Multiple Failures. Diverse Routing for Maximizing Survivability. Maximum Survivability Models. Minimum-Color (SRLG) Diverse Routing

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1 Mulple Falure Dvere Roung for Maxmzng Survvably One-falure aumpon n prevou work Mulple falure Hard o provde 100% proecon Maxmum urvvably Maxmum Survvably Model Mnmum-Color (SRLG) Dvere Roung Each lnk ha one or more SRLG aocaed wh Each SRLG ha falure probably: p Correpondng SRLG urvval probably: = 1 - p Specal cae: Each SRLG ha eual falure probably Each lnk ha ngle unue SRLG Lnk falure are ndependen p = lnk falure probably = 1 - p = lnk urvval probably Poble goal and objecve: Fnd ngle pah wh maxmum urvvably Fnd wo lnk-djon pah uch ha urvvably maxmzed Each SRLG repreened by a color Mnmze oal color va lnk djon pah Conran Lnk djon under SRLG Objecve Mn-Sum, Mn-Overlappng Problem Saemen: Mnmum Toal Color NP-Compleene for Sngle Pah Problem Gven G = (N,L), e of color C = {c1, c2, c3,, ck}, color cl C for every lnk l L A ource and a denaon are gven a,, N Fnd one or more djon pah from ource node o denaon node uch ha he oal number of color on he pah mnmum Applcaon: Fndng he mo relable pah() n nework wh mulple falure Fndng pah() raverng he mnmum number of doman (e.g., nework operaor) Reducon from he Mnmum Se Coverng Problem Gven a fne e S = {a 1, a 2,, a n }, and a collecon C = {C 1, C 2,, C m } uch ha each elemen n C conan a ube of S, here a mnmum ube C of C uch ha every member of S belong o a lea one member of C? c c 5 2 c 4 c 6 c 4 c 4 1

2 NP-Compleene for Sngle Pah Problem NP-Compleene for Djon Pah Problem Example: S = {a 1, a 2, a 3, a 4 }, C = {C 1, C 2, C 3, C 4, C 5 }, C 1 = {a 1, a 2 }, C 2 = {a 2, a 3 }, C 3 = {a 1, a 3 }, C 4 = {a 3, a 4 }, C 5 = {a 1, a 4 }. Conruc graph G uch ha G ha a mnmum color ngle pah f and only f C ha a mnmum e coverng all elemen n S. Reducon from he Mnmum Se Coverng Problem. Gven a fne e S = {a 1, a 2,, a n }, and a collecon C = {C 1, C 2,, C m } uch ha each elemen n C conan a ube of S, here a mnmum ube C of C uch ha every member of S belong o a lea one member of C? Two cae: Node djon conran Lnk djon conran NP-Compleene for Djon Pah Problem (Node-Djon) NP-Compleene for Djon Pah Problem (Lnk-Djon) Example: S = {a 1, a 2, a 3, a 4 }, C = {C 1, C 2, C 3, C 4, C 5 }, C 1 = {a 1, a 2 }, C 2 = {a 2, a 3 }, C 3 = {a 1, a 3 }, C 4 = {a 3, a 4 }, C 5 = {a 1, a 4 }. Conruc graph G uch ha G ha wo mnmum color nodedjon pah f and only f C ha a mnmum e coverng all elemen n S. Example: S = {a 1, a 2, a 3, a 4 }, C = {C 1, C 2, C 3, C 4, C 5 }, C 1 = {a 1, a 2 }, C 2 = {a 2, a 3 }, C 3 = {a 1, a 3 }, C 4 = {a 3, a 4 }, C 5 = {a 1, a 4 }. Conruc graph G uch ha G ha wo mnmum color lnkdjon pah f and only f C ha a mnmum e coverng all elemen n S. c 2 c 2 c3 a 1 c 3 a 3 u 1 c 3 a 1 u 3 c 3 a 3 c 4 c 5 c 4 d c 5 v 1 c 4 c 4 d c 2 a 2 c 5 a 4 u 2 a 2 u 4 a 4 c 2 c 5 Djon Pah Color Reducon Algorhm Djon Pah All Color Opmzaon Algorhm Sep 1. Run Suurballe algorhm and fnd wo lnk-djon pah p1 and p2. Aume he collecon of all he color on p1 and p2 e Cp. Sep 2. Go hrough every color n Cp. Selec he color uch ha, afer he lnk of ha color are removed from he nework, we run Suurballe algorhm agan and oban wo lnk-djon pah wh he mnmum number of oal color whch alo le han Cp. Remove he lnk of he eleced color. Sep 3. Repea Sep 1 and 2 unl he number of color on he lnk-djon pah canno be furher reduced. Runnng me O(m 2 n 2 logn). n he number of node, m he oal number of color n he nework. Sep 1. Run Suurballe algorhm and fnd wo lnk-djon pah p1 and p2. Aume he oal number of color on he wo pah Cp. Sep 2. Se he lnk co o zero on he lnk of one color, hen run Suurballe algorhm and fnd wo lnk-djon pah. Repea for all he color n he nework and elec he one ha reul n wo lnk-djon pah wh he mnmum number of color whch alo le han Cp. Keep he co o zero on he lnk of he eleced color. Sep 3. Repea Sep 1 and 2 unl he number of color on he lnk-djon pah canno be furher reduced. Runnng me O(m 2 n 2 logn). n he number of node, m he oal number of color n he nework. 2

3 Numercal Reul Problem Saemen: Mnmum Overlappng Color Gven G = (N,L), e of color C = {c1, c2, c3,, ck}, color cl C for every lnk l L A ource and a denaon are gven a,, N Fnd wo djon pah from ource node o denaon node uch ha hey hare he mnmum number of common color c c 5 2 c 4 c 6 c 4 c 3 Proof of NP-Compleene Solvng he mnmum overlapped color djon-pah problem Reducon from he pah proecon problem under he rk djon conran. Replace every rk ID wh an unue color. If we were able o olve he MOCDP problem and fnd wo lnk-djon pah from o d wh he mnmum overlapped color, hen he pah hould alo be color-djon (.e., rk djon) f uch colordjon pah ex n he nework. ILP formulaon Heurc: Smple Two Sep Algorhm Mnmum Color Fr-Pah Algorhm Jon-Search Mnmum Overlapped Color Algorhm Smple Two Sep Algorhm Mnmum Color Fr-Pah Algorhm Sep 1. Run Djkra algorhm and fnd pah p1. Sep 2. Increae he co of a lnk f he color of he lnk on p1. The addonal co proporonal o he number of lnk of ha color on p1. Sep 3. Remove all lnk on p1. Run Djkra algorhm agan and fnd he econd hore pah p2. Runnng me O(nlogn). n he number of node. May fal n he rap opology. Sep 1. Run Sngle Pah All Color Opmzaon Algorhm and fnd he fr pah p1. Sep 2. Increae he co of a lnk f he color of he lnk on p1. The addonal co proporonal o he number of lnk of ha color on p1. Sep 3. Remove all lnk on p1. Run Djkra algorhm agan and fnd he econd hore pah p2. Runnng me O(m2nlogn). n he number of node, m he oal number of color n he nework. May fal n he rap opology. 3

4 Jon-Search Mnmum Overlapped Color Algorhm Numercal Reul Sep 1. Run Suurballe algorhm and fnd wo lnk-djon pah p1 and p2. Sep 2. Increae he co of a lnk f he color of he lnk on p1. The addonal co proporonal o he number of lnk of ha color on p1. Sep 3. Remove all lnk on p1. Run Djkra algorhm agan and fnd he econd hore pah p1. Sep 4. Increae he co of a lnk f he color of he lnk on p2. The addonal co proporonal o he number of lnk of ha color on p2. Sep 5. Remove all lnk on p2. Run Djkra algorhm agan and fnd he econd hore pah p2. Compare he oal color of pah p1 and p1 wh he oal color of pah p2 and p2. The pah wh fewer overlappng color are reurned a oupu. Runnng me O(n2logn). n he number of node. Maxmum Survvably Ung Two Djon Pah Objecve Funcon Globecom 2006 Survvably probably aocaed wh each lnk Maxmze end-o-end urvvably va lnk djon pah Conran Lnk djon Objecve Max-Surv Survvably of - par: = Maxmze he end-o-end urvvably Problem Saemen: Maxmum Djon Pah Survvably NP-Compleene Gven G = (N,L), urvval probably on lnk. A ource and a denaon are gven a,, N Maxmze Q 1 + Q 2 Q 1 Q 2 where Q1 =, lnk pah 1, Q2 = j, lnk j pah Reduce 3SAT problem o he decon veron of he Max-Surv problem. 3-afably Se of boolean varable v1, v2, v3,, vn Collecon of claue C =C1 C2...Cm each claue C logcal OR of 3 varable or NOT of varable e.g., C1 = v1 OR ~v2 OR v4 C2 = ~v2 OR v5 OR ~v7 Where ~ ndcae logcal NOT E = C1 AND C2 AND C3 AND Cm Doe here ex a ruh agnmen for v1, v2,, vn uch ha E TRUE? 4

5 Tranformaon Lemma 1 Tranform he objecve funcon For any conan, 0 < < 1, we have log Q1 + Q2 Q1Q 2 = a b = + a+ b = log + j log j j log + log j Gven oal wegh d, he larger he wegh dfference beween he wo pah, he hgher he urvvably of wo djon pah proecon e.g. gven oal wegh of wo pah 8, we have for any < < > where = j a log b = log j Lemma 2 Maxmum Survvably Algorhm If he wegh dfference beween he wo pah fxed, hen he horer he oal wegh of he wo pah, he hgher he urvvably of wo djon pah proecon e.g. gven wegh dfference beween wo pah 4, we have > + for any 0 < < 1 Sep1: Agn wegh log o each lnk Sep2: Run Suurballe algorhm o fnd wo djon pah wh mnmum oal wegh Sep3: Reconruc he wo djon pah o make wegh dfference beween wo pah a much a poble Theorem 1 Theorem 2 Gven a nework and - par, we can fnd he mnmum oal wegh of he wo pah, denoed a l, and we can alo fnd he mnmum wegh of a ngle pah, denoed a The upper bound of he MAX-SURV problem l + l Gven a nework and - par, we can fnd he mnmum oal wegh of he wo pah, denoed a l The lower bound of he MSA algorhm l /2 l /2 l + 5

6 Smulaon Aumpon Numercal Reul: 14-node rap nework 24-node NSFNET and 14-node rap nework Unformly drbued falure probably on each lnk beween [0, p], 0<p<1 Numercal Reul: 24-node NSFNET Numercal Reul: Reue falure rae and Hop coun Table I Reue Falure Rae Table II Hop Coun Generc Rk A Graph Model for Maxmum Survvably n Meh Nework under Mulple Generc Rk Mulple mulaneou rk n he nework A Generc rk correpond o a ngle-lnk falure, a mulple-lnk falure, or a node falure. Uneual falure probable No lm on he number of rk on each lnk and node Dffcul o fnd wo rk-djon pah Fnd wo lnk-djon pah wh mnmum oal falure probably 6

7 Problem Defnon Tranformaon Gven G = (N,L), where N he e of node and L he e of Lnk n he nework; Rk drbuon, urvvable probably for rk A ource and a denaon are gven a,, N Maxmze Q3 ( Q1 + Q2 Q1Q 2) where Q 1 =, only pah 1, Q 2 = j, j only pah 2, Q 3 = k, rk k pah 1 and pah 2. NP-Compleene Rercng o MAX-SURV problem when allowng each rk o be aocaed wh only one lnk Tranform he objecve funcon For any conan, 0 < < 1, we have a = log Q ( Q + Q Q Q = a b = + where = log ) log + log k + log k k k + a + b c j log j + k b = log j log k j + log k j log + log + j k k log k c = log k k Auxlary Graph Model Theorem 1 Bac dea: add horcu edge beween adjacen lnk o avod he cae n whch he ame rk occur more han once on he ame pah The hore pah from ource o denaon generaed on he auxlary graph he pah wh he maxmum urvvably 0 < < 1 Maxmum Survvably Graph Algorhm Maxmum Survvably Graph Algorhm (con.) Sep1: Conruc he auxlary graph G (V,E) baed on he nework logcal opology and he rk drbuon Sep2: Run he hore pah algorhm o fnd he fr hore pah from ource TV o denaon TV d n he auxlary graph Sep3: Remove all he edge along he hore pah ha are dreced oward d and mulply he lengh of lnk on he revere drecon of he hore pah wh a facor ε Sep4: Run he hore pah algorhm agan o fnd he econd hore pah Erae he nerlacng edge o ge wo djon pah P 1 and P 2 Sep5: Map P 1 and P 2 n auxlary graph G back o he pah P1 and P2 n he orgnal graph G Snce CE keep he connecon nformaon of he orgnal graph, we ju need o map CE n he auxlary graph back o her correpondng orgnal lnk whle gnorng he TE and SCE n he auxlary graph 7

8 Smulaon Aumpon Numercal Reul 24-node NSFNET, and 14-node rap nework Unformly rk drbuon, falure probably for each rk beween [0, p], 0<p<1 Numercal Reul (con.) Numercal Reul (con.) A hop coun of 7 he mnmum average hop coun obaned by Suurballe algorhm In he 14-node rap nework, Reue Falure Rae, a hgh a 9.89% when runnng he wo-ep hore-pah algorhm Oulne IP over Opcal Nework Survvably n Mul-Layer Nework Opcal Layer Survvably IP Layer Dual-Homng Proecon Survvably n Mul-Layer Nework Sac Dual-Homng Proecon Problem Dynamc Dual-Homng Proecon Problem Summary 8

9 IP over Opcal Core Nework and Wrele Acce Nework Opcal Nework Survvably Survvably enable a nework o connue carryng raffc n he even of a falure Type of falure Opcal core nework lnk/node falure IP rouer falure/congeon Acce nework falure Proecon Reerve back-up reource n advance Guaranee urvvably for any ngle lnk falure Coly n erm of reource ued Reoraon Fnd pare reource afer falure ha occurred No need o reerve exra reource n advance No guaranee ha reource wll be avalable Pah v. Lne Proecon Dedcaed v. Shared Proecon Pah proecon Reerve end-o-end lnk-djon back-up wavelengh pah More reource effcen Lne proecon Reerve back-up fber pah around each lnk May allow mulaneou proecon of mulple workng pah Dedcaed proecon Reerved back-up reource are dedcaed o a ngle workng connecon Eaer managemen Shared proecon Reerved back-up reource may be hared by mulple workng connecon More reource effcen IP Layer Survvably Survvably Mechanm n Mul-Layer Nework Dual homng Each ho conneced o wo edge rouer If one rouer fal or congeed, ue he oher rouer IP roung updae If a lnk fal n he IP nework, roung able are updaed Opcal layer Pah proecon IP layer Dual homng Wrele layer Dual homng 9

10 IP-over-WDM Dual-Homed Nework Traffc Reue Model Sac: Gven: All reue beween all ource and denaon ho Phycal opology Fnd: Dual home for each ho Workng and back-up pah for all reue n he opcal nework Objecve: Mnmze oal amoun of reource ued Soluon approach: ILP formulaon Dynamc: Gven: A new arrvng reue Phycal opology Reource ued by eablhed connecon reue Fnd: Dual home for new reue Workng and back-up pah for new reue Objecve: Mnmze reource ued for new reue Soluon approach: varou heurc Sngle-Layer Survvably Problem Dual Homng: Survvably n he Acce Nework Unproeced Dual-Homng Archecure No neceary o be djon w u v d Dual homng archecure (Unproeced) Proecon: Survvably n he Core Nework Prmary Pah djon w u Backup Pah d Agn wo cloe/random IP rouer o be he dual home of a parcular ho Pah from dual home o denaon are no necearly djon Survvable agan acce node falure No urvvable agan lnk falure n he opcal layer Proecon Archecure Unproeced Dual-Homng Archecure Mul-Layer Survvably Problem Dual Homng wh Proecon: Survvably n he Acce and Core Nework djon No neceary o be djon w u v d Agn wo cloe/random IP rouer o be he dual home of a parcular ho Selec djon pah from dual home o denaon Survvable agan ngle acce node falure OR ngle lnk falure n he opcal layer NOT urvvable agan wo mulaneou ndependen falure, ha one n he acce layer (IP rouer falure) and one n he WDM layer (WDM lnk falure) djon No neceary o be djon 10

11 Mul-Layer Survvably Problem Decrpon Proeced Dual-Homng Archecure Objecve: Provde urvvably agan wo ndependen falure, one node (or lnk) falure n he acce nework and one lnk falure n he core nework, ung Dual-Homng and Proecon Dual-Homng provde urvvably agan a ngle node (or lnk) falure n he IP acce nework Proecon provde urvvably agan a ngle lnk falure n he WDM core nework Coordnaed IP-over-WDM Dual-Homng Proecon RULES OF SHARING - p 1 a and p 1 b (Prmary and econdary from 1 o d) mu be djon - p 2 a and p 2 b (Prmary and econdary from 2 o d) mu be djon - p 1 a & p 2 a, p 1 b & p 2 b, p 1 a & p 2 b, and p 2 a & p 1 b no necearly djon Sac Dual-Homng Proecon ILP Formulaon Sac Dual-Homng Proecon ILP Formulaon A WDM nework can be modeled a a dreced graph G =<V,E>, where V e of OXC and E e of WDM lnk c(e): Wavelengh co of a WDM lnk e E W: Maxmum number of wavelengh n each lnk R be he e of all ac reue n G Le each reue, r k = {{ k1, k2 }, d k }, where k1 and k2 are wo OXC conneced o he dual homed edge rouer of ho k, and d k he denaon OXC ha n urn conneced o an IP rouer whch connec o he denaon ho Prmary lghpah from k1 o d k be denoed by p a1 (k) Backup lghpah from k1 o d k be denoed by p b1 (k) Prmary lghpah from k2 o d k denoed by p a2 (k) Backup lghpah from k2 o d k denoed by p b2 (k) L k e of all lnk ued n he prmary and backup lghpah for reue r k L k defned a p a1 (k) p b1 (k) p a2 (k) p b2 (k) x an (k, e) 1 f pah p an (k) ue lnk e, 0 oherwe, when n = 1,2 x bn (k, e) 1 f pah p bn (k) ue lnk e, 0 oherwe, when n = 1, 2 y ek 1 f any pah for reue r k ue lnk e, 0 oherwe y e oal number of wavelengh ued n lnk e. In(v) e of lnk ha end a node v Ou(v) e of lnk ha ar from node v Objecve: mnmze Sac Dual-Homng Proecon ILP Formulaon Sac DHP ILP Reul Subjec o: 11

12 Sac DHP ILP Reul Dual-Homng Proecon Heurc Heurc Mn-Co Nework Flow Shore Pah Mn Sener Tree MCNFH MDSPH MCSPH MSTH MCNFH: Mnmum Co Nework Flow Heurc MDSPH: Mnmal Djon Segmen-Par Heurc MCSPH: Mnmum Co Shore Pah Heurc MSTH: Mnmum Sener Tree Heurc Mnmum Co Nework Flow Heurc Mnmum Co Nework Flow Heurc Fnd opmal lnk-djon prmary and backup pah from one of he ource dual home o he denaon (ue Suurballe Algorhm) Reduce co of edge ued for above prmary and backup pah Fnd opmal lnk-djon prmary and backup lghpah from he oher ource dual home o he denaon, baed on he rule of harng Repea he proce by wchng he order of he dual home, and elec he fnal oluon o be he one wh mnmum co The oluon obaned, n he wor cae, a mo 4/3 me he co of he opmal oluon Tme Complexy: O (N 2 ) Mnmal Djon Segmen-Par Heurc Mnmal Djon Segmen-Par Heurc Generalzed veron of Mn-Co Nework Flow Heurc Selec a node o be he branchng node and fnd a par of lnkdjon pah (ung Suurballe Algorhm) from: Fr dual-home o branchng node Second dual-home o branchng node Branchng node o denaon Repea he prevou ep wh all poble node a branchng node Selec he oluon uch ha oal wavelengh co ued n boh prmary pah and backup pah mnmum Tme Complexy: O (N 3 ) 12

13 Mnmum Co Shore Pah Heurc Mnmum Co Shore Pah Heurc Fnd lnk-djon hore pah from he dual home o he denaon Then, compue wo mnmum co lnk-djon pah beween he dual home hemelve Tme Complexy : O (N 2 ) Mnmum Sener Tree Heurc Fnd he opmal co ree connecng he dual home wh he denaon, h erve a he prmary pah beween he dual-home and he denaon Mnmum Sener Tree Heurc Then, fnd a lnk-djon backup pah from each ource o he denaon Alhough he mnmum Sener ree problem NP-hard n he general cae, polynomal-me olvable when here are only hree ermnal (Sener) node Tme Complexy : O (N 3 ) Smulaon Aumpon Soluon Co V Nework Sze Connecon reue arrval are Poon Gven nework ze and maxmum (ougong) nodal degree, we randomly generae a nework Ougong nodal degree of each node unformly drbued n [1, 2,..., D (Max)] Co of each lnk = 1 13

14 Soluon Co V Nework Sze Soluon Co V Nework Sze (4/3 OPT) Soluon Co V Nework Sze Soluon Co V Nework Sze Summary Coordnaed urvvably mechanm n mul-layer nework can provde varyng degree of urvvably wh a rade-off beween he level of urvvably and he co n reource Coordnaed IP dual homng and opcal proecon can provde urvvably agan wo ndependen falure n IP-over-WDM nework Four heurc, namely MCNFH, MDSPH, MCSPH, and MSTH for he dynamc dualhomng proecon problem are evaluaed MCNFH he be choce for dene nework and MCSPH he be choce for pare nework The coordnaed approach can gnfcanly reduce he co ncurred o provde proecon n he WDM core nework compared o ndependen urvvably oluon a each layer (IP and WDM) Fuure work Survvably n IP over wrele nework Pah proecon problem n opcal nework under SRLG conran 14