Survivable Rouing Mee Diveriy Coding Alija Pašić, Jáno Tapolcai, Péer Babarczi, Erika R. Bérczi-Kovác, Zolán Király, Lajo Rónyai MTA-BME Fuure Inerne Reearch Group, Budape Univeriy of Technology and Economic (BME) Deparmen of Operaion Reearch, Eövö Univeriy, Budape, Hungary Deparmen of Compuer Science, Eövö Univeriy, Budape, Hungary and MTA-ELTE Egerváry Reearch Group Compuer and Auomaion Reearch Iniue Hungarian Academy of Science and BME {paic, apolcai, babarczi}@mi.bme.hu, koverika@c.ele.hu, kiraly@c.ele.hu, ronyai@zaki.hu Abrac Survivable rouing mehod have been horoughly inveigaed in he pa decade in ranpor nework. However, he propoed approache uffered eiher from low recovery ime, poor bandwidh uilizaion, high compuaional or operaional complexiy, and could no really provide an alernaive o he widely deployed ingle edge failure reilien dedicaed 1 + 1 proecion approach. Diveriy coding i a candidae o overcome hee difficulie wih a relaively imple echnique: dividing he connecion daa ino wo par, and adding ome redundancy a he ource node. However, a miing link o make diveriy coding a real alernaive o 1+1 in ranpor nework i finding i minimum co urvivable rouing, even in pare opologie, where previou approache may fail. In hi paper we propoe a polynomial-ime algorihm wih O( V E log V ) complexiy for hi rouing problem. On he oher hand, we how ha he ame rouing problem urn o be NP-hard a oon a we limi he forwarding capabiliie of ome node and he capaciie of ome link of he nework. Index Term urvivable rouing, diveriy coding, fa recovery, ranpor nework I. INTRODUCTION A urvivable rouing cheme in ranpor nework ha hree umo imporan requiremen: low recovery ime, impliciy (i.e., low compuaional and operaional complexiy) and efficien capaciy allocaion. Bu which one i he mo imporan for ervice provider, and wha are hey willing o acrifice in order o reach ha? To anwer hi, we have o look wha i ued in pracice. In nework, he mo commonly ued urvivable rouing cheme i he o called dedicaed 1+1 pah proecion, which end he uer daa along wo dijoin pah (primary and backup). Alhough i conume wice a much capaciy a he primary capaciy, here are efficien algorihm o calculae 1 + 1 rouing (i.e., dijoin pah-pair [1]), while i provide inananeou recovery from any ingle edge failure. Alhough 1 + 1 i ill he mo commonly ued proecion cheme, in bandwidh uilizaion here i ill room for improvemen. On he oher hand, baed on he previou obervaion, impliciy and ulra fa recovery ime eem o be he mo imporan feaure, and efficien capaciy allocaion come only afer hem. Several urvivable rouing cheme were inroduced [2], [3] in he pa decade which could ignificanly reduce he bandwidh uilizaion [3], [4], bu hey acrifice eiher he ulra fa recovery ime, he low compuaional complexiy, or mo imporanly he imple operaion. Following hi argumen we how ha wih a ISBN 978-3-901882-68- c 201 IFIP careful deign we can keep hee meri while near opimal capaciy allocaion can be reached. In ranpor nework, opimal capaciy efficiency can be achieved wih in-nework modificaion of daa (called nework coding) [4] [6] while low recovery ime (i.e., < 40-0 m) i mainained. However, he applicaion of complex nework coding operaion acrifice impliciy. On he oher hand, here are ome imple pecial cae of coding which could aify all hree requiremen in ranpor nework, e.g., diveriy coding (DC) [7], which pli he daa a he ource node ino wo par A and B, and creae redundancy daa A B, oo ( denoe he excluive OR (XOR) operaion on he daa), hen end hee on hree edge-dijoin pah. Diveriy coding can reduce he capaciy conumpion of 1+1 (from 2 o 1. uni), while i complexiy and recovery ime are he ame. On he oher hand, hi mehod i applicable only in 3 edge-conneced nework which i a crucial drawback. Recenly, he paper [8] and [9] made imporan ep o remedy he conneciviy problem of diveriy coding, and enable i o be an alernaive of 1 + 1 in ranpor nework proecion. They formulae he urvivable rouing problem of 1 + 1 and diveriy coding more generally, while conidering he ame rouing cenario: ingle edge failure reilience [10], while he connecion daa i divided ino wo par a he ource node. Alhough hey provide polynomial-ime nework (and diveriy) coding algorihm in a minimum co urvivable rouing (called coding ubgraph), he iue of finding a minimum co urvivable rouing i no addreed in hee work. However, hi would be he la ep oward he pracical implemenaion of a capaciy efficien, imple and low-complexiy diveriy coding baed urvivable rouing mehod wih ulra-fa recovery ime a a counerpar of 1 + 1 proecion; which will be made in hi paper. Alhough he general verion of minimum co urvivable rouing have hown o be NP-complee [11], [12], urpriingly, he compuaional complexiy of opimal capaciy allocaion for hi pracically relevan pecial cae i an open queion. However, uing he ae-of-he-ar reul [9] i i known ha a minimum co urvivable rouing wih diveriy coding can be decompoed ino hree direced acyclic graph (DAG), which preerve one of he mo imporan feaure of 1 + 1, i.e., impliciy. Thu, in order o olve he opimal capaciy allocaion of diveriy coding, we formulae he minimum co urvivable rouing problem a finding hree appropriae DAG.
TABLE I NOTATION LIST FOR THE SURVIVABLE ROUTING PROBLEM Noaion Decripion G = (V, E, k, c) direced graph wih node e V, edge e E, edge co c(e) R +, capaciie k(e) N D = (,, d) connecion wih ource node, arge node wih bandwidh d uni demand urvivable rouing of a connecion wih node R = (V R, E R, f) e V R V, edge e E R E, and flow value e E R : f(e) k(e) direced muli-graph wih edge e E, where G = (V, E, c) all edge in G = (V, E, k, c) are replaced by k(e) parallel edge each wih co c(e) R urvivable rouing of a connecion in = (V R, E R ) G = (V, E, c), which i a DAG A, B wo par in which he connecion daa i decompoed A B redundancy XOR daa creaed a ource from connecion daa par E A, E B, E A B rouing DAG for A, B and A B We will demonrae ha wih he help of hi formulaion he problem i polynomial ime olvable, which reul make diveriy coding an alernaive o 1 + 1 in 2 edge-conneced nework a well. The re of he paper i organized a follow. In Secion II we formulae he minimum co urvivable rouing problem wih diveriy coding (SRDC), and reveal ome rucural properie of he hree DAG, which will be ued in he algorihm. A he main conribuion of he paper, in Secion III a polynomial-ime algorihm wih O( V E log V ) complexiy i preened for he SRDC problem. In Secion IV we how ha he SRDC problem i NP-hard wih ome boleneck link and wih limied node capabiliie. Finally, in Secion V we how ome imulaion reul, which reveal ome nework cenario where SRDC can be a real alernaive of 1 + 1 proecion, and in Secion VI we conclude he paper. II. PROBLEM FORMULATION AND RELATED WORK A. Survivable Rouing A ranpor nework i a collecion of rouer, wiche (referred o a node) and high bandwidh communicaion link (referred o a edge) beween hem. I may be repreened by a direced graph G = (V, E, k, c) wih node e V and edge e E. Each e E edge ha wo aribue, namely i capaciy k(e) N, i.e., number of bandwidh uni available for daa ranmiion, and i co c(e) R +, which i defined a he co of uing one uni of bandwidh along edge e. Given a connecion D = (,, d), wih informaion ource V, wih informaion ink V, and he number of bandwidh uni d requeed for daa ranmiion. Definiion 1. We ay ha R = (V R, E R, f) i a urvivable rouing of a connecion D = (,, d) in G (where V R V, E R E, and e E R : f(e) k(e)), if here i an flow of value F d in R, even if we delee any ingle edge of R. On he oher hand, a rouing i vulnerable if i i no urvivable. Noe ha, afer he edge failure i idenified any rouing mehod could be adoped and reend he flow on he inac edge of a urvivable rouing R, clearly reuling in low recovery. However, everal nework coding heorem [4], [], [13] enure ha a ufficien amoun of informaion reache he deinaion afer a failure occurred wihou failure idenificaion; bu for he price of complex in-nework operaion. We will demonrae ha we can bring ogeher hee meri (i.e., fa recovery and impliciy) of urvivable rouing approache wih he help of curren reearch reul on nework coding [8] and on diveriy coding [9]. Thi i dicued in Secion II-B. We ay ha a urvivable rouing R i criical, if we can no furher decreae he flow value f(e) along any edge in e E R wihou making rouing R vulnerable. Our goal i o find a urvivable rouing R for connecion D wih minimum bandwidh co from he poible e of urvivable rouing R D, formally: min c(e) f(e). (1) R R D e E R Claim 1. The minimum co urvivable rouing in erm of Eq. (1) i criical. Thi opimizaion problem ha been inveigaed for decade in he lieraure, and i wa hown ha finding he opimal urvivable rouing for a connecion wih d > 2 daa par, or finding he opimal urvivable rouing for muliple edge failure are NP-complee problem [11], [12], [14]. However, in curren ranpor nework ingle edge failure are he mo relevan failure cenario [10], while dividing uer daa ino more han wo par i impracical from an operaional poin of view. Surpriingly, he complexiy of hi pracically relevan pecial cae of ingle edge failure minimum co urvivable rouing when d = 2 i an open queion. Thu, in hi paper we will inveigae he urvivable rouing problem when d = 2, i.e., he connecion can be roued a wo par of equal ize, denoed by A and B (for he ake of impliciy, he problem can be caled o have boh wih rae 1). Thi rericion i moivaed by he fac ha he minimum co rouing oluion in mo real word nework can be reached by dividing he inpu flow ino 2 ubflow [2]. Furhermore, auming wo daa par a urvivable rouing oluion preerve he impliciy of 1 + 1 proecion. For he ake of eaier preenaion of our reul, he auxiliary graph G = (V, E, c) i inroduced. The node e of G i he ame a he node e of G, and each e E i replaced by k(e) (parallel) edge which have he ame ail and head node a e, each wih co c(e). Noe ha ingle edge failure e in G correpond o he failure of all k(e) edge in G. A criical urvivable rouing R = (V R, E R ) form a direced acyclic graph (DAG) in G (dicued in Secion II-B), repreening he rouing of he connecion, where V R V, E R E, while he objecive funcion in Eq. (1) can be rewrien a: min e E R c(e). (2) The noaion i ummarized in Table I.
E A p E B E A B Fig. 1. A urvivable rouing R = (V R, E R ) for connecion D = (,, 2) wih he correponding rouing DAG E A, E B and E A B. B. Diveriy Coding In urvivable rouing beide he heoreically good properie like low bandwidh uilizaion and fa recovery, from a pracical poin of view impliciy and eay deploymen are eenial a well. Thu, complex daa proceing a core node (i.e., oher han and ) like nework coding i no deired. Thu, all complex operaion have o be moved o he edge of he nework (i.e., o node and ). We will refer o he urvivable rouing approache which aify hi propery a diveriy coding baed mehod, or imply diveriy coding. Luckily, for ingle edge failure and wo daa par, he range of urvivable rouing providing hi impliciy i quie wide: Theorem 1. [9, Theorem 2] Suppoe ha urvivable rouing R i criical. Then here are dijoin edge e E A, E B, E A B of R, called rouing DAG, uch ha for an arbirary edge e E R, afer removing he correponding edge() from E R a lea wo of he rouing DAG connec o. For example, in radiional diveriy coding, for a connecion D = (,, 2) he redundancy daa A B i calculaed a he ource, and A, B and A B i en along hree dijoin end-o-end pah beween and. The edge e ued by he dijoin pah are denoed a E A, E B, E A B, repecively. Alo 1 + 1 proecion could be reaed a an implemenaion of diveriy coding: A and B are en along wo dijoin pah E A and E B, while he redundancy daa A B i en along boh pah. Noe ha boh of hee rouing are urvivable. We will refer o rouing aifying Theorem 1 a Survivable Rouing wih Diveriy Coding (SRDC) hroughou hi paper. Noe ha wih he help of diveriy coding he hree rouing DAG can be operaed independenly from each oher (E A E B E A B = E R, E A E B =, E A E A B =, E B E A B = ), i.e., hey carry he ame daa par regardle of he failure, while involved operaion are performed only a he end node of he connecion. However, hi general implemenaion of diveriy coding require pliing and merging of he pah a he core node. Le δ (v) and δ + (v) denoe he in-degree and he ou-degree of a node, repecively. Definiion 2. Node p V R i called plier, if δ (p) = 1 and δ + (p) = 2, i.e., i receive daa on a ingle edge, while forward he ame copy on wo ougoing edge. Similarly, node m V R i called merger, if δ (m) = 2 and δ + (m) = 1, i.e., i receive he ame daa on wo incoming edge, while forward one of hem (or upon failure he inac one) on i ingle ougoing edge. The e of available plier and merger m v are denoed a P V and M V, repecively. We believe ha pliing and merging operaion are imple enough in he ene ha every node in he nework can perform hem wihou any complicaed ofware updae. Furhermore, in curren neworking paradigm, uch a Sofware Defined Neworking (SDN), a plier can be eaily deployed by applying imple flow rule, while a merger funcionaliy can be implemened a a imple nework funcion a well [14]. In [8], [9] i wa hown ha a criical urvivable rouing ha a well defined rucure (creaed by a maximal erie of cu or by a block decompoiion). In a criical oluion, we have an alernaing erie of plier and merger node (if here are any a all). An example of a urvivable rouing wih diveriy coding i preened in Figure 1. For rouing DAG E A B node p i a plier and node m i a merger, while for rouing DAG E A node v and node ac a a plier and merger, repecively. Definiion 3. Node p and m are pliing and merging nodepair (or pm-pair horly) of a urvivable rouing R (given wih i rouing DAG E A, E B, E A B ), if here exi a rouing DAG, wihou lo of generaliy, E A, which pli a node p and merge back a m. The edge e of he correponding dijoin pah-pair beween p and m in he urvivable rouing i denoed by Ep,m, R and referred o a an iland in DAG E A. Noe ha, here alway exi an edge dijoin pair of pah beween a plier and i correponding merger in a criical urvivable rouing R [8], [9]. Claim 2. Each rouing DAG in a criical oluion coni of a erie of pah and iland from o. Furhermore, a pm-pair could be par of a mo one rouing DAG [8], [9]. For example, in radiional diveriy coding all of E A, E B, E A B are pah. In Figure 1 E A coni of an v pah and a v iland, E B i an pah, while E A B coni of a pah p, an iland p m, and a pah m. The algorihm o find he opimal rouing DAG wih heir compuaional complexiy are hown in Table II. Noe ha, diveriy coding correpond o he cae when only he ource and deinaion node can be a pm-pair, which can be olved by Suurballe algorihm 1. We are inereed in he general cae a well, when more pm-pair exi ill he oher exreme, i.e., all node-pair can ac a a pm-pair. In he re of he paper, we inroduce hee capaciy allocaion approache, and conduc imulaion o demonrae heir benefi. III. SURVIVABLE ROUTING WITH INFINITE CAPACITIES In hi ecion we how ha he minimum co urvivable rouing problem wih diveriy coding i olvable in polynomial ime, if he edge capaciie k(e) are infinie, i.e., here are no boleneck link in he nework. Noe ha for he demand D = (,, 2) in urvivable rouing wih diveriy coding (SRDC) we are earching hree rouing DAG, each 1 Noe ha he augmening pah echnique of Suurballe algorihm can be ued o find 3 edge-dijoin pah.
TABLE II OPTIMAL SURVIVABLE ROUTING ALGORITHMS PROPOSED IN THIS PAPER (INFINITE: e E : k(e) = 2, CONSTRAINED: e E : k(e) = 1) Infinie Capaciie Capaciy Conrained P = {}, Suurballe Suurballe M = {} O( V log 2 V + E ) O( V log 2 V + E ) P V, - - M V NP-complee P = V, SRDC-I Ineger Linear Program M = V O( V E log 1+ E / V V ) - forwarding one uni of capaciy (eiher A, B or A B). Thu, in a criical urvivable rouing one would hink ha infinie edge capaciy mean e E : k(e) = 3, i.e., he cae when all DAG may ue he ame edge. However, i wa hown in [9] (a a conequence of Theorem 1) ha wih he applicaion of diveriy coding (called reilien flow decompoiion) in a criical urvivable rouing for a connecion wih d = 2 he flow value are e E R : f(e) 2. Thu, wihou lo of generaliy, we can reric he available capaciie o k(e) = 2 for every edge e, wihou ruling ou he minimum co urvivable rouing (which i criical, obviouly). Hence, in SRDC infinie edge capaciie mean e E : k(e) = 2 (inead of 3). Furher noe ha, in hi cae he auxiliary graph G ha 2 E edge, a each edge of G i duplicaed. Claim 3. Le R be a criical urvivable rouing, decompoed ino 3 rouing DAG E A, E B, and E A B according o Theorem 1 and le Ep,m R be an iland for a given pm-pair in E A. Le Ep,m G denoe an arbirary edge-dijoin pah-pair beween node p and m in G, wih he correponding edge Ep,m G in G. If he nework ha infinie edge capaciie, he rouing R = R \ Ep,m R Ep,m G i alo urvivable. Proof: Noe ha in R = R \ Ep,m R Ep,m G he rouing DAG E A ha iland Ep,m G inead of iland Ep,m. R In order o demonrae ha R i urvivable again ingle edge failure of G, we have o how ha a lea wo rouing DAG connec o upon hee failure. For an edge failure no in Ep,m, G DAG ha remained conneced in R will be alo conneced in R. If he failed edge e Ep,m, G hen we know ha in R here were wo rouing DAG which conneced o. The only DAG ha changed i E A, bu noe ha i alo remain conneced, becaue he failure i in an iland. Thi prove he claim. Corollary 1. Le R be a minimum co urvivable rouing and Ep,m R an iland for a given pm-pair. If he nework ha infinie edge capaciie, hen Ep,m R i a minimum co dijoin pah-pair beween node p and m in G. Such a pair can be calculaed wih Suurballe algorihm in O( E + V log 2 V ) ep [1]. Noe ha Ep,m G urvive a ingle edge failure, a i correpond o a dijoin pah-pair in G. Thu, we can ubiue i wih a fail-afe edge beween p and m in E A. Thi give he baic idea for he algorihm, earching for a urvivable rouing in a racable form. Claim 4. Le R be a criical urvivable rouing, decompoed ino 3 rouing DAG E A, E B, and E A B. Replacing every iland Ep,m G wih an edge (p, m) reul in hree edge-dijoin -pah. Now, we are ready o preen he main reul of he paper. Theorem 2. If he nework ha infinie edge capaciie on all edge, he minimum co urvivable rouing R can be compued in O( V E log 1+ E / V V ) ep. Proof: Le co(u, v) denoe he um of he edge co of a minimum co dijoin pah-pair beween node u and v in G. We conruc he following auxiliary (muli-)graph Ĝ = (V, Ê, ĉ). The node e of Ĝ i he ame a he node e of G, and he edge of Ĝ are he edge of G wih co ĉ(e) = c(e) and we add an edge e n = (u, v) for every pair of diinc node-pair wih co ĉ(e n ) = co(u, v). We refer o he newly added edge a virual edge. Claim. Le P A, P B, P A B be hree edge-dijoin -pah in Ĝ. By replacing every virual edge (p, m) wih an iland Ep,m G of minimum co we ge DAG E A, E B, and E A B in G ha form a urvivable rouing. Moreover, he co of hee DAG in G equal he co of he pah in Ĝ, and vice vera. Proof: Equaliy of co i raighforward. Since P A, P B, P A B are edge-dijoin in Ĝ, every edge e in E i conained in a mo one pah a a non-virual edge, and may be conained in oher iland() ued for ubiuing virual edge. In cae of he failure of e, he laer one remain conneced, o a mo one of he DAG can be diconneced, which prove he claim. Since a minimum co urvivable rouing i criical, from Claim 2 and Claim 4 we ge ha i correpond o hree edgedijoin -pah in Ĝ. In our algorihm, we earch for 3 edgedijoin pah wih minimum bandwidh co in Ĝ, which in urn correpond o hree DAG in G. According o Claim, he obained rouing i urvivable, and he co of he minimum co 3 edge-dijoin pah equal o he co of he minimum co urvivable rouing R. Finding minimum co 3 edgedijoin pah could be done in O( E ). In he conrucion, finding he pair of hore edgedijoin pah from a ingle ource o every deinaion i O( E log 1+ E / V V ) ime [1], which hould be launched for every ource node, reuling O( V E log 1+ E / V V ) ep, which prove he heorem. The proof i conrucive, and give he polynomial-ime algorihm o find an opimal urvivable rouing, deailed in Algorihm 1. A naural queion i why he algorihm canno cope wih nework wih ome edge capaciie k(e) = 1. The problem i ha in uch a capaciy conrained cae Ep,m G depend on he roue of he oher wo rouing DAG, i.e., anoher rouing DAG may ue he ingle available capaciy uni along an edge e Ep,m G of he minimum co dijoin pah-pair. For example, Figure 2(a) how a nework wih an opimal urvivable rouing of co 20. Noe ha, he virual edge e n = (v 1, ) ha co ĉ(e n ) = becaue co(v 1, ) i
v 9 2 4 v 7 v 1 v 2 v 4 v 3 v 1 v 2 v 3 v 3 v 6 v 4 (a) The opimal urvivable rouing wih he rouing DAG. The edge co oher han 1 are hown. v (b) The 3 pah in Ĝ of Theorem 2. For he ake of impliciy, only virual edge (v 1, ) i hown. (c) The 2 edge-dijoin pah wih minimum co. (d) A poible vulnerable rouing, where he failure of (v 3, ) dirup wo rouing DAG. Fig. 2. An example nework G = (V, E, c) wih capaciy conrain on he edge (remember from he conrucion of G ha k(e) = 2 edge in G are parallel edge in G ), where c(e) = 1, or wrien nex o he edge oherwie. The edge of he rouing DAG E A, E B and E A B are denoed a dahed, doed and denely doed line, repecively. Here Algorihm 1 fail for connecion D = (,, 2). Algorihm 1: Survivable Rouing wih Diveriy Coding - Infinie Capaciie (SRDC-I) Inpu: G = (V, E, c), D = (,, 2) Reul: R = (V R, E R ), in pecific, rouing DAG E A, E B, and E A B 1 begin 2 Define co ĉ : E R + and edge e Ê =, E = ; // Creae graph Ĝ = (V, Ê, ĉ). 3 Add e E o Ê wih ĉ(e) = c(e); 4 for u V : do Find he pair of hore edge-dijoin pah from ource u o all oher node v V, u v in G wih Suurballe algorihm (denoe heir co wih co(u, v)); 6 Add virual edge e n = (u, v) o Ê wih ĉ(e n ) = co(u, v); // Find 3 edge-dijoin pah in Ĝ. 7 Find minimum co 3 edge-dijoin pah beween and in Ĝ wih Suurballe algorihm; 8 Add he ravered edge (i.e., heir correponding edge in G ) o E ; 9 for e = (u, v) E do 10 if e i a virual edge hen 11 Replace virual edge e wih minimum co iland Eu,v G in E ; // Save opimal urvivable rouing R. 12 for e = (u, v) E do 13 Add node u, v o V R (if u, v / V R ); 14 Add edge e o E R ; he co of he hore pah-pair v 1 v 2 v 3 and v 1 v i 3 + 2 =. The minimum co 3 edge-dijoin pah in Ĝ are hown in Figure 2(b). Clearly, hi i no a valid oluion in he capaciy conrained cae, a edge e = (v 2, v 3 ) ha only k(e) = 1 available capaciy in G, while wo rouing DAG hould ue i in he opimal oluion. Anoher direcion i o find he minimum co 3 edgedijoin pah wih Suurballe algorihm uing he augmening pah echnique. Applying hi echnique o SRDC, he virual edge are only ravered by he 3 rd augmening pah, only afer 2 edge-dijoin pah were already found. A naural exenion of Algorihm 1 may be o run he dijoin pah earch for each virual edge (e.g., o (v 1, )) a a dijoin pah-pair beween node v 1 and. During hi earch he revere edge of he already found 2 edge-dijoin pah can be ued (hown in Figure 2(c)) imilarly a in Suurballe algorihm, and addiionally i can ue he revere edge of he hird edge-dijoin pah egmen beween and v 1 (which i v 7 v 2 v 3 v 6 v v 1 ). Thi could reul in an augmening pah beween plier v 1 and merger of v 1 v v 6 v 3. In hi cae he econd augmening pah beween plier v 1 and merger would be v 1 v 4 v. Thi in fac reul in a vulnerable rouing hown in Figure 2(d) wih co 16. Thu, Algorihm 1 i no applicable o he capaciy conrained cae. IV. SURVIVABLE ROUTING WITH CAPACITY CONSTRAINTS In hi ecion, we will inveigae he capaciy conrained cenario, i.e., when here are ome boleneck link wih k(e) = 1. Alhough a polynomial ime algorihm exi for he infinie capaciy cae (Secion III), he urvivable rouing wih diveriy coding (SRDC) problem i more complex in hi cenario. In Secion IV-A we preen an Ineger Linear Program (ILP) for finding he hree rouing DAG for SRDC, while Secion IV-B preen our efficien heuriic oluion for he problem when all node can ac a a pm-pair. Furhermore, we how ha if he plier and merger node are rericed in he opology from e.g., echnological conideraion, he SRDC problem urn o be NP-hard (Secion IV-C). A. Ineger Linear Program for Minimum Co SRDC In hi ecion, we preen an ILP o obain an opimal urvivable rouing R in erm of bandwidh co. The ILP formulaion provide he hree rouing DAG for SRDC. To do o, we need o o inroduce he o called reduced 1. if k(e) 2 capaciy funcion [8]: k (e) = 1 if k(e) = 1. 0 oherwie
Theorem 3. In [8, Theorem 2] he auhor how ha a urvivable rouing exi in a given graph G = (V, E, k, c) if and only if here i a flow of value hree in G = (V, E, k, c). Theorem 3 will be ued boh in our ILP formulaion and in he heuriic approach decribed in Secion IV-B. Noe ha, given a rouing DAG E A in a criical urvivable rouing, a funcion x A which i half on he edge of an iland and 1 on all oher (pah) edge in E A, form an flow of value 1, according o Claim 2. Armed wih hi fac, we inveigae he benefi which diveriy coding can provide for urvivable rouing. Our goal i o obain he (criical) flow value f(e) in he inpu graph G = (V, E, k, c) which minimize he bandwidh co in erm of Equaion 1 for he connecion D = (,, 2). The hree flow are denoed a w {A, B, A B} = W, repecively, wih correponding (real) flow variable x w (e) and indicaor variable f w (e). Baed on Theorem 3 he reduced capaciy value k (e) enure ha he failure of an arbirary edge e diconnec a mo one rouing DAG, hu, a lea wo rouing DAG remain which connec and, i.e., he daa can be decoded a he deinaion. Our objecive i o minimize he bandwidh co of he SRDC problem in erm of Equaion 1: min e E c(e) f(e). The following conrain are required: (i,j) E w W, i V : x w (i, j) 1, if i = x w (j, i) = 1, if i = (j,i) E 0, oherwie, (3) e E: x w (e) k (e), (4) w W w W, e E: x w (e) f w (e), () e E: f w (e) f(e) k(e), (6) w W w W, e E: 0 x w (e) 1, (7) w W, e E: 0 f w (e) are ineger variable. (8) The conrain in Equaion (3) formulae he flow conervaion for each rouing DAG w. Conrain (4) e he maximal flow value baed on he reduced capaciy funcion, while Conrain () e he indicaor variable f w (e) of edge uage for he rouing DAG in G. Conrain (6) e he flow value in G = (V, E, k, c), i.e., if edge e wa ued in an arbirary rouing DAG w, we have o include i in he final oluion wih value f(e) = W f w (e). Conrain (7)-(8) e he bound for he flow variable, and e he ineger conrain for he indicaor variable f w (e). Noe ha he f w (e) variable correpond o he edge e w in he oluion, i.e., provide he hree end-o-end DAG. Since x A +x B +x A B give an flow of value 3 in G, from Theorem 3 we ge ha f i indeed urvivable. B. Heuriic Approach for Finding he Opimal Coding Graph A he ILP in Secion IV-A i NP-hard, he running ime of i oluion can be really long, epecially in large nework. In hi ecion, we preen a fa heuriic approach for finding a urvivable rouing in he capaciy conrained cae. Algorihm 2: Survivable Rouing wih Diveriy Coding - Capaciy Conrained (SRDC-C) Inpu: G = (V, E, k, c), D = (,, 2), α Reul: R = (V R, E R, f) 1 begin 2 Define capaciy k, k : E R +, co c : E R + and edge e E n = ; E = ; E = ; // Creae graph G = (V, E, k, c ). 3 Add e E o E wih k (e) = min {1, k(e)} and c (e) = c(e); 4 for e = (u, v) E : 2 k(e) do Add exra edge e n = (u, v) o E n wih k (e n ) := 0. and c (e n ) := c(e) α; 6 E := E E n ; // Phae 1: Find flow in G = (V, E, k, c ). 7 Find a minimum co flow f of value 3 in G wih repec o he reduced capaciy funcion k ; 8 for e = (u, v) E do 9 Se k (e) := 0; 10 Se e n = (u, v) E n : k (e n ) := 0; 11 if 0 < f (e) hen 12 if e n = (u, v) E n : 0 < f (e n ) hen 13 Add edge e o E and e k (e) := 1; 14 for e 1 = (u 1, v 1 ), e 2 = (u 2, v 2 ) E, e 1 e 2 do 1 if no pair of edge-dijoin pah in G \ (e 1 e 2 ) hen 16 e 1 n = (u 1, v 1 ), e 2 n = (u 2, v 2 ) E n : k (e 1 n) = 0.; k (e 2 n) = 0.; // Phae 2: Find flow in G = (V, E, k, c ). 17 Find a minimum co flow f of value 3 in G wih repec o he capaciy funcion k ; // Save urvivable rouing R. 18 for e = (u, v) E : 0 < f (e) do 19 Add node u, v o V R (if u, v / V R ); 20 Add edge e o E R ; 21 Se flow value f(e) o 2 if e n = (u, v) E n : 0 < f (e n ), and o 1 oherwie; The deailed decripion i given in Algorihm 2. The inpu of he algorihm i a graph G = (V, E, k, c) and he connecion reque D = (,, 2). The oupu i he urvivable rouing R = (V R, E R, f). In Sep (4)-(6) he auxiliary graph G = (V, E, k, c ) i creaed uing he reduced capaciie (k (e) = min {1, k(e)}). In he ranformaion, he e E edge of G are added o E wih heir original co c(e). Baed on i capaciy k(e) of e E, we add a parallel edge e n = (u, v) (called exra edge) o E if 2 k(e) wih reduced capaciy k (e n ) = 0. and wih co c (e n ) = c(e) α. The main idea of he algorihm i o find a minimum
co flow wih value of 3 in he reduced capaciy graph G = (V, E, k, c ) in Sep (7)-(13) 2, in uch way ha creaing iland (i.e., uing he exra edge e n wih k (e n ) = 0. capaciy) i penalized via a caling facor (i.e., he exra edge have a higher co c (e n ) = c(e) α, α 1). In order o avoid fale iland creaion (reuling from an imprecie elecion of α), baed on he reul f (e) of he fir phae, we prepare he inpu of he econd phae of Algorihm 2. Fir, we iniialize he reduced capaciy for he nex age e E : k (e) = 0. We creae an edge e E, o ha he original edge e = (u, v) i in E, if boh e = (u, v) and he correponding exra edge e n = (u, v) i ued in he oluion of he minimum co flow, in oher word if 0 < f (e) and e n = (u, v) E n : 0 < f (e n ). Addiionally, we e he capaciy of all ued original edge (e = (u, v) E : 0 < f (e)) o k (e) = 1. In Ieraion (14)-(16) we check for every edge-pair e 1 = (u 1, v 1 ), e 2 = (u 2, v 2 ) E, e 1 e 2, wheher here are wo edge-dijoin pah in G\(e 1 e 2 ) or no. If here i no uch flow, hen we conider he correponding exra edge a poenial iland in he urvivable rouing, and we e heir capaciy value o k (e 1 n) = 0. and k (e 2 n) = 0., repecively. Noe ha, he co c (e) remain he ame. In Phae 2 in Sep (17), we earch again for a minimum co flow wih value of 3 beween and, now in graph G = (V, E, k, c ). Finally, he flow value f (e) of he oluion give he urvivable rouing R. Hence, in Sep (18)- (21) he urvivable rouing i aved in a way ha if boh e = (u, v) and e n = (u, v) were ued in he oluion (f (e) > 0 and f (e n ) > 0), hen we e f(e) = 2. If f (e) > 0 and f (e n ) = 0, hen f(e) = 1, and ele f(e) = 0, which give R = (V R, E R, f). The rouing DAG in R can be conruced a hown in [9]. Selecion of Scaling Facor α: We have een ha chooing a proper α for a nework i eenial. For hi purpoe, inuiively we ued he raio α = E V, which i correponding o he deniy (o be pecific, i i half of he average nodal degree) of he nework. In a dener nework i i more likely o have 3 edge-dijoin pah wih a relaively hor hird pah, which migh be he opimal rouing DAG. Therefore, he co of he exra edge i relaively high (α i high) in order o avoid creaion of iland. Meanwhile in pare nework, i i more likely ha he hird pah (if i exi a all) i really long, and no beneficial o ue. Noe ha, if any of he edge e E i ued (f(e) > 0), we have o reerve f(e) = 1 flow on i correponding edge in G (depie he fac ha we ued only 0. in he reduced capaciy graph G ), a he flow value of he rouing DAG are ineger in he opimal urvivable rouing R. Noe ha, heoreically hi could reul in a higher capaciy conumpion han 1 + 1 for ome connecion in pecial opologie. However, a i i hown in Secion V, in real-like nework wih uch a carefully choen α i perform much beer han 1 + 1. 2 Noe ha baed on [8, Theorem 2] hi i equivalen wih finding a urvivable rouing in G = (V, E, k, c). a b 1 G 0 2 2 Fig. 3. The reducion of 2-DPP o he urvivable rouing problem (graph G = (V, E, c) i hown). C. Finding an SRDC in Direced Graph wih Rericed pm- Pair i NP-Hard Here, we how ha if from echnological conideraion no all node can perform pliing and merging, he urvivable rouing problem urn o be NP-hard. In our proof we will ue he well-known 2-Dijoin Pah Problem (2-DPP), ha ha been proven o be NP-complee, boh for he edge- and nodedijoin cae by Forune e al. [1]. Definiion 4. 2-Dijoin Pah Problem (2-DPP) Inpu: Direced graph G 0 = (V 0, E 0 ) and four diinc node 1, 2, 1, 2 V 0. Queion: I here an edge-dijoin pah-pair P 1 : 1 1 and P 2 : 2 2 in G 0? Theorem 4. I i NP-complee o decide he exience of an SRDC in a direced graph G = (V, E, k, c) if he candidae plier (P V ) and merger (M V ) node are rericed o a given ube of he node. Proof: Fir, we give a polynomial-ime reducion of he 2-DPP problem o our urvivable rouing problem. For hi, we add ix new node,, a, b, c, and d o G 0 and define he nework G = (V, E, k, c) a follow: V = V 0 {,, a, b, c, d}, E = E 0 {(, 1 ), (, a), (, b), (a, 2 ), (a, c), (b, d), ( 1, c), ( 2, d), (c, ), (d, )}, k(e) = 2 if e {(c, ), (d, )}; and 1 oherwie. Beide he inpu graph, i i given, ha P = {a} can only be a plier node, and M = {} can only be a merger. See alo Figure 3 for he ranformaion of G 0 o G = (V, E, c). G wa conruced in polynomial ime. Therefore, i i ufficien o how ha here exi wo direced dijoin pah beween node 1 1 and 2 2 if and only if here exi a urvivable rouing R for G. If here exi wo direced edge-dijoin pah beween node 1 1 and 2 2, hen he flowing hree rouing DAG give a urvivable rouing: E A : i he pah b d, E B : conain pah 1 1 c, and E A B :coni of pah a and iland a beween pm-pair a and wih edge-dijoin pah a 2 2 d and a c. 1 c d
The proof i by conradicion. Aume here i an SRDC in G, bu here i no direced edge-dijoin pah-pair beween node 1 1 and 2 2. Noe ha (c, ) and (d, ) form a 2 link cu in G, while (, 1 ), (, a) and (, b) i a 3 edge cu, hu here hould be a plier node beween hee cu; however a i he only node ha can be a plier. Thu a i a plier node and wo pah are ravering hrough 1 and 2. A he rouing i urvivable, wo pah have o go from 1 and 2 o 1 and 2 in G. Becaue every edge in G 0 ha k(e) = 1, he wo pah can no ue he ame (muli-)edge in G ; hu, hee wo pah are edge-dijoin in G 0. However, owing o he indirec aumpion, hee are no 2 edge-dijoin pah 1 1 and 2 2. Therefore, hee edge-dijoin pah could be 1 2 and 2 1 only. In hi cae edge (c, ) i ravered by wo copie of he ame daa, one hrough a c and an oher hrough a 2 1 c. Thu he rouing i vulnerable of edge failure (d, ) becaue afer deleing hi edge here will be a cu of capaciy 1 in he SRDC (coniing of edge (, a)). V. EXPERIMENTAL RESULTS In hi ecion we inveigae he bandwidh co in erm of Eq. (1) of he differen urvivable rouing approache hrough imulaion. We compare our mehod o he heoreical lower bound on he bandwidh co of urvivable rouing mehod (auming he connecion daa can be pli ino arbirary many par), called lower bound [14] on he char. We alo compare our algorihm o he bandwidh co of he mo common deployed urvivable rouing cheme, he 1 + 1 proecion, which i he 2-approximaion of he urvivable rouing problem [12] wih auming wo daa par 3. In he char SRDC-I refer o Algorihm 1, SRDC-C refer o Algorihm 2, and ILP refer o he Ineger Linear Program preened in Secion IV-A. We inveigaed random generaed real-like planar G = (V, E, k, c) opologie wih differen ize and deniie, and ome real world ranpor nework opologie, oo. The edge capaciie were e high enough o enure infinie edge capaciie for all demand. Thi i imporan in he fair comparion of he bandwidh co of differen mehod a we can ge rid of blocking becaue of lack of reource. Noe ha, raffic demand D = (,, 2) were generaed beween all pair, and all he arc have uni co ( e E : c(e) = 1). A. Gap o he Theoreical Lower Bound The imulaion reul in pare nework are preened in Figure 4a. Thi i an excellen example why 1 + 1 i ill he mo ofen deployed proecion cheme, a he gap beween he bandwidh co of 1 + 1 and he heoreical lower bound for urvivable rouing i mall. However, our SRDC mehod ouperform 1 + 1 even in hi cenario, when 1 + 1 perform well. In fac, he SRDC-I and he ILP reach he heoreical lower bound, a opimal urvivable rouing can be reached wih dividing connecion daa ino wo par. 3 Noe ha he 2-approximaion algorihm for feaible coding graph preened in [8] give 1 + 1 for mo pracical cenario. Figure 4b how he imulaion reul in maximal planar graph. I can be oberved ha all of our mehod approaching he heoreical lower bound. However, in hee opologie he heoreical lower bound require ha connecion daa i divided ino more han wo daa par, which i no alway feaible from a pracical poin of view. The gap beween 1+1 and our SRDC mehod i ignifican, epecially if we ake ino conideraion he fac, ha he bandwidh co reducion of 1 uni mean ha every SRDC connecion in he nework ue one le bandwidh uni han 1 + 1. B. Scalabiliy in Term of Nework Size In Figure 4c and in Figure 4d, we inveigaed he performance of SRDC in larger nework. One can oberve ha he performance gap beween he minimum co SRDC-I oluion and 1 + 1 grow a he nework ize increae, which clearly how he benefi of SRDC, while he impliciy of 1 + 1 i mainained. Furhermore, he SRDC-C heuriic mehod ouperform 1 + 1 in boh cenario. In pecific, i perform cloe o he minimum co SRDC-I in maximal planar graph, while i running ime i 10 ime faer. Thi i becaue he maximal planar graph are relaively dene nework. Thu, here are a number of dijoin pah beween and, which likely reul 3 edge-dijoin pah for boh algorihm. C. Performance Analyi in Real-World Topologie Finally, in hi ecion we inveigaed he performance of our mehod in real nework opologie (SNDLib [16] and Rockefuel [17]). In hi cenario he edge capaciie are conrained, i.e., we idenified a cerain number of edge which are mo prone o congeion baed on heir beweenne cenraliy value. We conidered hee edge a boleneck in he imulaion (i.e., only a ingle capaciy uni k(e) = 1 i available on hee edge), hu, 1 + 1 and SRDC-I canno ue hem in he urvivable rouing. One can oberve in Figure ha a he number of boleneck edge increae, he average bandwidh co of 1 + 1 and SRDC-I increae dramaically, while he average bandwidh co of SRDC-C cale much beer in erm of he number of boleneck link. VI. CONCLUSIONS Survivable rouing wih diveriy coding (SRDC) i a novel, eaily deployable rouing cheme in ranpor nework which keep he ulra-fa recovery and impliciy (boh in compuaion and operaion) of 1 + 1. Furhermore, SRDC can reduce he bandwidh co of 1 + 1 in mo nework cenario wih 1 uni per connecion, which could lead o a ignifican capaciy aving in ranpor nework wih exceive number of connecion. Thi require only ome minor modificaion in he curren operaion of he widely deployed dedicaed 1 + 1 proecion from he ervice provider. A a miing link of he pracical implemenaion of SRDC, we inveigaed i opimal capaciy allocaion. We howed ha a minimum co rouing for SRDC can be compued in polynomial ime wihou capaciy conrain on he link, while he problem urn o be complex when boh boleneck link and limied node capabiliie coexi in he nework.
average co per connecion 9. 9 8. 8 7. 1+1 7 lower bound 6. SRDC-C 6 ILP. SRDC-I 10 1 20 2 30 3 40 #node (a) Opimaliy gap in pare graph 6. 6. 4. 4 3. 3 10 1 20 2 30 3 40 #node (b) Opimaliy gap in maxplan graph 14 13 12 11 10 9 1+1 8 SRDC-C SRDC-I 7 20 40 60 80 100 #node (c) Performance in pare graph 10 9 8 7 6 4 20 40 60 80 100 #node (d) Performance in maxplan graph Fig. 4. Bandwidh co in pare (average nodal degree beween 2.4 and 3.2) and maximal planar (maxplan) graph (average nodal degree beween 4.2 and.7) wih infinie capaciie. The legend for all figure are hown in Fig. 4a and Fig. 4c. average co per connecion 36 34 32 30 28 26 24 22 20 18 16 1+1 SRDC-C SRDC-I 10 1 20 2 30 3 40 #of boleneck (a) Bandwidh co in NSFNET (79 node, 108 link wih diameer 16) [16] 11 10 9 8 7 6 1+1 SRDC-C SRDC-I 10 1 20 2 30 #of boleneck (b) Bandwidh co in Abovene (17 node, 37 link, wih diameer 4) [17] Fig.. The bandwidh co in real-world opologie in he capaciy conrained cenario, depending on he number of boleneck link. ACKNOWLEDGMENTS Reearch of J. Tapolcai and P. Babarczi wa parially uppored by he Hungarian Scienific Reearch Fund (OTKA gran K108947). P. Babarczi wa uppored by he Jáno Bolyai Reearch Scholarhip of he Hungarian Academy of Science (MTA). Z. Király and E. Bérczi-Kovác received gran no. CNK 77780 and no. K 109240 from he Naional Developmen Agency of Hungary, baed on a ource from he Reearch and Technology Innovaion Fund. L. Rónyai wa uppored by he Hungarian Reearch Fund (OTKA gran NK1064). Thi documen ha been produced wih he financial aiance of he European Union under he FP7 GÉANT projec gran agreemen number 60243 a par of he MINERVA Open Call projec. REFERENCES [1] J. W. Suurballe, Dijoin pah in a nework, Nework, vol. 4, pp. 12 14, 1974. [2] K.-S. Sohn, S. Y. Nam, and D. K. Sung, A diribued LSP cheme o reduce pare bandwidh demand in MPLS nework, IEEE Tranacion on Communicaion, vol. 4, no. 7, pp. 1277 1288, 2006. [3] P.-H. Ho, J. Tapolcai, and T. Cinkler, Segmen hared proecion in meh communicaion nework wih bandwidh guaraneed unnel, IEEE/ACM Tran. on Neworking, vol. 12, no. 6, pp. 110 1118, 2004. [4] R. Koeer and M. Médard, An algebraic approach o nework coding, IEEE/ACM Tran. on Neworking, vol. 11, no., pp. 782 79, 2003. [] R. Ahlwede, N. Cai, S. Li, and R. Yeung, Nework informaion flow, IEEE Tran. on Informaion Theory, vol. 46, no. 4, pp. 1204 1216, 2000. [6] S. Jaggi, P. Sander, P. Chou, M. Effro, S. Egner, K. Jain, and L. Tolhuizen, Polynomial ime algorihm for mulica nework code conrucion, IEEE Tran on IT, vol. 1, no. 6, pp. 1973 1982, 200. [7] E. Ayanoglu, I. Chih-Lin, R. Gilin, and J. Mazo, Diveriy coding for ranparen elf-healing and faul-oleran communicaion nework, IEEE Tran. on Communicaion, vol. 41, no. 11, pp. 1677 1686, 1993. [8] S. Rouayheb, A. Sprinon, and C. Georghiade, Robu nework code for unica connecion: A cae udy, IEEE/ACM Tranacion on Neworking, vol. 19, no. 3, pp. 644 66, 2011. [9] P. Babarczi, J. Tapolcai, L. Rónyai, and M. Médard, Reilien flow decompoiion of unica connecion wih nework coding, in Proc. IEEE Inl. Symp. on Informaion Theory (ISIT), 2014, pp. 116 120. [10] A. Markopoulou, G. Iannaccone, S. Bhaacharyya, C. Chuah, and C. Dio, Characerizaion of failure in an IP backbone, in Proc. IEEE Infocom, vol. 4. Cieeer, 2004, pp. 2307 2317. [11] G. Ellina, E. Bouille, R. Ramamurhy, J. Labourdee, S. Chaudhuri, and K. Bala, Rouing and reoraion archiecure in meh opical nework, Opical Nework Magazine, vol. 4, no. 1, pp. 91 106, 2003. [12] G. Brighwell, G. Oriolo, and F. B. Shepherd, Reerving reilien capaciy in a nework, SIAM journal on dicree mahemaic, vol. 14, no. 4, pp. 24 39, 2001. [13] C. Fragouli and E. Soljanin, Informaion flow decompoiion for nework coding, IEEE Tranacion on Informaion Theory, vol. 2, no. 3, pp. 829 848, 2006. [14] P. Babarczi, A. Paic, J. Tapolcai, F. Némeh, and B. Ladóczki, Inananeou recovery of unica connecion in ranpor nework: Rouing veru coding, acceped o Elevier Compuer Nework, 201. [1] S. Forune, J. Hopcrof, and J. Wyllie, The direced ubgraph homeomorphim problem, Theoreical CS, vol. 10, no. 2, pp. 111 121, 1980. [16] S. Orlowki, M. Pióro, A. Tomazewki, and R. Weäly, SNDlib 1.0 Survivable Nework Deign Library, in Proc. INOC, 2007. [17] N. Spring, R. Mahajan, and D. Weherall, Meauring ISP opologie wih rockefuel, ACM SIGCOMM Compuer Communicaion Review, vol. 32, no. 4, pp. 133 14, 2002.