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On Disjoint Path Pairs with Wavelength Continuity Constraint in WDM Networks

Reid Andersen Fan Chung Arunabha Sen Guoliang Xue Department of Computer Science and Engineering

University of California, San Diego. CA 90005 Email: {randerse. fan} @math.ucsd.edu

Department of Computer Science and Engineering Arizona State University, Tempe. A 2 85287

Email: {asen. xue} @asu.edu

Abstract-In a WDM optical network, each fiher link can carry a certain set of wavelengths A = {,\,. X?, . . . ~ ,AIr.}. One scheme for tolerating a single link failure (or node failure) in the network is the path prolection scheme, which establishes an active path and a link-dajnint (or nude-disjnint) backup path, so that in the event of a link failure (node failure) on the active path, data can be quickly re-routed through the backup path. We consider a dynamic scenario, where requests to estahlish active-hackup paths between a specified source- destination nude pa i r arrive sequentially. I f a link-disjoint (nude-disjoint) active-hackup path pair can he found at the time of the request, the paths are established; otherwise, the request is hlucked. In this scenario, at the time a request arrives, not every fiber link wi l l have al l 1V wavelengths availahle for new cal l estahlishment, as some of the wavelengths may already have been allocated to earlier requests and communication through these paths may s t i l l be in progress. We assume that the network nudes do not have any wavelength converters. This paper studies the existence of a pair o f link-disjoint (node-disjoint) active-hackup paths satisfying the wavelength continuity constraint hetween a specified source-devtination node pair. First we prove that hoth the link-disjoint and ncde-disjoint versions of the prohlem are NP-Complete. Then we f ixus on the link-rlisjoint version and present an apprx imat i im algorithm and an exact algorithm fnr the problem. Finally, through our experiniental evaluations, we denllrnstrate that our approximation algorithni produces near-

routers are also possible. In the dedicated path protection wavelength scheme. an alternate path is lnaintained in a Stand- We refer to the problem under study in this paper a.. by mode Sor every source-destination path used for the Disjoint putli Problem wider Wuveleiigth CoiitirruiQ data transmission. These paths are referred to the Construint (DPPWCC). Informally, it can be stated as secondap or backup path and the priiiiur? or active follows: Given a set A(e) A of wavelengths available path. respectively. Fig. I shows a WDM network (with on each link e, is it possible to establish an active and a 8 nodes and 10 links) and 2 existing connections. The hackup path hetween a specified source-destination node active and the hackup paths between the nodes a and d pair satisfying the wavelength continuity constraint? are (1.-6-d on wavelength X I and cr-f- t l on waveleng(h A?. AS discussed earlier, for link survivability (node respectively. The active and the backup paths between survivability, respectively), the active and the backup the Ilodes f and h We f-g-11 on WaVekngth and f- paths should be link-disjoint (node-disjoint, respec- ri-h on wAVdWglh XI. ~eSpectiVely. Clearly, in order tO tivcly). Thcrcforc there is a link vcrsion (LDPPWCC) tolerate any single link (node. respectively) Failure. the and a node version (NDPPWCC) of the DPPWCC backup path should not be sharing any fiber link (node. problem. me following approach may be considered for respectively) with its corresponding active path. Thus solving the 1)PPWCC problem. First. consider a network the backup path should he link-disjoint (node-disjoint. comprising of only those l i nk where wavelength XI is respectively) with the active path. available for call establishment. If this network h&s two

In this paper we study the hlbwing problem: Con- link-disjoint (node-disjoint, respectively) paths between sider a WDM network where each fiber can carry Mi the specified source-destination node pair, then we have wavelengths , X w ) . A lightpath WI. a solution for the DPPWCC problem >. If this network 1151 is established between a SoUrce-deStination node does not have two link-disjoint (node-disjoint, respec- pair when a request for such a path arrives and ap- tively) paths between the specified source-destination propriate network resources are available. For reasons node pair. then this process can be repeated for other of survivability. k)lloWing the dedicated path protection wavelengths X2 and X3 etc. If one of them finds the strategy, we try to establish a primary (active) as well as: a disjoint paths, we have a solution for the DPPWCC secondary (hackup) path. If sufficient network resources problem. It may be noted that if any one of the attempts are availahle at the time the request arrives. the active- succeeds. both the active and the backup paths will be backup path pair is established. otherwise the call request establishaj using the SUWE wavelength. However, it may is blocked. We assume that the nodes do not have wave- be noted that it is no[ ilecessuF that the active and length converters and a a consequence. the aclive path the backup paths use the Same wavelength. It may be must maintain the Same wavelength throughout the entire possible to establish the active path using wavelength path. The Same is true for the backup path, although Xz and the hackup path using wavelength XI, as in the it way he using a different wavelengh. In the WDM case of connection between nodes f and h in Fig. 1. research community, this is known as the wuveleiigtli n i s situation is certainly more complex than the one contiriuir); construint. We point out that computing a pair where both the active path and the backup path use of active-backup paths wilh shared proleclion is mOre the Same wavelength. As will be seen in this paper. complicated than the case with dedicated protection. under this sifuation both the I.DPPWCC problem and Therefore the NP-completeness of the prohleill with the NDPPWCC problem are intractable. dedicated protection &es a strong indication of the of this paper is.to specifically tackle hardness of the problem with shared protection. Ihe DPPWCC problems. We show that the disjoint

Consider a situation where a request to establish path coInputation probleIll with wavelength continuity an active-backup path pair arrives at time T. At time constraint is NP-complete, a coIllrllonly held belief in T, several active-backup path pairs may already be in file WDM researckl c(>mInunity. ~ " s forInal cxistcncc. Accordingly. not cvcry fihcr link Will have proof. validates the study of heuristics and integer linear all Mi wavelengths available for the establishment of prograInIning (ILP) formulations. we also design an the new paths. It is possible that link I may have only enhancd version coin1i>onIy used active path wavelengths {XI . A,} available. link 2 may have only first heuristic and present simulation results showing

that our new heuristic finds optimal solutions in 99.8%

available. and so on.

= ( X l ? X a ,

me

'TIE use uf wavclrnqth cwvritrrs is cuiisidcrcd expensive in cu~rrnt WUM networks. Also. as s1mu.n in the appendix. the disjoint pntlls prohlems c m be solved in polynomial lime in WDM networks with wavelength converters. while the disjoint paths problems are

'The existence of a pair of disjoint paths on IRe sume wnwlenglll can he solved in polynomial time usins Suurhalle's algonlhm [281,

NP-complete in WDM networks withoul wavelength convcrlers. [29].

0-7803-8355-9/04/$20.00 02004 IEEE. 525

of the cases tested. while using only a fraction of the time requircd by the integcr linear programnung hased algorithm. ' h e rest of the paper is organized as follows. Section I1 provides some hackground in related areas. Section 111 states the problems in a formal setting. Sec- tion IV presents the complexity result for the prohlems. Sections V and VI provide an approximate and an exact solution for the link version of the LDPPWCC problem. Section VI1 compares the results between the exact and the approximate solutions and Section VI11 concludes the paper.

11. RELATED WORK Although several researchers in the last few years have

published a significant nuuiher of papers on survivability issues in WDM optical networks [I]. [21. [51. [71, 1141. 1161. [ IX] . [191. [22], [23], [27]. to the hest of our knowledge, the topic OC this paper, the complexity of the disjoint paths problem with wavelength continuity constraint. remains open. Although many researchers in the area helieve that the problem is NP-complete [6]. [3]. there is no formal proof available in the literature. A major contribution of this paper is to provide for the first time a formal proof that the prohlein indeed is NP- Complete [XI .

Recently. Hu [ 121 puhlishcd NP-Comnplctcness results related to diverse routing in optical mesh networks [3]. Although at a first glance it inay appear that the problems discusscd in 1121 arc the samc as the problems disc.ussed in this paper. they are in het significantly different. In this paper we consider the case where each link can carry only a certain suhsct of thc wavclcngths, and ask whether a disjoint pair of active-hackup lightpaths satisfying the wavelength continuity constraint can he established bctwccn a specified pair of nodes. In thc prohlein studied in [12]. the logical topology of the network is given as part of the input. This implies that lightpaths havc already hccn established hctwccn the appropriate pairs of nodes. The objective of the problem studied in [I21 is not to try establish a new active- backup lightpath pair. but to use the already esruhlished lighfpatlrs to find link-disjoint active-backup paths he- tween the specified .sou