Robust Network Coding for Bidirected Networks

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1 Rou Nework Coding for Bidireed Nework A. Sprinon, S. Y. El Rouyhe, nd C. N. Georghide Ar We onider he prolem of nding liner nework ode h gurnee n innneou reovery from edge filure in ommuniion nework. Wih innneou reovery, lo d n e reovered he deinion wihou he need for ph re-rouing or pke re-rnmiion. We fou on peil l of idireed nework. In uh nework, for eh edge here exi orreponding edge in he revere direion of equl piy. We ume h mo one pir of idireed edge n fil ny ime. For uni onneion, we elih n upper ound of O( h ) on he minimum required eld ize nd preen n lgorihm h onru liner nework ode over GF ( h ). For muli onneion, we how h he minimum required eld ize i ounded y O( h ), where i he numer of erminl. We lo diu link- nd ow-yli idireed oding nework wih innneou reovery. I. INTRODUCTION In p yer, mjor effor h een pen on improving he reiliene nd urviviliy of ommuniion nework. The mjor hllenge i o ope wih he filure of nework edge. Suh filure re frequen due o he inheren vulneriliy of he underlying ommuniion infrruure []. Wih he drmi inree in he re of d rnmiion, even ingle edge filure my reul in v d lo nd ue mjor ervie dirupion o mny uer. Aordingly, mny ervie provider re inereed in providing innneou reovery from edge filure. Wih innneou reovery, he lo d n e reovered he deinion wihou he need for ph re-rouing or pke re-rnmiion. The ndrd pproh for gurneeing innneou reovery i o proviion wo dijoin ph eween he oure nd he deinion node nd end opy of eh pke over oh ph (ee Figure ()). Thi pproh, however, my reul in inefien ue of nework reoure. An lernive pproh, referred o diveriy oding [], i o proviion everl dijoin ph, ome of hem re ued for ending he originl pke nd oher for ending priy hek pke (ee Figure ()). Thi pproh, however, i no lwy feile, i require lrge numer of dijoin ph eween he oure nd he deinion node. Reenly, i w propoed o ue he nework oding ehnique for innneou reovery from edge filure [], [4]. The nework oding ehnique generlize he dijoin ph nd diveriy oding pprohe nd llow minimize he moun of nework reoure h need o e lloed o uppor innneou reovery. Nework oding w inrodued in he eminl pper y Ahlwede e l. [5]. The i ide The uhor re wih he Deprmen of Eleril nd Compuer Engineering, Tex A&M Univeriy, College Sion, Tex, USA. Emil: {lim,plex,georghide}@ee.mu.edu of nework oding i o llow inermedie nework node o genere new pke y performing lgeri operion on pke reeived over heir inoming edge (ee Figure ()). Thi i in onr o he rdiionl pproh in whih he inermedie node n only forwrd nd duplie heir inoming pke. v v v v v v () () () (d) (e) v v v v + Fig.. Differen mehod for hieving innneou reovery (ll operion re performed over GF ()). () The dijoin ph pproh. A opy of eh of eh pke i en over wo dijoin ph h onne nd. () The diveriy oding pproh wih hree dijoin ph eween nd. The r wo ph re ued for ending he originl pke, while he hird ph i ued for ending he priy hek pke +. () A nework oding heme wih h =. (d) A nework oding heme wih h =. (e) Pke rnmied y he edge of nework () upon filure of edge (, v 5 ). We ume h eh edge in he nework h n ineger piy; he piy of n edge peie he numer of pke h n e rnmied over hi edge in eh ommuniion round. We lo ume h pke re no frgmened nd re of xed lengh.

2 A mjor prmeer in he deign of he nework oding heme i he numer of pke h en y he oure in eh ommuniion round. Thi prmeer deermine he hpe of he rouing opology ued for d rnmiion. For exmple, for h =, he only wy o gurnee innneou reovery i o ue wo dijoin ph, depied in Figure (). For h = we n ue he me opology for h =, i.e., end wo pke over wo dijoin ph. In hi e, ll edge h elong o eh of he dijoin ph mu hve piy of le wo. We n lo ue hree dijoin ph, wo of whih re ued for ending originl pke, nd, nd he hird i ued for rnmiing he priy hek pke + (ll operion re performed over GF ()). Alernively, we n ue he nework oding pproh, depied in Figure (). Thi nework onin n enoding node, v whih reeive wo pke, nd, nd genere new pke, +. A more omplex nework oding opology wih h = i depied in Figure (d). In generl, he lrger i he vlue of h, he more exiiliy we hve in hooing he rouing opology. Deign of relile uni onneion h provide innneou reovery from link filure w inveiged in [] nd [4]. I w hown h relile ommuniion n e hieved y uing liner nework ode, in whih ll operion re performed over nie eld. Pril implemenion of he nework oding heme i preened in [6]. In our previou work [7] we onidered he prolem of elihing relile uni ommuniion for he peil e of h =. We howed h in hi e he underlying rouing opology h erin ominoril ruure whih enle he deign of efien nework ode over mll eld (GF ()). Wih liner nework oding, eh pke in he nework i liner ominion of he pke en y he oure node. Thi liner ominion n e pured y he glol enoding veor. The glol enoding veor n e inluded in he heder of eh pke wih only mll overhed [6]. We ume h in he e of filure of n edge ll pke rnmied over hi edge re idenilly equl o zero, i.e., hve zero glol enoding veor. Thu, in he e of n edge filure, only he node iniden o hi edge hnge heir ehvior, while ll oher node perform he me operion during he norml operion. For exmple, Figure (e) how he pke rnmied y he edge of he nework depied on Figure () upon filure of edge (, v 5 ). In hi work, we fou on he deign of uni oding nework wih h >. We onider peil l of idireed nework. In uh nework, he piy of eh ommuniion hnnel i eqully pli in oh direion, i.e., for ny edge in he underlying ommuniion grph here exi n edge in he revere direion of equl piy. We ume h mo one pir (e (v, u), e (u, v)) of idireed edge n fil ime. Indeed, in mny pril eing filure of ommuniion hnnel i frequen enough in order o wrrn oniderion. On he oher hnd, proeion from We onider only miniml opologie, i.e., opologie h do no onin redundn edge. muliple filure inur exeively high o in erm of nework uilizion, whih, ypilly, i no juied y he rre ourrene of imulneou filure. Our work mke he following onriuion. Fir, we inveige he innneou reovery from edge filure for uni onneion. We elih n upper ound of O( h ) on he required eld ize in idireed nework. Our ound only depend on he numer of pke h en eh ommuniion round nd doe no depend on he ize of he underlying ommuniion nework. Nex, we preen n lgorihm h onru feile nework ode over GF ( h ). Nex, we exend our reul for muli onneion nd how h he required eld ize i ounded y O( h ), where i he numer of erminl. Finlly, we diu rou oding nework wih yle. II. MODEL AND PRELIMINARIES A. Communiion Nework We model ommuniion nework y direed grph G(V, E), where V i he e of node nd E i he e of edge. Eh edge i oied wih prmeer e h peie he piy of he edge, i.e., he numer of pke h n e rnmied y he edge per ommuniion round. We ume h he grph G(V, E) i idireed, i.e., for eh edge e(v, u) G here exi n edge e (u, v) in he oppoie direion uh h e = e. We ume h eh pke in he oding nework i n elemen of nie eld F. Eh node in he oding nework n ree new pke y performing lgeri operion on he inoming pke over F. An inne I(G,,,, h) of uni nework oding prolem inlude grph G(V, E), he piy funion, he oure node, he deinion node nd he numer h of pke h need o e rnmied from o per ommuniion round. For lriy of preenion, we ue noion of oding nework N(G L,,, h) whih inlude link grph G L (V, L) formed from G(V, E) y uiuing eh edge e E y up o e prllel link, eh link n rnmi one pke per ommuniion round. We denoe y L e he e of link h orrepond o edge e. An inne I(G,,, T, h) o he muli nework oding prolem nd muli oding nework N(G L,, T, h) re dened in imilr mnner. Here, T denoe he e of erminl node. B. Nework Code We denoe y H i = {H, i..., Hh i } he e of pke en y he oure node round i. For eh link l L we denoe y yl i he pke rnmied on hi link round i. A nework ode F(N) for oding nework N(G L,,, h) (N(G L,, T, h)) i dened y e of lol enoding funion F(N) = {f l l L}. If l L i n ougoing link of he oure node, hen f l i mpping from F h o F. Oherwie, f l i mpping from F in(v) o F, where in (v) he ol numer of he inoming link of he il node v of l.

3 The enoding funion f l of link l(, v) deermine he pke yl i en on link l liner ominion of pke Hi en y he oure node round i. Similrly, he enoding funion f l of link l(v, u) peie he pke yl i rnmied on link l funion of pke h rrive node v round i. A menioned in he Inroduion, our gol i o deign nework ode h provide n innneou reovery upon filure of pir of idireed edge. When pir (e (v, u), e (u, v)), e, e E of idireed edge fil, ll link l L e L e nno rnmi pke. Aordingly, we ume h he lol enoding funion f l of link in L e L e re idenilly equl o zero. Le N(G L,,, h) e uni oding nework nd le F(N) e nework ode for N. We y h F(N) i rou if he deinion node n deode he pke en y he oure node fer xed numer of round even if ny wo idireed edge (e (v, u), e (u, v)) in E fil. A ode F(N) for muli nework N(G L,, T, h) i id o e rou if he ove requiremen hold for eh erminl T. C. Cu nd Flow We dene u C(V, V \ V ) i priion of V ino wo ue, V nd V \ V. We y h n edge e(v, u) E ( link l(v, u) L) i forwrd edge (link) of C in G (G L ) if v V nd u V \ V. A edge e(u, v) E ( link l(v, u) L) i referred o kwrd edge (link) of C if u V nd v V \ V. Noe h he me u C(V, V \ V ) in wo differen grph G(V, E ) nd G(V, E ) my inlude differen e of forwrd nd kwrd edge. The previou work on rou nework ode [], [9] elihed ondiion on he underlying nework opology G L neery for elihing rou nework ode. Theorem : ([], [9]) Le I(G,,,, h) e n inne of he uni nework oding prolem nd le N(G L,,, h) e orreponding uni oding nework. Then, here exi feile nework ode F(N) for N if nd only if for eh u C(V, V \V ) h epre nd nd for eh forwrd edge e of C in G i hold h he ol numer of forwrd link of C in G L i le h + e. We refer o he oding nework N(G L,,, h) h ify he ondiion of Theorem ondiion feile nework. A imilr ondiion hold for he muli oding nework. In ome pr of our pper we ue noion of nework ow [8]. Deniion (Flow): An inegrl (, )-ow θ i funion θ : L R h ie he following wo properie: ) For eh link l(u, v) L, i hold h θ(l) {, } ) For eh inernl node v V, v, v / i hold h w:(w,v) L θ((w, v)) = w:(v,w) L θ((v, w)). The vlue θ of ow θ i dened θ = θ((, v)). The o ω(θ) of ow θ i v:(,v) L dened ω(θ) = θ(l). l L A minimum o (, )-ow θ n e deompoed ino e of θ link-dijoin ph eween nd in G L (V, L) [8]. D. Min Reul In hi eion, we prove he min reul of our pper. Theorem : Le G(V, E) e idireed grph, G L (V, L) e he orreponding link grph, nd le N(G L,,, h) e feile uni nework over G L (V, L). Then, here exi rou nework ode F(N) for N over GF ( h ). Moreover, uh nework ode n e onrued in polynomil ime in V nd E. Thi heorem elihe n upper ound on he minimum required eld ize of rou nework ode. The heorem implie h he minimum required eld ize doe no depend on he ize of he nework. We lo prove he exiene of nework ode h n e implemened wih pke of lengh mo h i. Our proof i onruive nd employ he ehnique nd ool of he heory of nework ow [8]. III. NETWORK CODING ALGORITHM The min ide of our lgorihm i o idenify e Θ of (, )-ow in G L (V, L) uh h for eh pir of idireed edge (e (v, u), e (u, v)), e, e E here exi ow in θ Θ h doe no ue neiher e nor e, i.e., θ(l) = for eh l {L e L e }. We y h uh ow θ proe (e, e ). Given e Θ h proe every idireed edge pir in he nework, we n ue modiion of he lgorihm preened in [9] in order o nd feile oding nework N(G L,, T, h) nd feile nework ode F(N) for N. Theorem 4: Le N(G L,, T, h) e idireed uni nework nd Θ e e of (, )-ow in N, eh ow of vlue h uh h for ny pir of idireed edge (e (v, u), e (u, v)), e, e E, here exi ow θ Θ h proe i. Then, here exi rou nework ode F(N) for N over eld of ize Θ. Moreover, uh ode n e found in polynomil ime in V nd Θ hrough rndomized lgorihm. Proof: The proof follow he me line in [9, Theorem ]. In wh follow, we preen n lgorihm h nd, for ny feile uni nework N(G L,,, h), e Θ of (, )-ow h proe eh edge in N, uh h Θ h. The lgorihm inlude he following ep: ) Find minimum o inegrl (, )-ow θ r of vlue h in G L (V, L). ) Θ {θ r }. ) Deompoe θ r ino h link-dijoin (, )-ph R = {P r, P r,..., Ph r }. Thee ph re referred o red ph. 4) For eh ue S of R (exep empy e) do: ) Denoe y W S he ue of edge in E, eh edge e W S ie he following ondiion: () Eh red ph h elong o S inlude link in L e ; () Eh red ph h doe no elong o S doe no inlude link in L e. ) Denoe y L(W S ) he e of link h orrepond o edge in W S, i.e., L(W S ) = e WS L e. Noe h

4 ine he originl grph i idireed, for ny link in L(W S ) here exi link in he revere direion. ) Conru n uxiliry grph Ĝ(S) whih i oined from G L (V, L) y removing ll link in L(W S ). d) Find h link-dijoin ph eween nd. We denoe hee ph y B = {P, P,..., Ph } nd refer o hem lue ph. e) Denoe y B,..., B h he e of ll ue of B. f) Divide he edge in W S ino h ue WS,..., W S h, h orrepond o ll ue of B. Speilly, given ue B i of B, he e WS i inlude ll edge e in W S h ify he following ondiion: () Eh ph h elong o B i inlude link in L e () Eh ph h elong o B \ {B i } doe no inlude link in L e. Here, e i revere edge of e. g) For eh ue WS i do i) Conru n uxiliry grph Ĝ(S, i) formed from y G L (V, L) y removing ll link in L e nd in L e, where e denoe he e W i S e W i S revere edge of e. ii) Find ow θ(s, i) in Ĝ(S, i) nd dd i o Θ. Exmple : Conider he nework N(G L,,, ) depied in Figure (). Figure () how ow θ r h inlude hree red ph {P r, P r, P r }. Figure () nd (d) demonre he ierion of Sep 4 h orrepond o S = {P r, P r }. In hi e, he e W S inlude edge (v, v ), (v, v 5 ), (v 5, v 6 ), (v 6, v 8 ), (v 8, v 9 ), nd (v 9, v ). The uxiliry grph Ĝ(S) onrued in Sep 4 pper in Figure (). The e B = {P, P, P } of lue ph idenied in Sep 4d i hown in Figure (d). The uxiliry nework Ĝ(S, i) h orrepond o he ue B i = {P } onrued in Sep 4(g)i i hown in Figure (e). The gure lo how ow θ(s, i) h proe edge (v 5, v 6 ), (v 6, v 5 ), (v 8, v 9 ), nd (v 9, v 8 ). Our lgorihm preened n e exended for he e of muli. Thi n e hieved y invoking he uni lgorihm for eh of he erminl. The ol numer of ow i ounded y O( h ), where i he numer of he erminl. IV. CORRECTNESS PROOF We need o prove h ow e Θ proe ll pir of idireed edge of G. Fir, noe h he ow θ r proe ll e of idireed edge (e (v, u), e (u, v)), e E, e E for whih i hold h θ r (l) = for eh l {L e L e }. I i ey o ee h for ny oher pir of idireed edge (e, e ) i hold h eiher e or e elong o W S for ome ue S of R. Indeed, for eh idireed pir (e, e ) here exi le one link in {L e L e } h elong o red ph. Nex, noe h eh edge e W S elong o one of he ue WS,..., W S h dened in Sep 4f. Finlly, ow θ(s, i), dded o Θ Sep 4(g)ii of he lgorihm, proe ll edge of W S nd ll edge whoe orreponding revere edge elong P P, P r PP r, r P r PP r, r v v v v v v v v v P v v v v v v () () () Fig.. () An exmple of oding nework N(G L,,, ) nd he uxiliry grph Ĝ(S). Bidireed link re hown wihou line end. () A ow θr = {P r, P r, P r } (mrked y dhed line). () The uxiliry grph Ĝ(S) for S = {P r, P r}. (d) The e of lue ph {P, P, P }. (e) The uxiliry nework Ĝ(S, i) h orrepond o he ue B i = {P } nd ow θ(s, i) h proe edge (v 5, v 6 ), (v 6, v 5 ), (v 8, v 9 ), nd (v 9, v 8 ). o W S. I i ey o verify h he ol numer of ow in Θ i ounded y h. We proeed o how h he lgorihm never fil. The following lemm prove h Ĝ(S) onin h link-dijoin ph eween nd. Sine Ĝ(S) i ugrph of G L(L(E), V ), he lemm implie h he lue ph re feile ph in G L (L(E), V ). Lemm 5: Grph Ĝ(S) onin h link-dijoin ph eween nd. Proof: We prove he lemm y howing h eh u C(V, V \V ) h epre nd onin le h forwrd link Ĝ(S). Thi implie, y he Mx-Flow Min-Cu heorem [8], h here exi h dijoin ph eween nd in Ĝ(S). Fir, we noe h if C inlude mo one forwrd edge h elong o W S, hen, y Theorem, hi u onin le h forwrd link in Ĝ(S). Nex, we denoe y g he numer of edge of W S whih (d) (e) v v v v v v v v v v 4

5 re forwrd edge of C. Noe h eh red ph in S roe C in he forwrd direion le g ime. Thi implie h eh red ph in S roe C in he kwrd direion le g > ime. Sine for eh link h elong o red ph here exi link in Ĝ(S) in he revere direion, he red ph in S orrepond o le S forwrd link of C in Ĝ(S). In ddiion, he red ph h do no elong o S inlude le h S forwrd link of C in Ĝ(S). Thu, he ol numer of forwrd link in ny (, )-u C of Ĝ(S) i le h. Thi implie, y he Mx-Flow Min-Cu heorem, h here re h link-dijoin ph in Ĝ(S). The nex lemm prove h Ĝ(S, i) onin h link-dijoin grph eween nd. Lemm 6: Le WS i e ue of W S, dened in Sep 4f of he lgorihm. Le Ĝ(S, i) e ugrph of Ĝ(S) onrued in Sep 4(g)i y deleing ll link h orrepond o edge in WS i (in oh forwrd nd revere direion). Then, Ĝ(S, i) onin h link-dijoin grph eween nd. Proof: We prove he lemm y howing h eh u C(V, V \ V ) h epre nd in Ĝ(S, i) onin le h link in he forwrd direion. By Mx-Flow Min- Cu heorem [8], hi implie h here exi h dijoin ph eween nd in Ĝ(S, i). Le g f e he numer of edge in WS i h re forwrd edge of C in Ĝ(S, i) nd y g he numer of edge in WS i h re kwrd edge of C. We onider he wo following e: ) Ce : g f g. Le Pi e lue ph h elong o B i. Noe h Pi roe C le g f ime in he kwrd direion (eue for eh e WS i ph Pi onin link in he revere direion). Hene, Pi roe C le g f + ime in he forwrd direion. Noe when Pi roe C in he forwrd direion, i my ue edge h whoe orreponding revere edge elong o WS i, u he numer of uh edge i ounded y g. Thi, eh ph in B i inlude le one forwrd link of C in Ĝ(S, i). Nex, we noe h ll link of eh ph in B \ B i exi in Ĝ(S, i), hene eh uh ph inlude le one forwrd link of C in Ĝ(S, i). We onlude C onin le h forwrd link. ) Ce : g > g f. In hi e eh red ph Pi r in S roe C le g ime in he kwrd direion, hene i roe C le g + ime in he forwrd direion. Noe when Pi r roe C in he forwrd direion, i my ue edge h elong o WS i, u he numer of uh edge i ounded y g f. Thu, eh red ph in S inlude le one forwrd link of C in Ĝ(S, i). In ddiion, ll link of he red ph h do no elong o S exi in Ĝ(S, i), hene eh of hee ph inlude le one forwrd link. We onlude C onin le h forwrd link in Ĝ(S, i). V. NETWORKS WITH CYCLES In generl, yle re undeirle in oding nework. Brero nd Yrehu [] lify yli nework ino wo v v v v v v Fig.. () An exmple of oding nework whih i link-yli, u no ow-yli. () An exmple of ow-yli nework. le, link-yli nework nd ow-yli nework. A nework N(G L,,, ) i referred o link-yli if he link grph he link grph G L (V, L) onin le one yle. A nework N(G L,,, ) i referred o ow-yli if for ny e Θ of ow h proe ll edge in he underlying ommuniion grph G(V, E), here exi yle C uh h ny wo oneuive link of C elong o one ow. For exmple, Figure () depi nework whih i link-yli, u no ow-yli, while Figure () depi ow-yli nework. Ahlwede e l. [5] howed h nework oding pproh n e pplied o nework wih yle. If he oding nework i link-yli, u no ow-yli, hen feile nework ode n found y modiion of LIF lgorihm []. Furhermore, uh nework n e implemened dely-free nework, i.e., nework wih zero-dely link. For ow-yli nework, feile nework ode n e found y uing he rndom oding mehod propoed in [9], nd, in ome peil e, y he LIFE-CYCLE lgorihm preened in []. In generl, rou idireed nework n e link-yli. To ee hi, onider he nework depied in Figure () nd ume h ll edge re idireed nd edge (v, v 4 ) nd (v 4, v ) re pir of idireed edge. I remin n open queion wheher idireed nework n e ow-yli. In he following lemm, we prove h for h = ny inne I(G,,,, h) of he nework oding prolem here exi oding nework whih doe no onin yle. The lemm pplie oh for idireed nd direed nework. Lemm 7: Le I(G,,,, h) e n inne of he nework oding prolem for h =. Then, here exi oding nework N(G L,,, ) uh h N i no link-yli (nd no owyli). Proof: (keh) In [7] i w hown h I(G,,,, h) () v v () 5

6 i feile if nd only if he orreponding ow nework Ĝ nework dmi n (, )-ow of vlue, where he ow nework Ĝ i formed from G y reduing eh edge of piy or more y n edge of piy.5. Sine minimum-o ow in Ĝ doe no inlude yle, he orreponding oding nework N(G L,,, ) i yli. VI. CONCLUSION AND FUTURE DIRECTIONS In hi pper, we ddreed he prolem of nding rou nework ode for idireed ommuniion nework. In idireed nework, for eh edge here exi n edge in he revere direion of equl piy. We ume h mo one pir of idireed edge n fil ny ime. Exending he reul of our previou work [7], we hve onidered he generl e of h, where h i he numer of pke en eh ommuniion round. For uni onneion we elihed n upper ound of O( h ) on he minimum eld ize nd preened n lgorihm h onru feile nework ode over GF ( h ). For muli onneion, we howed h he mximum ize of he nie eld i ounded y O( h ), where i he numer of erminl. A fuure direion, we would like o ddre n open queion of wheher for eh inne I(G,,,, h) of uni nework oding prolem wih idireed nework here exi oding nework N(G L,, T, h) whih i no ow-yli. In ddiion, we would like o exend our ound for generl direed nework. REFERENCES [] W. D. Grover. Meh-Bed Survivle Trnpor Nework: Opion nd Sregie for Opil, MPLS, SONET nd ATM Neworking. Prenie- Hll, New York, NY, USA,. [] E. Aynoglu, C.-L. I., R. D. Gilin, nd J.E. Mzo. Diveriy Coding for Trnpren Self-Heling nd Ful-Tolern Communiion Nework. IEEE Trnion on Communiion, 4,(): , 99. [] R. Koeer nd M. Medrd. An Algeri Approh o Nework Coding. IEEE/ACM Trnion on Neworking, (5):78 795,. [4] S. Jggi, P. Snder, P. A. Chou, M. Effro, S. Egner, K. Jin, nd L. M. G. M. Tolhuizen. Polynomil Time Algorihm for Muli Nework Code Conruion. IEEE Trnion on Informion Theory, 5(6):97 98, 5. [5] R. Ahlwede, N. Ci, S.-Y. R. Li, nd R. W. Yeung. Nework Informion Flow. IEEE Trnion on Informion Theory, 46(4):4 6,. [6] P. A. Chou, Y. Wu, nd K. Jin. Pril Nework Coding. In Proeeding of Alleron Conferene on Communiion, Conrol, nd Compuing, Moniello, IL, Ooer. [7] S. El Rouyhe, A. Sprinon, nd C. Georghide. Simple Nework Code for Innneou Reovery from Edge Filure in Uni Conneion. In Proeeding of Proeeding of he UCSD Workhop on Informion Theory nd i Appliion, Sn Diego, CA (Invied Pper), Ferury 6. [8] R. K. Ahuj, T. L. Mgnni, nd J. B. Orlin. Nework ow: heory, lgorihm, nd ppliion. Prenie-Hll, In., Upper Sddle River, NJ, USA, 99. [9] T. Ho. Neworking from Nework Coding Perpeive. PhD hei, MIT, My 4. [] A. I. Brero nd Ø. Yrehu. Cyle-logil Tremen for Cylophi Nework. IEEE Trnion on Informion Theory, 5(6):795 84, 6. 6

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