Robust Network Coding Using Diversity through Backup Flows

Transcription

1 Robus Nework Coding Using Diversi hrough ackup Flows Hossein ahramgiri Farshad Lahoui Wireless Mulimedia Communicaions Laboraor School of ECE, College of Engineering Universi of ehran Absrac- We inroduce algorihms o design robus nework codes in he presence of link failures for mulicas in a direced acclic nework Robusness is achieved hrough diversi provided b he nework links flows, while he imum mulicas rae due o -flow min-cu bound is mainained he proposed scheme is a receiver-based robus nework coding, which explois he diversi due o he possible gap of he specific receivers min-cu wih respec o he nework mulicas capaci An improved version of his scheme guaranees mulicas capaci for a cerain level of failures n a mulicas session, failure of a flow ma no necessaril reduce he capaci of he nework as oher useful branches wihin he nework could sill faciliae back up roues (flows) from he source o he sinks We inroduce a scheme o emplo backup flows in addiion o he main flows o mulicas daa a imum rae h, when possible n a limiing case, he scheme guaranees he rae h, for all link failure paerns, which do no decrease he imum rae below h Here, he number of link failures ma in general exceed he refined singleon bound ndex erms- Nework coding, join nework-channel coding, link failure, mulicas NRODUCON Nework coding is a field of much recen ineres in informaion heor aiming a designing codes for nework nodes in order o improve he performance b devising an enhanced communicaion ssems nework laer n paricular, nework coding has been iniiall moivaed for improving nework hroughpu in mulicas applicaions his aricle deals wih he design of robus nework codes in he presence of link failures, while achieving he mulicas rae bound due o -flow min-cu heorem he proposed scheme can be viewed as a join nework channel coding approach in a direced acclic nework Ahlswede e al inroduced a new arge for mulicas in a nework b evaluaing he capaci showing ha i can be achieved onl b nework coding [] Nex, Li e al showed ha he capaci can be achieved even b linear coding finie alphabe size [] Koeer Medard demonsraed an algebraic scheme for nework coding sudied informaion flows [3] Ho e al proposed a simple rom scheme for nework coding [4], in which he probabili of success is enhanced b increasing finie field size he approach is decenralized consequenl more proper for neworks wih unknown or ime varian opologies Jaggi e al in [5] inroduced a cenralized algorihm o find he coding soluion in polnomial ime he inroduced deerminisic linear nework coding (DLF), a compleel deerminisic algorihm, rom linear nework coding (RLF), which is rom in he middle sages Fragouli e al in [6], develop a disribued deerminisic mehod for nework code design b ranslaing he problem as a graph coloring problem More recenl, Jabbarihagh Lahoui presened a decenralized approach based on learning o design nework codes [7] he scheme ma be viewed as a smar rom search which faciliaes he code design especiall for large neworks small field sizes n he perspecive of separae nework channel coding, ergodic errors in differen links are combaed b channel coding echniques in he daa link laer Some join nework-channel coding schemes have been presened in he lieraure for robus nework code design hese schemes comprise wo caegories n he firs caegor, link failure is considered Koeer Medard in [3], showed ha here is a nework code which can suppor a deermined rae, k, for all failure paerns ha do no reduce he capaci below his value n [5], Jaggi e al inroduced a nework code for his case, in which all such failure paerns are o be recognized checked during code design hus, he scheme is complicaed for pracical siuaions Ho e al in [8], showed he benefis of rom nework coding for robusness in he presence of nework changes link failures Chou e al in [9] suggesed a pracical nework coding algorihm, based on he rom scheme of [4], presening he concep of daa packe generaions, aiming o increase he robusness o packe loss, dela variaions in nework opolog n [0], receiver-based nework-wide recover schemes are analzed nework managemen is quanified b some resuls Also, i is shown ha here is no a single coding scheme for all recoverable single-link failures a leas he receivers mus reac o he failure model n [], El Rouaheb e al invesigae nework codes ha faciliae insananeous recover from single edge failure for unicas connecions n [], using a rade off of mulicas rae diversi, a coding scheme referred o as robus nework coding, RNC(h,k), is proposed where for a nework wih mulicas capaci, h, a mulicas rae of k is guaraneed in presence of a mos h-k flow failures he scheme is flow-based hus he design complexi is manageable he second caegor is iniiaed b he work of Cai Yeung [3], in which nework error correcing codes are presened he arge is o correc (or deec) a deermined number of edge errors he work is elaboraed in [4][5] he Hamming, Singleon Gilber-Varshamov bounds for nework codes are sudied Zhang in [6] Yang e al in [7] independenl presen a refined singleon bound his bound is sinkdependen acknowledging he possible difference in he min-cu value of differen sinks Oher works in his caegor includes [8]-[] n a recen work, Koeer Kschischang sud error correcion in rom nework codes [] he code is designed a he ransmier, based on vecor spaces which are preserved hrough he nework, he srucure of he nework does no affec he design his work falls in he firs caegor n his paper, we presen algorihms o design robus nework codes in he presence of link failures, using inrinsic diversi provided b he nework branches flows he diversi exploied does no reduce he achievable mulicas rae below -flow min-cu bound We inroduce a receiver-based robus nework

2 coding scheme, RNC(h+,h), which explois he diversi due o he possible gap of he specific receivers unicas capaciies (receiver min-cu) wih respec o he nework mulicas capaci An improved version of RNC, RNC(h+,h), is also presened o guaranee he ransmission rae h, for a cerain level of failures wihin he nework o design he local encoding vecors, we emplo he scheme of [5] he idea of using he diversi hrough he beond-mulicas-capaci-flows, is similar o he refinemen in singleon bound in he works of he [6] [7] he proposed algorihms could be viewed as exploiing a muli-user diversi wihin he nework for robusness in he presence of link failure Failure of a roue (flow) ma no necessaril reduce he mulicas capaci of he nework due o exisence of oher useful branches wihin he nework Once carefull considered in he nework code design process, such branches in fac could faciliae back up flows from he source o he sinks in face of link failures We inroduce RNC3, o emplo backup flows in addiion o he main flows o ransmi he daa a rae h, when possible n a limiing case, he scheme guaranees he ransmission rae h, for all link failure paerns, which do no decrease he imum ransmission rae below h As eviden from he above discussion, in he firs caegor he proposed scheme, he design objecive is o faciliae sufficien inac flows o mainain he capaci following a failure, as opposed o he number of edge errors erasures he refined singleon bound, which is he crierion in he second caegor Noe ha, for he proposed scheme, he number of failures in acive links of receiver, wih he min-cu value of h ma in general, be much more han h -h; which is he ulimae number of olerable failures for nework error correcing codes based on achieving refined singleon bound he res of his paper is organized as follows: n secion, we describe he ssem in deail presen our moivaions for robusness hrough diversi wihin he nework n he nex hree secions, RNC, RNC RNC3 are inroduced, respecivel, heir complexi field size are analzed Finall he schemes are compared discussed in secion V n an error-free nework, one can design a nework code o achieve he ransmission rae h = cmulicas n his case, i is enough o find h flows from he ransmier o each receiver he res of he links, which are no presen in hese flows are no required for his session As we shall demonsrae, i is possible o emplo he res of he links for diversi hus, he robusness of he ransmission is increased b emploing wha we refer o as he redundan flows in a mulicas session Such flows ma be idenified b considering he -flow of each receiver denoed b h, for receiver, which is greaer han or a leas equal o h Figure shows a simple example he node s mulicass daa o he nodes z We can define he following flows for he receivers: f = [( s, ), (, )] f = [( s, u), ( u, w), ( w, x), ( x, )] f z = [( s, ), (, w), ( w, x), ( x, z)] f z = [( s, u), ( u, z)] f z = [( s, v), ( v, z)] 3 We can see ha Maxflow()=, Maxflow(z)=3 he mulicas capaci of he session is equal o he minimum -flow, or hus, he sender can send wo linear combinaions of he wo smbols in each ime uni u here is an exra pah from he source s o z, which can be emploed o increase robusness he source can send anoher linear combinaion of he wo smbols in he hird pah, such ha an wo of he hree linear combinaions are independen hus, receiving wo combinaions is enough o exrac he daa he degree of diversi is equal o h h for he receiver n he example of figure, h= We can send wo combinaions of source smbols o, hree combinaions o z ased on he concep of redundan flows, wo robus nework coding schemes are presened in secions V MOVAONS A Ssem Descripion he nework can be represened b a graph G=(V,E), which is assumed o be direced acclic, wih V E represening he se of nodes he se of links or edges, respecivel All links have uni capaci; hus a link wih a posiive ineger capaci of c, is shown b c parallel links he link e=(u,v,i) is he i-h link from u=ail(e) o v=head(e), which ma be shown b e=(u,v) where i= A node v V has a se of inpus links, denoed b Γ (v) he sequence of links f = [( s, b ),( a, b ),, ( a, b ),( a, )] is a k k k flow of he receiver if for i=,,k, b i = a Here, we assume he i nework is snchronous We consider mulicas session in he nework wih a ransmier, s V a se of receivers, V he nework encouners link failures, in which he failed links can no ransmi an daa in he failure ime he objecive is o increase he robusness of he nework agains link failures Robusness hrough Diversi of Redundan Flows Consider a mulicas session in a nework, modeled b graph G he capaci of he session is compued, appling -flow mincu heorem, as follows [] c = Min{ Maxflow( )} () mulicas Figure - A simple example for mulicas: he node s mulicass daa o z C Robusness hrough ackup Flows Consider a nework wih mulicas capaci of h n he classic nework coding, one could find h flows from he source o each receiver design he code for he deermined se of flows he oher links of he nework are no used in he ransmission scheme Here, we wish o emplo hese links, as backup links, for increased robusness is possible ha he original se of flows fail o ransmi he rae h in he even of link failure, bu one can sill find a leas h flows from he inac links backup links, which can suppor he ransmission rae of h For a receiver, i is possible ha more han h -h original flows fail, bu one can find h or more flows from s o n he example nework of figure, failure of he edges (s,) (u,z) fails he flows f z f z, respecivel u here are sill wo flows from he node s o he node :

3 z f = [( s, ), (, w), ( w, x), ( x, )] f 3 he flow f is presen, bu no considered, in he error-free nework hus, he second approach o robus nework code design, elaboraed in secion V, is o use backup links for consrucing backup flows o replace hose ha possibl fail wihin he nework ROUS NEWORK CODNG HROUGH DVERSY OF REDUNDAN FLOWS n his secion, we inroduce a robus nework code, RNC(h+,h), designed o uilize redundan flows for improved performance in he presence of link failure as described in secion he algorihm is based on a cenralized approach, is inspired b he work of Jaggi e al in [5] However, he ideas echniques presened for code design could also be cas ino oher design algorihms he scheme is cenralized consiss of wo seps he firs sep is o find he flows for each of he receivers hus, we find Φ = { f,, f }, he se of h h flows from he source o he receiver is possible ha he flows of wo receivers have common edges, bu he flows of one receiver are edge-disjoin An edge ha is presen in a leas one flow is referred o as an acive edge n he second sep, we consruc a scheme o ransmi he rae h from he source o each of he receivers emplo redundan flows he idea is o ransmi h linear combinaion of h independen source smbols in h flows of he receiver, such ha ever h smbol subse is enough o exrac he source smbols he source s generaes h independen smbols, X = { x,, x h }, which is o be ransmied o he receivers in Ever acive edge carries a smbol, which is a linear combinaion of he elemens of X he global encoding vecor, b(e), is a vecor of lengh h, which consiss of he combinaion coefficiens of he smbol in he edge e he global encoding vecor of an acive edge, e, is a linear combinaion of he global encoding vecors of he edges in he se Γ ( ail( he combinaion coefficiens are deermined b local encoding vecor, m e, as follows b ( e) = m ( e ) b( e ) () e e Γ ( ail ( he se C is defined for receiver, consising of he las considered edges in h edge-disjoin flows, ransmiing h smbols o his receiver he se of he global encoding vecors of hese edges, corresponding o he ransmiing smbols, is denoed b he se mus have he proper ha each of is h-elemen subses forms an independen se of vecors We refer o his proper as h-independence o iniialize he algorihm, we add a new node s o he nework connec i o s wih h = ( h ) direced edges from s o s herefore, we have he new graph G = ( V, E ) wih V = V { s } E = E {( s, s, ),,( s, s, h )} he se se of added edges, {( s, s, ),, ( s, s, )}, C is iniialized b he h he algorihm considers he acive edges, one b one he edge e is considered, onl afer all he edges in Γ ( ail( have been processed he graph is acclic, so all he links will be considered b his sraeg he edge e replaces Γ (e) in C for each receiver Q(e), where Q(e) denoes he se of all receivers, for which he edge e is in one of heir flows Also, Γ (e) is he previous edge of he edge e in he flow of Afer replacemen, h-independence is verified for each updaed he algorihm concludes when he edges ending a he receivers are processed o process edge e, m e is seleced roml b(e) is compued using equaion Nex, he resuling is verified for h-independence for each Q( e) he following heorem shows he condiion for he presence of a leas one valid local encoding vecor heorem : A local encoding vecor for each ever nework edge exiss in RNC(h+,h), if he size of he emploed finie field F saisfies he following condiion F (3) Proof- We show ha in each edge, local encoding vecor, m e, can be compued, or equivalenl here is a linear combinaion of he global encoding vecors of he acive edges in Γ ( ail( such ha h- independence is saisfied afer replacing Γ (e) b e in C, for all of he receivers in Q(e) For he h-independence verificaion of he receiver, independence ess are done hus, he number of independence ess, denoed b N, is a mos Q ( e) We prove he heorem b inducion on he number of independence ess denoed b n he vecor m e is of size Q(e) We order roml he receivers in Q(e) as,, For n=, consider he receiver Q( e) Q( e) () C, an h elemen subse of C, which includes he edge e Consider e i as a Q(e) elemen vecor of zeros wih onl a a posiion i f m e, =e, we have b(e,)=b( Γ ( e ) ), which saisfies he () () independence of he updaed, relaed o C Now, assume ha m e,n saisfies he independence for n ess Assume ha he (n+)-h es is one of he ess of he h-independence of he receiver k for he subse ( n+ ) ( +) C n his sep, he independence of n is esed f m e,n saisfies ( n+) independence of k, hen m e,n+ = m e,n ; Oherwise, he resuled b e,n+ ( n+ ) k is in he subspace of \ b ( Γ ( hus, we consider m k e,n+ = am e,n + e k, which saisfies independence for he (n+)-h es, for all a F We now check he independence for each of he previous ess Each es fails for a mos one value of a Since, if he es fails for wo combinaions of u= a m e,n + e k(n+) v= a m e,n + e k, he es fails for u-v=(a - a ) m e,n, which conradics he inducion hpohesis hus, if he parameer a is seleced from a field of size F n +, hen here exiss a linear combinaion ha passes all he ess As we know, N, hus he equaion 3 is obained n his scheme, m e is seleced roml he following heorem deermines a bound for he probabili of success in designing m e in each rom rial as a funcion of he field size heorem : n RNC(h+,h), if he coefficiens of m e are seleced roml from he finie field F, he probabili of h-independence of he ses Q e, is given b, ( ) k

4 p h, (4) F a a! where h = ( h ) b = b!( a b)! Proof- he flows of a receiver are disjoin, hus Q() e herefore, he coefficien cidaes for m e are a mos checked for receivers Consider he receiver Q() e, C ha are o be updaed for he edge e n C, Γ (e) is replaced b e he vecor m e is seleced roml Using equaion, we examine a cidae of b(e) for h-independence for he updaed Specificall, he vecor b(e) wih ever h- elemen subse of he global encoding vecors of h - oher edges in C mus consruc an independen se he probabili ha b(e) fails each es of h-independence is p = F, which is resuled using Lemma 4 in [5] he number of ( h )-elemen subses under consideraion is Appling union bound, he probabili ha m e fails a leas one of he ess for h-independence is p h Considering he receivers in Q (e), he F probabili ha m e fails a leas one of he verificaions for h- independence is p 3 herefore, he probabili Q ( e) F of h-independence of C, Q() e, is given b p = p 3 As we know, Q( e) Q ( e) F h h hus p h F heorem 3: he expeced number of operaions of RNC(h+,h) is O( E {, h }) h F Proof- he algorihm is execued edge b edge here are a mos E acive edges Ever edge is a mos in flows For edge e, a mos Q( e) verificaion of h-independence is o be performed, each of which consiss of a mos ess he expeced number of operaions of each es is O ( h ) hus, he expeced number of operaions for each es of local encoding vecor is O ( k ) he probabili of failure of a es of he local encoding vecor is q h, due o heorem F hus, he expeced number of operaions of he scheme o find proper local encoding vecor in an edge is O({, k q}) h, he expeced number of operaions of he scheme is O( E {, k }) h F Noe ha b increasing he field size, he expeced number of operaions decreases, bu in conras he complexi of each operaion is increased n he nex secion, we inroduce an exension of RNC o guaranee he rae h for h h flow failures for each receiver V RAE-GUARANEED ROUS NEWORK CODNG n RNC(h+,h), receivers can exrac all daa b receiving smbols of each h flows of h edge-disjoin flows A flow fails if a leas one link of he flow fails Using RNC(h+,h), if here is onl one receiver, failing an se of h -h flows, he receiver can exrac all he daa using he remaining flows However, in he scenario wih more han one receiver, receiving h flows for each receiver is no sufficien for exracing all he daa, since he failed flows of oher receivers ma have affeced he conens of he received smbols n figure, we can see ha Failure of link (s, ) desros z f has common edges wih f f f z f, bu does no direcl fails an oher flow However, his sill affecs he flow, in he sense ha i ma lead o collapsing he h-independence of he a receiver (decoding failure), since he flow z f joins ino f f, anoher flow of herefore, o guaranee h-independence in he presence of h -h flow failures for each receiver, we mus examine hese join flows A joinflow is a flow ha has a leas a common edge wih a leas one flow of anoher receiver A join-failure is a failure paern in which a leas one join-flow fails Here, we presen an improvemen o RNC, referred o as RNC(h+,h), o overcome join-failures n RNC(h+,h), in each edge e, he local encoding vecor, m e, is verified for h-independence for all Q( e) for failure-free all join-failure paerns ha do no reduce he capaci below k herefore, he consruced code resiss all such join-failure paerns For each edge e, he node v=ail(e) sends zero, if all he flows passing he edge e fail, or oherwise, send a packe based on he compued local encoding vecor Also, he receivers reac o failure paerns, b changing he decoding marix Now, assume n flows of h flows of he receiver are join-flows R denoes he se of all join-failure paerns for he flows of receiver ha do no reduce he capaci below h R, i denoes he se of join-failure n paerns for receiver in which i join-flows fail We have R, i =, i h h n R = (5) i = 0 i a Noe ha = 0, if b > a f R is he se of all join-failure paerns b ha do no reduce he capaci below h, we have h h n = R = R = (6) i 0 i n RNC, he local encoding vecor of an edge is examined for all joinfailure paerns one b one

5 heorem 4: A local encoding vecor for each ever nework edge exiss in RNC(h+,h), if he size of he finie field F saisfies he following F R (7) Proof- he argumen follows from he proof of heorem Here, he number of independence ess, is a mos Q ( e) R, since he ess are repeaed for each of R join-failure paerns Also, for each paern, he elemens of m e relaed o he edges of he failed flows are se o zero heorem 5: n RNC(h+,h), if he coefficiens of m e are seleced roml from he finie field F, he probabili ha he se, Q(e) as follows,, is h-independen for all he specified failure paerns is p R h (8) F Proof- he argumen follows direcl from he proof of heorem he deails are omied for brevi heorem 6: he expeced number of operaions of RNC(h+,h) is O( E {, h R }) h F Proof- n each edge e, h-independence is o be verified for all he specified failure paerns hus, he number of ess is muliplied b R he argumen follows from he proof of heorem 3 he deails are omied for brevi V ROUS NEWORK CODNG VA ACKUP FLOWS n his secion, we inroduce RNC3, in which backup flows are emploed when some main flows fail o ransmi daa An example was presened in secion C he scheme is no limied o one se of flows, as we design one se of local encoding vecors for all flow selecions n he limiing case he scheme guaranees he ransmission rae h for ever failure paern, which does no reduce he capaci below h We consider Ψ as he se of all flows from s o he flows ma have common edges Each h-elemen subse of he se Ψ, which consiss of h edge-disjoin flows, can provide a mulicas rae h he se Ω has all such subses as elemens f a leas one of hese subses is compleel safe, he -flow from he node s o he node is equal o h A complee-flow-se = ω,, ω } is a se of subses ω Ω for all { he se of all possible complee-flow-ses is denoed b Θ can be easil shown ha Θ = Ω Afer finding all subses composed of h disjoin flows for all, we design he coding scheme for all edges o iniialize he scheme, we add he node s h edges from s o he node s, find h independen linear combinaion of h source smbols, as he global encoding vecors of hese edges Here, an acive edge is an edge, which is presen in a leas one of he h disjoin-flow subses he algorihm considers he acive edges, one b one he edge e is considered, onl afer all acive edges in Γ ( ail( have been processed he edge e replaces Γ (e) in C for each receiver ω Q(e) each subse ω Ω, for which he edge e is in one of heir flows Afer replacemen, independence is verified for each updaed he algorihm concludes when he edges ending a he receivers,ω are processed heorem 7: A local encoding vecor for each ever nework edge exiss in RNC3, if he size of he finie field F saisfies he following condiion F Ω (9) Proof- n his case, he local encoding vecor is esed for all compleeflow-ses hus, he number of independence ess is equal o N = Q c ( e) We know ha N Θ = Ω Here, he c C size of m e is equal o he number of inpu acive edges of he node ail(e) For each complee-flow-se, he elemens of m e relaed o he edges ha are no presen in he complee-flow-se, is se o zero he argumen follows from he proof of heorem heorem 8: n RNC3, if he coefficiens of m e are seleced roml from he field F, he probabili ha he ses of all h-disjoin flows, ψ ω Ω for all Q(e) are independen, is as follows, p Ω (0) F Proof- he argumen follows from he proof of heorem he deails are omied for brevi heorem 9: he expeced number of operaions of RNC3 o guaranee he rae h for all failure paerns ha do no reduce he capaci below h is O( E {, h [ Ω ] }) F Proof- n each edge e, h-independence is o be verified for all compleeflow-ses hus, he number of ess is muliplied b Θ = Ω he deails are omied for brevi Usuall, here are more han one se of h disjoin flows for each receiver We selec he se roml he oher remaining flows do no paricipae in he ransmission scheme he opimizaion of he se selecion is no considered here, remains open o new ideas he pseudo-code of RNC3 is presened in figure For each edge e, he node v=ail(e) sends zero, if all he flows passing he edge e fail, or oherwise, send a packe based on he compued local encoding vecor Also, he receivers reac o failure paerns, b changing he decoding marix is possible o consider onl some h disjoin flow subses of he se for each, find he coding scheme for hem For example, here ma be some sensiive flows in he nework, which can be suppored b some backup flows he scheme finds a unique coding scheme for he failurefree case he cases in which some backup flows replaces some main flows Noe ha, he heorems 7, 8 9 are resuled for he limiing, Ψ

6 case, where all he h-disjoin-flow subses of he se Ψ for each are considered in he scheme RNC3: h is calculaed from Max-flow Min-cu heorem for all F is a finie field saisfing equaion 9 for each do find Ψ :he se of all flows from s o find Ω : he se of all h-disjoin-flows of he receiver find Θ : he se of all complee-flow-ses Θ for each do for each do iniialize C, ω,, ω, for each acive edge e in opological order do for each Θ do for each Q () e, C, ω C, \{ ( e)} { e}, Γ ω, selec m e roml for each Θ do if e E ( E is he se of acive edges of he complee-flow-se ) b ( e) = m ( e ) ( e ) b e e Γ ( ail (, = \ b { Γ ( e)} { b ( e)},, if, is no h-independen Go o selec =,, Figure - he pseudo-code for RNC3 V CONCLUDNG REMARKS We presened hree flow-based nework coding schemes o increase he robusness of he nework agains link failure he RNC(h+,h) is he simples scheme, in which diversi hence robusness is achieved hrough redundan flows he scheme can be caegorized as receiver-based [3],[] he RNC(h+,h) guaranees he rae h for he failure of h h failures for each receiver of he deermined flows he advanage is provided a he cos of increased design complexi b a facor of R, he need o inform he flow failure o he nodes in he failed flows n RNC(h+,h), for each edge e, he node v=ail(e) sends zero, if all he flows passing he edge e fail, or oherwise, send a packe based on he compued local encoding vecor he flow failure is announced o he nodes in he failed flows, which requires feedback is possible ha he flows fail o ransmi he rae h, due o failure of more han h h flows for he receiver, bu here are sill oher flows presen ha can subsiue he failed flows hus, we presened RNC3 scheme o emplo backup flows n a limiing case, he scheme can guaranee he rae h for all failure paerns, which do no decrease he capaci below h he RNC3 can be applied o increase he robusness b compensaing he sensiive poins of he nework via backup flows n RNC3, in each sae, we can selec one h-disjoin se for each receiver remove all 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