Finding Minimum-Cost Paths with Minimum Sharability

Transcription

1 Finding Minimum-Co Pah wih Minimum Sharabiliy S.Q. Zheng Dep. of Compuer Science Unieriy of Texa a Dalla Richardon, TX 75083, USA izheng@udalla.edu Bing Yang Cico Syem 2200 E. Preiden G. Buh HW Richardon, TX 75082, USA binyang@cico.com Mei Yang Dep. of Elec. and Compu. Engg. Unieriy of Neada La Vega, NV 89154, USA meiyang@egr.unl.edu Jianping Wang Dep. of Compuer Science Ciy Unieriy of Hong Kong Ta Chee Ae., Kowloon, HK jianwang@ciyu.edu.hk Abrac In communicaion nework, muliple communicaion pah haring minimum number of link or/and node may be deirable for improed performance, reource uilizaion and reliabiliy. We inroduce he noion of link harabiliy and node harabiliy, and conider he problem of finding minimum-co k pah ubjec o minimum link/node harabiliy conrain. We idenify 65 differen link/node harabiliy conrain, and conider he fundamenal problem of finding minimum-co k pah beween a pair of node under hee conrain. We preen a unified polynomial-ime algorihm cheme for oling hi problem ubjec o 25 of hee differen harabiliy conrain. Keyword: Nework, graph, rouing, nework planning, proocol, algorihm, proecion, reliabiliy, uriabiliy, dijoin pah, muliple pah, nework flow, complexiy. I. INTRODUCTION In a communicaion nework he connecion beween a ource node and a deinaion node i a pah beween hem. Currenly, mo nework employ proocol baed on hore pah rouing algorihm which deermine a ingle pah of minimum co. Finding muliple pah beween a ource and a deinaion ha been propoed. Poenial benefi of muliple pah include improed reliabiliy (e.g. [7], [12], [15], [17], [18], [19], [20], [21], [22]), load balancing (e.g. [6], [16]), higher nework hroughpu (e.g. [8], [16]), and alleiaion of congeion (e.g.[2], [7]). I i deirable ha muliple pah are link or/and node dijoin. Aume ha a nework i modeled a a weighed graph G =(V,E), where V i he e of node, E i he e of link connecing node, and each link i aociaed wih a nonnegaie co. In he lieraure, he complexiie of ariou erion of he problem of finding opimal dijoin pah beween wo node, G hae been ineigaed. Ford and Fulkeron propoed a polynomial-ime algorihm for finding wo pah wih minimum oal co (named he Min- Sum 2-Pah Problem) baed on minimum-co nework flow model [4]. Suurballe and Tarjan proided a differen reamen, and preened algorihm ha are more efficien [13], [14]. Li e al. proed ha he problem of finding wo dijoin pah uch ha he co of he longer pah i minimized (named he Min-Max 2-Pah Problem) are rongly NP-complee [10]. They alo conidered a generalized min-um problem (referred a he G-Min-Sum k-pah Problem) auming ha each link i aociaed wih k differen lengh. The objecie of hi problem i o find k dijoin pah uch ha he oal co of he pah i minimized, where he jh link-co i aociaed wih he jh pah. They howed ha he G-Min-Sum k-pah Problem are rongly NP-complee for k 2 [11]. In [18]- [21], a e of opimal dijoin 2-pah problem wih differen objecie funcion, including he Min-Min 2-pah problem,he α-min-sum 2-pah problem, and he MinSum-MinMin 2-pah problem, are conidered and proed o be NP-complee. Gien a pair of node, finding k, k > 1, dijoin pah, hough deirable, may no alway be poible in pracical nework applicaion for a lea wo reaon. Fir, if he nework i oo pare, uch pah may no phyically exi. Second, if ome link are oerly auraed, addiional raffic on hee link may be prohibied o ha wo dijoin pah wihou uing hee prohibied link do no exi. When k dijoin pah do no exi, alernaiely k pah from he ource o he deinaion wih minimum hared link/node hould be found. In he conex of nework reliabiliy, hee pah can proide parial proecion[22]. In he lieraure, limied work on muliple pah wih minimum number of hared link/node ha been repored. In [3], an algorihm baed on minimum-co nework flow (MCNF) i gien for finding he k-be pah (i.e., k pah wih minimum node haring). Howeer, he algorihm can only be applied o relli graph. In [12], an algorihm i proided o ranform an arbirary graph o a relli graph and hen o obain he k- be pah by he algorihm hown in [3]. A analyzed in [9], hi oluion i only a heuriic one and he complexiy of he ranformaion i quie high. In [9], an MCNF-baed algorihm i propoed for finding k-be pah in arbirary nework. Howeer, only be link-dijoin pah are conidered in [9]. In hi paper, we inroduce he noion of link harabiliy and node harabiliy, which are a ariaion of he concep of ulnerabiliy defined in [15]. We ue hi noion o characerize he degree of link/node haring among differen pah. Larger link/node harabiliy implie more link/node haring among a e of pah. A e of pah are link-dijoin if he link harabiliy of he pah i 0, and hey are node-dijoin if heir node harabiliy i 0 (in hi cae, heir link harabiliy i alo 0). We define fie baic harabiliy conrain: minum link harabiliy conrain, min-um node harabiliy conrain, min-max link harabiliy conrain, min-max node harabiliy conrain, and empy conrain (no rericion on harabiliie). Baed on hee baic harabiliy conrain, we idenify 65 compoie harabiliy conrain, which are obained by elecion and permuaion of baic harabiliy conrain. We ineigae he problem finding minimum-co X/07/$ IEEE 1532

2 pah ubjec o hee harabiliy conrain by conidering he problem of finding minimum-co k pah from node o node in nework G. Thi i a generalizaion of he claical fundamenal problem of finding minimum-co k pah from node o node which ha receied coniderable aenion in he conex of proecing a nework again link/node failure. By preening a general algorihm cheme, we how ha 25 erion of he problem finding minimum-co k pah from node o node in nework G ubjec o link or/and node harabiliy conrain are olable in polynomial ime. The re of he paper i organized a follow. In Secion II, we define link harabiliy and node harabiliy, inroduce 65 differen harabiliy conrain. We how ha a ube of 25 harabiliy conrain are muually inequialen. Secion III preen a unified algorihm cheme ha can be ued o generae differen algorihm for oling he problem of finding minimum-co k pah from o ubjec o he 25 differen harabiliy conrain in polynomial ime. In Secion IV, we generalize our algorihm cheme o ole he problem of finding minimum-co k pah ubjec o conrain on allowable indiidual link and node harabiliie, and he problem of finding minimum-co one-o-many and many-o-one pah ubjec o ariou harabiliy conrain. Secion V conclude he paper. II. LINK AND NODE SHARABILITY CONSTRAINTS We reric our dicuion o direced graph. All our algorihm and claim are applicable o undireced graph, ince undireced graph can be conered o direced graph eaily. In he re of he paper, he erm graph and nework are ued inerchangeably. So are he erm of edge and link. Le G = (V,E) be a direced graph wih non-negaie co l(e) or l(u, ) defined for link e = (u, ). Furher, we aume ha G i imple; i.e. i ha no elf-loop and parallel link. Gien a ource and a deinaion in G, le P = {P 1,P 2,,P k } be a e of k pah (- pah) in graph G =(V,E) from o. We define { 1, e Pi δ(e, P i )= 0, e / P i and δ(e, P )= k δ(e, P i ). i=1 Then δ(e, P ) repreen he number of ime e appear in P. We define (P )= e P(δ(e, P ) 1), and δ(p ) = max(δ(e, P ) 1). e P (P ) i called oal link harabiliy of P, and δ(p ) i called he maximum link harabiliy of P. Clearly, (P )=0if and only if δ(p )=0, and eiher (P )=0or δ(p )=0indicae ha all pah in P are link-dijoin. and Similarly, we define { 1, Pi γ(, P i )= 0, / P i γ(, P) = k γ(, P i ). i=1 Then γ(, P) repreen he number of ime node i hared among pah in P. We define Γ(P )= (γ(, P) 1), and γ(p ) = P,, max (γ(, P) 1). P,, Γ(P ) i called he oal node harabiliy of P and γ(p ) i called he maximum node harabiliy of P. Clearly, Γ(P )=0 if and only if γ(p )=0, and eiher Γ(P )=0or γ(p )=0 indicae ha all pah in P are node-dijoin. The oal co of k pah in P i defined a l(p )= k l(p i ), i=1 where l(p i )= e P i l(e). The problem udied in hi paper hae a common feaure of finding a e of pah P = {P 1,P 2,,P k } in G wih minimum l(p ), ubjec o ariou combinaion of minimum (P ), minimum Γ(P ), minimum δ(p ) and minimum γ(p ) a conrain. Le C = C q,c q 1,,C 1 be an ordered li of conrain. An opimizaion problem Π ubjec o ordered conrain li C i o find a oluion S(I) for an inance I of Π uch ha (1) : S(I) aifie C 1 ; (2) : S(I) aifie C 2 ubjec o condiion (1); (q) : S(I) aifie C q ubjec o condiion (q 1); and (q +1): S(I) ha he opimal oluion alue among all oluion ha aify C q. We call C an ordered compoie conrain, and C i he componen conrain of C. Conrain are defined recuriely. An empy li i an ordered compoie conrain; i i alo called empy conrain. A ingle conrain i an ordered compoie conrain. Then, an ordered compoie conrain i an ordered pair of wo ordered compoie conrain. Haing defined hi recurie rucure, we refer o ordered compoie conrain imply a conrain. Define empy conrain, minimum, minimum Γ, minimum δ, and minimum γ (denoed by, min, min Γ, min δ, and min γ, repeciely) a baic harabiliy conrain. min and min Γ are called min-um conrain, and min δ and min γ are called min-max conrain. In general, we can hae he following e of conrain: Z = { X i,,x 1 X 1,, X i {, min δ, min γ, 1533

3 min, min Γ }, 1 i 4}. Clearly,, =,, X = X and X, = X. LeS(n, r) denoe he number of ordered r-uple of diinc elemen from an n-elemen e. Then, we hae Z = 4 S(4,i)= i=0 4 i=0 4! i! =65 differen conrain. We are able o how ha all of hee 65 erion of he problem of finding minimum-co k pah ubjec o harabiliy conrain are polynomial-ime olable. In hi paper, we focu on 25 conrain lied in Table I, i.e. he compoie conrain wih min-max componen conrain (if any) haing prioriy higher han min-um componen conrain (if any). The erion correponding o conrain C 0 can be reduced o a problem of finding minimum-co nework flow (MCNF) of flow alue k. Thu, uch a problem of finding minimum-co k-pah can be oled by an MCNF algorihm, uch a he ucceie hore pah algorihm [1], in O(k ( E + V log V )) ime. Each conrain C in he re of Table I can be pariioned ino wo componen conrain C and C uch ha C conain min-max conrain and C conain min-um conrain. Then, C can be conidered a a conrain C,C formed by concaenaing C and C. For eay reference, we call C,C he normal form of C. Baed on normal form, we diide he conrain C 1 o C 24 in Table I ino fie clae a follow. Cla 1 : C i empy. Thi cla conain C 1 o C 4. Cla 2 : C i min. Thi cla conain C 5 o C 9. Cla 3: C i min Γ. Thi cla conain C 10 o C 14. Cla 4 : C i min Γ, min. Thi cla conain C 15 o C 20. Cla 5 : C i min, min Γ. Thi cla conain C 20 o C 24. For example, he normal form of conrain C 22 = min, min Γ, min γ i C 20, C 2, and i i in Cla 5. The erion correponding o conrain C 22 i o find a e P of k pah from o uch ha (1) P ha min-max node harabiliy; (2) P ha min-um node harabiliy ubjec o condiion (1); (3) P ha min-um link harabiliy ubjec o condiion (2); and (4) l(p ) = min{p P i a oluion ha aifie (3)}. Thi claificaion i ueful in he analyi of our algorihm cheme. Remark 1: The problem we are conidering i a problem wih a prioriized hierarchy of opimizaion objecie (he oal co of he pah coming he la). The objecie of a higher prioriy narrow feaible oluion pace of he objecie of lower prioriie. According o hi feaure of hierarchical opimizaion objecie conrain, we ue ordered conrain o refer o ordered opimizaion objecie only for he ake of eay underanding. Reader hould keep in mind ha, ricly peaking, hi i no a conrained opimizaion problem in he claical ene. Conider wo erion Π 1 and Π 2 of he minimum-co k- pah problem ubjec o ordered compoie conrain X and Y, repeciely. For he ame graph G, heir oluion can be differen if X and Y are differen. The following aemen i alway rue: if Y i a ubli of X, i.e. all conrain in Y are in X and hey are in he ame order a hey appear in X, hen he co of he oluion ubjec o Y i no larger han he co of he oluion ubjec o X. Clearly, aifying more harabiliy conrain end o reduce link/node haring wih increaed co. Thu, here i a radeoff beween co and harabiliy beween chooing X and Y. The 25 differen erion of he minimum-co k-pah problem defined by he conrain of Table I proide a wide pecrum of conrain prioriie and co-harabiliy radeoff for he ame nework. In he conex of finding minimum-co k- pah, we ay ha wo conrain X and Y are equialen if and only if any opimal oluion obained under X i alo an opimal oluion obained under Y for any nework. One queion arie: are all he 25 compoie conrain gien in Table I muually inequialen? The following heorem gie a definie anwer. Theorem 1: Conrain of Table I are muually inequialen. Proof: See Appendix. III. AN ALGORITHM SCHEME In hi ecion, we preen a unified algorihm cheme for all erion of he problem of finding opimal k-pah ubjec o he conrain defined in Table I. Thi cheme, which i ued o generae lighly differen algorihm by reducing he problem of finding a e P of minimum-co k pah in G o finding a minimum-co flow f in G uing differen co and capaciy funcion, ha hree ep. Sep 1 : Compue min-max link harabiliy k L or/and minmax node harabiliy k N if needed, and conruc flow nework G = (V,E ) from G = (V,E) according o harabiliy requiremen. Sep 2 : Find a minimum-co flow f of flow alue k from o in G. Sep 3 : Conruc a e P of k-pah in G from he flow f in G. For Sep 1, wo ranformaion, TRANSFORM-1 and TRANSFORM-2, are inroduced. TRANSFORM-1: Obain G =(V,E ) from G =(V,E) by node pliing a follow: replace each node ha i neiher nor by wo node and uch ha all link ending a in G alo end a in G and all link originaing from in G originae from, and hen add a link (, ) (ee Figure 1 and Figure 3). TRANSFORM-2: Obain graph G =(V,E ) from G = (V,E ) by link pliing a follow: replace each link e in G by wo parallel link wih he ame direcion of e. We denoe he wo link generaed from a link (u,) in G correponding o link (u, ) in G by (u,) and (u,), which are called he primary link-generaed u link (or imply, primary u link) and he econdary link-generaed u link (or imply, econdary u link), repeciely (noe: u can be ource node ). We denoe he wo link generaed from a link (, ) in 1534

4 conrain conrain C 0 C 1 min δ = C 0, C 1 = C 13 min Γ, min γ,min δ = C 10, C 3 =, min δ min Γ, min γ,min δ C 2 min γ = C 0, C 2 = C 14 min Γ, min δ,min γ = C 10, C 4 =, min γ min Γ, min δ,min γ C 3 min γ,min δ = C 0, C 3 = C 15 min Γ, min = C 15, C 0 =, min γ,min δ min Γ, min, C 4 min δ,min γ = C 0, C 4 = C 16 min Γ, min, min δ = C 15, C 1 =, min δ,min γ min Γ, min, min δ C 5 min = C 5, C 0 = C 17 min Γ, min, min γ = C 15, C 2 = min, min Γ, min, min γ C 6 min, min δ = C 5, C 1 = C 18 min Γ, min, min γ,min δ = C 15, C 3 = min, min δ min Γ, min, min γ,min δ C 7 min, min γ = C 5, C 2 = C 19 min Γ, min, min δ,min γ = C 15, C 4 = min, min γ min Γ, min, min δ,min γ C 8 min, min γ,min δ = C 5, C 3 = C 20 min, min Γ = C 20, C 0 = min, min γ,min δ min, min Γ, C 9 min, min δ,min γ = C 5, C 4 = C 21 min, min Γ, min δ = C 20, C 1 min, min δ,min γ min, min Γ, min δ C 10 min Γ = C 10, C 0 = C 22 min, min Γ, min γ = C 20, C 2 = min Γ, min, min Γ, min γ C 11 min Γ, min δ = C 10, C 1 = C 23 min, min Γ, min γ,min δ = C 20, C 3 min Γ, min δ min, min Γ, min γ,min δ C 12 min Γ, min γ = C 10, C 2 = C 24 min, min Γ, min δ,min γ = C 20,, C 4 = min Γ, min γ min, min Γ, min δ,min γ TABLE I 25 DIFFERENT CONSTRAINTS FOR THE MINIMUM-COST k-path PROBLEM. co and capaciy cco and capaciy C 0 [(l(e),k),(, 0)], [(0,k),(, 0)] C 1 [(l(e),k L +1), (, 0)], [(0,k),(, 0)] C 13 [(l(e),k L +1), (, 0)], [(0, 1), (M, k N )] C 2 [(l(e),k),(, 0)], [(0,k N +1), (, 0)] C 14 [(l(e),k L +1), (, 0)], [(0, 1), (M, k N )] C 3 [(l(e),k L +1), (, 0)], [(0,k N +1), (, 0)] C 15 [(l(e), 1), (M + l(e),k 1)], [(0, 1), (M, k 1)] C 4 [(l(e),k L +1), (, 0)], (0,k N +1), (, 0)] C 16 [(l(e), 1), (M + l(e),k L )], [(0, 1), (M, k 1)] C 5 [(l(e), 1), (M + l(e),k 1)], [(0,k),(, 0)] C 17 [(l(e), 1), (M + l(e),k 1)], [(0, 1), (M, k N )] C 6 [(l(e), 1), (M + l(e),k L )], [(0,k),(, 0)] C 18 [(l(e), 1), (M + l(e),k L )], [(0, 1), (M, k N )] C 7 [(l(e), 1), (M + l(e),k 1)], [(0,k N +1), (, 0)] C 19 [(l(e), 1), (M + l(e),k L )], [(0, 1), (M, k N )] C 8 [(l(e), 1), (M + l(e),k L )], [(0,k N +1), (, 0)] C 20 [(l(e), 1), (M + l(e),k 1)], [(0, 1), (M,k 1)] C 9 [(l(e), 1), (M + l(e),k L )], [(0,k N +1), (, 0)] C 21 [(l(e), 1), (M + l(e),k L )], [(0, 1), (M,k 1)] C 10 [(l(e),k),(, 0)], [(0, 1), (M, k 1)] C 22 [(l(e), 1), (M + l(e),k 1)], [(0, 1), (M,k N )] C 11 [(l(e),k L +1), (, 0)], [(0, 1), (M, k 1)] C 23 [(l(e), 1), (M + l(e),k L )], [(0, 1), (M,k N )] C 12 [(l(e),k),(, 0)], [(0, 1), (M, k N )] C 24 [(l(e), 1), (M + l(e),k L )], [(0, 1), (M,k N )] TABLE II (co, capaciy) ASSIGNMENTS IN G FOR THE 25 VERSIONS OF THE k-path PROBLEM OF TABLE I. link (, ) u u primary u- link (u,) (u,) econdary u- link Fig. 1. Node pliing. node in G. i replaced by wo node and an edge (, ). Fig. 2. Link pliing. Original link in G. Two link obained for he link of. G correponding o node in G by (, ) and (, ), which are called he primary node-generaed link (or imply, primary link) and he econdary node-generaed link (or imply, econdary link), repeciely (ee Figure 2 and Figure 3). TRANSFORM-1 i ued o conruc G from G and compue min-max link harabiliy k L or/and min-max node harabiliy k N, which are ueful in defining he capaciie of link in G. Fir, le u conider he erion of he minimum-co k-pah problem correponding o he conrain in Cla 1 (i.e. { C 1, C 2, C 3, C 4 }). For hee erion, we need o 1535

5 u (u, u ) u (, u) (u, ) (, ) (, ) (, ) u u (u, u ) u (, u) (u, ) (u, u ) (, u) (u, ) (, ) (, ) (, ) (, ) (c) Fig. 3. An example. Original graph. Graph G conruced from G by TRANSFORM-1. (c) Graph G conruced from from G by TRANSFORM-2. Capaciie for link of, and (co, capaciy) pair of (c) are no hown. (, ) (, ) ued o compue min-max link harabiliy k L or/and min-max node harabiliy k N for C. Each link in G i aigned a co-capaciy pair. More pecifically, for each link e = (u, ) in G, i correponding primary u link (u,) and econdary u link (u,) in G are aigned pair (co(u,), capaciy(u,)) and (co(u,), capaciy(u,)), repeciely. Similarly, for each node in G, i correponding primary link (, ) and econdary link (, ) are aigned pair (co(, ), capaciy(, )) and (co(, ), capaciy(, )), repeciely. We ue an ordered pair of co-capaciy pair [(co(u,), capaciy(u,)), (co(u,), capaciy(u,))], [(co(, ), capaciy(, )), (co(, ), capaciy(, ))] compue min-max link harabiliy k L or/and min-max node harabiliy k N. The following procedure MinMax-Sharabiliy reurn k L and k N wih flag e o 0 and 1, repeciely. procedure MinMax-Sharabiliy(G,flag,k,k ) begin E 1 := he e of all edge in E ha are reuled from node pliing in he conrucion of G by TRANSFORM-1; E 2 := E E 1 ; if flag = 0 hen E := E 2 ele E := E 1 ; aign capaciy k o all link in E E ; low := 0; high := k; lim := k/2 ; while low high do for each edge e E do aign e capaciy lim ; find he maximum flow f in G from o by running a maximum flow algorihm; if f <khen low := lim and lim := low + (high low)/2 ; ele high := lim and lim := low + (high low)/2 ; end-while reurn lim 1; end The min-max harabiliie for he problem of Cla 1 are compued a follow: C 1 : flag := 0; k := k; k L := MinMax-Sharabiliy(G,flag,k,k ); C 2 : flag := 1; k := k; k N := MinMax-Sharabiliy(G,flag,k,k ); C 3 : flag := 0; k := k; k L := MinMax-Sharabiliy(G,flag,k,k ); flag := 1; k := k L ; k N := MinMax-Sharabiliy(G,flag,k,k ); C 4 : flag := 1; k := k; k N := MinMax-Sharabiliy(G,flag,k,k ); flag := 0; k := k N ; k L := MinMax-Sharabiliy(G,flag,k,k ); I i imporan o noe ha procedure MinMax-Sharabiliy i inoked wice for C 3 and C 4.For C 3 (rep. C 4 ), k L (rep. k N ) i compued fir, and he alue of k L (rep. k N ) affec he alue of k N (rep. k L ) ha i compued by he econd call o MinMax-Sharabiliy. Thu, he k L and k N alue compued for C 3 and C 4 for he ame nework G may be differen. For all conrain of Table I in he normal form C,C wih nonempy min-max harabiliy componen conrain C, procedure MinMax-Sharabiliy i o repreen he co and capaciy aignmen for he link of G.Lel max = max e E l(e). Wihou lo of generaliy, aume ha l max > 1. If hi i no rue, we imply chooe a conan c o cale up he co of each link e from l(e) o c l(e) o ha l max > 1. Define M = k V l max and M = M 2 = k 2 V 2 lmax. 2 Then, all erion of he minimum-co k-pah problem conidered are reduced o finding minimum-co flow, and we preen algorihm by only preening he co and capaciie aigned o he link in G. We li (co, capaciy) aignmen for all he 25 conrain of Table I in Table II. We ue o repreen a don care alue. In Sep 2, an MCNF algorihm i applied o he flow nework G conruced in Sep 1 o find minimum-co - flow f of flow alue k. Inuiiely, for a conrain normal form C,C, he alue of k L or/and k N found by MinMax- Sharabiliy and ued in capaciy funcion guaranee ha C i aified, he alue M and M ued in co funcion are ued o aify C ubjec o C. For Sep 3, we inroduce he following procedure PATH- RECOVER. procedure PATH-RECOVER(G =(V,E ),,,f,k) begin P := ; for each e E do if f (e) 0 hen remoe e from G ; for i =1o k do find a hore pah P i from o in G ; obain an - pah P i in G from P i by replacing (, ) and (, ) by node and replacing (u,) and (u,) by link (u, ); P := P {P i } for each e in G uch ha e P i do f (e) :=f (e) 1; if f (e) =0hen remoe e from G ; reurn P ; end Gien a flow f in G uch ha f = k and f (e) i an ineger flow alue for edge e in G, procedure PATH- RECOVER conruc a unique e of k - pah correponding o f. Baed on flow coneraion propery, hee k pah exi and can be enumeraed ieraiely by PATH-RECOVER. 1536

6 Theorem 2: For any graph G =(V,E) wih non-negaie edge co, ource and deinaion in V, if here exi an - pah in G, hen for each of he harabiliy conrain C 0 o C 24 our algorihm cheme compue a e of k- pah P = {P 1,P 2,,P k } uch ha l(p ) i minimum ubjec o he harabiliy conrain. Proof: See Appendix. Boh G and G hae O( V ) node and O( E ) link, where V and E are he e of node and link of G, repeciely. Conrucing G and G ake O( V + E ) ime. Compuing maximum flow on G can be done in O(k E ) ime by applying Breadh Fir Search on reidual graph. Thu, procedure MinMax-Sharabiliy for compuing k L or/and k N ha ime complexiy O(k log k E ). Finding a minimum-co flow on G ake O(k ( E + V log V )) ime. Thu, he oal ime for compuing minimum-co k - pah under any of he 25 conrain lied in Table I i O(k ( E log k + V log V )). Summerizing aboe dicuion, we hae he following reul. Theorem 3: Our algorihm cheme ake O(k ( E log k + V log V )). ime for compuing minimum-co k- pah in G ubjec o any harabiliy conrain of Table I. In nework applicaion, k < V, and he complexiy of our algorihm cheme i acually O(k E log V ). IV. GENERALIZATIONS A. Nonuniform Maximum Allowable Sharabiliy In he problem we conidered o far, we aumed ha all link and node (excep and ) hae he ame maximum allowable harabiliy k 1. For many applicaion, we may wan o aign differen maximum allowable harabiliie o indiidual link or/and node. For example, if we know ha a link i highly reliable (rep. unreliable), we may wan o aign maximum allowable harabiliy k 1 (rep. 0) o he link. For opical nework, he number of aailable waelengh on link may be differen, and conequenly we may aume ha he maximum allowable harabiliy of a link o be he number of i aailable waelengh le 1. We generalize he minimum-co k - pah problem wih harabiliy conrain by adding wo more conrain: each link e (rep. node, ), i allowed o be hared by a mo (e) +1 (rep. () +1) pah, where (e) (rep. ()) i he maximum allowable link (rep. node) harabiliy of e (rep. ). Clearly, he problem (wih uniform maximum allowable harabiliy conrain) conidered in he preiou ecion are pecial cae of he problem (wih nonuniform maximum allowable harabiliy conrain) we are dicuing. The algorihm cheme for he conrain lied in Table I in Secion III can be eaily modified o ole he problem wih exra nonuniform maximum allowable harabiliy conrain. For finding he minimum δ (rep. γ), binary earch of procedure MinMax-Sharabiliy can be lighly modified o find minimum k L (or k N ) uch ha a flow f of alue k exi wihou iolaing he capaciy (e)+1 (rep. ()+1) of each link (rep. node). Then, wih repec o Table II, he capaciy of a (co, capaciy) pair for a link in G for finding opimal (c) 1 Fig. 4. Tranformaion ued o ole relaed problem. A gien graph G. GraphG for finding opimal pah from o deinaion in T = { 1, 2, 3 }.GraphG for finding opimal pah from o deinaionpair ( 1, 2 ) and ( 2, 3 ). k pah under a pecific compoie conrain i modified a follow: alue k i replaced by (e) +1 (rep. () +1), and alue k L (rep. k N ) i replaced by min{(e),k L } (rep. min{(),k N }). I i eay o ee ha minimum-co k - pah ha aify he gien (compoie) conrain of Table I and nonuniform maximum allowable indiidual link/node harabiliie can be compued by finding a minimum-co flow f of flow alue k from o in G. Hence, all 25 erion of he problem of finding minimum-co k - pah wih nonuniform maximum allowable harabiliie can be oled in he ame amoun of ime a heir counerpar wih uniform maximum allowable harabiliie. I i poible ha a feaible oluion doe no exi. Bu uch a iuaion can be deeced eaily. B. Minimum Co One-o-Many Pah wih Minimum Sharabiliy Our algorihm cheme can be eaily generalized o ole oher problem. We menion wo problem. The fir i o find minimum-co pah P = {P 1, P 2,,P k } from o a e of deinaion T = { 1,, k } in a graph G = (V,E) ubjec o minimum harabiliy conrain C. Clearly, he pah in P are ueful for mulica communicaion. Auming ha all node of T are reachable from, we can reduce hi problem o finding k- pah a follow: We conruc a graph G =(V,E ) from G by inroducing a new node, and inroducing a link from each node i o he new node wih co 0 and maximum allowable link harabiliy 0 (ee Figure 4 for an example). Then, we apply our algorihm cheme o find minimum-co k - pah from o in G aifying C. By reering he direcion of link in G, hi mehod can alo be ued o compue minimumco k pah wih ariou harabiliy conrain for many-oone communicaion. The econd relaed problem i defined a follow: gien a graph G =(V,E) wih V = n, E = m, a ource node, a e T of k pair ( i, j ) wih 1, 2 V {}, and harabiliy conrain C, find wo pah from o eery pair ( i, j ) of node in T uch ha C i aified and he oal co of he pah i minimum. Thi i he problem of finding minimumco proecion of dual homing archiecure conidered in [15], [17], [22]. Thi problem can be reduced o finding k- pah a follow: We conruc a graph G =(V,E ) from G by inroducing a new node i,j for each pair ( i, j ), wo link from node i and j o he new node i,j wih co 0 and 1 1,2 2,3 1537

7 maximum allowable link harabiliy 0, a new node, and a link from each i,j o wih co 0 and maximum allowable link harabiliy 2 (ee Figure 4(c) for an example). Then, all we need o do i o find minimum-co 2k pah from o aifying C in G. Clearly, 25 erion of each of hee wo problem can be oled in polynomial ime. V. CONCLUDING REMARKS In hi paper, we characerized he degree of link haring and node haring by he noion of link harabiliy and node harabiliy. We idenified 65 differen harabiliy conrain for he problem of finding minimum-co k - pah in a direced or undireced graph G, and howed ha 25 of hem are no muually equialen. We howed ha hee 25 erion of he problem finding minimum-co k pah are polynomialime olable by reducing i o he minimum-co nework flow problem. We alo howed ha finding minimum-co oneo-many pah ubejec o ariou harabiliy conrain are polynomial-ime olable. Our algorihm can be ued o find link-dijoin and node-dijoin pah if hey exi by checking he min-max link harabiliy and min-max node harabiliy in he reuling oluion. The algorihm preened in hi paper are ery ueful for many nework applicaion. For all 65 erion of he problem of finding minimumco k pah wih minimum harabiliy, we hae hown ha hey are pair-wie inequialen and deeloped a new algorihm cheme wih lighly higher ime complexiy. We will repor hee new reul in a ubequen paper. ACKNOWLEDGEMENT Thi work wa parially uppored by NSF gran CCF and CNS REFERENCES [1] R.K. Ahuja, T.L. Magnani, and J.B. Orlin, Nework Flow, Prenice-Hall, [2] B. Bank and M.E. Zarki, Dynamic muli-pah rouing and how i compare wih oher dynamic rouing algorihm for high peed wide area nework, Proc. of SIGCOMM, [3] D. Caanon, Efficien algorihm for finding he K be pah hrough a relli, IEEE Tran. AES, ol.26, pp , [4] L.R. Ford and D.R. Fulkeron, Flow in Nework, Princeon, [5] S. Forune, J. Hopcrof, and J. Wyllie, The direced ubgraph homeomorphim problem, Theore. Compu. Sci., ol 10, pp , [6] P. Georgao and D. Griffin, A managemen yem for load balancing hrough adapie rouing in mulierice ATM nework, Proc. of SIG- COMM, [7] K. Ihida, Y. Kakuda, and T. Kikuno, A rouing proocol for finding wo node-dijoin pah in compuer nework, Proc. of Inernaional Conference on Nework Proocol, pp , [8] R. Krihnan and J. Sileer, Choice of allocaion granulariy in muli-pah ource rouing cheme, Proc. of IEEE INFOCOM, [9] S.-W. Lee and C.-S. Wu, A k-be pah algorihm for highly reliable communicaion nework, IEICE Tran. Commun. E82-B.4, pp , [10] C.L. Li, S.T. McCormick, and D. Simchi-Lei, The complexiy of finding wo dijoin pah wih min-max objecie funcion, Dicree Appl. Mah. ol. 26, no. 1, pp , [11] C.L. Li, S.T. McCormick, and D. Simchi-Lei, Finding dijoin pah wih differen pah-co: complexiy and algorihm, Nework, ol. 22, pp , 1 22(1992), pp [12] S.D. Nikolopoulo, A. Piillide, and D. Tipper, Addreing nework uriabiliy iue by finding he K-be pah hrough a relli graph, Proc. IEEE INFOCOM, pp ,1997. Fig. 5. A graph for which pair {C i,c j } of differen1 conrain in { C 0, C 1, C 2, C 3, C 4 } uch ha {C i,c j } {C 2,C 4 } are muually inequialen. A graph for which conrain C 2 and C 4 are muually inequialen. Fig. 6. A graph for which conrain C 0, C 5, C 10, C 15 and C 20 are muually inequialen. The graph ued for Cae (3) in he proof of Theorem 1. [13] J. W. Suurballe, Dijoin pah in a nework, Nework, ol. 4, pp , [14] J. W. Suurballe and R. E. Tarjan, A quick mehod for finding hore pair of dijoin pah, Nework, ol. 14, pp , [15] J. Wang, M. Yang, X. Qi, and R. Cook, Dual-homing mulica proecion, Proc. IEEE Globecom, pp , [16] H. Suzuki and F.A. Tobagi, Fa bandwidh reeraion cheme wih muli-link and mul-pah rouing in ATM nework, Proc. IEEE INFO- COM, [17] J. Wang, M. Yang, B. Yang, and S.Q. Zheng, Dual-homing baed calable parial mulica proecion, IEEE Tran. on Compuer, ol. 55, no. 9, pp , [18] B. Yang, S.Q. Zheng, and S. Kaukam, Finding wo dijoin pah in a nework wih Min-Min objecie funcion, Proc. of he 15h IASTED In l Conf. on Parallel and Dir. Compuing Syem, pp , [19] B. Yang, S.Q. Zheng, and E. Lu, Finding wo dijoin pah in a nework wih normalized α + -Min-Sum objecie funcion, Proceeding of he 16h Inernaional Sympoium on Algorihm and Compuaion (Lecure Noe in Compuer Science, 3827, Springer), pp , [20] B. Yang, S.Q. Zheng, and E. Lu, Finding wo dijoin pah in a nework wih normalized α -Min-Sum objecie funcion, Proceeding of he 17h IASTED In l Conf. on Parallel and Dir. Compuing Syem, pp , [21] B. Yang, S.Q. Zheng, and E. Lu, Finding wo dijoin pah in a nework wih MinSum-MinMin objecie Funcion, Technical Repor, UTDCS-16-05, Dep. of Compuer Science, Uni. of Texa a Dalla, Apr [22] M. Yang, J. Wang, X. Qi, and Y. Jiang, On finding he be parial mulica proecion ree under dual-homing archiecure, Proc. IEEE HPSR, Proof of Theorem 1: APPENDIX Gien graph G, node and, conan k, and conrain X and Y, les X and S Y denoe he feaible oluion pace, he e of e of k - pah, for X and Y, repeciely. To how ha X and Y are no equialen, i.e. X Y, we only need o find a graph G and a k alue for which S X S Y. For a gien graph G and a gien conrain X, we ue δ X o denoe he δ alue of feaible - k-pah oluion under X. Since here can be muliple poible δ alue for differen feaible oluion under X, we ue range(δ X ) =[δ 1,δ 2 ] o denoe he cloed ineger ineral of hee δ alue, where δ 1 and δ 2 are minimum and maximum poible alue wih repec o X, repeciely. A cloed ineger ineral of a ingle alue δ i denoed by 1538

8 conrain C range(γ C ) range( C ) [k 1, 6k 6] [2k 5, 7k 7] min [k, k +1] [2k 5] min Γ [k 1] [2k 4, 2k 2] min Γ, min [k] [2k 5] min, min Γ [k 1] [2k 4] TABLE IV POSSIBLE Γ AND VALUES FOR FEASIBLE k-path SOLUTIONS UNDER DIFFERENT MIN-SUM CONSTRAINTS FOR NETWORK OF FIGURE 6. [δ]. Wih repec o a gien conrain X, range( X ), range(γ X ), and range(γ X ) are imilarly defined for a graph G. Then, finding G and k for which S X S Y i equialen o finding G and k for which or range(λ X ) range(λ Y ) range(λ X ) range(λ Y ), where λ {δ, γ} and Λ {, Γ}. We aume ha all raional number x y ued are ineger; i.e. x i a muliple of y for x y. We compare wo conrain X and Y in Table I by comparing heir normal from X = X, X and Y = Y, Y. For any wo diinc conrain X and Y, we hae hree cae. Cae (1): X = Y = and X Y. Conider he graph hown in Figure 5 and Figure 5. We li δ C and γ C alue of Figure 5 in he op par of Table III. Clearly, for ome k eiher range(δ X ) range(δ Y ) or range(γ X ) range(γ Y ) if { X, Y } {C 2,C 4 } = { min γ ), δ, min γ }. For Figure 5, range( min γ ) range( min δ, min γ ), a indicaed in he boom par of Table III. Cae (2): X = Y = and X Y. Conider he graph hown in Figure 6. We li C and Γ C alue for hi graph in Table IV. For any wo diinc X = X and Y = Y of he fie conrain, eiher range( X ) range( Y ) or range(γ X ) range(γ Y ) for ome k. Cae (3): { X, Y } { } and { X, Y } { }. If X = Y, hen by Table III range( X ) range( Y ) or range(γ X ) range(γ Y ). If X Y, hen conider he graph hown in Figure 6. For hi graph, range(δ X )=range(γ X )=range(δ Y )=range(γ Y )=k 1. By Cae (2), we know ha eiher range( X ) range( Y ) or range(γ X ) range(γ Y ) for ome k. Proof i compleed. Proof of Theorem 2: For conrain C 0, he heorem obiouly hold. We proe he heorem by conidering four remaining clae. For conrain in Cla 1, our algorihm fir find min-max link harabiliy k L or/and min-max node harabiliy k N, which are ued o reric he feaible oluion pace. The MCNF algorihm find an opimal oluion wihin hi rericed pace. For Cla 2, he oluion hae o aify min-max conrain, which are enforced by k L or/and k N. Suppoe for he ake of conradicion he claim i no rue, hen here exi a differen e of k pah from o, P = {P 1,P 2,,P k } in G, uch ha one of he following condiion hold: (1) (P ) < (P ); (2) (P )= (P ), and l(p ) <l(p ). We creae he unique nework flow f in G according o P a follow: for eery e E do f (e) :=0 for i =1o k do for each link (u, ) in P i do cae :f (u,)=0: f (u,):=1; :f (u,)=1: f (u,):=f (u,)+1; end-cae Le f denoe he flow correponding o P in G.Noe ha δ(e, P ) 1 i exacly he alue of f on e = (u, ). Le c(f ) denoe he co of flow f in G.Wehae c(f ) = l(e)+ (δ(e, P ) 1) (M + l(e)) e P e P = δ(e, P )l(e)+m (δ(e, P ) 1) e P e P = δ(e, P )l(e)+m (P ) e P Similarly, we hae (P )= e P (δ(e, P ) 1), and c(f )= e P δ(e, P )l(e)+m (P ). Then, c(f ) c(f ) = ( δ(e, P )l(e) δ(e, P )l(e)) e P e P +M ( (P ) (P )). Since 0 e P δ(e, P )l(e) < M and 0 e P δ(e, P )l(e) < M, e P δ(e, P )l(e) e P δ(e, P )l(e) > M. Now we conider he wo poibiliie. Cae (1): (P ) < (P ). Then (P ) (P ) 1. Thu, c(f ) c(f )=( e P δ(e, P )l(e) e P δ(e, P )l(e))+ M ( (P ) (P )) > 0, which conradic he aumpion ha f i a minimum-co flow. Cae (2): (P )= (P ) and l(p ) <l(p ). Then, we hae c(f ) c(f )=l(p ) l(p ) > 0, which conradic he aumpion ha f i a minimum-co flow. Thi complee he proof for Cla 2. The proof for Cla 3 i abou he ame a ha for Cla 2, excep ha c(f )= e P l(e) + P,, (γ(e, P ) 1) = e P l(e) +Γ(P ) and c(f ) = e P l(e) + P,, (γ(, P ) 1) = e P l(e)+γ(p ). For Cla 4, he oluion hae o aify min-max conrain, which are enforced by k L or/and k N. Suppoe for he ake of conradicion he claim i no rue, hen here exi a differen e of k pah from o, P = {P 1,P 2,,P k } uch ha one of he following condiion mu hold: 1539

9 conrain C range(γ C ) range(δ C ) range(γ C ) range( C ) graph [ k 2 1,k 1] [ k 1,k 1] [3k 7, 6k 6] [4k 12, 7k 7] Fig. 5 3 min δ [ 2k 3 1,k 1] [ k 19k 1] [3k 7, 4k 11] [4k 12, 19] Fig min γ [ k 2 1] [ k 2 1] [ 9k 11k 11] [ 15] Fig min γ,min δ [ 2k 3 1] [ k 1] [4k 11] [5k 15] Fig. 5 3 min δ, min γ [ k 2 1] [ k 2 1] [ 9k 11k 11] [ 15] Fig min γ [ k 2 1] [ k 4 1, k 1] [k 2, 3k 10] [2k 8, 4k 16] Fig. 5 2 min δ, min γ [ k 2 1] [ k 1] [2k 6] [3k 12] Fig. 5 4 TABLE III POSSIBLE γ AND δ VALUES FOR FEASIBLE k-path SOLUTIONS UNDER DIFFERENT MIN-MAX CONSTRAINTS FOR NETWORK OF FIGURE 5. (1) (P ) < (P ); (2) (P )= (P ), and Γ(P ) < Γ(P ); (3) (P ) = (P ), Γ(P ) = Γ(P ), and l(p ) < l(p ). We creae a nework flow f in G according o P a follow: for eery e E do f (e) :=0 for i =1o k do for each node ha i eiher nor in P i do cae :f (, )=0: f (, ):=1; :f (, )=1: f (, ):=f (, )+1; end-cae for each link (u, ) in P i do cae :f (u,)=0: f (u,):=1; :f (u,)=1: f (u,):=f (u,)+1; end-cae We hae c(f ) = (γ(, P ) 1) M + l(e) P,, e P + (δ(e, P ) 1) (M + l(e)) e P = Γ(P ) M + δ(e, P )l(e)+ e P (δ(e, P ) 1) M e P = M Γ(P )+M (P )+ Similarly, we hae Then, e P δ(e, P )l(e). c(f )=M Γ(P )+M (P )+ e P δ(e, P )l(e). c(f ) c(f ) = M (Γ(P ) Γ(P )) + M ( (P ) (P )) + ( δ(e, P )l(e) δ(e, P )l(e)). e P e P For any e of k- pah wih minimum oal co, here i no loop of poiie co in any pah. Then, 0 Γ(P ) k ( V 2) and 0 Γ(P ) k ( V 2), and Γ(P ) Γ(P ) > k ( V 2) and M (Γ(P ) Γ(P )) > M + 2kM. Since e P δ(e, P )l(e) k ( V 1)l max <M and e P δ(e, P )l(e) k ( V 1)l max <M,wehae e P δ(e, P )l(e) e P δ(e, P )l(e) > M. Nowwe conider he hree poibiliie. Cae (1): (P ) < (P ). Then (P ) (P ) 1, and c(f ) c(f ) = M (Γ(P ) Γ(P )) + M ( (P ) (P )) + ( δ(e, P )l(e) δ(e, P )l(e)) e P e P M (Γ(P ) Γ(P )) + M +( δ(e, P )l(e) e P δ(e, P )l(e)) e P > ( M +2kM)+M M = (2k 1)M >0, which conradic he aumpion ha f i a minimum-co flow. Cae (2): (P )= (P ), and Γ(P ) < Γ(P ). Then c(f ) c(f ) = M (Γ(P ) Γ(P )) + ( e P δ(e, P )l(e)) > 0, e P δ(e, P )l(e) which conradic he aumpion ha f i a minimum-co flow. Cae (3): (P ) = (P ), Γ(P ) = Γ(P ), and l(p ) < l(p ). Then, c(f ) c(f )= δ(e, P )l(e) δ(e, P )l(e) > 0, e P e P which conradic he aumpion ha f i a minimum-co flow. Thi complee he proof for Cla 4. The proof for conrain in Cla 5 i abou he ame a ha for Cla 4, excep ha c(f )=M Γ(P )+M (P )+ e P δ(e, P )l(e) and c(f )=M Γ(P )+M (P )+ e P δ(e, P )l(e) are ued. Proof i compleed. 1540