Network Coding and its Applications in Communication Networks

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1 Network Coding nd its Applictions in Communiction Networks Alex Sprintson Texs A&M University, College Sttion, Texs, USA Astrct. The network coding technique generlizes the trditionl routing pproch y llowing the intermedite network nodes to crete new pckets y comining the pckets received over their incoming edges. This technique hs severl importnt enefits such s n increse in throughput nd n improvement in the reliility nd roustness of the network. The gol of this chpter is to present tutoril review of the network coding technique, the prcticl implementtion of network coding, s well s its pplictions in severl res of networking. We egin y presenting the encoding model nd the lgeric frmework for network code construction. Next, we discuss efficient deterministic nd rndomized lgorithms for construction of fesile network codes in multicst networks. Next, we present prcticl implementtion schemes nd discuss the pplictions of network coding in content distriution networks, peer-to-peer networks, nd wireless networks. Introduction. Motivtion Communiction networks re designed to deliver informtion from source to destintion nodes. The trditionl wy of delivering dt employs pths for unicst connections nd trees for multicst connections. When the dt is routed over unicst pth, ech intermedite node forwrds the pckets received over its incoming edges to its outgoing edges. In multicst connection over tree, the intermedite nodes my duplicte pckets nd forwrd them to severl outgoing edges. The network coding pproch [] llows the intermedite nodes to generte new pckets y comining the pckets received on their incoming edges. This technique offers severl enefits, such s n increse in throughput nd n improvement in reliility nd roustness of the network. To demonstrte the dvntge of the network coding technique, consider the network depicted on Figure (). The network includes two informtion sources, s nd s 2, nd two terminls, t nd. We ssume tht ll edges of the network re of unit cpcity, i.e., ech edge cn trnsmit one pcket per time unit. With the trditionl pproch, the pckets re forwrded over two Steiner trees, such tht the first tree forwrds the pckets generted y source s, while the second tree forwrds pckets generted y node s 2. However, the network does not contin two edge-disjoint Steiner trees A Steiner tree is tree tht connects the source node with the terminls nd my include ny numer of other nodes.

2 s s 2 s s 2 s s 2 s s 2 v v v v v 2 v 2 v 2 v 2 t t t t () () (c) (d) Fig.. Bsic network coding exmple. with roots in s nd s 2, hence the multicst connection with two informtion sources cnnot e implemented using trditionl methods. For exmple, the trees depicted in Figures () nd (c) shre the ottleneck edge (v,v 2 ). Figure (d) shows tht this conflict cn e resolved y employing the network coding technique. To demonstrte this pproch, let nd e the pckets generted y the informtion sources s nd s 2, respectively, t the current communiction round. Both pckets re sent to the intermedite node v which genertes new pcket, which is then sent to oth nodes t nd. It is esy to verify tht oth terminl nodes cn decode the pckets nd from the pckets received over their incoming edges. The network coding technique cn lso e useful for minimizing the dely of dt delivery from the source to the terminl nodes [0]. For exmple, consider the network depicted on Figure 2(). Suppose tht ech edge cn trnsmit one pcket per time unit nd tht the dely of ech edge is lso one time unit. Figures 2() nd (c) show two edge-disjoint Steiner trees tht connect s to the terminls t,, nd t 3. However, one of the trees is of depth three, nd, s result, terminl will receive one of the pckets fter dely of three time units. It cn e verified tht ny scheme tht does not employ network coding results in dely of three time units. Figure 2(d) shows network coding solution which delivers the dt with the dely of just two time units. The network coding technique cn lso e employed to minimize the numer of trnsmissions in wireless networks [36]. For exmple, consider the wireless network depicted in Figure 3. The network contins two nodes s nd s 2 tht wnt to exchnge pckets through n intermedite rely node v. More specificlly, node s needs to send pcket to s 2 nd node s 2 needs to send pcket to s. Figure 3() shows trditionl routing scheme tht requires four trnsmissions. Figure 3() shows network coding scheme in which the intermedite node v first otins two pckets, nd from s nd s 2 nd then genertes new pcket, nd rodcsts it to oth s nd s 2. This scheme requires only three trnsmissions. The exmple shows tht the network coding 2

3 s s s s t t 3 t t 3 t t 3 t t 3 () () (c) (d) Fig. 2. Dely minimiztion with network coding. technique cn tke dvntge of the rodcst nture of wireless networks to minimize the numer of trnsmissions. As demonstrted y the ove exmples, network coding hs severl enefits for rod rnge of pplictions in oth wired nd wireless communictions networks. The gol of this chpter is to descrie the network coding fundmentls s well s to show rod rnge of pplictions of this technique..2 Relted work Network coding reserch ws initited y seminl pper y Ahlswede, Ci, nd Yeung [] nd hs since then ttrcted significnt interest from the reserch community. Mny initil works on the network coding technique focused on estlishing multicst connections. It ws shown in [] nd [28] tht the cpcity of the network, i.e., the mximum numer of pckets tht cn e sent from the source s to set T of terminls per time unit, is equl to the minimum cpcity of cut tht seprtes the source s nd terminl t T. In susequent work, Koetter nd Médrd [25] developed n lgeric frmework for network coding nd investigted liner network codes for directed grphs with cycles. This frmework ws used y Ho et l. [8] to show tht liner network codes cn e efficiently constructed through rndomized lgorithm. Jggi et l. [2] proposed deterministic polynomil-time lgorithm for finding fesile network codes in multicst networks. Network coding for networks with cycles hs een studied in [2] nd [4]. Network coding lgorithms resilient to mlicious interference hve een studied in [20], [35], nd [24]. While there re efficient polynomil-time lgorithms for network code construction in multicst settings, finding efficient network codes in non-multicst scenrios is more difficult prolem [33]. The complexity of severl generl network coding prolems hs een nlyzed y Lehmn nd Lehmn [27]. Dougherty el l. [] showed tht liner network codes re insufficient for chieving cpcity of network with multiple unicst connections. 3

4 s v s 2 s v s 2 s v s 2 s v s 2 s v s 2 s v s 2 s v s 2 () () Fig. 3. Reducing energy consumption with network coding: () trditionl pproch () network coding pproch. The pplictions of network coding in wired nd wireless communiction networks hve een the suject of severl recent studies. Chou nd Wu [0] discussed implementtion of network coding in content distriution networks. They discussed severl issues such s synchroniztion, vrying dely, nd trffic loss. The dvntges of network coding in lrge scle peer-to-peer content distriution systems hve een studied in [4,5]. Network coding techniques for improving the performnce of wireless networks hve een studied in [23], [6], nd [22]. Comprehensive surveys on the network coding techniques re ville in the recent ooks [3, 9, 37]. 2 Network Coding Bsics In this section we descrie the network model nd present the sic definitions for the network coding technique. Then, we present n lgeric frmework for multicst connections. Finlly, we present deterministic nd rndomized lgorithms for construction of efficient network codes. 2. Network Model We model the communiction network y directed grph G(V,E), where V is the set of nodes nd E the set of edges in G. The informtion etween network nodes is trnsmitted in pckets. We ssume tht ech pcket is n element of some finite field 2 F q = GF (q) nd cn e represented y inry vector of length n = log 2 (q) its. We ssume tht the communiction is performed in rounds, such tht t ech round, every 2 For definition of finite field see e.g., [3]. 4

5 edge in the network cn trnsmit single pcket. Note tht this ssumption implies tht ll edges of the network hve the sme cpcity of one unit. This ssumption, however, does not result in loss of generlity since edges of lrger cpcity cn e represented y multiple prllel edges of smller cpcity. We define the multicst coding network N(G, s, T ) s triple tht includes the grph G(V,E), source node s V, nd set T V of terminls. We define the cpcity of the multicst coding network N(G, s, T ) to e the tightest upper ound on the mount of informtion tht cn e trnsmitted from the source node s to ll destintion nodes T per communiction round. More specificlly, let h(i) e the mximum numer of pckets tht cn e delivered from s to ll terminls in T in i rounds. Then, the cpcity h of the network is defined s h = lim sup i h(i). () i For exmple, the network depicted in Figure () cn deliver two pckets per time unit to ech terminl, hence its cpcity is equl to two. Indeed, the network coding scheme depicted in Figure (d) cn deliver two new pckets t every communiction round. For this network, it holds tht h() = h, which, in turn, implies tht h = h(i) i. The lst property holds for ny cyclic communiction network, ut it does not necessrily hold for network tht contins cycles. To see this, consider the network N(G, s, T ) depicted on Figure 4(). For this network, it is esy to verify tht one round of communiction is insufficient for delivering two pckets to oth terminls t nd. Figure 4() shows network coding scheme tht cn trnsmin pckets over n + rounds, hence the cpcity of the network is equl to two. In prticulr, t the first round, node v 3 forwrds the pcket i received over its incoming edge (v,v 3 ), i.e., x = Then, for ech round i, i>, node v 3 genertes new pcket y computing itwise XOR etween i nd y i. Node v 4 lso genertes new pcket y computing itwise XOR etween i nd x i. It is esy to verify tht the destintion nodes t nd cn decode the pckets sent y the source node fter dely of one round. s s i i v v 2 v 3 v 4 v 5 v 6 i v v 2 i i x i = v 3 v 4 y i x i y i x i v 5 v 6 i i if i = i y i otherwise y i = i if i = i x i otherwise x i y i t t () () Fig. 4. A coding network with cycles 5

6 2.2 Encoding Model In this section, we present forml definition of liner network code. For clrity, we ssume tht the underlying network grph G(V,E) is cyclic. As discussed ove, such networks re esier to nlyze, ecuse we only need to consider single communiction round. We lso ssume tht exctly one pcket is sent over ech edge in the network nd tht ech node must receive ll pckets from its incoming edges efore sending pcket on its outgoing edges. Suppose tht we would like to trnsmit h pckets R =(p,p 2,,p h ) over the multicst network N(G, s, T ). We ssume tht the source node s hs exctly h incoming edges, indexed y e,e 2,,e h, nd ech terminl t T lso hs h incoming edges nd no outgoing edges. Note tht these ssumptions cn e mde without loss of generlity. Indeed, suppose tht the second ssumption does not hold for some terminl t T. In this cse, we cn dd new terminl t, connected with t y h prllel edges, resulting in n equivlent network. Figure 5() depicts n exmple network tht stisfies these ssumptions. For ech edge e E we denote y p e the pcket trnsmitted on tht edge. Ech incoming edge e i, i h, of the source node s trnsmits the originl pcket p i. Let e(v, u) E e n edge of the coding network N(G, s, T ) nd let M e e the set of incoming edges in G of its til node v, M e = {(w, v) (w, v) E)}. Then, we ssocite with ech edge e M e locl encoding coefficient β e,e F q = GF (q). The locl encoding coefficients of the edges tht elong to M e determine the pcket p e trnsmitted on edge e s function of pckets trnsmitted on the incoming edges M e of e. Specificlly, the pcket p e is equl to p e = β e,e p e, (2) e M e where ll opertions re performed over finite field F q. We note tht if node v V hs in-degree one, then the locl encoding coefficient {β e,e} for every pir of edges e (u, v) nd e(v, w) cn e set to one. Definition (Liner network code) Let N(G, s, T ) e coding network nd let F q = GF (q) e finite field. Then, the ssignment of encoding coefficients {β e,e GF (q) e (v, u),e(u, w) E} is referred to s liner network code for N(G, s, T ). Figure 5(c) demonstrtes the locl encoding coefficients tht form liner network code for the coding network depicted in Figure 5(). Our gol is to find set of network coding coefficients {β e,e} tht llows ech terminl to decode the originl pckets R from the pckets otined through its incoming edges. The ssignment of {β e,e} tht stisfies this condition is referred to s fesile network code for N(G, s, T ). For exmple, consider the network depicted in Figure 5() nd suppose tht ll opertions re performed over field F 2 = GF (2). Then, the ssignment of encoding coefficients β e,e 3 = β e2,e 4 = β e6,e 9 = β e7,e 9 = 6

7 e e 2 s e 3 e 4 β (e,e 3) s β (e2,e 4) p p s p 2 p 2 v v 2 v v 2 v v 2 e 6 e 7 p p 2 v 3 v 3 v 3 e 5 e 9 e 8 β (e6,e 9) β (e7,e 9) p p p 2 p 2 e 0 v 4 v 4 p p 2 p p 2 e v 4 t t t () () (c) Fig. 5. Encoding nottion nd β e,e 4 = β e2,e 3 =0results in fesile network code. The pckets trnsmitted y the edges of the network re shown in Figure 5(c). Note tht ech pcket trnsmitted over the network is liner comintion of the originl pckets R = {p,p 2,,p h } generted y the source node s. Accordingly, for ech edge e E we define the glol encoding vector Γ e =[γ e γh e] F q h, tht cptures the reltion etween the pcket p e trnsmitted on edge e nd the originl pckets in R: h p e = p i γi e. (3) i= Note tht if e i is n outgoing edge of the source node s, then Γ e is equl to Γ ei =[β (e,e i) β (e2,e i) β (eh,e i)]. (4) For ny other edge e i E, Eqution (2) implies tht Γ ei = e M ei β e,e i Γ e. (5) We note tht for n edge e E ech component γ e i of Γ e is multivrite polynomil on the locl encoding coefficients {β e,e}. For exmple, for the network depicted in Figure 5() it holds tht Γ e3 = Γ e5 = Γ e6 =[β e,e 3 β e2,e 3 ]; Γ e4 = Γ e7 = Γ e8 =[β e,e 4 β e2,e 4 ]; Γ e9 = Γ e0 = Γ e =[β e,e 3 β e6,e 9 + β e,e 4 β e7,e 9 β e2,e 3 β e6,e 9 + β e2,e 4 β e7,e 9 ]. (6) 7

8 2.3 Coding dvntge Let N(G, s, T ) e multicst network. Recll tht its cpcity h is defined y Eqution (). Let h e the mximum mount of informtion tht cn e sent from the source s to the set of terminls T per communiction round without network coding, i.e., in communiction model in which ech intermedite node cn only forwrd incoming pckets. We define the coding dvntge s the rtio etween h nd h. The coding dvntge cptures the enefit of the network coding techniques for incresing the overll throughput of the network. Agrwl nd Chrikr [7] hve shown tht the coding log V dvntge of multicst networks cn e s lrge s Ω( log log V ) nd Ω( T ). For undirected networks, the coding dvntge is upper ounded y two [29]. For multiple unicst connections in directed networks it is esy to show tht the coding dvntge cn e s lrge s the numer of unicst pirs. 3 Algeric Frmework In this section, we present the lgorithmic frmework due to [25] for liner network coding in cyclic multicst networks nd estlish its connection to the min-cut-mxflow theorem. Let N(G, s, T ) e coding network nd let t e one of the terminls in T. We denote y E t = {e t,,e h t } e the set of incoming edges of terminl t. We define the h h mtrix M t s follows: M t = Γ e t Γ e 2 t Γ e h t, (7) Tht is, ech row of M t contins the glol encoding vector of one of the incoming edges e i t of t. We refer to M t s the trnsfer mtrix. The trnsfer mtrix cptures the reltion etween the originl pckets R nd the pckets received y the terminl node t T over its incoming edges. For exmple, for the network depicted in Figure 5() the trnsfer mtrix M t for the terminl t is equl to M t = β e,e 3 β e2,e 3 β e,e 3 β e6,e 9 + β e,e 4 β e7,e 9 β e2,e 3 β e6,e 9 + β e2,e 4 β e7,e 9 Similrly, the trnsfer mtrix M t2 for the terminl is equl to βe,e M t2 = 3 β e6,e 9 + β e,e 4 β e7,e 9 β e2,e 3 β e6,e 9 + β e2,e 4 β e7,e 9 β e,e 4 β e2,e 4. (8). (9) Terminl t cn decode the originl pckets in R if nd only if the trnsfer mtrix M t is of full rnk, or equivlently, the determinnt det(m t ) is not zero. Thus, the purpose of the network coding scheme is to find the ssignment of the coefficients {β e,e} tht results in full-rnk trnsfer mtrix M t for ech terminl t T. 8

9 For exmple, for the network depicted in Figure 5(), the determinnt of the trnsfer mtrix M t is equl to det(m t )=β e,e 3 (β e2,e 3 β e6,e 9 + β e2,e 4 β e7,e 9 ) β e2,e 3 (β e,e 3 β e6,e 9 + β e,e 4 β e7,e 9 ). (0) Similrly, the determinnt of the trnsfer mtrix M t2 is equl to det(m t2 )=β e2,e 4 (β e,e 3 β e6,e 9 + β e,e 4 β e7,e 9 ) β e,e 4 (β e2,e 3 β e6,e 9 + β e2,e 4 β e7,e 9 ). It is esy to verify tht the ssignment β e,e 3 = β e2,e 4 = β e6,e 9 = β e7,e 9 =nd β e,e 4 = β e2,e 3 =0results in non-zero vlues of the determinnts of oth mtrix, det(m t ) nd det(m t2 ). We oserve tht the determinnt det(m t ) of the trnsfer mtrix M t is multivrite polynomil with vriles {β e,e}. Let P = t T det(m t) e the product of the determinnts of the trnsfer mtrices for ech terminl t T. Clerly, if P is identiclly equl to zero, then there is no fesile network code for N(G, s, T ). However, it turns out tht if P is not identiclly equl to zero, then it is possile to find fesile ssignment of coefficients {β e,e}, provided tht the field F q is sufficiently lrge. Specificlly, the size q of F q must e lrger thn the mximum degree of P with respect to ny vrile β e,e. Figure 6 presents procedure, referred to s Procedure FINDSOLUTION, tht finds non-zero solution for multivrite polynomil P. The procedure receives, s input, non-zero polynomil P (x,x 2,,x n ) nd finite field F q = GF (q). The procedure itertively finds the ssignments x i = ζ i such tht P (ζ,ζ 2,,ζ n ) = 0. At itertion i, the procedure considers polynomil P i otined from P y sustituting x j = ζ j for j i. Then, we consider P i to e multivrite polynomil in x i+,,x n whose coefficients re (univrite) polynomils in x i. Next, we pick monomil P of P i nd consider its coefficient P (x i ). Since the size q of the finite field is lrger thn the mximum degree of vrile in P, there exists vlue ζ i F q, such tht P (ζ i ) is not zero. Hence, oth P xi=ζ i nd, in turn, P i xi=ζ i re non-zero polynomils. For exmple, suppose we would like to find solution for polynomil P (x,x 2,x 3 )= x x 2 2x 3 +x 2 x 2 2x 3 +x 2 x 2 2x 2 3 over F 3 = GF (3). We consider P (x,x 2,x 3 )=P (x,x 2,x 3 ) to e polynomil in x 2 nd x 3 whose coefficients re polynomils in x. Specificlly, we write P (x,x 2,x 3 )=(x + x 2 )x 2 2x 3 + x 2 x 2 2x 2 3 = P (x )x 2 2x 3 + P (x )x 2 2x 2 3, where P (x )=x + x 2 nd P (x )=x 2. Next, we select monomil P (x ) n find ζ F q such tht P (ζ ) = 0. Note tht ζ =would e good choice for F q = GF (3). Next, we set P 2 (x 2,x 3 )=P (x,x 2,x 3 ) x= =2x 2 2x 3 + x 2 2x 2 3 nd egin the next itertion. The following lemm shows the correctness of Procedure FINDSOLUTION. Lemm. Let P e non-zero polynomil in vriles x,x 2,,x n over F q = GF (q) nd let d e the mximum degree of P with respect to ny vrile. Let F q e finite () 9

10 Procedure FINDSOLUTION (P (x,x 2,,x n), F q) Input: P (x,x 2,,x n) - non-zero polynomil in vriles x,x 2,,x n; F q = GF (q) - finite field; Output: ζ,ζ 2,,ζ n GF (q) such tht P (ζ,ζ 2,,ζ n) = 0 P (x,x 2,,x n) P (x,x 2,,x n) 2 For ech i =to n do 3 Consider P i to e multivrite polynomil in x i+,,x n whose coefficients re univrite polynomils in F q[x i]. 4 Select monomil P of P i which is not identiclly equl to zero 5 Denote P (x i) e coefficient of P. 6 Choose ζ i F q such tht P (ζ i) = 0 7 Sustitute x i = ζ i in P i nd denote the resulting polynomil s P i+(x i+,,x n) 8 Return ζ,ζ 2,,ζ n Fig. 6. Procedure FINDSOLUTION field of size q such tht q>d. Then, Procedure FINDSOLUTION {P (x,x 2,,x n ), F q } returns ζ,ζ 2,,ζ n F q such tht P (ζ,ζ 2,,ζ n ) = 0. Proof: (sketch) We only need to show tht t ech itertion i, i n there exists ζ i F q such tht P (ζ i ) = 0. This follows from the fct tht P (x i ) is polynomil of mximum degree d, hence it hs t most d roots. Since F q includes q>delements, there must e t lest one element ζ i F q tht stisfies P (ζ i ) = 0. Theorems nd 2 (elow) show the reltion etween the lgeric properties of the trnsfer mtrices M t,t T, comintoril properties of G(V,E), nd the existence of fesile network code {β e,e} We egin with the nlysis of the unicst connections, i.e., the cse in which T contins single terminl node. Theorem. Let N(G, s, T ) e coding network, with T = {t}, nd h e the numer of pckets tht need to e delivered from s to t. Then, the following three conditions re equivlent.. There exists fesile network code for N(G, s, T ) nd h over GF (q) for some finite vlue of q; 2. The determinnt det(m t ) of the trnsfer mtrix M t is (multi-vrite) polynomil not identiclly equl to zero. 3. Every cut 3 tht seprtes s nd t in G(V,E) includes t lest h edges. Proof: (sketch) ) 2) Suppose tht there exists fesile network code {β e,e} for N(G, s, T ) nd h over GF (q). This implies tht det(m t ) is not zero for {β e,e}, 3 A cut in grph G(V,E) is prtition of the nodes of V into two susets V nd V \ V.We sy tht cut C =(V,V \ V ) seprtes nodes s nd t if s V nd t V \ V. 0

11 which, in turn implies tht det(m t ) s polynomil in {β e,e} is not identiclly equl to zero. 2) ) Lemm implies tht there exists non-zero ssignment of the locl encoding coefficients {β e,e} for N(G, s, T ) over sufficiently lrge field F q. This ssignment constitutes vlid network code for N(G, s, T ). ) 3) Suppose tht there exists fesile network code {β e,e} for N(G, s, T ) nd h over GF (q). By the wy of contrdiction, ssume tht there exists cut C tht seprtes the source s nd terminl t tht includes h <hedges. Let Γ,Γ 2,,Γ h e the set of glol encoding vectors for the edges tht elong to C. Then, for ech incoming edge e of t it holds tht the glol encoding vector of e is liner comintion of Γ,Γ 2,,Γ h. This, in turn, implies tht the glol encoding vectors tht correspond to incoming edges of t spn suspce of F h q of dimension h or smller. This implies tht t lest two rows of M t re linerly dependent nd, in turn, tht det(m t ) is identiclly equl to zero, resulting in contrdiction. 3) ) The Min-Cut Mx-Flow theorem implies tht there exist h edge-disjoint pths tht connect s nd t. Let {β e,e} e n ssignment of the locl encoding coefficients such tht β e (v,u),e(u,w) =only if oth e (v, u) nd e(u, w) elong to the sme pth. It is esy to verify tht this ssignment constitutes fesile network code. The next theorem extends these results for muticst connections. Theorem 2. Let N(G, s, T ) e multicst coding network nd let h e the numer of pckets tht need to e delivered from s to ll terminls in T. Then, the following three conditions re equivlent.. There exists fesile network code for N(G, s, T ) nd h over GF (q) for some finite vlue of q; 2. The product t T det(m t ) of the determinnts of the trnsfer mtrices is (multivrite) polynomil which is not identiclly equl to zero. 3. Every cut tht seprtes s nd t T in G(V,E) includes t lest h edges. Proof: (sketch) ) 2) Similr to the cse of unicst connections, the existence of fesile network code {β e,e} for N(G, s, T ) nd h over GF (q) implies tht the polynomil det(m t ) is not identiclly equl to zero for ech t T. 2) ) Lemm implies tht there exists non-zero ssignment of the locl encoding coefficients {β e,e} for N(G, s, T ) over sufficiently lrge field q. Since this ssignment stisfies det(m t ) = 0for ech t T, {β e,e} is fesile network code for N(G, s, T ). ) 3) Note tht fesile network code for the multicst connection N(G, s, T ) is lso fesile for ech unicst connection N(G, s, {t}), t T. Then, we cn use the sme rgument s in Theorem to show tht every cut etween s nd t includes t lest h edges. 3) 2) The Min-Cut Mx-Flow theorem implies tht for ech t T there exist h edge-disjoint pths tht connect s nd t. The rgument similr to tht used in Theorem implies tht for ech t T the polynomil det(m t ) is not identiclly equl to zero. This, in turn, implies tht t T det(m t) is lso not identiclly equl to zero.

12 Algorithm NETCODE (N(G, s, T ),h) Input: N(G, s, T ) - coding network; h - required numer of pckets; Output: A fesile network code {β e,e} for N(G, s, T ) For ech node v v in topologicl order 2 For ech outgoing edge e(v, u) of do 3 Write glol encoding vector Γ e of e s function of {β e,e} 4 For terminl t T do 5 Write the trnsfer mtrix M t of t s function of {β e,e} 6 Identify det(m t) s multi-vrite polynomil in {β e,e} 7 Identify Q t T det(mt) s multi-vrite polynomil in {β e,e} 8 Use Procedure FINDSOLUTION to find set of vlues of {β e,e} for which Q t T det(mt) = 0 9 Return {β e,e} Fig. 7. Algorithm NETCODE Theorem 2 implies tht the cpcity of undirected multicst network is equl to the minimum size of cut tht seprtes source s nd terminl t T. Algorithm NETCODE depicted in Figure 7 summrizes the steps required for finding fesile network code for multicst network. 4 Required field size One of the most importnt prmeters of network coding scheme is the minimum required size of finite field. The field size determines the numer of ville liner comintions. The numer of such comintions, nd, in turn, the required field size, is determined y the comintoril structure of the underlying communiction network. For exmple, consider the network depicted on Figure 8. Let Γ e,,γ e4 e the glol encoding vectors of edges e,,e 4. Note tht in this network ech pir of (v i,v j ) of the intermedite nodes is connected to terminl, hence ny two of the glol encoding vectors Γ e,,γ e4 must e linerly independent. Note lso tht with GF (2) there exist only three non-zero pirwise linerly independent vectors of size two: ( 0) T, (0 ) T, nd ( ) T, hence F 2 = GF (2) is insufficient for chieving network cpcity. However, it is possile to find network coding solution over GF (3) or lrger field. For exmple, over GF (3) the following glol encoding coefficients re fesile: ( 0) T, (0 ) T, ( ) T, nd (, 2) T. As mentioned in the previous section, fesile network code cn e found y identifying non-zero solution of multi-vrite polynomil P = t T det(m t). As shown in Lemm, such solution exists if the size q of the finite field F q is lrger thn 2

13 e e 2 s e 3 e 4 e 5 e 6 v v 2 v 3 v 4 t t 3 t 4 t 5 t 6 Fig. 8. A coding network the mximum degree of ny vrile β i of P. In this section, we show tht mximum degree of ny vrile in P = t T det(m t) is ounded y k = T, which implies tht field of size q k is sufficient for finding fesile solution to the prolem. In our model we ssumed tht ech edge of the network sends pcket only one time, when it receives pcket from ech incoming edge. In this section, for the purpose of nlysis, we ssume tht the communiction is performed in rounds s follows. Let e(v, u) E e n edge in the communiction network nd let M e e the set of prent edges of e. Then, t round i, edge e forwrds liner comintion of pckets received from edges in M e t round i. We ssume tht the originl pckets x,,x h re sent over the incoming edges of node s t round 0. For ech edge e E we denote y Γ i e the glol encoding coefficients of the pcket sent over edge e t round i. Intuitively, Γ i e cptures the new informtion delivered to edge e over pths of length i, while the glol encoding vector Γ e summrizes the informtion ville from ll communiction rounds: Γ e = d Γe, i (2) where d is the length of the longest pth in the network tht strts t node s. i=0 We define n E E mtrix T tht cptures the informtion trnsfer etween different communictions rounds. Mtrix T is referred to s n djcency mtrix. T (i, j) = βei,e j if e i is prent edge of e j 0 otherwise. (3) T: For exmple, the network depicted in Figure 9 hs the following djcency mtrix 3

14 e e 3 e 6 s e 5 t e 2 e 4 e 7 Fig. 9. A exmple of coding network 00β e,e 3 β e,e β e2,e 3 β e2,e β e3,e 5 β e3,e 6 0 T = β e4,e 7 (4) β e5,e We lso define h E mtrix A nd E vector B e for ech e E s follows: if i = j A(i, j) = 0 otherwise (5) We note tht if i = j B ei (j) = 0 otherwise For exmple, for edge e 3 in Figure 9 it holds tht (6) Γ i e = AT i B e. (7) 00β e,e 3 β e,e β e2,e 3 β e2,e Γe β e3,e 5 β e3,e = β e4,e 7 0 βe,e = β e5,e 7 0 β e2,e (8) Note for e 3 it holds tht Γe i 3 is non-zero only for i =. In contrst, edge e 7 hs two non-zero vectors, Γe 2 7 nd Γe 3 7. By sustituting Eqution (7) into Eqution (2) we otin: 4

15 Γ e = A (I + T + T T d )B e (9) We oserve tht mtrix T is nilpotent, 4 in prticulr it holds tht T d+ is zero mtrix. Thus, it holds tht Γ e = A (I + T + T 2 + )B e = A (I T ) B e. (20) Let t T e one of the terminls. We define E h mtrix B t s conctention of h glol encoding vectors Γ e tht correspond to the incoming edges of t. Then, the trnsfer mtrix M t cn e written s M t = A (I T ) B t. (2) The following theorem shows tht the determinnt of M t is equl to the determinnt of nother mtrix, M t, tht hs certin structure. Theorem 3. Let N(G, s, T ) e coding network nd let t e terminl in T. Then the determinnt of the trnsfer mtrix M t = A(I T ) B t for t is equl to det(m t ) = det(m t), where M A 0 t = I TBt T The proof of Theorem 3 involves sic lgeric mnipultions nd cn e found in [7]. The structure of mtrix M t implies tht the mximum degree of ny locl encoding coefficient β e,e in the multivrite polynomil M t, nd, in turn, M t is equl to one. As result, the degree of ech encoding coefficient β e,e in polynomil t T det M t is ounded y T. Thus, y Lemm, the field of size q T is sufficient for finding solution for the network coding prolem. We summrize our discussion y the following theorem. Theorem 4. Let N(G, s, T ) e multicst coding network. Then, there exists vlid network code {β e,e} for N on ny field GF (q), where q is greter thn the numer of terminls. 5 Rndom Network Coding One of the importnt properties of network coding for multicst networks is tht fesile network code cn e efficiently identified through rndomized lgorithm. A rndomized lgorithm chooses ech encoding coefficient t rndom with uniform distriution over sufficiently lrge field F q. To see why rndom lgorithm works, recll tht the min gol of the network coding lgorithm is to find set of encoding coefficients {β e,e} tht yield non-zero vlue of P = t T det M t. Theorem 5 elow ounds the proility of otining d solution s function of the field size. 4 A mtrix T clled nilpotent if there exists some positive integer n such tht T n is zero mtrix. 5

16 Theorem 5. (Schwrtz-Zippel) Let P (x,,x n ) e non-zero polynomil over F of totl degree t most d. Also, let r,,r n e set of i.i.d rndom vriles with uniform distriution over finite field F q of size q. Then, Pr(P (r,,r n ) = 0) d q The theorem cn e proven y induction on the numer of vriles. As discussed in the previous section, the degree of ech vrile β e,e in t T det M t is t most T. Let η e the totl numer of vriles. Thus, if we use finite field F q such tht q>2ηk, the proility of finding fesile solution is t lest 50%. In [7] tighter ound of ( T q )η on the proility of finding non-zero solution hs een shown. Rndom network coding hs mny dvntges in prcticl settings. In prticulr, it llows ech node in the network to choose suitle encoding coefficient in decentrlized mnner without prior coordintion with other nodes in the network. Rndom coding hs een used in severl prcticl implementtion schemes [9]. Rndom network coding cn lso e used to improve the network roustness to filures of network elements (nodes or edges) or to del with frequently chnging topologies. Let N(G, s, T ) e the originl coding network nd let N (G, s, T ) e the network topology resulting from filure of n edge or node in the network. Further, let P = t T det(m t) e the product of determinnts of the trnsfer mtrices in the N(G, s, T ) nd P = t T det(m t) e product of determinnts of trnsfer mtrices in N (G, s, T ). The network code {β e,e} tht cn e used in oth the originl network nd in the network resulting from the edge filure must e non-zero solution of the polynomil P P. Note tht degree of P P is ounded y 2η T hence for sufficiently lrge field size the rndom code cn e used for oth networks, provided tht fter edge filure the network stisfies the minimum cut condition. We conclude tht with the rndom network code, the resilience to filure cn e chieved y dding redundncy to the network to gurntee tht the min-cut condition is stisfied. Then, upon filure (filures) of n edge or node with high proility the sme network code cn e used. 6 Polynomil-Time Algorithm In this section we present polynomil time lgorithm for network code construction due to Jggi et l. [2]. The lgorithm receives, s input, coding network N(G, s, T ), numer of pckets h tht need to e delivered to ll terminls nd outputs fesile network code {β e,e} for N. In this section we ssume, without loss of generlity, tht the source node hs exctly h outgoing edges. Indeed, if this is not the cse, we cn lwys introduce new source s connected to the originl one with h edges. We ssume tht the outgoing edges of source node s re indexed y e,,e h. The lgorithm consists of two stges. In the first stge, the lgorithm finds, for ech terminl t T, set of edge-disjoint pths f t = {P, t,ph t } tht connect the source node s to tht terminl. This stge cn e implemented y using minimum-cost flow lgorithm (see e.g., [2]). Only edges tht elong to one of the pths {Pi t t T, i h} 6

17 Algorithm NETCODE2 (N(G, s, T ),h) Input: N(G, s, T ) - coding network; h - required numer of pckets; Output: A fesile network code {β e,e} for N(G, s, T ) For ech terminl t T do 2 Find h edge disjoint pths f t etween s nd t 3 For ech edge e i from i =to h do 4 Γ (e i)=[0 i,, 0 h i ] T 5 C i = {e,,e h } 6 B t = {Γ (e ),,Γ(e h )} 7 For ech node v v in topologicl order 8 For ech outgoing edge e(v, u) of do 9 Choose the vlues of locl encoding coefficients {β e,e e M e} such tht 0 For ech terminl t T (e) the mtrix B t formed from B t y sustituting Γ (P t (e)) y Γ (e) is linerly independent For ech terminl t T (e) 2 Sustitute P t (e) y e in C t 3 Sustitute Γ (P t (e)) y Γ (e) in B t 4 Return {β e,e} Fig. 0. Algorithm NETCODE2 re considered in the second stge of the lgorithm. Indeed, the minimum cut condition cn e stisfied y edges tht elong to {Pi t t T, i h}, hence ll other edges cn e omitted from the network. For ech edge e E we denote y T (e) the set of sink nodes such tht every sink t T includes edge e in one of the pths in f t. For ech terminl t T (e) we denote y P t (e) the predecessor of e on the pth Pi t f t tht e elongs to. 7

18 s s v v2 v3 v v2 v3 v4 v5 v4 v5 v6 v7 v6 v7 v8 v8 v9 v9 t t2 t t2 () () B t = s B t2 = v x v2 x2 x3 v v4 v5 v6 v7 v8 v9 t (c)? 0 s B t = 0? 0 B t2 = x x3 x2 0? v v2 x x2 x2 x3 v3 t2? 0 0? 0? 0 v4 v5 x x3 v6 v7 v8 v9 B t = t? 0? 0 0? (d) s B t2 = v x x2 x3 v2 x x2 x2 x3 v3 t2? 0? 0 0? v4 v5 x x + x2 x3 v6 v7 v8 v9 t (e) t2 Fig.. An exmple of lgorithm execution. 8

19 The gol of the second stge is to ssign the locl encoding coefficient β e,e for ech pir of edges (e(v, u),e (u, w)) in E. For this purpose, ll nodes in the network re visited in topologicl order. 5 Let (V,V ) e cut in G(V,E), where V includes suset of nodes in V lredy visited y the lgorithm nd y V the set of nodes tht were not visited yet. We refer to (V,V ) s the running cut of the lgorithm. At the eginning of the lgorithm, the running cut (V,V ) seprtes the source node s from the rest of the nodes in the grph. At the the end of the lgorithm, the running cut seprtes ll terminls in T from the rest of the network. At ny time during the lgorithm, ech pth Pi t, t T, i h, hs exctly one edge tht elongs to the cut. We refer to this edge s n ctive edge. For ech terminl t T we denote y C t the set of ctive edges of the disjoint pths {Pi t i h}. Also, we denote y B t, the h h mtrix whose columns re formed y the glol encoding vectors of edges in C t. The min invrint mintined y the lgorithm is tht the mtrix B t for ech t T must e invertile t every step of the lgorithm. In the eginning of the lgorithm we ssign the originl pckets R =(p,p 2,,p h ) to h outgoing edges of s. When the lgorithm completes, for ech terminl t T the set of ctive edges includes the incoming edges in t. Thus, if the invrint is mintined, then ech terminl will e le to decode the pckets in R. We refer to this lgorithm s Algorithm NETCODE2 nd present its forml description on Figure 0. An exmple of the lgorithm s execution is presented in Figure. Figures () nd () show the originl network nd two sets of disjoint pths f t nd f t2 tht connect the source node s with terminls t nd, respectively. Figures (c) shows the coding coefficients ssigned to edges (s, v ), (s, v 2 ), nd (s, v 3 ) fter node s hs een processed. Note tht this is one of severl possile ssignments of the coefficients nd tht it stisfies the invrint since oth B t nd B t2 re full-rnk mtrices. Nodes v, v 2, nd v 3 re processed in strightforwrd wy since ech of those nodes hs only one outgoing edge. Figure (d) shows the processing step for node v 4. This node hs one outgoing edge (v 4,v 6 ) nd needs to choose two encoding coefficients β = β (v,v 4),(v 4,v 6) nd β 2 = β (v2,v 4),(v 4,v 6). In order to stisfy the invrint, the vector β [ 0 0] T + β 2 [0 0] T must not elong to the two liner suspces, first defined y vectors [ 0 0] T nd [0 0] T, nd the second is defined y [0 0] T nd [0 0 ] T. Note tht if the finite field GF (2) is used, then the only fesile ssignment is β = β 2 =. For lrger field, there re severl possile ssignments of the encoding coefficients. Figure (e) demonstrtes the processing step for node v 5. The key step of the lgorithm is the selection of locl encoding coefficients {β e,e e M e } such tht the requirement of Line 0 of the lgorithm is stisfied. To descrie n efficient procedure for coefficient selection, we need the following nottion. Let e e n edge in E, let T (e) T e the set of destintion nodes tht depend on E nd let M e e the set of prent edges of e. Also, consider the step of the lgorithm efore edge e is processed nd let B t e the set of mtrices for ech t T (e). Since ech mtrix B t is of full rnk, there exists n inverse mtrix A t = Bt. For ech t T (e), let t e 5 A topologicl order is numering of the vertices of directed cyclic grph such tht every edge e(v, u) E stisfies v<u. 9

20 A t B t t Γ(P t (e)) = () A t B t t Γ(e) =? () Fig. 2. Dt structures row in A t tht stisfies t Γ (P t (e)) =, i.e., t is row in A t tht corresponds to column P t (e) in B t (see Figure 2()). We lso oserve tht if the column P t (e) in B t is sustituted y column Γ (e), the necessry nd sufficient condition for B t to remin of full rnk is tht t Γ (e) = 0(see Figure 2()). Thus, we need to select the locl encoding coefficients {β e,e e M e } such tht the vector Γ (e) = e M e β e,eγ (e ) will stisfy Γ (e) t =0for ech t T (e). The coding coefficients re selected through Procedure CODING depicted in Figure 3. The procedure receives, s input, n edge e for which the encoding coefficients {β e,e e M e } need to e determined. We denote y g the size of T (e) nd index terminls in T (e) y t,,,t g. Then, we denote e i = P ti (e). We lso denote y t i row in A t tht stisfies t i Γ (e i )=. The min ide is to construct sequence of vectors u,u 2,u g such tht for ll i, j with j i g it holds tht u i t j = 0. The lgorithm egins y setting u = Γ (e ). Then, for ech i, i g we perform the following opertions. First, if u i t i+ is not zero, then we set u i+ = u i. Otherwise, we note tht for ech α F q it holds tht (αu i + Γ (e i+ )) t i+) = 0. We lso note tht for ech j, j i it holds tht (αu i + Γ (e i+ )) t j )=0only if α = α j = Γ (ei+ ) t j u i. t j Thus, the set F q \{α j j i} is not empty. Thus, we choose α F q such tht α = α j for j, j i nd set u i+ = α u i +Γ (e i+ ) (y setting coefficients {β ej,e} ccordingly). By construction, it holds tht u i+ t j = 0for j i. 20

21 Procedure CODING (N(G, s, T ),h) Input: e E - n edge for which the encoding coefficients need to e determined; {Γ e e M e} - set of glol encoding vectors for prent edges of e; { t t T (e)} - set of normls Output: Coefficients {β e,e e M e} such tht P e M e β e,eγ (e ) stisfies Γ (e) t = 0 g T (e) 2 Index terminls in T (e) y t,,,t g 3 For i =to g do 4 e i = P ti (e) 5 β e i,e 0 6 β e,e 7 For i =to g do 8 u i P i j= β e j,eγ (e j ) 9 If u i t i+ =then 0 β e j,e 0 else 2 For ech j, j i do 3 α j Γ (ei+ ) t j u i t j 4 Choose α F q such tht α = α j for j, j i 5 For ech j, j i do 6 β e j,e β e j,e α 7 β e j,e 8 Return {β e,e e M e} Fig. 3. Procedure CODING 7 Network Coding in Undirected Networks So fr we hve considered coding networks represented y directed grphs. In ech edge e(v, u) of directed grph, the dt cn only flow in one direction, from v to u. In contrst, in undirected networks, tht dt cn e sent in oth directions, provided tht the totl mount of dt sent over n edge does not exceed its cpcity. Accordingly, to estlish multicst connection in n undirected network, we need first to determine the optimum orienttion of ech edge of the network. The edge orienttion is selected in such wy tht the resulting directed network hs mximum possile multicst cpcity. In some cses, to mximize the cpcity of the network n undirected edge needs to e sustituted y two directed edges with opposite orienttion. For exmple, consider the undirected network depicted in Figure 4(). Figure 4() shows possile orienttion of the edges in the network, resulting in directed network of cpcity one. The optiml orienttion is shown in Figure 4(c). In this orienttion, undirected edge (t, ) 2

22 s s s t t 0.5 t 0.5 () () (c) Fig. 4. An exmple of n undirected network. is sustituted y two i-directed edges of cpcity 0.5, resulting in directed multicst network of cpcity.5. For coding network N(G(V,E), s, T ) we define y λ(n) the minimum size of cut tht seprtes the source node s nd one of the terminls. As discussed ove, λ(n) determines the mximum rte of multicst coding networks over directed grphs. However, in undirected network λ(n) cn only serve s n upper ound on the trnsmission rte. For exmple, for the network N(G(V,E), s, {t, }) depicted in Figure 4() it holds tht λ(n) =2, while the mximum chievle multicst rte is equl to.5. A tighter upper ound cn e estlishing y considering the Steiner strength of the multicst network [30], defined s follows: Definition 2 (Steiner strength) Let N(G(V, E), s, T ) e multicst coding network over n undirected grph G, with the source s nd set T of the terminls. Let P e the set of ll possile prtitions of G, such tht ech prtition includes t lest one node in T {s}. Then, the Steiner strength η(n) of N is defined s η(n) =min p P E p ( p ), where p is the numer of components in p nd E p E is the set of edges tht connects different components of p. For exmple, the Steiner strength of the network depicted in Figure 4() is equl to.5, which is tight ound for this prticulr cse. It turns out tht η(n) determines the mximum rte of the multicst trnsmission in the specil cse in which the set T {s} includes ll nodes in the network. The following theorem is due to Li nd Li [29]. Theorem 6. Let N(G(V,E), s, T ) e multicst coding network over n undirected grph G. Let π(n) e the mximum rte of multicst connection using trditionl methods (Steiner tree pcking) nd let χ(n) e the mximum rte chievle y using the network coding pproch. Then for the cse of V = T {s} it holds tht λ(n) π(n) =χ(n) λ(n). 2 22

23 Otherwise it holds tht λ(n) π(n) χ(n) λ(n). 2 Theorem 6 shows tht the mximum coding dvntge of network coding in undirected networks is upper ounded y two. This is in contrst to the cse of directed networks, where the coding dvntge cn e significntly higher. 8 Prcticl Implementtion As discussed in the previous sections, network coding techniques cn offer significnt enefits in terms of incresing throughput, minimizing dely, nd reducing energy consumption. However, the implementtion of network coding in rel networks incurs certin communiction nd computtionl overhed. As result, thorough cost-enefit nlysis needs to e performed to evlute the pplicility of the technique for ny given network setting. For exmple, it is highly unlikely tht the network coding technique will e implemented t core network routers due to the high rte of dt trnsmission t the network core. Thus, finding the network setting tht cn enefit from the network coding technique is chllenging prolem y itself. In this section, we discuss the prcticl implementtion of the network coding technique proposed y Chou et l. [0,34]. The principles of this implementtion were dopted y mny susequent studies [6] nd y rel commercil systems such s Microsoft Avlnche. The content distriution system includes single informtion source tht genertes strem of its tht need to e delivered to ll terminls. The its re comined into symols. Ech symol typiclly includes 8 or 6 its nd represents n element of finite field GF (q). The symols, in turn, re comined into pckets, such tht pcket p i is comprised of N symols σ,σ i 2, i,σn i. The pckets, in turn, re comined into genertions, ech genertion includes h pckets. In typicl settings, the vlues of h cn vry etween 20 nd 00. Figure 5 demonstrtes the process of creting symols nd pckets from the it strem. The key ide of the proposed scheme is to mix the pckets tht elong to the sme genertion, the resulting pcket is then sid to elong to the sme genertion. Further, when new pcket is generted, the encoding is performed over individul symols rther thn the whole pcket. With this scheme, the locl encoding coefficients elong to the sme field s the symols, i.e., GF (q). For exmple, suppose tht two pckets p i nd p j re comined into new pcket p l with locl encoding coefficients β GF (q) nd β 2 GF (q). Then, for y N the y s symol of p l is liner comintion of the y s symol of p i nd y s symol of p j, i.e., σ l y = β σ i y + β 2 σ j y. The scheme is sed on the rndom liner coding technique descried in Section 5 tht chooses locl encoding coefficients uniformly over GF (q) (excluding the zero). For ech pcket sent over the network, it holds tht its symols re liner comintions of the corresponding symols of the originl pckets, i.e., the pckets generted y 23

24 Bit strem Symols σ σ σn σ 2 σ2 2 σn 2 2 σ h+ σ2 h+ σ h+ N Pckets 2 p p 2 p h+ Genertion Genertion 2 Fig. 5. Pcketiztion process: forming symols from its nd pckets from symols. the source node. Thus, ech pcket p l cn e ssocited with glol encoding vector Γ l = {γ l,,γ l h } tht cptures the dependency etween the symols of p l nd the symols of the originl pckets. Specificlly, symol σ l y of p l cn e expressed s σ l y = h γi l σy i i= Another key ide of this scheme is to ttch the glol encoding coefficients to the pcket. These coefficients re essentil for the terminl node to e le to decode the originl pckets. This method is well-suited for settings with rndom locl encoding coefficients. The lyout of the pckets is shown in Figure 6. Note tht ech pcket lso includes its genertion numer. Attching glol encoding incurs certin communiction overhed. The size of the overhed depends on the size of the underlying finite field. Indeed, the numer of its needed to store the glol encoding vectors is equl to h q. In the prcticl cse considered in Chou et l. [0], h is equl to 50 nd the field size q is equl to two ytes, resulting in totl overhed of 00 ytes for pckets. With pcket size of 400 ytes, the overhed constitutes pproximtely 6% of totl size of the pcket. If the field size is reduced to one yte, then the overhed is decresed to just 3% of the pcket size. Note tht the destintion node will e le to decode the originl pckets fter it receives h or more linerly independent pckets tht elong to the sme genertion. With rndom network coding, the proility of receiving linerly independent pckets is high, even if some of the pckets re lost. The mjor dvntge of the proposed scheme is tht it does not require ny knowledge of the networking topology nd efficiently hndles dynmic network chnges, e.g., due to link filures. The opertion of n intermedite network node is shown in Figure 7. The node receives, vi its incoming links, pckets tht elong to different genertions. The pck- 24

25 Genertion numer Glol encoding vectors 0 0 Originl vectors γ g γ g 2 γ g 3 σ 3 σ 2 σ σ g σ 3 2 σ 2 2 σ 2 σ g 2 σ 3 N σ 2 N σ N σ g N Fig. 6. Structure of the pcket. ets re then stored in the uffer, nd sorted ccording to their genertion numer. At ny given time, for ech genertion, the uffer contins set of linerly independent pckets. This is ccomplished y discrding ny pcket tht elongs to the spn of the pckets lredy in the uffer. A new pcket trnsmitted y the node is formed y rndom liner comintion of the pckets tht elong to the current genertion. The importnt design decision of the encoding node is the flushing policy. The flushing policy determines when new genertion pckets ecomes the current genertion. There re severl flushing policies tht cn e considered. One possiility is to chnge the current genertion s soon s pcket tht elongs to new genertion rrives vi some of the incoming links. An lterntive policy is to chnge genertion when ll incoming links receive pckets tht elong to the new genertion. The performnce of different flushing policies cn e evluted y simultion or n experimentl study. 8. Peer-to-peer networks Network coding cn lso enefit peer-to-peer networks tht distriute lrge files (e.g., movies) mong lrge numer of users [4]. The file is typiclly prtitioned into lrge numer, sy k, of chunks, ech chunk is disseminted throughout the network in seprte pcket. A trget node collects k or more pckets from its neighors nd tries to reconstruct the file. To fcilitte the reconstruction process, the source node typiclly distriutes prity check pckets, generted y using n efficient ersure correction code 25

26 Interfce Rndom liner comintion Interfce Interfce 3 uffer 3 3 Fig. 7. Opertion of n intermedite network node such s Digitl Fountin [5]. With this pproch, the trget node cn decode the originl file from ny k different pckets out of n>kpckets sent y the source node. 6 With the network coding technique ech intermedite node forwrds liner comintions of the received pckets to its neighors (see Fig. 8). This pproch significntly increses the proility of the successful decoding of the file t the trget node. For exmple, consider the network depicted on Fig. 9. In this exmple, the file is split into two chunks, nd. The source node then dds prity check pcket, c, such tht ny two of the pckets,, nd c re sufficient for reconstructing the originl file. Fig. 9() demonstrtes trditionl pproch in which ech intermedite node forwrds pckets,, nd c to its neighors. Since there is no centrlized control nd the intermedite nodes do not hve ny knowledge of the glol network topology, the routing decision is done t rndom. Suppose tht two trget nodes, t nd would like to reconstruct the file. Note tht node t otins two originl pckets, nd. However, node receives two copies of the sme pcket (), which re not sufficient for successful decoding opertion. Fig. 9() shows network coding pproch in which the intermedite nodes generte new pckets y rndomly comining the pckets received over their incoming edges. With this pproch, the proility tht ech destintion node receives two linerly independent pckets, nd hence the proility of successful decoding opertion is significntly higher. 8.2 Wireless networks The distinct property of wireless medium is the ility of sender node to rodcst pckets to ll neighoring nodes tht lie within the trnsmission rnge. In Section. we presented n exmple, tht shows tht the network coding technique llows us to 6 Some efficient coding schemes require slightly more thn k pckets to decode the file. 26

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