apply thi algorithm in wirele/dynamic multicat etting. The flooding meage paing approach i applicable then, a change in the network can be handled by new meage to the neighboring node. Latly, in the puhback algorithm, rate i the meage ent by node to their parent, i.e. each node ignal how many layer down-tream receiver can or want to receive. It may be poible to ext the meage to include other contraint, uch a power (decoding power, delay, and reliability. REFERENCES [1] A. Albanee, J. Blömer, J. Edmond, M. Luby, and M. Sudan, Priority encoding tranmiion, IEEE Tranaction on Information Theory, vol. 42, no. 6, pp , [2] N. Shacham, Multipoint communication by hierarchically encoded data, in Proceeding of INFOCOM 92, 1992, pp [3] M. Effro, Univeral multireolution ource code, IEEE Tranaction on Information Theory, vol. 47, no. 6, pp , [4] R. Ahlwede, N. Cai, S.-Y. R. Li, and R. W. Yeung, Network information flow, IEEE Tranaction on Information Theory, vol. 46, no. 4, pp , July [5] R. Kötter and M. Médard, An algebraic approach to network coding, IEEE/ACM Tranaction on Networking, vol. 11, no. 5, pp , [6] D. Silva and F. R. Kchichang, Rank-metric code for priority encoding tranmiion with network coding, in Proceeding of 10th Canadian Workhop on Information Theory (CWIT 07, June [7] J. M. Walh and S. Weber, A concatenated network coding cheme for multimedia tranmiion, in Proceeding of 4th Workhop on Network Coding, Theory and Application (NetCod 08, Jan [8] X. Wu, B. Ma, and N. Sarhar, Rainbow network flow of multiple decription code, IEEE Tranaction on Information Theory, vol. 54, no. 10, pp , [9] S. B. Niveditha Sundaram, Paramewaran Ramanathan, Multirate media tream uing network coding, in Proceeding of 43rd Annual Allerton Conference on Communication, Control, and Computing, [10] J. Zhao, F. Yang, Q. Zhang, Z. Zhang, and F. Zhang, Lion: Layered overlay multicat with network coding, IEEE Tranaction on Multimedia, vol. 8, no. 5, pp , Oct [11] C. Xu, Y. Xu, C. Zhan, R. Wu, and Q. Wang, On network coding baed multirate video treaming in directed network, in Proceeding of IEEE International Conference on Performance, Computing, and Communication, IPCCC 07, April [12] S. Dumitrecu, M. Shao, and X. Wu, Layered multicat with interlayer network coding, in The 28th IEEE Conference on Computer Communication (INFOCOM 09, April [13] T. Ho, R. Koetter, M. Médard, M. Effro, J. Shi, and D. Karger, A random linear network coding approach to multicat, IEEE Tranaction on Information Theory, vol. 52, no. 10, pp , October [14] D. S. Lun, N. Ratnakar, M. Médard, R. Koetter, D. R. Karger, T. Ho, E. Ahmed, and F. Zhao, Minimum-cot multicat over coded packet network, IEEE Tranaction on Information Theory, vol. 52, no. 6, pp , June 2006.