Minimal Coding Network With Combinatorial Structure For Instantaneous Recovery From Edge Failures


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1 Mnmal Codng Network Wth Combnatoral Structure For Instantaneous Recovery From Edge Falures Ashly Joseph 1, Mr.M.Sadsh Sendl 2, Dr.S.Karthk 3 1 Fnal Year ME CSE Student Department of Computer Scence Engneerng SNS College of Technology,Combatore. ashlyj89gmal.com 2 Professor,Department of Computer Scence Engneerng SNS College of Technology,Combatore. 3 HOD,Department of CSE, SNS College of Technology,Combatore, AbstractProtectng nodes aganst lnk falures n communcaton networks(both uncast and multcast) s essental to ncrease robustness, accessblty, and relablty of data transmsson. The use of network codng algorthm offers, establshng relable uncast connectons across a communcaton network wth nonunform edge capabltes. The Proposed System addresses the task of multcast communcaton usng usng the concept of network codng n the presence of passve eavesdroppers and actve jammers. Despte the complexty ntroduced by dstrbuted network codng, recently, t has been shown that ths rate can also be acheved for multcastng to several snks provded that the ntermedate nodes are allowed to reencode the nformaton they receve. Index Terms Instantaneous recovery, network codng, relable communcaton, uncast,multcast. I.INTRODUCTION In today's practcal communcaton networks such as the Internet, endtoend nformaton delvery s performed by routng,.e., by havng ntermedate nodes store and forward packets. Note though, routng does not encompass all operatons that can be performed at a node.the ssue of relably transportng data from a sngle orgnal source to multple nterested recpents, partcularly when the recever set s large, s one of growng promnence. Recently, the noton of network codng arses as a very mportant promsng generalzaton of routng.network codng refers to a scheme where a node s allowed to generate output data by encodng (.e., computng certan functons of) ts receved data. Thus, network codng allows nformaton to be mxed", n contrast to the tradtonal routng approach where each node smply forwards receved data. Potental advantages of ths generalty of network codng over routng are many fold: resource effcency, computatonal effcency, and robustness to network dynamcs.network codng s potentally applcable to many forms of network communcatons. Up to now, the best understood scenaro where network codng offers unque advantages s multcastng n a communcaton network. A sgnfcant effort has been devoted to mprovng the reslence of communcaton networks to falures and ncreasng ther survvablty. Edge falures are frequent n communcaton networks due to the nherent vulnerablty of the communcaton nfrastructure. Wth the dramatc ncrease n data transmsson rates, even a sngle falure may result n vast data losses and cause major servce dsruptons for many users. Accordngly, there s a sgnfcant nterest n mprovng network recovery mechansms that enable a contnuous flow of data from the source to the destnaton wth mnmal damage n the event of a falure. Edge falures may occur due to several reasons, such as physcal damage, msconfguraton, or a human error. Networks are typcally desgned to be wthstand aganst a sngle edge falure. Indeed, protecton from multple falures ncurs hgh costs n terms of network utlzaton, whch s usually not justfed by the rare occurrence of such falures. we consdered the problem of establshng relable uncast (snglesource sngledestnaton) connectons across a communcaton 674
2 network wth nonunform edge capactes. Our goal s to provde nstantaneous recovery from sngle edge falures. Here focus on two cases of practcal nterest: 1) Backup protecton of a sngle flow that can be splt up to two subflows; and 2) shared backup protecton of two uncast flows. The nstantaneous recovery mechansms ensure a contnuous flow of data from the source to the destnaton node, wth no nterrupton or data loss n the event of a falure. Such mechansms elmnate the need of retransmsson and reroutng. Instantaneous recovery s typcally acheved by addng redundant packets and by routng packets over multple paths n a way that ensures that the destnaton node can recover the data t needs form the receved packets. Instantaneous recovery s typcally acheved by addng redundant packets and by routng packets over multple paths n a way that ensures that the destnaton node can recover. Fg.2. DverstyCodng method for h=2. The dversty codng technque extends the dedcated path protecton scheme by usng multple dsjont paths for sendng the data. Ths technque can be used for protectng a sngle uncast connecton or for shared backup protecton of two or more uncast connectons (Fg.2). In the case of a sngle uncast connecton, the nformaton flow s splt nto two or more subflows, and each subflow uses a separate path to reach the destnaton. Fg.1. Dedcated path protecton method There are several technques to acheve nstantaneous recovery. A standard approach s to use the 1+1 dedcated path protecton scheme. Ths approach requres provsonng of two dsjont paths p1 and p2 between s and t. Each packet generated by the source node s sent over both paths, p1 and p2(fg. 1). In the case of a sngle edge falure, at least one of the paths remans operatonal, hence the destnaton node wll be able to receve the data wthout nterrupton. Wth ths scheme, both p1 and p2 must be of capacty at least h. Sngle edge falure, at least one of the paths remans operatonal, hence the destnaton node wll be able to receve the data wthout nterrupton. Wth ths scheme, both p1 and p2 must be of capacty at least h. Whle the dedcated path protecton scheme s smple and easy to mplement, t ncurs hgh communcaton overhead due to the need to transmt two copes of each packet. The dedcated path protecton scheme s smple and easy to mplement, t ncurs hgh communcaton overhead due to the need to transmt two copes of each packet. In addton, t requres the exstence of two dsjont paths, each of capacty h between s and t. some examples shows a dversty codng scheme that uses three dsjont pathsp1, P2, and P3 of capacty one between s and t to protect a uncast flow of rate two packets per communcaton round. In ths example, the flow s splt nto two subflows that use paths P1 and P2 to send data to the destnaton, whle path P3 serves as a shared backup protecton for these subflows. The network codng approach generalzes both the dedcated protecton and the dversty codng approach. In partcular, t enables shared backup protecton of multple uncast connectons and does not requre multple lnkdsjont paths between the source and destnaton nodes. The network, edges (s,v1) and (s,v2)are of capacty two, whle all other edges are of unt capacty. Our goal s to establsh a uncast connecton that delvers two packets from s to t per communcaton round. Thus ths paper ncludes a robust network code for both uncast and multcast networks can be establshed through the standard network codng algorthm.by dvdng the smple network nto blocks,t s possble to route the packets effcently n the case when h=2 and also when h>2.some mprovements on the exstng network codng algorthm s done for effcently handlng the above mentoned functonaltes. We also address the problem of effcent allocaton of network resources for a robust codng network. 675
3 by removng an edge or decreasng the capacty of an edge s no longer feasble. It s easy to verfy that the capacty of each edge n a mnmal network s bounded by h. Related work The network codng technque has been ntroduced n the semnal paper of Ahlswede et al. [3]. Intal work on network codng has focused on multcast connectons. It was shown n [3] that the maxmum rate of a multcast network s equal to the mnmum total capacty of a cut that separates the source from a termnal. Ths maxmum rate can be acheved by usng lnear network codes [8]. Ho et al. [9] showed that the maxmum rate can be acheved by usng random lnear network codes. Jagg et al. [10] proposed a determnstc polynomaltme algorthm for fndng feasble network codes n multcast networks. They showed that f the network has a suffcent capacty to recover from each falure scenaro (e.g., by reroutng) then nstantaneous recovery from each falure scenaro can be acheved by employng lnear network codes. A. Our contrbuton In ths paper we propose effcent algorthms for constructon of robust network codes over small fnte felds. We consder two major cases. In the frst case, we assume that all edges of the network have unform capacty, whle n the second case the capacty of network edges can vary. For the frst case we present an effcent network codng algorthm that dentfes a robust network code over a small feld. The algorthm takes advantage of specal propertes of Maxmum Rank Dstance (MRD) codes [13]. For the second case, we focus on settngs n whch the source node needs to delver two packets per tme unt to all termnals. We show that n ths case, a specal topologcal propertes of robust codng networks can be exploted for constructng a network code over a small fnte feld. II MODEL A. Multcast Network A multcast network N that uses a drected acyclc graph G(V,E) to send data from source s to a set T of k destnaton nodes {t1,..., tk} V. The data s delvered n packets. We assume that each packet s an element of a fnte feld Fq = GF(q). We also assume that the data exchange s performed n rounds, such that each edge e E can transmt c(e) packets per communcaton round. We assume that c(e) s an nteger number and refer to t as the capacty of edge e. At each communcaton round, the source node needs to transmt h packets R = (p1, p2,..., ph) T from the source node s V to each destnaton node t T. We refer to h as the rate of the multcast connecton. It was shown n [3] and [8] that the maxmum rate of the network,.e., the maxmum number of packets that can be sent from the source s to a set T of termnals per tme unt, s equal to the mnmum capacty of a cut that separates the source s from a termnal t T. Accordngly, we say that a multcast network N s feasble f any cut that separates s and a termnal t T has at least h edges. We say that a codng network N s mnmal f any network formed from N B. Codng Networks For clarty of presentaton, we defne an auxlary graph ˆG(V,A) formed by the network graph G(V,E) by substtutng each edge e E by c(e) parallel arcs that have the same tal and head nodes as e; each arc can transmt one packet per communcaton round. We denote by A(e) A the set of arcs that correspond to edge e. In what follows we only refer to packets sent at the current communcaton round. The packets sent n the subsequent rounds are handled n a smlar manner. C. Robust Codng Networks Snce a faled edge e cannot transmt packets, we assume that the encodng functon fa of each arc a A(e) s dentcally equal to zero,.e., fa 0. To guarantee nstantaneous recovery, t s suffcent to ensure that for each edge falure there exsts a set of h lnearly ndependent packets receved by t. We dstngush between two types of robust networks codes. In strongly robust network codes the local encodng coeffcents of all arcs n A reman the same, except for the arcs A(e) that correspond to the faled edge e whch are assgned zero encodng coeffcents. In weakly robust network codes, the arcs that are located downstream of the faled edge e are allowed to change ther encodng coeffcents, whle all the encodng coeffcents that correspond to other edges must reman the same. III. STRONGLY ROBUST CODES FOR NETWORKS WITH UNIFORM CAPACITIES All edges are of the network have unform capacty c,.e., each edge can send exactly c packets per tme unt. We present an effcent algorthm that can construct a robust network code over a fnte feld of sze O(k). We observe that wthout loss of generalty, we can assume that the capacty of each edge s one unt. Indeed, a feasble network code for unt capacty edges can be extended nto the case n whch the capacty of each edges s equal to c by combnng c communcaton rounds nto a sngle round. In [10] t was shown that communcaton at rate h wth nstantaneous recovery from sngle edge falures s possble f and only f for each edge e E, t holds that the network G (V,E ) formed from G(V,E) by removng e, contans at least h edgedsjont paths from source s to each termnal node t T. Ths mples that a necessary and suffcent condton for the feasblty of network N s the exstence of h+1 edgedsjont paths between s and each t T. Our approach can be summarzed as follows. Frst, we generate a specal party check packet, referred to as ph+1. Ths packet s a lnear combnaton of the orgnal packets and s constructed as descrbed n Secton IIIA. Then, we use a standard network codng algorthm due to Jagg et al. [10] for sendng ˆR = {p1, p2,..., ph, ph+1} 676
4 IV. NETWORK WITH NONUNIFORM CAPACITIES packets from s to T. The standard algorthm wll treat the packets n ˆR as generated by ndependent random processes. The algorthm ensures that n the normal network condtons, each destnaton node receves h + 1 ndependent lnear combnaton of the packets n ˆR. The followng lemma shows that after a sngle edge falure, each destnaton node receves at least h lnearly ndependent combnatons of packets n ˆR. Lemma 1: Upon an edge falure, each termnal t T receves at least h lnear combnatons of packets n ˆR. Proof: Snce we assume that all edges are of unt capacty, each edge n the network can be represented by a sngle arc. For each arc a A we defne the global encodng vector that captures the relaton between the packet pa transmtted on arc a and the orgnal packets n ˆR: h + 1 = 1 p = p γ e. e Let t be a termnal n T. We defne the transfer matrx Mt that captures the relaton between the orgnal packets R and the packets receved by the termnal node t T over ts ncomng edges. The dfferent network edges have dfferent capactes. We focus on a specal case n whch only two packets need to be delvered from the source to all termnals at each communcaton round. The desgn of robust network codes for h = 2 n the context of uncast connectons has been studed n [11]. In ths work, we buld on the results of [22] for constructng a robust network code for multcast connectons. Let G(V,E) be a mnmum robust codng network,.e, a feasble robust network such that the removal of an edge or a reducton n the capacty of an edge results n a volaton of ts feasblty. Note that the capacty of any edge e E s at most two. For each termnal t T let Gt(Vt,Et) to be a subgraph of G(V,E) that contans a mnmum codng network wth respect to termnal t. That s, Gt(Vt,Et) only contans edges of G(V,E) that are necessary to guarantee the condtons defned by Equaton 1 for termnal t. Furthermore, any reducton of the capacty of edges n Gt(Vt,Et) wll result n a volaton of ths condton for at least one of the (s, t) cuts. V. CONCLUSION A. Creatng party check packet Lemma 1 mples that n the event of any sngle edge falure, each termnal node receves at least h ndependent lnear combnatons of the packets n {p1, p2,..., ph, ph+1}. Snce packet ph+1 s a lnear combnaton of h orgnal packets R = {p1, p2,..., ph}, each destnaton nodes receves, n fact, h lnear combnatons of R. Accordngly, our goal s to construct packet ph+1 n such a way that each destnaton node receves h ndependent lnear combnatons of R. Ths wll allow each destnaton node to decode the orgnal packets. For clarty of presentaton we frst focus on the case of h = 2. In ths case we have two orgnal packets, p1 and p2, and one party check packet p3 = γ1 p1+γ2 p2. Suppose that a termnal t T receves two lnearly ndependent combnatons of p1, p2, and p3. B. General case We turn to consder a more general case of h > 2. Snce we need to only recover from a sngle falure, we need to fnd a (h + 1, h) MRD code. For ths case, we can use the party check matrx [13]: 2 h H = α, α, Kα,1 over the feld GF(qh+1), wth h p h + 1 = α p = 1 We summarze our results by the followng theorem: Theorem 1: The proposed scheme acheves an nstantaneous recovery from any sngle edge falure. Proof: Follows drectly from the propertes of MRD codes [13]. Ths paper addressed the problem of constructng robust network codes for uncast networks,.e., codes that enable nstantaneous recovery from sngle edge falures. There s need to obtan the same results for multcast connectons also. But the networks where the achevable rates obtaned by codng at ntermedate nodes are arbtrarly larger s not allowed. An effcent network codng algorthm based on the MRD codes that requres a small fnte feld O(k)can be used. For the case of nonunform capactes, we focused on a specal case of h = 2 and showed that t s also possble to construct a robust code over a small feld. Future research ncludes the constructon of network codes wth nonunform capactes and transmsson of more than two packets per communcaton round. REFERENCES [1] W.D.Grover.,Ho. lugmeshbased Survvable Transport Networks:Optons and Strateges ntented for Optcal,MPLS,SONET and ATM Networkng.Prentce Hall,New York,NY,USA,2003 [2] E.Ayanoglu,C.L.I.R.D.Gtln And J.E.Mazo.Dversty codng for transparent selfhealng and faulttolerent communcaton networks. IEEE Transactons on communcatons,41,(11):1677, 1686,1993. [3] R.Ahlswede,N.Ca,S.Y.R.L,and R.w.Yeung.Network Informaton Flow,IEEE Transactons on Informaton Theory,46(4): ,
5 [4] R.Kotter and M.Medard.An Algebrac Approach to Network Codng.IEEE/ACM Transactons On Networkng, 11(5): ,2003. [5] T.Ho,M.Medara, and R.Koetter.An Informaton Theoretc vew of Network Management.IEEE Transactons on Informaton Theory,51(4),Aprl [6] Desmond S Lun,Murel Medard,Ralf Kotter, and Mchelle Effros.Further results on codng for Relable communcaton over packet networks. In IEEE Internatonal Symposum on Informaton Theory (ISIT 05),2005. [7] Desmond S.Lun,Murel Medard, and Mchelle Effros, On codng for relable communcaton Over packet networks.in Proc.42 nd Annual Allerton Conference on Communcaton, Control,And Computng,Sept Oct.2004,Invted,2004. [8] S.Y.R.L,R.W.Yeung,and N.Ca.Lnear Network Codng, IEEE Transactons on Informaton Theory,49(2): ,2003 [9] T.Ho,R.Kotter,M.Medard,D.Karger, and M.Effros. The Benefts of Codng Over Routng n a Randomzed Settng, In Proceedngs of the IEEE Internatonal Symposum on Informaton Theory,2003. [10] S. Jagg, P. Sanders, P. A. Chou, M. Effros, S. Egner, K. Jan, and L. Tolhuzen. Polynomal Tme Algorthms for Multcast Network Code Constructon. To appear n IEEE Transactons on Informaton Theory, [11] S. Jagg, M. Langberg, S. Katt, T. Ho, D. Katab, R.C.N.M. Medard, and M. Effros. Reslent Network Codng n the Presence of Byzantne Adversares. IEEE Transactons on Informaton Theory, 54(6): , June [12] D. Slva, F.R. Kschschang, and R. Koetter. A Rank Metrc Approach to Error Control I Random Network Codng. Informaton Theory, IEEE Transactons on, 54(9): , Sept [13] E. Gabduln. Theory of codes wth maxmum rank dstance. Problems of Informaton Transmsson, 21(1):3 14, July
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