Lightpath routing for maximum reliability in optical mesh networks

Transcription

1 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 449 Lightpath routing for maximum reliability in optial mesh networks Shengli Yuan, 1, * Saket Varma, 2 and Jason P. Jue 2 1 Department of Computer and Mathematial Sienes, University of Houston Downtown, One Main Street, Houston, Texas 77002, USA 2 Department of Computer Siene, University of Texas at Dallas, 800 West Campbell Road, Rihardson, Texas , USA *Corresponding author: yuans@uhd.edu Reeived Otober 19, 2007; revised Marh 4, 2008; aepted Marh 4, 2008; published April 16, 2008 Do. ID We onsider the problem of maximizing the reliability of onnetions in optial mesh networks against simultaneous failures of multiple fiber links that belong to a shared-risk link group (SRLG). We study the single-lightpath, parallel-lightpaths, and lightpath protetion problems for onnetions between two end nodes, as well as the lightpath-ring problems for onnetions of three or more end nodes. We first study the speial problems where all SRLGs have the same failure probability. In these problems, every SRLG is represented by a distint olor and every fiber link is assoiated with one or more olors, depending on the SRLGs to whih the link belongs. We formulate the problems as minimum-olor lightpath problems. By minimizing the number of olors on the lightpaths, the failure probability of the lightpaths an be minimized. We prove the problems to be NP-hard. We then extend the results to the general problems where the failure probabilities of the SRLGs may differ. Heuristi algorithms are proposed for larger instanes of the problems, and the heuristis are evaluated through simulations Optial Soiety of Ameria OCIS odes: , , Introdution In mesh WDM networks, users ommuniate with eah other via end-to-end lightpaths [1,2]. Three types of onnetions are ommonly used for ommuniation between two end nodes: a single lightpath, multiple parallel lightpaths, and a working lightpath protetion lightpath pair. A single lightpath may reah a data rate of 40 Gbits/ s or even higher [3,4]. Two or more parallel lightpaths are often used between bakbone routers to arry traffi simultaneously, resulting in a higher aggregate data rate, with the additional benefits of load balaning [5]. The parallel lightpaths are link disjoint so that the failure of one fiber link does not disonnet the end nodes entirely. Sine a single lightpath is vulnerable to fiber link failures, it is often required to provide a high degree of reliability for this type of onnetion, whih leads to the third onfiguration, i.e., a working lightpath protetion lightpath pair. In this onfiguration, a link-disjoint protetion lightpath is preomputed and provisioned for every working lightpath [6 8]. Suh protetion shemes provide 100% reliability against any single-link failure in the network. However, due to various risk fators suh as natural and man-aused atastrophes, disjoint fiber links may belong to the same sharedrisk link group (SRLG) and fail simultaneously [9,10]. Therefore it is insuffiient for the lightpaths to be merely link disjoint. Rather, it is neessary to find a working and a protetion lightpath that are SRLG disjoint [11]. A drawbak of SRLG-disjoint protetion shemes is that if many SRLGs exist in the network it may be diffiult, or even impossible, to find a working and a protetion lightpath between two end nodes that are ompletely SRLG disjoint [12]. Thus, it may be diffiult to provide 100% reliability against ertain failure events. For ases in whih 100% reliability is not possible, the objetive should be to find one or more lightpaths for eah onnetion, suh that the reliability for eah onnetion is maximized, or equivalently, the failure probability of the onnetion is minimized. For onnetions of three or more end nodes, suh as those in multiparty video onferening, internet telephony, and online gaming [13 15], a ommon pratie for ahieving a high degree of reliability is establishing lightpaths between the end nodes /08/ /$ Optial Soiety of Ameria

2 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 450 suh that the lightpaths form a ring, whih keeps the end nodes onneted even when a single fiber link fails [16,17]. However, if two of more fiber links on the ring belong to the same SRLG, a failure of that SRLG may still disonnet the end nodes. Therefore it is neessary to minimize the failure probability of the lightpaths on the ring. In this work, we introdue and study the single-lightpath problem, the parallellightpaths problem, and the lightpath protetion problem for onnetions between two end nodes. We also study the lightpath-ring problem for onnetions of three or more end nodes. For eah of the problems, we first onsider the speial ase in whih all SRLGs have uniform failure probability. We then disuss the problems in the more generi ase in whih the SRLGs an have different failure probabilities. When all SRLGs have uniform failure probability, we use a unique olor to distintly represent eah SRLG. If multiple links belong to the same SRLG, then all of these links are marked with the orresponding olor. If a link belongs to multiple SRLGs, then that link has multiple olors. Sine all SRLGs have the same failure probability, minimizing the failure probability of a lightpath is equivalent to minimizing the number of SRLGs, or olors, on the lightpath. For the single-lightpath problem, by minimizing the number of olors on the lightpath, the failure probability of the lightpath an be minimized. For the parallel-lightpaths problems, by minimizing the total number of olors on the parallel lightpaths, the probability that a failure ours to one of the lightpaths is minimized. For the lightpath protetion problem, by minimizing the number of overlapping olors on the working lightpath and the protetion lightpath, the probability that a single failure event will ause both lightpaths to fail simultaneously is minimized. For the lightpath-ring problem, if we minimize the total number of olors on the lightpath ring, we minimize the failure probability of any part of the ring. On the other hand, if we minimize the overlapping olors of all the lightpaths of onseutive end nodes along the ring, we also minimize the probability of simultaneous failures of lightpaths that would disonnet the end nodes. In this study, we evaluate the omputational omplexity of various minimum failure problems and prove them to be NP-hard [18]. NP-hard problems have the property that solutions that yield optimal results with polynomial time omplexity have never been found [19]; thus, solutions for these problems have high omputational omplexity and tend to be infeasible for large networks. A problem may be found NP-hard if an existing NP-hard problem an be redued to it using an algorithm of polynomial time omplexity [19,20]. For a proven NP-hard problem, attempts an then be made to develop effiient heuristi algorithms. Despite the seeming similarities between minimum-olor lightpath problems and various versions of minimum-ost path problems, the minimum-olor lightpath problems are muh harder to solve. For many minimum-ost path problems, there exist effiient algorithms that solve the problems with polynomial omplexity. For instane, Dijkstra s algorithm an be used to find a single minimum-ost path between two end nodes [20], and Suurballe s algorithm an be used to find two link-disjoint paths with the minimum total ost [21,22]. However, for the minimum-olor lightpath problems, many of them are NP-hard, as shown later in this study. A thorough searh of existing literature yields limited results in the area of interest. One relevant study looked into the problem of establishing a spanning tree using the minimum number of labels (i.e., olors) [23]. It proved the problem to be NP-hard and also proposed two heuristi algorithms. The first heuristi is named the edge replaement algorithm. The algorithm first forms an arbitrary spanning tree, then tries to replae eah edge with a different edge that an redue the total number of olors in the tree. The seond heuristi is named the maximum vertex overing algorithm. This algorithm starts with an empty spanning tree, then sans through all the olors and hooses the one that overs the most unovered verties. This proedure is repeated until a spanning tree is formed. The studies in [24] analyzed the performane of the two heuristis and showed that the first an be arbitrarily bad while the seond ahieves a logarithmi approximation ratio. The work in [25] further investigated the seond algorithm with slightly better approximation. It should be pointed out that the minimum-olor lightpath problems disussed in this study also apply to other areas of networking. For instane, a nationwide optial network may onsist of fiber links that belong to different network arriers. A lightpath is often less expensive to establish and operate if it involves fewer arriers, even if the length of that lightpath is suboptimal. The problem of minimizing the number of

3 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 451 different network arriers along a lightpath thus beomes a minimum-olor lightpath problem if we represent every arrier with a distint olor. Other instanes of network heterogeneousness also exist and may raise similar issues that add to the relevane of the study in this work. The rest of the paper is organized as follows. In Setion 2, we study the singlelightpath problem, the parallel-lightpaths problem, and the lightpath protetion problem for onnetions between two end nodes, as well as the lightpath-ring problems for onnetions of three or more end nodes. We prove these problems to be NP-hard and develop heuristi solutions for them. In Setion 3, we present omputer simulations and results for the heuristis. In Setion 4, we onlude the paper. 2. Lightpath Routings for Maximum Reliability For eah of the problems in this setion, we first onsider the speial ase in whih all SRLGs have uniform failure probability, and SRLGs are represented distintly by different olors. One the NP-hardness of the problems has been proved for this speial ase, we disuss the general ase in whih the SRLGs have different failure probabilities. In the remaining disussions, we assume that all fiber links are bidiretional and that full wavelength-onversion apability is available at every network node; thus, the wavelength-ontinuity onstraint is not a onern in our study. The latter assumption does not ompromise the NP-hardness of the problems, sine adding the wavelength-ontinuity onstraint only inreases the hardness of the problems. 2.A. Single-Lightpath with Minimum Failure Probability For the speial ase in whih all SRLGs have uniform failure probability, eah SRLG is distintly represented by a olor. We refer to this instane of the single-lightpath problem as the minimum-olor single-lightpath (MCSL) problem of finding a single lightpath between two end nodes suh that it has the minimum number of olors. For the purpose of NP-hardness analysis, if we prove the MCSL problem to be NP-hard for networks ontaining only single-olor links, then it will be straightforward to onlude that the MCSL problem is also NP-hard for networks possibly ontaining multiolor links, sine the former is a speial ase of the latter. The MCSL problem is defined as follows. Given network G= N,L, where N is the set of nodes and L is the set of fiber links, and given the set of olors C = 1, 2, 3,..., K, where K is the maximum number of olors in G, and given the olor l C on every link l L, find one lightpath from soure node s to destination node d suh that it uses the minimum number of olors. 2.A.1. Proof of NP-hardness We redue a known NP-hard problem to the MCSL problem. The known NP-hard problem in this ase is the minimum set overing problem [19]. This problem is stated as follows. Given a finite set S= a 1,a 2,a 3,...,a n, and a olletion C= C 1,C 2,...,C m suh that eah element in C ontains a subset of S, is there a minimum subset, C C suh that every member of S belongs to at least one member of C? We onstrut a graph G for an arbitrary instane of the minimum set overing problem, suh that the graph ontains one path from s to d with the minimum number of olors, if and only if C ontains a minimum set over C. The following are the steps for the graph onstrution: Step 1. For every element a i in S, reate a network node a i. Step 2. For every subset C j to whih a i belongs, reate a network link a i 1 a i of olor j. For element a 1, the link is sa 1. Also reate a single link between a n and d with olor 0. An example is given in Fig. 1. In this example, we onstrut graph G for a minimum set overing problem 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. It is apparent that if there is a minimum-olor path from s to d, then the olors on that path are mapped diretly to a minimum set overing all the elements in S. Conversely, if there is a minimum set overing all the elements, then a minimum-olor

4 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 452 Fig. 1. Redution of the minimum set overing problem to the MCSL problem. path an be derived by going through every node and seleting the link with the olor representing the set that overs the orresponding element. Therefore, it is an NP-hard problem to find a single lightpath between two end nodes with the minimum number of SRLGs. Subsequently the general problem is also NP-hard where the failure probabilities of the SRLGs in the network may differ and eah fiber link may belong to multiple SRLGs. 2.A.2. Heuristi Algorithms We introdue two simple greedy heuristis to solve the MCSL problem. We name the first heuristi the single-lightpath olor-redution algorithm (SLCRA). The details are as follows: Step 1. Find a minimum-hop-ount path p from s to d. Let the olletion of all the olors on p be set C p = 1, 2,..., k. Step 2. Go through every olor in C p. Selet the olor suh that removing all links of that olor results in a minimum-hop-ount path with the minimum number of olors, whih is also less than C p. Remove all links of the seleted olor from the network. Step 3. Repeat steps 1 and 2 until the number of olors on the new minimum-hopount paths annot be further redued. Step 1 of the algorithm may use one of the minimum-ost-path algorithms suh as Dijkstra s algorithm by setting all link osts to 1. The running time in this step is O n log n where n is the number of nodes [20]. The number of olors on this path is used as the upper bound for lightpaths, whih we find in the next step. Step 2 selets one olor at a time from the olors on the path obtained in step 1 (i.e., C p ) and tries to find a new minimum-hop-ount path after temporarily eliminating all links of that partiular olor. After going through all the olors in C p, we identify the olor whose elimination results in the minimum number of olors on the new path. The links of that olor are then permanently removed from the network graph before going to step 3. The worst-ase running time of this step is O mn log n, where m is the total number of olors in the network. Step 3 repeats the previous two steps until no new path an be found with fewer olors. In the worst ase, the number of iterations is m, whih results in the total running time of O m 2 n log n for this entire algorithm. We name the next heuristi the single-lightpath all-olor-optimization algorithm (SLACOA). The details are as follows: Step 1. Initialize link ost to 1 on all links in the network. Step 2. Find a minimum-ost lightpath p. Let the number of olors on p be C p. Step 3. Pik one olor at a time, set the link ost to zero on all links of that olor, and find a new minimum-ost path. Repeat this proedure for all olors in the network, and selet the olor that results in the path with the minimum number of olors, whih is also less than C p. Keep the osts to zero on the links of that seleted olor. Step 4. Repeat steps 2 and 3 until the number of olors on the minimum-ost paths annot be further redued. By setting all link osts to 1 in the first step, the minimum-ost path found in step 2 is effetively the minimum-hop-ount path from the soure s to the destination d. The running time of step 2 is O n log n, where n is the number of nodes. The number of olors on this path is used as the upper bound for lightpaths that we find in the next step. Step 3 selets one olor at a time from all olors in the network, sets the link ost to zero on all links of that partiular olor, and finds a new path between s and d. The intent is to determine whether we an redue the number of olors on that path if we make the links of a olor more attrative by setting their ost to zero. After going through all the olors, a olor is identified when we find the new path with the

5 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 453 minimum number of olors. We then permanently set the ost of the links of that olor to zero before going to step 4. The running time of this step is O mn log n, where m is the total number of olors in the network. Step 4 repeats the previous two steps until no new path an be found with fewer olors. The number of iterations is m, whih results in a total running time of O m 2 n log n for this entire algorithm. 2.B. Parallel Lightpaths with Minimum Failure Probability For the speial ase in whih all SRLGs have uniform failure probability, eah SRLG is distintly represented by a olor. We refer to this instane of the parallel-lightpaths problem as the minimum total-olor disjoint-lightpaths (MTCDL) problem of finding parallel lightpaths that use the minimum number of total olors. For the purpose of NP-hardness analysis, if we prove the MTCDL problem to be NP-hard for networks ontaining only single-olor links, it will be straightforward to onlude that the MTCDL problem is also NP-hard for networks possibly ontaining multiolor links, sine the former is a speial of ase of the latter. A onnetion of parallel lightpaths uses at least two disjoint lightpaths. If more than two lightpaths are used, the problem beomes even harder. Here we study the two-path version of the MTCDL problem, whih is defined as follows. Given network G= N,L, where N is the set of nodes and L is the set of fiber links, and given the set of olors C= 1, 2, 3,..., K, where K is the maximum number of olors in G, and given the olor l C on every link l L, find two disjoint lightpaths from soure node s to destination node d suh that the total number of olors on the two lightpaths is minimal. 2.B.1. Proof of NP-hardness There are two variations of the MTCDL problem based on the requirement of lightpath disjointness. In the first variation, the two lightpaths are node disjoint. In the seond variation, the two lightpaths are link disjoint. We redue the minimum set overing problem to both variations of the problem to prove their NP-hardness. For the minimum set overing problem, assume the given finite set S is a 1,a 2,a 3,...,a n and the olletion C is C 1,C 2,...,C m. Variation 1. MTCDL problem with node-disjoint requirement. We onstrut a graph G for an arbitrary instane of the minimum set overing problem, suh that the graph ontains two node-disjoint paths from s to d with the minimum total number of olors, if and only if C ontains a minimum set over C. The following are the steps for the graph onstrution: Step 1. For every element a i in S, reate a network node a i. Step 2. For every subset C j to whih a i belongs, reate a network link a i 2 a i of olor j. For elements a 1 and a 2, the links are sa 1 and sa 2, respetively. Also reate single links a n 1 d and a n d with olor 0. An example is given in Fig. 2. In this example, we onstrut graph G for a minimum set overing problem 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. In the onstruted graph G, a n 1 d and a n d are single links. Therefore any two nodedisjoint paths from s to d must have one path going along s a 1 a 3 d and the other path going along s a 2 a 4 d. If two node-disjoint paths p 1 and p 2 have the Fig. 2. Redution of the minimum set overing problem to the MTCDL problem with node-disjoint requirement.

6 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 454 minimum total number of olors, sine eah olor exept for 0 is assoiated with a member in C, the olletions of all the olors on p 1 and p 2 map to a minimum subset of C that overs all the elements. Conversely, if there is a minimum subset C C that overs all the elements, then for eah node a i in G, there is at least one member C j in C that ontains a i, and we hoose a link a i 2 a i (or sa 1, sa 2 ) of the olor j. All the links, together with the single links with olor 0, form two node-disjoint paths from s to d that have the minimum total number of olors. Variation 2. MTCDL problem with link-disjoint requirement. We onstrut a graph G for an arbitrary instane of the minimum set overing problem, suh that the graph ontains two link-disjoint lightpaths from s to d with the minimum total number of olors, if and only if C ontains a minimum set over C. The following are the steps for the graph onstrution: Step 1. For every element a i in S, reate network nodes a i and u i. Step 2. For every element a 2i in S (exept for a n if n is even, and a n 1 if n is odd), reate a network node v i. Step 3. For every subset C j to whih a i belongs, reate a network link u i a i of olor j. Step 4. Create a single link su 1, su 2. If n is even, reate single link a 1 v 1, a 2 v 1, v 1 u 3, v 1 u 4,..., a 2i 1 v i, a 2i v i, v i u 2i+1, v i u 2i+2,..., a n 3 v n/2 1, a n 2 v n/2 1, v n/2 1 u n 1, v n/2 1 u n, a n 1 d, a n d. If n is odd, reate a single link a 1 v 1, a 2 v 1, v 1 u 3, v 1 u 4,..., a 2i 1 v i, a 2i v i, v i u 2i+1, v i u 2i+2,..., a n 2 v n 1 /2, a n 1 v n 1 /2, v n 1 /2 u n, v n 1 /2 d, a n d. All of the links are of olor 0. An example is given in Fig. 3(a). In this example, we onstrut graph G for a minimum set overing problem that has an even number of elements in S, i.e., 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. Another example is given in Fig. 3(b). In this example, S has an odd number of elements, i.e., S= a 1,a 2,a 3,a 4,a 5, C= C 1,C 2,C 3,C 4,C 5, C 1 = a 1,a 2,a 5, C 2 = a 2,a 3, C 3 = a 1,a 3,a 5, C 4 = a 3,a 4,a 5, C 5 = a 1,a 4. If there are two link-disjoint paths p 1 and p 2 in the onstruted graph G, every network node a i must be on exatly one of the paths. If the two paths have the minimum total number of olors, sine eah olor exept for 0 is assoiated with a member in C, the olletions of all the olors on p 1 and p 2 map to a minimum subset of C that overs all the elements. Conversely, if there is a minimum subset C C that overs all the elements, then for eah node a i in G, there is at least one member C j in C that ontains a i, and we hoose a link u i a i of the olor j. All the links, together with the single links with olor 0, form two link-disjoint paths from s to d that have the minimum total number of olors. (a) (b) Fig. 3. Redution of the minimum set overing problem to the MTCDL problem with link-disjoint requirement. (a) With an even number of elements. (b) With an odd number of elements.

7 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 455 Therefore, the problem of finding two parallel lightpaths with the minimum total number of SRLGs that are either link or node disjoint is NP-hard. Subsequently, more general versions of the problem, suh as the ase in whih the failure probabilities of the SRLGs in the network may differ, the ase in whih the number of parallel lightpaths is more than two, or the ase in whih eah fiber link may belong to multiple SRLGs, are also NP-hard. 2.B.2. Heuristi Algorithms Here we introdue two simple greedy heuristis to solve the MTCDL problem with the link-disjoint requirement. Node disjointness is often not a onern due to built-in redundany in most optial swithes. We name the first heuristi the disjointlightpaths olor-redution algorithm (DLCRA). The details are as follows: Step 1. Run Suurballe s algorithm and find two link-disjoint paths p 1 and p 2 with the minimum-total-hop-ount. Assume the olletion of all the olors on p 1 and p 2 is set C p = 1, 2,..., k. Step 2. Go through every olor in C p. Selet the olor suh that, after the links of that olor are removed from the network, Suurballe s algorithm yields two link-disjoint minimum-total-hop-ount paths with the minimum number of total olors, and the number of olors is also less than C p. Remove the links of the seleted olor. Step 3. Repeat steps 1 and 2 until the number of olors on the link-disjoint minimum-total-hop-ount paths annot be further redued. In step 1, we run Suurballe s algorithm to find two link-disjoint lightpaths from the soure node s to the destination node d with running time O n 2 log n [21]. The total number of olors on the paths is used as the upper bound for lightpaths, whih we find in the next step. Step 2 selets one olor at a time from the olors on the two paths obtained in step 1 (i.e., C p ) and finds two new disjoint paths after temporarily eliminating all links of that partiular olor from the network graph. After repeating this proedure for all those olors, we identify the olor whose elimination results in two disjoint paths with the minimum total number of olors. The links of that olor are then permanently removed from the network graph before going to step 3. The worstase running time of this step is O mn 2 log n. Step 3 repeats the previous two steps until no new disjoint paths an be found with fewer total olors. In the worst ase, the number of iterations is m, whih results in the total running time of O m 2 n 2 log n for this entire algorithm. We name the next heuristi the disjoint-lightpaths all-olor-optimization algorithm (DLACOA). The details are as follows: Step 1. Initialize link ost to 1 on all links in the network. Step 2. Run Suurballe s algorithm and find two link-disjoint paths. Assume that the total number of olors on the two paths is C p. Step 3. Pik one olor at a time, set the link ost to zero on all links of that olor, then run Suurballe s algorithm and try to find two new link-disjoint paths. Repeat this for all the olors in the network and selet the one that results in two link-disjoint paths with the minimum total number of olors that is also less than C p. Keep the osts to zero on the links of the seleted olor. Step 4. Repeat steps 2 and 3 until the number of olors on the link-disjoint paths annot be further redued. By setting all link osts to 1 in the first step, the link-disjoint paths found in step 2 are effetively the minimum-total-hop-ount lightpaths from the soure s to the destination d. The running time of step 2 is O n 2 log n, where n is the number of nodes. The total number of olors on the two paths is used as the upper bound for lightpaths, whih we find in the next step. Step 3 selets one olor at a time from all olors in the network and tries to find two new disjoint paths after temporarily setting the link ost to zero on all links of that partiular olor. The intent is to determine whether we an redue the total number of olors on the disjoint paths if we make the links of that olor more attrative by setting their ost to zero. After going through all the olors, a olor is identified if the two disjoint paths we found have the minimum total number of olors. We then permanently set the ost of the links of that olor to zero before going to step 4. The running time of this step is O mn 2 log n. Step 4 repeats the pre-

8 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 456 vious two steps until no new disjoint paths an be found with fewer total olors. The number of iterations is m, whih results in the total running time of O m 2 n 2 log n for this entire algorithm. 2.C. Proteted-Lightpaths with Minimum Failure Probability For the speial ase in whih all SRLGs have uniform failure probability, eah SRLG is distintly represented by a olor. This ase of the lightpath protetion problem is referred to as the minimum overlapping-olor disjoint-lightpaths (MOCDL) problem of finding the working lightpath and the protetion lightpath that have the minimum number of overlapping olors. For the purpose of NP-hardness analysis, if we prove the MOCDL problem to be NP-hard for networks ontaining only single-olor links, it will be straightforward to onlude that the MOCDL problem is also NP-hard for networks possibly ontaining multiolor links sine the former is a speial ase of the latter. The MOCDL problem is defined as follows. Given network G= N,L, where N is the set of nodes and L is the set of fiber links, and given the olors C= 1, 2, 3,..., K, where K is the maximum number of olors in G, and given the olor l C on every fiber link l L, find two link-disjoint lightpaths from soure node s to destination node d suh that they share the minimum number of overlapping olors. 2.C.1. Proof of NP-hardness We redue a known NP-hard problem to the MOCDL problem. The known NP-hard problem in this ase is the problem of finding two link-disjoint paths from soure node s to destination node d that are ompletely SRLG-disjoint [12,26]. We replae every SRLG with a different olor. If we were able to solve the MOCDL problem and find two link-disjoint paths from s to d with the minimum overlapping olors, then the paths should also be SRLG-disjoint if suh paths exist in the network. Therefore, the problem of finding a working lightpath and a protetion lightpath with the minimum number of ommon SRLGs is NP-hard. Subsequently, more general versions of the problem, where the failure probabilities of the SRLGs in the network may differ or eah fiber link may belong to multiple SRLGs, are also NP-hard. 2.C.2. Heuristi Algorithms We introdue three greedy heuristi algorithms for solving the MOCDL problem. We name the first heuristi the minimum-olor-first-lightpath algorithm. The details are as follows: Step 1. Run the SLACOA from Subsetion 2.A.2 and find the first lightpath p 1. Then set the link ost bak to 1 on all links in the network. Step 2. Inrease the ost of a link if the olor of the link is on p 1. The additional ost is proportional to the number of links of that olor on p 1. Remove all links on p 1. Step 3. Run Dijkstra s algorithm and find lightpath p 2. In the first step, we attempt to find a minimum-olor single-lightpath from s to d. Based on the disussion in Subsetion 2.A.2, the running time of this step is O m 2 n log n where n is the number of nodes and m is the total number of olors in the network. In the seond step, we inrease the ost of all the links in the network whose olors overlap with those on the first lightpath. The more links of a partiular olor are on the first lightpath, the higher we inrease the ost of all the links of that olor. The intent is to make those links less attrative when we exeute step 3. We also remove all the links on the first lightpath to ensure that the new path we find in step 3 is link disjoint from the first path. The running time of step 3 is O n log n. So the total running time for the entire algorithm is O m 2 n log n. However, this algorithm may fail to find two link-disjoint lightpaths in networks ontaining a trap topology [27,28]. An example of suh a network is depited in Fig. 4. This network has only one wavelength. Eah fiber link has a unique olor. One the first lightpath from soure node s to destination node d is found along s a b d, another link-disjoint lightpath annot be found, even though two link-disjoint lightpaths do exist (s e b d and s a f d). Hene we propose the seond heuristi for the MOCDL problem and name it the joint-searh minimum-overlapping-olor algorithm, whih resolves the trap topology issue. The details are as follows:

9 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 457 Fig. 4. Example of a trap topology. The number next to eah fiber link is the link ost. Step 1. Set link ost to 1 on all links in the network. Then run Suurballe s algorithm and find two link-disjoint lightpaths p 1 and p 2. Step 2. Inrease the ost of a link if the olor of the link is on p 1. The additional ost is proportional to the number of links of that olor on p 1. Step 3. Remove all links on p 1. Run Dijkstra s algorithm and find path p 1. Restore links on p 1. Set ost bak to 1 on all links in the network. Step 4. Inrease the ost of a link if the olor of the link is on p 2. The additional ost is proportional to the number of links of that olor on p 2. Step 5. Remove all links on p 2. Run Dijkstra s algorithm and find path p 2. Step 6. From the two pairs of disjoint paths p 1 /p 1 and p 2 /p 2, selet the pair that has fewer overlapping olors as the result. We start with Suurballe s algorithm in step 1. Suurballe s algorithm always finds two link-disjoint lightpaths as long as they exist. Let the two paths be p 1 and p 2. The running time of this step is O n 2 log n, where n is the number of nodes in the network. In steps 2 and 3, we try to find a new path p 1 that is link disjoint from p 1 and is also less likely to share ommon olors with p 1. The running time of this step is O n log n. In steps 4 and 5, we try to find a new path p 2 that is link disjoint from p 2 and is also less likely to share ommon olors with p 2. The running time of this step is O n log n. From these two pairs, we selet the pair of paths that has fewer overlapping olors in step 6. The total running time for this entire algorithm is O n 2 log n. To establish an upper bound on the number of overlapping olors for the results obtained from the two previous heuristis, we develop a simple two-step algorithm. Details of this algorithm are as follows: Step 1. Set link ost to 1 on all links in the network. Run Dijkstra s algorithm and find the shortest lightpath p 1. Step 2. Inrease the ost of a link if the olor of the link is on p 1. The additional ost is proportional to the number of links of that olor on p 1. Step 3. Remove all links on p 1. Run Dijkstra s algorithm again and find the seond shortest path p 2. Paths p 1 and p 2 are link disjoint. Sine the osts on the links of those olors on p 1 are inreased, it is expeted that p 1 and p 2 share fewer olors. The running time of this heuristi is the same as that of Dijkstra s algorithm, whih is O n log n. 2.D. Lightpath Ring with Minimum Failure Probabilities For the speial ase in whih all SRLGs have uniform failure probability, the lightpath-ring reliability problem beomes the minimum total-olor lightpath-ring (MTCLR) problem of finding a lightpath ring that has the minimum number of total olors on all the onstituent lightpaths, and the minimum overlapping-olor lightpathring (MOCLR) problem of finding a lightpath ring that has the minimum number of overlapping olors on all the onstituent lightpaths. For the purpose of NP-hardness analysis, if we prove the MTCLR problem and MOCLR problem NP-hard for networks ontaining only single-olor links, it will be straightforward to onlude that these problems are also NP-hard for networks possibly ontaining multiolor links, sine the former are speial ases of the latter.

10 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING D.1. Lightpath Ring with Minimum Failure Probability on Any Constituent Lightpaths The MTCLR problem is defined as follows. Given network G= N,L, where N is the set of nodes and L is the set of fiber links, and given the olors C= 1, 2, 3,..., K, where K is the maximum number of olors in G, and given the olor l C on every fiber link l L, find the lightpaths that onnet node n 1,n 2,...,n m suh that the total number of olors on the lightpaths is minimized. This problem is reduible to the Hamiltonian-yle problem [19], whih is a known NP-hard problem, by the following steps: Step 1. For the network graph G of a Hamiltonian-yle problem, onstrut an idential network graph G. Step 2. Assign a unique olor to every link in G. Hene, a minimum-olor ring onneting all nodes in G is redued to a Hamiltonian-yle in G and this problem is proven NP-hard. Therefore, the problem of finding a lightpath ring with the minimum number of total SRLGs when the SRLGs have uniform failure probability is NP-hard. Subsequently the general problem is also NP-hard where the failure probabilities of the SRLGs in the network may differ and eah fiber link may belong to multiple SRLGs. Due to the page limit, we have omitted the heuristis for this problem. They will be studied in a separate paper. 2.D.2. Lightpath Ring with Minimum Probability of Simultaneous Failures of Multiple Constituent Lightpaths The MOCLR problem is defined as follows. Given network G= N,L, where N is the set of nodes and L is the set of fiber links, and given the olors C= 1, 2, 3,..., K, where K is the maximum number of olors in G, and given the olor l C on every fiber link l L, find the lightpaths that onnet node n 1,n 2,...,n m (an be in any order) suh that the number of overlapping olors on the lightpaths is minimized. This problem is an NP-hard problem beause it ontains the MOCDL problem of Subsetion 2.C as a speial ase when m=2. Subsequently the general problem is also NP-hard where the failure probabilities of the SRLGs in the network may differ and eah fiber link may belong to multiple SRLGs. Due to the page limit, we have omitted the heuristis for this problem. They will be studied in a separate paper. To summarize Setion 2, we disussed five lightpath reliability problems with the objetives of minimizing the failure probability of the onnetions. For eah of the problems, it was first shown that the problem is NP-hard for the speial ase in whih all SRLGs have uniform failure probability and eah fiber link belongs to a single SRLG. Then it beomes evident that the general problems are also NP-hard when the SRLGs have different failure probabilities and eah fiber link may belong to multiple SRLGs. Heuristi solutions were developed for the first three problems. 3. Computer Simulations We implement omputer simulations to evaluate the heuristis on networks that are randomly generated using LEDA [29]. The network size ranges from 10 to 40 nodes. The nodal degree ranges from 2.6 to 3.0. Eah fiber link is assumed to support an unlimited number of lightpaths and is assumed to belong to any SRLG with equal probability. The olor intensity of the network ranges from 1 to 20. The network olor intensity is defined as the average number of links of the same olor. When the olor intensity is 1, every network link has a different olor and therefore belongs to a different SRLG. When the olor intensity inreases, more links have the same olor, i.e., the links belong to the same SRLG. Subsequently more fiber links may fail simultaneously. If the olor intensity is equal to the number of fiber links in a network, then all links have the same olor and they always fail simultaneously. In addition to the heuristis, we also developed integer linear programming (ILP) formulations (Appendix A) in order to obtain the optimal solutions using CPLEX [30]. We observed that the differene between the amount of time required for the exeution of the heuristis and the amount of time required for solving the ILPs was signifiant. For example, using an Intel-based personal omputer, it took only a few minutes to exeute eah of the

11 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 459 heuristis for 10,000 end-node pairs on a 40-node network, whereas it took several hours to solve the ILPs for the same end-node pairs and network. 3.A. Simulations for the Minimum-Color Single-Lightpath Problem We developed two heuristis for the MCSL problem, i.e., the SLCRA and the SLACOA. For every randomly hosen pair of end nodes, we solved the ILP and used the heuristis to obtain the minimum-olor lightpath. We then ompared the average numbers of olors on all the lightpaths. To establish an upper bound for the results of the heuristis, we also ran Dijkstra s algorithm to obtain the minimum-hop-ount paths for all the end-node pairs. Two sets of the results are depited in Figs. 5 and 6. Results on other network topologies are similar. We note that, as the nodal degree inreases, the number of olors on the lightpath dereases. The reason for this behavior is that an inrease in nodal degree results in a wider hoie of available routes for eah onnetion. Furthermore, an inrease in nodal degree redues the average hop distane for eah onnetion, thereby reduing the number of olors. We also note that olor intensity has an impat on the number of olors on the lightpaths as well. When the olor intensity is 1, all fiber links in the network have a distint olor; hene, the number of olors for a given lightpath is simply the hop distane of that lightpath, and the average number of olors for eah lightpath is simply the average hop distane in the network. As the olor intensity inreases, the number of links with the same olors also inreases. As a result, the number of unique olors on the lightpaths dereases. The network topology and the size of the network also affet the number of olors on the lightpath. Larger networks with more nodes result in a higher average hop ount for lightpaths; hene, for the same nodal degree and olor intensity, lightpaths in a network with more nodes will have a greater number of olors than lightpaths in a network with fewer nodes. 3.B. Simulations for the Parallel-Lightpaths Problem We developed two heuristis for the MTCDL problem, i.e., the DLCRA and the DLA- COA. For every randomly hosen pair of end nodes, we solved the ILP and used the heuristis to obtain two parallel lightpaths with a minimum total number of olors. We then ompared the average total numbers of olors on the lightpaths. To establish Fig. 5. Average number of olors on lightpaths of all soure destination pairs versus network olor intensity. Network nodal degree=2.6.

12 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 460 Fig. 6. Average number of olors on lightpaths of all soure destination pairs versus network olor intensity. Network nodal degree=3.0. an upper bound for the results of the heuristis, we ran Suurballe s algorithm on all the end-node pairs. Two sets of the results are depited in Figs. 7 and 8. Results on other network topologies are similar. Based on the simulation results, the parallel lightpaths obtained from DLACOA are losest to the optimal ILP solutions in the average number of total olors on the lightpaths. This is beause DLACOA selets optimal olors from all olors in the network while DLCRA is restrited to the olors on the two initial lightpaths. Fig. 7. Average total number of olors on two parallel lightpaths of all soure destination pairs versus network olor intensity. Network nodal degree=2.6.

13 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 461 Fig. 8. Average total number of olors on two parallel lightpaths of all soure destination pairs versus network olor intensity. Network nodal degree=3.0. The network nodal degree, the olor intensity, and the network size have similar impats on the number of total olors on the parallel lightpaths as in the simulations in Subsetion 3.A. Higher nodal degree, or greater olor intensity, or smaller network size all ontribute to fewer total olors on the parallel lightpaths. 3.C. Simulations for the Proteted-Lightpaths Problem We developed two heuristis for the MOCDL problem, i.e., the minimum-olor-firstlightpath algorithm (MCFLA) and the joint-searh minimum-overlapping-olor algorithm (JSMOCA). We used the simple two-step algorithm (STSA) to establish an upper bound on the number of overlapping olors for the two heuristis. For every randomly hosen pair of end nodes, we solved the ILP and used the heuristis to obtain working and protetion lightpaths with a minimum number of overlapping olors. We then ompared the average numbers of overlapping olors on all the lightpath pairs. Two sets of the results are depited in Figs. 9 and 10. Results on other network topologies are similar. Based on the simulation results, the proteted lightpaths obtained from the JSMOCA are losest to the optimal ILP solutions in the average numbers of overlapping olors on the lightpaths. This is beause JSMOCA selets optimal olors from all olors in the network while MCFLA is restrited to the olors on the initial pair of lightpaths. The number of overlapping olors is diretly related to how suseptible the two lightpaths are to a single SRLG failure event. As the olor intensity inreases, more fiber links belong to the same SRLG and it is more likely that the two lightpaths belong to a greater number of ommon SRLGs. Thus, the lightpaths beome more suseptible to simultaneous failures. 4. Conlusions In this paper we have disussed five lightpath reliability problems. For onnetions between two end nodes, we disussed the single-lightpath problem, the parallellightpaths problem, and the proteted-lightpaths problem. For the interonnetion of three or more end nodes, we disussed the lightpath-ring problems. When all SRLGs have uniform failure probability, these problems are formulated as minimum-olor lightpath problems (MCSL, MTCDL, and MOCDL) or minimum-olor lightpath-ring problems (MTCLR and MOCLR). We proved these problems NP-hard and extended

14 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 462 Fig. 9. Average number of overlapping olors on proteted lightpaths of all soure destination pairs versus network olor intensity on a 20-node network. Nodal degree =2.6. Fig. 10. Average number of overlapping olors on proteted lightpaths of all soure destination pairs versus network olor intensity on a 40-node network. Nodal degree =3.0. the onlusions to the general problems where the SRLGs may have different failure probabilities and eah fiber link may belong to multiple SRLGs. We proposed various heuristis that exeute in polynomial time and that may be suitable for large-sale networks. Despite the simpliity of the heuristis, omputer simulations demonstrated that the heuristis yield solutions that are lose to optimal. From the simulations, we also observed that various fators affet the number of olors (i.e., SRLGs) on the lightpaths, inluding the nodal degree, the olor intensity, and the number of nodes in the network. An inrease in the nodal degree helps redue the number of olors on the lightpaths for the problems that attempt to minimize the total number of olors on the lightpaths (MCSL, MTCDL). This is due to the fat that there is a greater hoie of routes for the lightpaths. The heuristi algorithm SLACOA performs the best for reduing the total number of olors on a single lightpath (MCSL), whereas DLACOA is the most suessful with the parallel-lightpaths problem (MTCDL). The olor intensity also affets the number of olors on the lightpaths. An inrease in the olor intensity inreases the number of overlapping olors between a working

15 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 463 lightpath and its protetion lightpath, thereby making both lightpaths more suseptible to simultaneous failures (MOCDL). JSMOCA redues the number of overlapping olors and is reasonably lose to the ILP s optimal solution. While the emphasis of this paper is to identify the problems and to prove their NP-hardness, future work may involve the development of approximation algorithms for eah of the problems. For instane, the minimum set overing problem is reduible to the MCSL problem with polynomial omplexity as shown in Subsetion 2.A. Based on the signifiant number of studies already done on the approximation algorithms for the minimum set overing problem [19,31], similar work an be extended for the MCSL problem. Another interesting topi for future work would be to onsider the multiplelightpath problems (MTCDL and MOCDL) without the link-disjoint onstraint. In this ase, the lightpaths may share a ommon fiber link if the failure probability of that link is low. Appendix A All ILP formulations developed below are for stati traffi. The problem with a single onnetion is a speial ase when there is only one soure destination pair in. 1. ILP Formulation for the Minimum-Color Single-Lightpath Problem The following are given as inputs to the problem: N: number of nodes in the network. L: number of links in the network. olor ij : 1 if link ij has olor ; 0 otherwise. = s 1 d 1,s 2 d 2,...,s m d m,...s M d M, M m 1: All the soure destination pairs of the onnetion requests. The ILP solves for the following variables: ij m : 1 if link ij is used on the lightpath between soure destination pair s m d m in ; 0 otherwise. m : 1 if olor is on the lightpath between soure destination pair s m d m in ; 0 otherwise. Objetive: minimize the average number of olors on the lightpaths, m M. A1 m Constraints: Eqs. (A2) (A4) desribe the flow onstraints, i.e., eah lightpath goes through a fiber link at most one: xj m = 1, where x = s m, m, A2 j ik m i j m kj =0, k s m,d m, m, A3 jy m = 1, where y = d m, m. A4 j Inequalities (A5) and (A6) desribe the upper and lower bounds of the number of ourrenes of a partiular olor on a lightpath: m i,j olor ij m ij, m, A5 L m i,j olor ij m ij, m. A6

16 Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING ILP Formulation for the Parallel-Lightpaths Problem The following are given as inputs to the problem: N: number of nodes in the network. L: number of links in the network. olor ij : 1 if link ij is of olor ; 0 otherwise. = s 1 d 1,s 2 d 2,...,s m d m,...s M d M, M m 1: All the soure destination pairs of the onnetion requests. The ILP solves for the following variables: ij m : 1 if link ij is used on lightpath p 1 between soure destination pair s m d m in ; 0 otherwise. ij m : 1 if link ij is used on lightpath p 2 between soure destination pair s m d m in ; 0 otherwise. 1m : 1 if olor is on lightpath p 1 between soure destination pair s m d m in ; 0 otherwise. 2m : 1 if olor is on lightpath p 2 between soure destination pair s m d m in ; 0 otherwise. overlap m : 1 if olor is on both lightpaths p 1 and p 2 between soure destination pair s m d m in ; 0 otherwise. Objetive: minimize the average total number of olors on the parallel-lightpaths, 1m + 2m overlap m M. A7 m Constraints: Eqs. (A8) (A13) desribe the flow onstraints for the two parallel lightpaths, i.e., eah lightpath goes through a fiber link at most one: xj m = 1, where x = s m, m, A8 j ik m i j m kj =0, k s m,d m, m, A9 jy m = 1, where y = d m, m, A10 j xj m = 1, where x = s m, m, A11 j ik m i j m kj =0, k s m,d m, m, A12 jy m = 1, where y = d m, m. A13 j Inequalities (A14) (A17) desribe the upper and lower bounds of the number of ourrenes of a partiular olor on eah of the two lightpaths: 1m olor ij ij m, m, A14 i,j L 1m olor ij ij m, m, A15 i,j 2m olor ij ij m, m, A16 i,j L 2m olor ij ij m, m. A17 i,j