Efficient Network Coding Algorithms For Dynamic Networks

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1 Efficient Network Coding Algorithm For Dynmic Network Mohmmd Ad R. Chudhry, Slim Y. El Rouyhe, nd Alex Sprinton Deprtment of Electricl nd Computer Engineering Tex A&M Unierity, College Sttion, Tex emil:{mdch, lim, Atrct Network coding i new prdigm tht llow the intermedite node in network to crete new pcket y comining the pcket receied on their incoming edge. The centrl prolem in the deign of network coding cheme i to ign locl encoding coefficient for the intermedite edge in wy tht llow eery terminl node to decode the pcket generted y the ource node. The min ppliction of the network coding technique include content ditriution, peer-topeer network, nd wirele d-hoc network. Such network re chrcterized y highly dynmic et of uer nd frequent topologicl chnge. In thi pper we focu on the deign of efficient multict network code for dynmic network. Firt, we conider the prolem of mintining the feiility of gien network code upon chnge in the network topology or the ddition of new uer. Our gol i to minimize the numer of encoding coefficient tht need to e modified to keep the network code feile. Second, we preent new network coding lgorithm tht ue pth-ed coding ignment to efficiently hndle frequent chnge in the network topology nd the multict group. I. INTRODUCTION Network coding [1] i new promiing technique with mny potentil ppliction in networking nd ditriuted computing. The centrl prolem in network coding i to deign n efficient lgorithm tht ign the locl encoding coefficient in wy tht llow ech terminl node to decode the required pcket. Network coding h utntil potentil for improing throughput, reliility, nd routne of wired nd wirele network. Currently, the min ppliction of the network coding technique re in the re of content-ditriution network [2], peer-to-peer network [3], nd wirele network [4]. Such network typiclly he highly dynmic topologie nd frequently chnging et of uer. Accordingly, there i need to deelop network coding lgorithm tht cn efficiently hndle frequent network chnge. In thi pper, we focu on mintining the feiility of gien network code upon n ddition of new uer or chnge in the network topology. Our gol i to minimize the numer of encoding coefficient tht need to e modified to keep the network code feile. A mller numer of required chnge in the encoding coefficient will llow the coding network to Thi work w upported in prt y NSF grnt ECCS nd y the Qtr Telecomm (Qtel), Doh, Qtr. djut for new uer or chnge in the network topology more efficiently nd lo reduce the diruption for exiting uer. We lo preent new lgorithm tht ue pth-ed ignment of encoding coefficient to contruct feile network code. While the pth-ed pproch typiclly require lrger field ize thn the tndrd lgorithm uch the one due to Jggi et l [5], it cn hndle network chnge through ft nd efficient procedure, with the limited numer of chnge required in the network code. Relted Work. Network coding reerch h een initited y the eminl pper y Ahlwede et l. [1], nd ince then ttrcted ignificnt interet from the reerch community. Koetter nd Médrd [6] deeloped n lgeric frmework for network coding. Thi frmework w ued y Ho et l. [7] to how tht liner network code cn e efficiently contructed through rndomized lgorithm. Jggi et l. [5] propoed determinitic polynomil-time lgorithm for finding feile network code in multict network. Network coding lgorithm for dynmic network he een tudied in reference [8], [9], nd [10]. Ho et l. [8] howed tht the network coding pproch proide utntil enefit in dynmiclly rying enironment. Zho nd Médrd [9] conidered the prolem of modifying network topology in wy tht minimize the numer of required code rerrngement. Their lgorithm ue liner progrmming pproch nd relie on cot function tht penlize edge whoe ddition might require chnge in the network code. Ho et l. [10] preented frmework for network mngement ed on the network coding pproch nd conidered prolem of minimizing the numer of network code required for hndling ll ingle edge filure. II. NETWORK MODEL We conider multict network N tht ue directed cyclic grph G(V,E), with ertex et V nd edge et E, to end dt from ource to et T of terminl node. The numer of terminl node i denoted y k, i.e., k = T. The dt i deliered in pcket, ech pcket i n element of finite field F q = GF (q). We ume tht the communiction i performed in round, uch tht ech edge e E trnmit one pcket per round. Note tht thi doe not reult in ny lo of generlity ince edge of higher cpcity cn e utituted y multiple prllel edge. At ech communiction round, the /09/$25.00 (c)2009 IEEE 1

2 t t 3 t 3 t 1 t 4 t 1 t 4 t 1 () () (c) t 4 Fig. 1. for ˆN. () Originl network N nd feile network code C for N. () A new network ˆN tht include new terminl node. (c) A new network code Ĉ ource node need to trnmit h pcket R =(p 1,p 2,...,p h ) T from the ource node V to ech detintion node t T. We refer to h the rte of the multict connection. It w hown in [1] nd [11] tht the mximum rte of the network, i.e., the mximum numer of pcket tht cn e ent from the ource to et T of terminl per time unit, i equl to the minimum cpcity of cut tht eprte the ource from terminl t T. Accordingly, we y tht multict network N i feile if ny cut tht eprte nd terminl t T h t let h edge. Without lo of generlity, we ume tht the ource node h exctly h incoming edge, ech incoming edge trnmit one of the originl pcket in R. We lo ume tht ech terminl t T lo h h incoming edge nd no outgoing edge. For ech edge e E we denote y p e the pcket trnmitted on tht edge. Let e(, u) e n edge in E nd let M e e the et of incoming edge of it til node, M e = {(w, ) (w, ) E)}. Then, we ocite with ech pir of edge {(e,e) e M e } locl encoding coefficient β e,e F q. The locl encoding coefficient of the edge tht elong to M e determine the pcket p e trnmitted on edge e function of pcket trnmitted on the incoming edge of e. Specificlly, the pcket p e i equl to p e = e M e β e,e p e, (1) where ll opertion re performed oer F q. We y tht edge e i djcent to edge e if the hed node of e i identicl to the til node of e. We denote y S the et of the djcent pir of edge in the network. We refer to the et of locl encoding coefficient C = {β e,e (e,e) S} network code for N. Note tht ech pcket trnmitted oer the network i liner comintion of the originl pcket p 1,p 2,...,p h generted y the ource node. Accordingly, for ech edge e E we define the glol encoding ector Γ e =(γ e 1,...,γ e h )T F h q, tht cpture the reltion etween the pcket p e trnmitted on edge e nd the originl pcket in R: h p e = γi e p i. (2) i=1 For ech terminl t in T we define the trnfer mtrix M t tht cpture the reltion etween the originl pcket R nd the pcket receied y the terminl node t T oer it incoming edge. The mtrix M t i defined follow: M t = [ Γ e 1 t Γ e 2 t... Γ e h t ], (3) where E t = {e 1 t,...,e h t } i the et of incoming edge of terminl t. Our gol i to find et of network coding coefficient C = {β e,e (e,e) S} tht llow ech terminl to decode the originl pcket R from the pcket otined on it incoming edge. Thi cn e ccomplihed only if the mtrix M t i full-rnk mtrix for ech terminl t T. The ignment of C tht tifie thi condition i referred to feile network code. III. NETWORK CODING IN DYNAMIC NETWORKS In thi pper we focu on network coding lgorithm for dynmic network. When new uer join the network or when the network undergoe topologicl chnge (e.g., due to filure of n edge), the exiting network code might no longer e feile. For intnce, conider the exmple depicted in Figure 1. The originl network N together with the correponding network code C i depicted in Figure 1(). The network delier two pcket, nd, to four terminl, t 1,,t 3, nd t 4. Suppoe tht new terminl join the network. In order to mintin the feiility requirement, two new edge, ( 6, ) nd (, ) he een dded to the network. The reulting multict network ˆN i depicted in Figure 1(). Since the glol encoding coefficient of edge ( 4, 6 ) nd ( 5, ) re linerly dependent, nd ince node 6 nd 7 cn only forwrd their incoming pcket, we need to chnge ome of the encoding coefficient for edge in S o the new terminl will e le to decode the pcket ent y ource. A trightforwrd pproch i to compute new network code Ĉ for ˆN from crtch. Howeer, with thi pproch the 2

3 locl encoding coefficient for ll pir of edge in S might chnge. In prctice, thi cn incur utntil oerhed, ocited with determining, ditriuting, nd chnging the encoding coefficient for lrge numer of network node. Howeer, in mny ce, we only need to chnge mll numer of encoding coefficient to mke the new network feile. For exmple, Figure 1(c) depict new network code Ĉ formed from C y chnging ingle encoding coefficient (β (2, 4),( 4, 6)). Let N e n originl multict network oer grph G(V,E) withets of djcent edge nd correponding feile network code C = {β e,e}. LetˆN e modified multict network oer grph Ĝ( ˆV,Ê), V ˆV, with et of djcent edge Ŝ nd feile network code Ĉ. Then, the et of edge with modified locl encoding coefficient i defined S = { (e,e) (e,e) S, (e,e) Ŝ,β(e,e) ˆβ(e },e). We refer to S the numer of chnge of locl encoding coefficient required y Ĉ. For exmple, the network code Ĉ depicted in Figure 1 require chnge of only one coefficient. Dynmic network cn lo experience frequent chnge in the network topology. In prticulr, ome of the network edge my fil nd new edge nd node my e dded to the network to mintin it feiility. For exmple, conider the network N with the feile network code C depicted in Figure 2(), nd ume tht edge (, 7 ) h filed. The modified network topology i depicted in Figure 2(). In the new topology, node nd t 3 re connected to node 6 nd. Howeer, with the network code C, the glol encoding ector of the edge ( 4, 6 ) nd ( 4, ) re linerly dependent. The new feile network code Ĉ for ˆN cn e contructed y modifying the locl encoding coefficient β (2, 4),( 4, 6) for the pir of edge (, 4 ) nd ( 4, 6 ). IV. ADDING A NEW USER TO THE MULTICAST GROUP In thi ection we focu on the cenrio in which new uer join the exiting multict group T. Firt, we etlih lower nd upper ound on the numer of modified network coding coefficient. Let N e n originl multict network oer the grph G(V,E), with et of terminl node T nd correponding feile network code C = {β e,e}. Let e new terminl node, nd ˆN e modified multict network oer grph Ĝ( ˆV,Ê). We egin with the lower ound. Lemm 1: Let M e the trnfer mtrix for with repect to the code C. Then, feile network code Ĉ for ˆN require t let h rnk(m ) chnge of the locl encoding coefficient in C. Proof: It i ufficient to how tht chnge in one encoding coefficient cn incree the rnk of the trnfer mtrix y t mot one. Suppoe tht we chnge the locl encoding coefficient for pir (e,e) of djcent edge from β e,e to β e,e.letm nd M e the trnfer mtrice efore nd fter the chnge, repectiely. Then M cn e expreed : M = M +(β e,e β e,e)(γ e M e ), (4) where Γ e i the glol encoding coefficient for edge e (efore the chnge) nd M e i 1 h mtrix tht tie the pcket ent on edge e nd the pcket receied y the incoming edge of. Note tht the rnk of Γ e M e i t mot one. The udditiity property of the rnk function implie tht the rnk of M i ounded y the rnk of M plu one. We proceed to etlih n upper ound. Lemm 2: Let ˆP e et of h dijoint pth etween nd in Ĝ( ˆV,Ê). LetŜ e et tht contin pir of djcent edge in S uch tht ech pir (e,e) Ŝ elong to one of the pth in ˆP. Then, feile network code cn e contructed y chnging the locl encoding coefficient of the edge tht elong to Ŝ. Note tht the lemm implie tht it i ufficient to only chnge Ŝ encoding coefficient. Proof: Firt, we define new network code Ĉ follow: β e,e +Δβ e,e if (e,e) S nd (e,e) ˆP ˆβ e,e = β e,e if (e,e) S nd (e,e) / ˆP Δβ e,e otherwie. (5) We how tht it i poile to ign the lue of {Δβ e,e} uch tht the reulting network code Ĉ i feile. We how tht for ech t ˆT the determinnt det( ˆM t ) of the trnfer mtrix ˆM t i not identiclly equl to zero with repect to the new network code Ĉ. To thi end, we utitute the lue of coefficient in {β e,e} ccording to their ignment in C nd lee {Δβ e,e} to e rile. Then, for ech t ˆT the determinnt of the trnfer mtrix ˆM t i multirite polynomil in {Δβ e,e}. We oere tht for ech t T thi polynomil i not identiclly equl to zero. Indeed, with the ignment of Δβ e,e = 0 for ech pir of djcent edge (e,e) in ˆP the trnfer mtrix ˆM t for ech t T i identicl to the trnfer mtrix M t for the me terminl under code C.For terminl the multirite polynomil det(m ) will include n dditie term (e,e) ˆP Δβ e,e, hence thi polynomil i lo not identiclly equl to zero. Therefore, for ufficiently lrge field (q ˆT ) it i poile to elect the lue of {Δβ e,e} uch tht the trnfer mtrix for ech terminl re inertile. The modified network code cn e contructed through imple modifiction of the lgorithm due to Jggi et. l. [5]. In ddition, rndom lgorithm for the network code ignment cn e ued. Figure 3 how n intnce of the dynmic network coding prolem for which the ound etlihed y Lemm 2 i tight. The initil network include node, 1,...,. The exiting network code C i hown in Figure 3(). While the preented coding network i not miniml, ll redundnt edge cn e jutified y dding dditionl terminl. When new terminl 3

4 t 1 t 3 t 4 t 1 t 3 t 4 () () Fig. 2. () Originl network N nd feile network code C for N. () A new network ˆN contructed fter filure of edge (, 7 ) with new network code Ĉ for ˆN () () Fig. 3. An intnce of coding network. The rc how locl encoding coefficient etween djcent edge. () Originl network code. () Modified network code. join the network, it need to receie t let one liner comintion oer pth {, 1, 3, 4, 6, 7,, } tht include three pir of djcent edge. Figure 3() how modified network code tht require t let three chnge. V. PATH-BASED ASSIGNMENT OF ENCODING COEFFICIENTS In thi ection we preent pth-ed pproch for the ignment of the locl encoding coefficient. We firt preent our pproch in the context of ttic network nd then dicu it opertion in dynmic network. Let N e the originl multict network tht need to delier h pcket per communiction round from ource node to et of terminl node t i T oer communiction grph G(V,E). The firt tep of our lgorithm i to determine, for ech terminl t i T,etF i = {P1,...,P i h i } of h edgedijoint pth etween nd t i. Then, we ocite ech terminl t i with the encoding prmeter ϕ i. For eery two pir of djcent edge e =(u, ), e =(, w) we define uet T (e,e) of T which include ll terminl t i T for which it hold tht oth e =(u, ) nd e =(, w) elong to 1 the me pth in F i. Then, the locl encoding coefficient β e,e i defined follow: β e,e = t i:t i T (e,e ) ϕ i. (6) Tht i, the locl encoding coefficient β e,e i defined to e the um of the encoding prmeter ϕ i tht correpond to the terminl in T (e,e). We demontrte our pproch through the following exmple. Conider the communiction network preented in Figure 4(). The network include ource node nd et of terminl T = {t 1,,t 3 }. Firt, we identify three dijoint pth to ech terminl in T (ee Figure 4()-(d)). Then, we ocite ech terminl t i with n encoding prmeter ϕ i. Then, for ech pir of djcent edge e nd e we ign the correponding locl encoding coefficient β e,e to e the um of the coefficient tht correpond to the terminl which include oth edge (, u) nd (u, w) on one of their dijoint pth from the ource. Figure 4(e) how the locl encoding coefficient igned y our cheme for the network depicted in Figure 4(). Our gol i to elect the lue of ϕ i tht yield feile network code. For the network depicted in Figure 4() it i ey to erify tht ϕ 1 = ϕ 2 = ϕ 3 =1yield feile olution oer F 3. Uing the rgument imilr to tht ued in Lemm 2, we cn how for ufficiently lrge field ize F q there lwy exit feile ignment of the encoding coefficient. Moreoer, when new terminl join the network, it i poile to find feile lue of ϕ proided tht the totl numer of ctie terminl i ounded. Howeer, finding feile ignment of ϕ require full knowledge of the network topology. In wht follow we preent n ignment of the encoding coefficient tht doe not require full knowledge. VI. CODES BASED ON PRIME NUMBERS Let F i = {P1,...,P i h i } e the et of h edge-dijoint pth etween nd t i determined in the firt phe of the lgorithm. 4

5 () () (c) (d) 1 3 ϕ 4 3 ϕ ϕ ϕ 2 + ϕ 3 2 ϕ 1 ϕ 3 ϕ p (4, 6 ) = ϕ 3 p (1, 4 ) + p (5, 7 ) = ϕ 3 p (2, 5 ) + p (8, ) =(ϕ 2 + ϕ 3 ) p (6, ) + (ϕ 1 + ϕ 2 ) p (2, 4 ) ϕ 1 p (3, 5 ) ϕ 1 p (7, ) (e) (f) (g) Fig. 4. () An intnce to the network coding prolem with three terminl t 1,,ndt 3 ; () Three dijoint pth to terminl t 1 ; (c) Three dijoint pth to terminl ; (d) Three dijoint pth to terminl t 3 ; (e) Locl encoding coefficient for node 4, 5 nd. For exmple the pcket p (4, 6 ) ent on edge ( 4, 6 ) i equl to p (4, 6 ) = ϕ 3 p (1, 4 ) +(ϕ 1 + ϕ 2 ) p (2, 4 ). Let lo θ j denote the j th prime numer (i.e., θ 1 =2,θ 2 = 3,θ 3 =5,...). Letπ k e the product of the firt k numer, i.e., π k = k θ j. j=1 Define the encoding prmeter ϕ i for terminl t i follow: ϕ i = π k /θ i = k j=1,j i θ j. (7) Figure 5 how n exmple of network code ed on prime numer for network with two terminl. In Theorem 3 elow we how tht the ignment of the locl encoding coefficient gien y Eqution (7) i feile oer ufficiently lrge primry field. Theorem 3: Let F q e primry field uch tht q>(2 π k ) E, where E i the numer of edge in network. Then, the pth-ed ignment of the locl encoding coefficient oer F q gien y Eqution (7) reult in feile network code. Proof: Let N(G(V,E),,T) e coding network nd let {β ej,e i } e et of locl encoding coefficient for the djcent edge in N. LetF denote flow compoed of h edge-dijoint pth {P 1,...,P h } etween nd terminl in T.Wey tht pir (e i,e j ) of djcent edge elong to F if there exit pth P i in F uch tht oth edge, e i nd e j elong to P i. We denote y g(f) = (e i,e j) djcent, (e i,e j) F β ei,e j the gin of F. Then, ed on the reult in [12, Th. 3], the network code {β ej,e i } i feile, if nd only if, for ech terminl t i T it hold tht B i = F:F i flow from to t i l i (F) g(f) 0, (8) where l i (F) {1, 1}, nd the ummtion i done oer ll flow etween nd t i. Next, we will how tht the choice of the ϕ i decried in Eq. (7) will led to B i 0, for ll t i T. For eery flow F from to the terminl node t i with non-zero gin, one of the two following condition hold: 1) Flow F i equl to flow F i. In thi ce g(f) will include n dditie term ϕ N i = (π k /θ i ) N for ome integer N. It ey to erify tht ll other multiple term in g(f) will he multiplictie fctor θ i. 2) Flow F nd F i re not identicl. In thi ce, there exit djcent edge e =(u, ), e =(, w) uch tht 5

6 1 3 p 1 p2 p p 1 +3p 2 2p 2 +3p 3 4p 1 +12p 2 +9p 3 t 1 () t 1 () Fig. 5. () Locl encoding coefficient, ϕ 1 = θ 2 =3nd ϕ 2 = θ 1 =2. () Glol encoding coefficient. oth e nd e elong to pth in F, ut there i no pth in F i tht include oth e nd e. In thi ce, eery dditie term in g(f) i diiile y θ i. We conclude tht B i will include one dditie term (π k /θ i ) N for ome integer N, while ret of the dditie term will include θ i multiplictie fctor. The Fundmentl Theorem of Arithmetic implie tht B i i not equl to zero when ddition i conidered oer the integer. We note tht thi lo hold if the ize q of the finite field i greter thn B i. Next, we how tht (k π k ) E i n upper ound on B i.thi will imply tht for ny prime field F q uch tht q (2 π k ) E, B i h non-zero lue. Firt, we oere tht ech encoding coefficient i ounded y π k. Second, ny flow F from to t contin t mot E edge, o the mximl gin g(f) of F i t mot (π k ) E.We lo note tht the numer of (, t) flow i ounded y 2 E. Thu, y Eqution 8, (2π k ) E i n upper ound on the lue of B i nd the theorem follow. Our lgorithm require finite field of ize (2π k ) E.Itcn e hown tht the product of k prime numer i ounded y 4 k. Then, the required ize of the field i equl to 2 (2k+1) E. Ech element in uch field cn e repreented y (2k +1) E it. Hence, the required pcket ize i liner in the numer of edge in the network nd the numer of terminl. VII. CONCLUSION AND FUTURE WORK REFERENCES [1] R. Ahlwede, N. Ci, S.-Y. R. Li, nd R. W. Yeung. Network Informtion Flow. IEEE Trnction on Informtion Theory, 46(4): , [2] C. Gkntidi nd P. R. Rodriguez. Network coding for lrge cle content ditriution. INFOCOM th Annul Joint Conference of the IEEE Computer nd Communiction Societie. Proceeding IEEE, 4: ol. 4, Mrch [3] Me Wng nd Bochun Li. Network coding in lie peer-to-peer treming. Multimedi, IEEE Trnction on, 9(8): , Dec [4] S. Ktti, H. Rhul, W. Hu, D. Kti, M. Medrd, nd J. Crowcroft. Xor in the ir: Prcticl wirele network coding. Networking, IEEE/ACM Trnction on, 16(3): , June [5] S. Jggi, P. Snder, P. A. Chou, M. Effro, S. Egner, K. Jin, nd L. M. G. M. Tolhuizen. Polynomil Time Algorithm for Multict Network Code Contruction. IEEE Trnction on Informtion Theory, 51(6): , [6] R. Koetter nd M. Medrd. An Algeric Approch to Network Coding. IEEE/ACM Trnction on Networking, 11(5): , [7] T. Ho, R. Koetter, M. Medrd, D. Krger, nd M. Effro. The Benefit of Coding oer Routing in Rndomized Setting. In Proceeding of the IEEE Interntionl Sympoium on Informtion Theory, [8] T. Ho, M. Medrd, nd R. Koetter. An informtion-theoretic iew of network mngement. Informtion Theory, IEEE Trnction on, 51(4): , April [9] Fng Zho nd Muriel Medrd. Online network coding for the dynmic multict prolem. Informtion Theory, 2006 IEEE Interntionl Sympoium on, pge , July [10] T. Ho, B. Leong, M. Medrd, R. Koetter, Yu-Hn Chng, nd M. Effro. On the utility of network coding in dynmic enironment. Wirele Ad- Hoc Network, 2004 Interntionl Workhop on, pge , My-3 June [11] S.-Y. R. Li, R. W. Yeung, nd N. Ci. Liner Network Coding. IEEE Trnction on Informtion Theory, 49(2): , [12] T. Ho. Networking from Network Coding Perpectie. PhD thei, MIT, My Thi pper focue on the deign of efficient network coding lgorithm for dynmic network etting. Firt, we etlih upper nd lower ound on the numer of required chnge. Second, we preent pth-ed lgorithm for igning locl network coding coefficient. The lgorithm efficiently hndle dynmic chnge in the multict group nd the network topology. 6