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A Polynomial-Time Algorithm for ComputingDisjoint Lightpath Pairs in Minimum

Isolated-Failure-Immune WDM Optical NetworksGuoliang Xue, Fellow, IEEE, Ravi Gottapu, Xi Fang, Student Member, IEEE, Dejun Yang, Student Member, IEEE,

and Krishnaiyan Thulasiraman, Fellow, IEEE

Abstract—A fundamental problem in survivable routing inwavelength division multiplexing (WDM) optical networks is thecomputation of a pair of link-disjoint (or node-disjoint) lightpathsconnecting a source with a destination, subject to the wavelengthcontinuity constraint. However, this problem is NP-hard whenthe underlying network topology is a general mesh network. As aresult, heuristic algorithms and integer linear programming (ILP)formulations for solving this problem have been proposed. Inthis paper, we advocate the use of 2-edge connected (or 2-nodeconnected) subgraphs of minimum isolated failure immune net-works as the underlying topology for WDM optical networks.We present a polynomial-time algorithm for computing a pair oflink-disjoint lightpaths with shortest total length in such networks.The running time of our algorithm is , where is thenumber of nodes, and is the number of wavelengths per link.Numerical results are presented to demonstrate the effectivenessand scalability of our algorithm. Extension of our algorithm to thenode-disjoint case is straightforward.

Index Terms— Disjoint lightpath pairs, minimum isolatedfailure immune networks, partial 2-trees, wavelength divisionmultiplexing (WDM) optical network.

I. INTRODUCTION

O PTICAL networks implemented using wavelengthdivision multiplexing (WDM) techniques are consid-

ered promising candidates for backbone high-speed widearea networks [23], [26]. Among other things, wavelengthrouting allows logical topologies to be built on top of physicaltopologies to reflect the traffic intensities between the variousnodes as well as to provide reliable services by allowing forreconfiguration in the event of failures [18], [22], [31].In a WDM optical network, each fiber link can carry many

wavelengths. A WDM network is modeled by an undirectedgraph , where is the set of vertices, denotingnodes in the network; is the set of edges, denoting links (or

Manuscript received November 29, 2011; revised August 06, 2012 andJanuary 24, 2013; accepted March 01, 2013; approved by IEEE/ACMTRANSACTIONS ON NETWORKING Editor A. Walid. This work was supportedin part by the NSF under Grants 0830739, 1115129, and 1115130, and theARO under Grant W911NF-09-1-0467. The information reported here doesnot reflect the position or the policy of the federal government.G. Xue, X. Fang, and D. Yang are with Arizona State University, Tempe, AZ

85287 USA (e-mail: [email protected]).R. Gottapu is with Amazon, Inc., Seattle, WA 98144 USA.K. Thulasiraman is with the University of Oklahoma, Norman, OK 73019

USA.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TNET.2013.2257180

optical fibers) in the network; is the set ofwavelengths, and is the set of wavelengths availableon link . The terms vertices and nodes are used interchange-ably, as well as edges and links. In this model, each undirectededge in the network represents a bidirectional link con-necting and . Whenever a link is used by a connection, it isoccupied in both directions.In WDM networks, data packets are transmitted along light-

paths [7]. A lightpath between nodes andon wavelength is an – path in that

uses wavelength on every link of path . We assumethat the nodes do not have wavelength converters, and as a con-sequence, each lightpath must maintain the same wavelengththroughout the entire path. This is known as thewavelength con-tinuity constraint [3], [36].Cuts in fibers are one of the most common failures in optical

networks, while failures of routers are also possible. Sinceoptical links carry a very high volume of data, survivabilityis important. Hence, survivable lightpath routing has beenextensively studied. In the rest of this section, we give a briefoverview of survivable lightpath routing, introduce minimumisolated failure immune networks, state the problem to bestudied and our contributions, and differentiate our work fromclosely related work.

A. Disjoint Lightpath Routing

To tolerate a single link (node, respectively) failure inthe network, the path protection scheme of fault manage-ment [2], [25] establishes an active lightpath and a link-disjoint(node-disjoint, respectively) backup lightpath, so that in theevent of a link failure (node failure, respectively) on the activelightpath, data can be quickly rerouted through the backuplightpath. In the dedicated path protection scheme, each activelightpath has a dedicated backup lightpath—no two backuplightpaths share a common link on a common wavelength. Inthe shared path protection scheme [24], two backup lightpathsmay share a common link on a common wavelength, as longas the corresponding active lightpaths do not share a commonlink. These are studied under the theme of Shared Risk LinkGroup (SRLG) [1], [10], [29]. In this paper, we concentrateon dedicated path protection. Also, we concentrate on singlefailures. For protection schemes against double failures, werefer the readers to [8]. The link-disjoint lightpath routingproblem is formally defined in the following.Definition 1.1 (LDLP): Let a WDM optical network be given

by , with node set and link set , whereis the set of wavelengths. For each link ,

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2 IEEE/ACM TRANSACTIONS ON NETWORKING

Fig. 1. Active and backup lightpaths in a WDM network are link-disjoint.

the set of available wavelengths on link is . Letbe the source node, and the destination node.

The link-disjoint lightpath routing problem (LDLP) asks for ashortest pair of – link-disjoint lightpaths (on a wavelength

) and (on a wavelength ). In other words, wewant to find a pair of – paths and such that we have thefollowing.1) Paths and do not share a common link.2) Wavelength is available on every link on .3) Wavelength is available on every link on .4) The path pair has the minimum total length (measured bythe total number of links used) among all – path pairssatisfying the above three conditions.

The LDLP problem deals with dynamic routing, where theconnection requests come in a sequential order. Static routingproblems aim to satisfy multiple connection requests simultane-ously. Dynamic routing algorithms are often used (as heuristics)for solving static routing problems, where we try to establish theconnection requests in a sequential manner.Fig. 1 shows a WDM network and 2 existing connections.

The active and backup lightpaths between nodes and areon wavelength and on wavelength ,

respectively. The active and the backup lightpaths betweennodes and are on wavelength and onwavelength , respectively. Note that the active lightpaths arelink-disjoint with their corresponding backup lightpaths.In order to compute a pair of link-disjoint active-backup

lightpaths, the current literature uses the shortest active pathfirst (SAPF) heuristic [11], [15], which first computes a shortestlightpath as the active lightpath, then finds a shortest lightpaththat is link-disjoint with the active lightpath as the backuplightpath. Computational studies show that the SAPF heuristicis quite effective in practice [3], [11]. However, there is noperformance guarantee for the SAPF heuristic. Thereforeresearchers proposed integer linear programming (ILP) for-mulations [4], [16], [25]. Although the ILP formulation of theproblem can be used to find optimal solutions, solving an ILPtakes exponential time in the worst case.In [3], Andersen et al. proved that the problem of computing

a pair of disjoint (either link-disjoint or node-disjoint) lightpathsin a WDM network is NP-hard, asserting a commonly held be-lief in the WDM networking research community [11]. Yuanand Jue [36] independently proved the hardness of the disjointlightpath routing problem. Hardness of a related problem wasproved by Hu [12]. These hardness results give us a strong be-lief that there do not exist efficient algorithms for computing apair of link-disjoint lightpaths in a WDM network with a gen-eral mesh underlying topology.

Fig. 2. (a) 2-tree on 11 nodes. This graph is IFI. (b) 2-connected partial 2-tree.This graph is not IFI.

B. Isolated Failure Immune Networks

In order to provide network survivability, a price to pay isredundancy. Complete graphs offer the most survivability, buthave the highest redundancy. Trees have no redundancy (amongconnected networks), but offer no survivability. A ring networkcan survive a single node failure or a single link failure, butcannot survive two link failures.Farley [13] studied an important class of networks, which

can survive many simultaneous node and link failures, as longas they are isolated. Here, two node failures are isolated if thenodes are not adjacent; two link failures are isolated if the linksdo not meet at a node; a node failure and a link failure are iso-lated if the failed link is not incident to the failed node.Definition 1.2 (IFI Networks): A network is said to be iso-

lated-failure-immune (IFI) if all pairs of functional nodes canstill communicate, as long as the failures are pairwise isolated.An IFI network is called a minimum IFI network if it has theminimum number of links among all IFI networks on the samenumber of nodes.The network in Fig. 2(a) is IFI. When nodes , , , and

links , fail simultaneously (note that these net-work elements are pairwise isolated), the functional nodes arestill connected. Simultaneous failures of links andleave node disconnected from the other nodes. However, thelinks and are not isolated, as they are both inci-dent to node . The network in Fig. 2(b) is 2-connected, butnot IFI. Simultaneous failures of nodes and leave nodesand disconnected from the other functional nodes. Note thatnodes and are isolated.Farley [13] gave a detailed discussion of both the merits and

the usage of IFI networks as survivable networks. In addition, heshowed that IFI networks on nodes have at least links.He further demonstrated that every 2-tree [27] is a minimum IFInetwork. Wald and Colbourn [33] proved that every minimumIFI network is a 2-tree. Therefore, minimum IFI networks areexactly 2-tree networks.Minimum IFI networks are closely related to a graph theory

technique that has been used by many researchers to designprotection schemes for networks. Medard et al. [21] presented aprotection scheme using red/blue trees, which has been furtherstudied by Lumetta and Medard [19] and Medard et al. [20]in the design of loop-back recovery schemes. Xue et al. [35]extended the protection scheme of red/blue trees of [21]and discovered a tight connection of the loops used in [20]and [21] with the ear decomposition in graph theory [34].Zhang et al. [37] further explored this connection and designedfaster algorithms to construct red/blue trees. Other researchworks along this line can be found in [14] and [32]. In reality,


XUE et al.: POLYNOMIAL-TIME ALGORITHM FOR COMPUTING DISJOINT LIGHTPATH PAIRS IN MINIMUM IFI WDM OPTICAL NETWORKS 3

a 2-tree has a special ear decomposition—the first ear is a tri-angle, and every other ear is a 2-hop path that makes a trianglewith an edge of earlier ears.We believe that minimum IFI networks (and 2-edge con-

nected or 2-node connected subgraphs of minimum IFInetworks) are an excellent candidate for the backbone of futuresurvivable optical networks. There are many reasons leadingus to this belief. First, minimum IFI networks are fault-tolerantin the sense that they are immune to multiple simultaneous iso-lated failures. Second, minimum IFI networks are sparse—fora network with nodes, the number of links is , a factwhich enhances scalability. Third, there is a close relationshipbetween minimum IFI networks and the loop-back recoveryschemes studied in [14], [19]–[21], [32], [35], and [37] and thelogical topology mapping scheme of Kurant and Thiran [17].Fourth, minimum IFI networks have a nice structure thatmakes routing decisions much easier than in general meshnetworks [9], [33].

C. Main Contributions

We advocate the use of minimum IFI networks (or their2-edge/2-node connected subgraphs) as the underlying topologyfor WDM optical networks. We present a polynomial-time al-gorithm for solving the LDLP problem when the underlyingtopology is a subgraph of a minimum IFI network. In particular,our algorithm has a worst-case time complexity of ,where is the number of nodes in the network, and isthe number of wavelengths per link. Our algorithm can beeasily generalized to the node-disjoint case. Given the hardnessresult discussed in Section I-A, our theoretical contributionis significant. In addition, extensive numerical results showthat our algorithm has a noticeable advantage over the shortestactive path first heuristic. Given the fault-tolerant properties ofminimum IFI networks, our work will have an impact on thefuture design of survivable networks.

D. Other Related Work

To make our review of the literature as complete as possible,we present a review of works that are related to the problem offinding link/node disjoint paths in a network. Here, the paths arenot required to be lightpaths.Given a pair of vertices, finding edge/vertex-disjoint paths

(not lightpaths) between these vertices is a fundamental problemin network survivability. Suurballe [30] presented efficient algo-rithms for computing disjoint paths for a given source–destina-tion pair with minimum total length. Zheng et al. [39] studiedthe computation of multiple paths that have minimum link andnode sharing. Banner and Orda [5] and She et al. [28] studiedmultipath routing with minimum sharing. The aforementionedworks all deal with paths (rather than lightpaths). Therefore, ourwork is different from these works.The rest of this paper is organized as follows. In Section II,

we present important properties of minimum IFI networks. InSection III, we present a linear-time algorithm for computing apair of link-disjoint source-to-destination lightpaths, when theunderlying network topology is a minimum IFI network andthe wavelengths for the two paths are fixed. In Section IV, wepresent a polynomial-time algorithm for computing a pair of

link-disjoint source-to-destination lightpaths, when the under-lying network topology is a subgraph of a minimum IFI net-work. In Section V, we present numerical results. We concludethe paper in Section VI.

II. MINIMUM IFI NETWORKS AND 2-TREE NETWORKS

In this section, we present some important properties ofminimum IFI networks that will be used in our algorithmdesign. Since minimum IFI networks are equivalent to 2-treenetworks [33], we will use these two terms interchangeably.

A. 2-Tree Networks

We use to denote an undirected link connecting nodesand . Since the link is undirected, and both de-note the same link. We use to denote the triangle withnodes and links , and . The nodes in atriangle are not ordered. Therefore, , , andall denote the same triangle.Definition 2.1 (2-Tree Networks): A 2-tree is defined recur-

sively as follows, and all 2-trees may be obtained in this way. Atriangle is a 2-tree. Given a 2-tree and a link of the 2-tree,we can add a new node adjacent to both and (to expandthe current 2-tree); the result is a 2-tree.Fig. 2(a) shows a 2-tree network. Initially, the triangle

is a 2-tree with node set and linkset . Next, we add a new nodeadjacent to link in the current 2-tree, leadingto the 2-tree with node set and link set

. To describe the aboveprocess, we say the 2-tree is expanded by triangleor node expands out of link . We can continue thisprocess to expand the 2-tree constructed so far to obtain the2-tree shown in Fig. 2(a).One way to obtain the 2-tree shown in Fig. 2(a) from the

2-tree on node set is to execute the following se-quence of expansion operations:1) expand by ;2) expand by ;3) expand by ;4) expand by ;5) expand by ;6) expand by ;7) expand by .Another way to obtain the 2-tree shown in Fig. 2(a) from the2-tree on node set is to execute the following se-quence of operations:1) expand by ;2) expand by ;3) expand by ;4) expand by ;5) expand by ;6) expand by ;7) expand by .Let be a triangle. We say link ( , as well

as ) is incident to triangle . Let be a link in a2-tree. The degree of link is the number of triangles in the2-tree that are incident to link . For example, the degree oflink in Fig. 2(a) is 1, while the degree of link inFig. 2(a) is 3.



Fig. 3. The 2-tree in Fig. 2(a) can be partitioned into three components thatpairwise intersect in : (a) the component that contains node ; (b) thecomponent that contains node ; (c) the component that contains node .

B. Separation Properties

Our algorithms rely on a separation property of 2-trees [27]:For every link , the graph can be partitioned into one ormore components, which pairwise intersect only in , andthe union of these components is the entire 2-tree. Moreover,each component so obtained is a 2-tree, and the removal of andfrom a component does not disconnect the component. When

this partition for link contains only a single component,the link is called a peripheral; otherwise, the link is called a2-separator. We have the following property [27].Lemma 2.1 ([27]): Let be a 2-separator in a 2-tree .

Let and be two of the components induced by the 2-sep-arator . Then, we have the following.1) Let be a path within that does not use link , and

be a path within that does not use link . Then,and do not have any common link and do not have

any common node other than and .2) Let be a node in other than and , and be a nodein other than and . Then, any – path in must gothrough a node in .

Example 1: We use the 2-tree in Fig. 2(a) to illustrate theabove concepts. The links , , , , ,

, and are 2-separators. All other links are peripherals.We see that every peripheral has degree equal to 1, while each2-separator has degree greater than or equal to 2. This fact canbe easily proved since a link has degree 1 if and only if it isincident to only one triangle in the 2-tree.The 2-tree can be partitioned into three components at

link , shown in Fig. 3(a)–(c). The three components pair-wise intersect only at link . Let be the pathand be the path . Clearly, the two paths do notshare any common link or any common node other than and. Since and are in two different components, any pathconnecting to must go through either or .

C. Reduced 2-Trees

Let and be two given nodes in a 2-tree . A 2-separatoris called a separator of and if none of the components

partitioned at contains both and . A separator of andis not incident to or . In Fig. 2(a), , , andare all the separators of and .Definition 2.2: Let be a 2-tree. Let and be two nodes

in . A subgraph of is called a reduced 2-tree of withrespect to and , if we have the following.1) is a 2-tree that contains nodes and .2) The set of 2-separators of are exactly the set of separatorsof with respect to and .

A reduced 2-tree with respect to and gives a nice pictureabout the – paths, as will be shown in Lemma 2.2. A naturalquestion is whether a reduced 2-tree exists and how to computeit. In the following, we present a simple algorithm to compute areduced 2-tree with respect to a pair of nodes.

Algorithm 1: Reduced2Tree

Input: A 2-tree , and two nodes and in .Output: A reduced 2-tree with respect to and .1: ;2: while is not a triangle and a degree-2 node otherthan and in do

3: delete from ;4: end while5: output ;

Theorem 2.1: Let and be two nodes in a 2-tree withnodes. Algorithm 1 computes a reduced 2-tree with respect

to and in time. If no triangle in contains both and, the reduced 2-tree with respect to and is unique.Proof: We use a queue to store the nodes (other than and

) in whose degree in is 1. With such an implementation,checking the condition in line 2 takes time. When deletingnode in line 3, we update the degrees of the two nodes ofwhich form a triangle with . If anyone of them has degree 1(and is different from and ), insert it into the queue. Hence,the algorithm takes total time.The resulting subgraph is a 2-tree since the deletion opera-

tion is the inverse of an expansion operation (Definition 2.1). Inthe next two paragraphs, we will prove that the resulting 2-treeis indeed a reduced 2-tree with respect to and .We first prove that the deletion in line 3 of the algorithm does

not change the set of separators of and . Assume that isa separator of and before the deletion. Since is a sep-arator of and before the deletion, and belong to differentcomponents that only intersect in . Since we are deleting adegree-2 node that is neither nor , both components remainnonempty after the deletion. Hence, the link remains a sep-arator of and after the deletion of node . Clearly, if isnot a separator of and before the deletion, it cannot becomea separator of and after the deletion, even if it remains a linkof after the deletion.We next prove that the resulting 2-tree at the end of the al-

gorithm does not contain any 2-separator that is not a separatorof and . To the contrary, assume that is not a separatorof and , but is a 2-separator of the resulting 2-tree at the endof the algorithm. Since is a 2-separator of the 2-tree, the2-tree can be decomposed into two or more parts that only in-tersect in . Since is not a separator of and , bothand belong to a single component. Hence, there is a compo-nent that contains neither nor (with and excluded). Sincethis component is a 2-tree, it must contain a degree-2 node otherthan and . Therefore, the algorithm should not have stopped.This contradiction proves our claim. Hence, the 2-tree at theend of the algorithm is a reduced 2-tree with respect to and .Finally, we prove the uniqueness of the reduced 2-tree with

respect to and , when no triangle contains both and . Sinceno triangle in contains both and , the resulting 2-treeat the end of the algorithm cannot be a triangle. Hence, the setof separators of and is nonempty. As a result, the reduced



Fig. 4. (a) Reduced 2-tree for Fig. 2(a) with respect to and . (b) Reduced2-tree with respect to and if is a link in .

2-tree computed by Algorithm 1 has and as the only de-gree-2 nodes, and has the set of separators of and as its only2-separators.Let be any reduced 2-tree with respect to and . Then,

cannot have a degree-2 node other than and . To the contrary,assume that (other than and ) is a degree-2 node, which iscontained in a triangle . Then, link is a 2-separatorof , but is not a separator of and . This is a contradiction.Therefore, the only degree-2 nodes of are and . Sinceand both have and as their only degree-2 nodes, and bothhave the same set of 2-separators, they must be identical. Thisproves the uniqueness of the reduced 2-tree with respect toand .Example 2: Given the 2-tree in Fig. 2(a), we can compute the

reduced 2-tree with respect to and using Algorithm 1. Ini-tially, the degree-2 nodes other than and are , , , and. Following the algorithm, node is deleted, followed by

and . When is deleted, becomes a degree-2 node. Thus,we continue to delete nodes and . Now the only degree-2nodes are and . Hence, the algorithm stops, producing thereduced 2-tree shown in Fig. 4(a).Since no triangle in Fig. 2(a) contains both and , the re-

duced 2-tree with respect to and is unique, in which andare the only degree-2 nodes. If there is a triangle containing

both and , the reduced 2-tree with respect to and will bea triangle [see Fig. 4(b)], which might not be unique. For the2-tree in Fig. 2(a), both and are reduced 2-treeswith respect to and .Remark 1: When the set of separators of and is nonempty,

the reduced 2-tree with respect to and is unique, whichwill be denoted by . In addition, there is a unique se-quence of deletions of the degree-2 node other than to con-vert to a triangle. A linear ordering of the separators ofand is formed in the order they change from a 2-separator

to a peripheral during the above deletion process. For the re-duced 2-tree in Fig. 4(a), we can delete node to get

, then delete to get , then delete to getthe triangle . Therefore, the three separators of andform a linear ordering of , , . Hence,is a predecessor of , and is a successor of .We will use this ordering and the concepts of predecessor andsuccessor in the design of our algorithm in Section III.A reduced 2-tree gives a nice picture about the– paths, as shown in the following lemma. It is this propertythat makes the LDLP problem tractable when the underlyingnetwork topology is a subgraph of a minimum IFI network.Lemma 2.2: Let be a 2-tree, and be the reduced

2-tree with respect to and . Let be any one of the 2-sep-arators in , and let and be two link-disjoint paths

Fig. 5. WDM network with a 2-tree topology. Link labelsindicate the available wavelengths on each link. Links are also color/type-coded:A black/dotted link has no wavelength available; a red/dashed link has wave-length available; a blue/dash-dot link has available; a green/solid link hasboth and available.

in connecting and . Then, at least one of the following fourstatements is true.1) Both and go through node .2) goes through node , and goes through node .3) goes through node , and goes through node .4) Both and go through node .Proof: By Lemma 2.1, both and have to go through a

node in . Therefore, at least one of the four combinationsin this lemma must be true.

D. Partial 2-Trees

Subgraphs of 2-trees are known as partial 2-trees [33]. Bothtrees and rings are partial 2-trees. Partial 2-trees have nice struc-tures, which have been exploited by many researchers in di-verse applications. Many otherwise intractable problems can besolved efficiently on such networks [9], [33].Wald and Colbourn [33] presented a linear-time algorithm to

verify whether a given graph is a partial 2-tree, and when thegraph is a partial 2-tree, to complete it to a 2-tree by addingartificial links. For ease of presentation, we first present our al-gorithm in Section III with the assumption that the underlyingnetwork topology is a 2-tree.We then show in Section IV how togeneralize our algorithm to the case where the underlying net-work topology is a partial 2-tree, making use of the linear-timealgorithm in [33].

III. COMPUTING A PAIR OF LINK-DISJOINT LIGHTPATHS IN AMINIMUM IFI NETWORK, WITH GIVEN WAVELENGTHS

Our goal is to design a polynomial-time algorithm for com-puting a shortest pair of link-disjoint – lightpaths in a WDMoptical network whose underlying topology is a partial 2-tree,i.e., a subgraph of a minimum IFI network. For easier under-standing of the overall algorithm, we present the core of ouralgorithm in this section, concentrating on a restricted versionof the problem where the underlying topology is a 2-tree, andthe wavelengths of the two paths are chosen/fixed.Before formally defining this restricted problem, we use an

example to illustrate link-disjoint lightpath routing. Fig. 5 showsa WDM network whose underlying topology is a 2-tree with11 nodes, and . The attributes of the links are de-scribed in the caption of the figure. We will also use the WDMnetwork in Fig. 5 as our running example while illustratingmajor steps of our algorithm.Example 3: The shortest – lightpath on wavelength 1 (or

2) is . However, there does not exist another– lightpath that does not share a common link with . Onecan verify that on wavelength 1 and

on wavelength 2 form a pair of link-



disjoint lightpaths, with a total length of 10. Hence, computingthe shortest active lightpath first may fail to find a pair of disjointlightpaths when such a pair exists.In this section, we study LDLP-CW, a restricted version of

the LDLP problem with the wavelengths on and prede-termined, and the underlying topology being a 2-tree.Definition 3.1 (LDLP-CW): Let a WDM optical network be

given by , where is a 2-tree, andis the set of wavelengths. For each link

, the set of available wavelengths on link is. Let be the source node, and the destina-tion node. Let be given. The link-disjoint lightpathrouting problem with chosen wavelengths (LDLP-CW) asksfor a shortest pair of – link-disjoint lightpaths on wave-length and on wavelength . In other words, we wantto find a pair of – paths and such that we have thefollowing.1) Paths and do not share a common link.2) Wavelength is available on every link on .3) Wavelength is available on every link on .4) The path pair has the minimum total length (measured bythe total number of links used) among all – path pairssatisfying the above three conditions.

The development of our algorithm for solving LDLP-CW isorganized in the remainder of this section. In Section III-A, wedescribe the link attributes, where each link is associated witha constant number of attributes. In Section III-B, we describethe contraction operation. We also show the time updateof the link attributes associated with each contraction operation.In Section III-C, we describe how to apply the contraction op-erations to obtain a reduced 2-tree with respect to and . InSection III-D, we describe how to apply the contraction opera-tions to convert the reduced 2-tree to a triangle. We also showhow to find an optimal solution to LDLP-CW at the triangle.In Section III-E, we assemble these building blocks together topresent our -time algorithm for LDLP-CW.

A. Link Attributes

Assume that wavelengths are chosen for theLDLP-CW problem. For each undirected link , there aretwo ordered node pairs and , one in each direc-tion. For each ordered pair , we associate three attributes:

, , and . The meaning and value of theseattributes are explained in the following. We will use todenote the subgraph currently represented by link . Beforeperforming any contraction operations, consists of thenodes and , and the link . When we perform acontraction operation, will expand to include the sub-graph that has been contracted to it. Wewill explain this in detailin Section III-B. We will use to denote that lightpathdoes not exist. In this case, we say the lightpath is empty. We use

to denote the length of lightpath .When , is thenumber of hops in . When , we have .The meaning and value of the link attributes are explained inTable I.Therefore, for each undirected link in , we have

defined six attributes: , , , ,, and . These six attributes correspond to

eight (possibly empty) lightpaths: , , ,, , , , . In general,

TABLE ILINK ATTRIBUTES

iff . When ,contains the links in in reverse order. Similarly,

iff . When ,contains the links in in reverse order. Given this rela-tionship between ( and , respectively)and ( and , respectively), we will onlyshow the value of one of them in our illustrations.Example 4: We use the network in Fig. 5 to illustrate the

initial link attribute values, assuming and . Sincefor each link the current value of is the subgraphwith nodes and and link , there does not exist a pair oflink-disjoint – lightpaths in . Therefore, for each ofthe 19 links in Fig. 5, we have ,i.e., .Since has no wavelength available, we have

, i.e., .The same is true for and . Link has bothand available. Hence, is the lightpath ,

and is the lightpath . The attributes at linksand can be computed similarly. Link has

only available. Hence, is the lightpath ,but . The attributes at links , ,

, , , and can be computed similarly.Link has only available. Hence, and

is the lightpath . The attributes at links ,, , , and can be computed similarly.

B. Contraction Operations

We have seen in Section II-A that a 2-tree is either a triangle,or obtained by expanding a degree-2 node out of a link in anexisting 2-tree. The reverse operation of expansion is the con-traction of a degree-2 node onto the link , with which itforms a triangle. The link attributes are updated in timeaccording to defined rules when a contraction operation is per-formed. The contraction operations are used in our algorithm tocompute a reduced 2-tree with respect to and , while updatingthe link attributes that will be used to compute the optimal so-lution. We make use of the concatenation operator and the opti-mization operator, defined in the following.We use to denote the concatenation operator. If

and are two lightpaths on wavelength which shareno common node except , the concatenation of and



Fig. 6. If is a degree-2 node that makes a triangle with link , we cancontract node to link . After the contraction, is expanded to

, which is the union of , , and .

is a lightpath from to on wavelength . We useto denote this concatenation, whose length

is . Clearly,if or .

The second operator is the optimization operator, denotedby . This operator can be applied to either two lightpathsconnecting nodes and or four pairs of lightpaths connectingnodes and . If and are – lightpaths on, then is if

, and otherwise. In other words,is the shorter of the two lightpaths, with

tie broken in favor of the lightpath with a smaller index. Letbe link-disjoint – lightpath pairs for

. Then, ,, is the shortest of

the four lightpath pairs, with tie broken in favor of the light-path pair with a smaller index. Algorithmically, the valueof can becomputed as follows:

if then

outputelseif then

outputelseif then

outputelse outputendif

Lemma 3.1: Assume that is a degree-2 node, and is atriangle. Then, we can contract node onto link as shownin Fig. 6. The attributes associated with are updated byperforming (3.1)–(3.5) in the given order

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

Proof: We use ( , , respectively) todenote the subgraph already contracted to link ( ,

, respectively) prior to the contraction of node , and useto denote the subgraph contracted to link after

the contraction of node . Note that and shareonly one common node and no common link; and

share only one common node and no common link;and share only one common node and no

common link. Fig. 6 illustrates the contraction operation.A shortest – lightpath on in is either a

shortest – lightpath on in , or the concatenationof a shortest – lightpath on in and a shortest– lightpath on in . Therefore, we have (3.1).Equation (3.2) can be similarly derived.Now we prove the correctness of (3.3). Let

be a shortest pair of link-dis-joint lightpaths in , where is onwavelength and is on wavelength . Since

is the union of , , and ,exactly one of the following four cases must be true.Case 1) Neither lightpath goes through node .Case 2) Both lightpaths go through node .Case 3) Only goes through node .Case 4) Only goes through node .In Case 1), both and are in .

Since is a shortest pair of link-disjoint– lightpaths in , the first option in (3.3) gives ashortest pair of link-disjoint – lightpaths in .In Case 2), both and go through node .

Hence, is the concatenation of a lightpathon wavelength in , and a lightpath onwavelength in ; is the concatenationof a lightpath on wavelength in and alightpath on wavelength in . Further-more, and ( and ,respectively) are link-disjoint. Since is ashortest pair of link-disjoint – lightpaths in , we have

. Sim-ilarly, .Therefore, is a shortestpair of link-disjoint – lightpaths in , which corre-sponds to the second option in (3.3).In Case 3), is in , but goes

through node . Note that is a shortest – lightpathon in ; is a shortest – light-path on in the union of and ; and that

and are link-disjoint. Hence,is a shortest pair of link-disjoint

– lightpaths in , which corresponds to the thirdoption in (3.3). Case 4) is symmetric to Case 3), which can besimilarly proved.

C. From 2-Tree to Reduced 2-Tree

We can use the contraction operation described inSection III-B to obtain from , one contraction ata time, and update the link attributes in time per contrac-tion. The order of contraction follows Algorithm 1. The updateof link attributes follows the rules stated in Lemma 3.1.Example 5: In the 2-tree in Fig. 5, node is a degree-2 node

that makes a triangle with link . After contracting node ,the attributes at link are updated so that remainsto be the lightpath , is changed from to the



TABLE IILINK ATTRIBUTES

lightpath , and is changed from to thelightpath pair .Since is a degree-2 node, we can contract node onto

link . After this operation, node becomes degree-2.Therefore, we can contract node onto link . At thistime, represents the subgraph induced by the nodes, , , and . We can continue to contract node and node. After these contraction operations, we have transformed

the graph in Fig. 5 to the reduced 2-tree shown inFig. 4(a), where and are the only degree-2 nodes.

D. From Reduced 2-Tree to a Triangle

When is not a triangle, we can perform furthercontraction operations to convert the reduced 2-tree intoa triangle, at which we can find an optimal solution toLDLP-CW in time. Here, we need new attributes

, and for link ,which is either a separator of and , or a peripheral in thereduced 2-tree such that and . These aredefined in Table II.Let be a 2-separator of the reduced 2-tree .

Link divides the reduced 2-tree into two parts, onecontaining the source node , and the other containing thedestination node . We will use to denote the part thatcontains the source node, with the link excluded (as atechnical convention). Let be a peripheral of the reduced2-tree , with but . We will useto denote with the link excluded. Assume that

and . For the reduced 2-tree in Fig. 4(a),consists of the nodes , and , and the links ,

, , and (but not , which is excluded).Let be a 2-separator of , or but. We associate with the following four new attributes

, ,, and . Given the relationship

, we will concentrate on only.

Fig. 7. Values of can be computed from the values of intime. (a), (b) 2-separator is the successor of 2-separator .

(c), (d) 2-separator makes a triangle with the destination node , whichcoincides with .

For a given , is either nonempty withthe following meaning, or empty (denoted by ) when thecorresponding lightpath pair does not exist.The computation of the attributes starts with the link

in the reduced 2-tree that makes a triangle with thesource node . Lemma 3.2 shows how to compute these at-tributes for the link in that makes a triangle withthe source node . Lemmas 3.3, 3.4, 3.5, and 3.6 show the rulesfor computing , , , and ,respectively, when are known, where is thepredecessor of in .Lemma 3.2: Let be the link in that makes a

triangle with the source node , where . Then,can be computed as follows:

(3.6)

(3.7)

(3.8)

(3.9)

where and are defined in Section III-A and com-puted according the rules in Section III-B.

Proof: In this case, contains the nodesand the links and . When nonempty,

, where is an – lightpath onin , and is an – lightpath on insuch that and are link-disjoint and the twoform a shortest link-disjoint lightpath pair. Therefore, we have(3.6). When nonempty, is a shortest – lightpath onin , is a shortest – lightpath on in. The two lightpaths are guaranteed to be link-disjoint.

Hence, we have (3.7). Equation (3.8) is symmetric to (3.7).Equation (3.9) is symmetric to (3.6). Hence, they can be provedsimilarly to (3.7) and (3.6).Having computed the values for 2-separator ,

we can compute the values for its succeeding 2-sep-arator (if is the 2-separator which makes a tri-angle with the destination node, we can compute the values

). The two cases are illustrated in Fig. 7(a) and (b) andFig. 7(c) and (d), respectively.The rules for computing from the values of

are given in the next four lemmas. Their proofs are straight-forward (illustrated in Figs. 8–11). We provide the proof forLemma 3.3 and omit those for Lemmas 3.4–3.6 because theyfollow similar logic.Lemma 3.3: Let be a 2-separator in the reduced

2-tree . Let be the successor of , as shownin Fig. 7(a), or is a peripheral where coincides with, as shown in Fig. 7(c). Then, is



Fig. 8. Computing the attribute by selecting the best among fourpossible cases. (a) c-1.1: using ; (b) c-1.2: using ; (c) c-1.3:using ; (d) c-1.4: using .

a shortest lightpath pair among the following four (possiblyempty) lightpath pairs, as illustrated in Fig. 8:

c-1.1: ;c-1.2: ;c-1.3: ;c-1.4: .Proof: By definition, is the

shortest link-disjoint lightpath pair in , with an– lightpath on , and an – lightpath on . Thereare exactly four possible cases, as shown in Fig. 8(a)–(d).In c-1.1, neither nor uses any link in , as

shown in Fig. 8(a). Hence, since nolinks in are used by either lightpath.In c-1.2, (on ) does not use any link in , but(on ) goes from to through , as shown

in Fig. 8(b). Hence, .Recall that is a shortest link-disjoint lightpath pairin such that is an – lightpath on , andis an – lightpath on , and that is a shortest –lightpath on in .In c-1.3 (symmetric to c-1.2), (on ) goes from tothrough , but (on ) does not use any link in

, as shown in Fig. 8(c). Hence,. Recall that is a shortest link-dis-

joint lightpath pair in such that is an – lightpathon , and is an – lightpath on , and that isa shortest – lightpath on in .In c-1.4, both and go from to through

, as shown in Fig. 8(d). Hence,. Recall that is

a shortest link-disjoint lightpath pair in such thatis an – lightpath on , and is an – lightpath on .Also recall that is a shortest link-disjointlightpath pair in such that is a – lightpathon and is a – lightpath on .Lemma 3.4: Let be a 2-separator in the reduced 2-tree

. Let be the successor of , as shown inFig. 7(a), or is a peripheral where coincides with ,as shown in Fig. 7(c). Then, is theshortest lightpath pair among the following four (possiblyempty) lightpath pairs, as illustrated in Fig. 9:

c-2.1: ;c-2.2: ;c-2.3: ;c-2.4: .Proof: is a shortest link-disjoint

lightpath pair in , with an – lightpath on , andan – lightpath on . There are exactly four possible

cases, as shown in Fig. 9(a)–(d). The rest of the proof followssimilar logic as used in the proof of Lemma 3.3.

Fig. 9. Computing the attribute by selecting the best among fourpossible cases. (a) c-2.1: using . (b) c-2.2: using . (c) c-2.3:using . (d) c-2.4: using .



Lemma 3.5: Let be a 2-separator in the reduced2-tree . Let be the successor of , as shownin Fig. 7(a), or is a peripheral where coincides with, as shown in Fig. 7(c). Then, isa shortest lightpath pair among the following four (possiblyempty) lightpath pairs, as illustrated in Fig. 10:

c-3.1: ;c-3.2: ;c-3.3: ;c-3.4: .Proof: is a shortest link-disjoint


cases, as shown in Fig. 10(a)–(d). The rest of the proof followssimilar logic as used in the proof of Lemma 3.3.Lemma 3.6: Let be a 2-separator in the reduced 2-tree

. Let be the successor of , as shown inFig. 7(a), or is a peripheral where coincides with ,as shown in Fig. 7(c). Then, is theshortest lightpath pair among the following four (possiblyempty) lightpath pairs, as illustrated in Fig. 11:

c-4.1:;

c-4.2: ;c-4.3: ;c-4.4: .Proof: is a shortest link-disjoint


cases, as shown in Fig. 11(a)–(d). The rest of the proof followssimilar logic as used in the proof of Lemma 3.3.Now we have most of the building blocks for our algorithm.

Let us use the reduced 2-tree in Fig. 4(a) as an example. Using



Fig. 12. Selecting the best among four possible cases. (a) c-F.1: using .(b) c-F.2: using . (c) c-F.3: using . (d) c-F.4: using .

Lemma 3.2, we can compute the values for for. Using Lemmas 3.3–3.6, we can compute the values

for , then for , and for . The solutionto LDLP-CW can be decided by the following lemma.Lemma 3.7: Suppose that we have computed for

, where and is a peripheral in the re-duced 2-tree . Then, an optimal solution to LDLP-CWis the lightpath pair , which is the shortest among thefollowing four (possibly empty) lightpath pairs, as illustrated inFig. 12(a)–(d):

c-F.1: ;c-F.2: ;c-F.3: ;c-F.4: .

The instance of LDLP-CW has no solution when all of theabove four lightpath pairs are .

Proof: The proof is similar to that of Lemma 3.3. As illus-trated in Fig. 12, there are four possible cases.

c-F.1: Both and go from to through .c-F.2: goes from to through , but doesnot use any link in .c-F.3: does not use any link in , but goes fromto through .

c-F.4: Neither nor uses any link in .In c-F.1, the solution uses and

. Since is a shortest pair of link-dis-joint – lightpaths in , and is a shortest pair oflink-disjoint – lightpaths in ,

is an optimal solution for LDLP-CW in this case.The other three cases can be similarly proved, where c-F.2 uses

and ; c-F.3 uses and ; c-F.4uses .

E. Linear-Time Algorithm for LDLP-CW

After the detailed descriptions of all the building blocks, wecan summarize our design of the algorithm in Algorithm 2. Ona high level, Algorithm 2 can be divided into three phases. Inthe first phase (lines 1–9), we perform contraction operationsto compute the reduced 2-tree from . The order of thecontraction operations follows that of Algorithm 1, while theupdate of the link attributes ( , , ) follows the rules inLemma 3.1. In the second phase (lines 10–19), we perform fur-ther contraction operations to convert the reduced 2-treeinto a triangle. The order of contraction operations follows thatdescribed in Remark 1, while the link attributes ( , , , )are initialized according to Lemma 3.2 (line 10) and updated ac-cording to Lemmas 3.3–3.6 (lines 11–19). Finally, the optimalsolution is computed according to Lemma 3.7 (lines 20–21).

Algorithm 2: Alg-LDLPCW

Input: A WDM optical network with a 2-tree topology.A source node . A destination node . Two chosenwavelengths and .Output: A shortest pair of link-disjoint – lightpaths onand on , if such a pair exists.1: for each link in do2: Initialize , , , ,

, , using the rules in Section III-A;3: end for4: while a degree-2 node other than and in do5: if the two nodes adjacent to are and then6: break;7: end if8: Assume makes a triangle with link . Contract

node onto link , and update ,and , using the rules in Lemma 3.1.

9: end while{By now we have computed the reduced 2-tree.}

10: Let be the triangle in the reduced 2-tree thatcontains . Initialize , , , and

according to Lemma 3.2.11: while a degree-2 node in the reduced 2-tree do12: if the reduced 2-tree is a triangle then13: break;14: end if15: Let be the 2-separator that forms a triangle with

. Let be the successor of , orbe a peripheral and coincides with .

16: Compute for according toLemmas 3.3–3.6.

17: Delete node in the reduced 2-tree.18: STOP if

;19: end while20: Compute the shortest pair of link-disjoint lightpaths

according to Lemma 3.7. STOP if such a pairdoes not exist;

21: Extract the link-disjoint lightpaths and using atop-down method;

Example 6: We use the network in Fig. 5 to illustrate thealgorithm. Here, we assume that and . By thetime the reduced 2-tree in Fig. 4(a) is constructed, we have thefollowing nonempty link attributes (for each link , we onlylist the attribute for one ordered pair): ,

; ;; , ;

; , ,; ,

, ;, ; .

The sequence of 2-separators of the reduced 2-treeis , , and .At , the values of are computed ac-

cording to Lemma 3.2: ,, , and. Note that we also know the values of

by the relationship .



At , the values of , , ,and are computed according to Lemmas 3.3,3.4, 3.5, and 3.6, respectively: ,

, ,and . Note thatwe also know the values of by the relationship

.At , the values of , ,

, and are computed according toLemmas 3.3, 3.4, 3.5, and 3.6, respectively:

, ,, and

. Note that we also have the values forby the relationship .

At , the values of , ,, and are computed according

to Lemmas 3.3, 3.4, 3.5, and 3.6, respectively:,,

, and.

Finally, in line 20 of the algorithm, we find that the shortestpair of – lightpaths is .Meanwhile in the illustrations, we have listed out the lightpathsduring each step. In actual implementations, we only need to re-member the length of the lightpath, as well as its composition(the concatenation of one or two lightpaths that are previouslycomputed). Once we reached the final answer, the pair of light-paths can be extracted in time.Our analysis in this section leads to the following theorem.Theorem 3.1: The worst-case time complexity of

Algorithm 2 is . If there exists a pair of link-disjoint– lightpaths and such that is available on each linkon and that is available on each link on , Algorithm 2finds such a pair of lightpaths with minimum total length.Otherwise, Algorithm 2 stops without outputting any lightpath.

Proof: Each contraction operation takes time. Thereare contraction operations. This leads to the time com-plexity of the algorithm.The correctness of the algorithm follows from Lemma 2.2

and Lemmas 3.1–3.7. Let and be the optimal solutionfor LDLP-CW, where is an – lightpath on , andis an – lightpath on such that the two are link-disjoint andform the shortest pair. Let be a separator of and (i.e.,

is a 2-separator in ). Lemma 2.2 implies that atleast one of the following four cases is true.1) Both and go through . In this case, the sub-path of from to can be replaced by (the firstelement of ) without losing optimality, and thesubpath of from to can be replaced by (thesecond element of ) without losing optimality.

2) goes through , and goes through . In thiscase, the subpath of from to can be replaced by

(the first element of ) without losing opti-mality, and the subpath of from to can be replacedby (the second element of ) without losingoptimality.

3) goes through , and goes through . In thiscase, the subpath of from to can be replaced by

Fig. 13. WDM network with a partial 2-tree topology. Linklabels indicate the available wavelengths on each link. Links are also color/type-coded: A red/dashed link has wavelength available; a blue/dash-dot link hasavailable; a green/solid link has both and available.

(the first element of ) without losing opti-mality, and the subpath of from to can be replacedby (the second element of ) without losingoptimality.

4) Both and go through . In this case, the sub-path of from to can be replaced by (the firstelement of ) without losing optimality, and thesubpath of from to can be replaced by (thesecond element of ) without losing optimality.

Lemma 3.1 ensures that lines 1–9 of Algorithm 2 correctly com-pute the reduced 2-tree and the link attributes ,and . Lemmas 3.2–3.6 ensure that lines 10–19 correctly com-pute the attributes , , , and , while converting the re-duced 2-tree into a triangle. Lemma 3.7 ensures thatAlgorithm 2 computes a pair of link-disjoint – lightpaths thatare as good as the optimal pair. Hence, Algorithm 2 solves theLDLP-CW problem correctly.

IV. POLYNOMIAL-TIME ALGORITHM FOR LDLP WITH APARTIAL 2-TREE TOPOLOGY

For any given WDM network with a partial 2-tree topology,we can apply the linear-time algorithm of [33] to complete thepartial 2-tree to a 2-tree. The algorithm of [33] may add artificiallinks to augment the partial 2-tree into a 2-tree. Note thatwe willnot change the network topology. To ensure this, as a technicalconvention, we mark these added artificial links as if they haveno wavelength available at all.The underlying topology of the WDM network in Fig. 13 is a

partial 2-tree, but not a 2-tree. It can be completed to the 2-tree inFig. 5 by adding artificial links , , and , whichhave no wavelength available. Note that any link added in thecompleting process has no wavelength available—because thelink simply does not exist in the physical network. Assuminga 2-tree topology simplifies the description of our algorithm inSection III.Now we are ready to present our polynomial-time algorithm

for solving the LDLP problem when the underlying networktopology is a partial 2-tree. The algorithm can be dividedinto two phases. In the first phase, we augment the networktopology from a partial 2-tree to a 2-tree using the algorithmof [33]. Since the links added in the process have no wave-length available, the augmenting process does not change thesolution to the LDLP problem. In the second phase, we solveLDLP-CW for all possible pairs of and choose theshortest lightpath pair computed as the overall solution. Thealgorithm is listed as Algorithm 3.



Algorithm 3: Alg-LDLP

Input: A WDM optical network with a partial 2-treetopology. A source node . A destination node .Output: A shortest pair of link-disjoint lightpaths (on

) and (on ).1: Apply the linear-time algorithm of [33] to augment thepartial 2-tree to a 2-tree. Mark all links added in thisprocess as having no wavelength available.

2: , , ;3: for do4: for do5: Apply Alg-LDLPCW to compute a shortest pair

of link-disjoint lightpaths: on , and on;

6: if then7: , , ,

;8: end if9: end for10: end for11: if then12: STOP: no disjoint lightpath pair exists;13: else14: OUTPUT and ;15: end if

Theorem 4.1: Assume that is a partial 2-tree. Then,Algorithm 3 solves the LDLP problem in time.

Proof: Augmenting to a 2-tree takes time.Since we are solving LDLP-CW for wavelength pairs

, the time complexity of Algorithm 3 is . Theinstance of LDLP problem has a solution if and only if one ofthe instances of LDLP-CW has a solution. Therefore,Algorithm 3 correctly solves the LDLP problem.For example, assume that the input to Algorithm 3 is the

WDM network shown in Fig. 13 with source node anddestination node . The algorithm will first augment thepartial 2-tree shown in Fig. 13 to the 2-tree shown in Fig. 5.The algorithmwill run Alg-LDLPCW three times, withbeing set to (1,1), (1,2), and (2,2), respectively.With , Alg-LDLPCW returns no light-

path pair. With , Alg-LDLPCW returnsthe lightpath pair .With , Alg-LDLPCW returns no light-path pair. Hence, Alg-LDLP returns the lightpath pair

, which is an optimalsolution.

V. NUMERICAL RESULTS

To verify the effectiveness of our proposed algorithm, we im-plemented our algorithm (denoted by LDLP) and compared itto the well-known shortest active path first heuristic (denotedby SAPF). The comparison was done on randomly generatednetworks whose underlying topologies are 2-edge connectedsubgraphs of minimum isolated failure immune networks. Thenumber of nodes varied from 100 to 500 in steps of 100.For each value of , we first randomly generated a 2-tree net-work on nodes. Then, for a given decimal number ,

Fig. 14. Improvement on the number of successful requests with .(a) . (b) .

we randomly deleted links in the 2-tree to obtain a par-tial 2-tree (we sequentially deleted links whose deletion doesnot destroy the 2-edge connectivity of the resulting graph). Theresulting 2-edge connected partial 2-tree was used as the un-derlying topology. We tested with each link having wave-lengths, with . We studied four different net-work load values , where of the networkresources are being used. Here, each available wavelength on alink counts as one unit of resource. Therefore, a network withlinks has resources before any of the wavelengths is

used. For a partial 2-tree with links, and any given value, we randomly reserved units

resource so that these resources become unavailable. For eachnetwork so obtained, we generated connection requestswith the source and destination nodes randomly chosen. Undereach of these combinations, we compared LDLP to SAPF. Allthe tests were performed on a 3.2-GHz Linux PC. The resultsreported here are the average over 50 test cases.To compare LDLP to SAPF, we used the same network

state information and the same connection request, given by asource–destination pair. For this reason, we did not reserve theresource (i.e., wavelength) after the requests are accepted. Foreach connection request, we used both algorithms to find a pairof link-disjoint lightpaths. There are four possible results.1) Neither algorithm finds a pair of link-disjoint lightpaths.2) LDLP finds such a pair, but SAPF fails.3) Both algorithms find a pair of lightpaths, but the pair foundby LDLP is shorter.

4) Both algorithms find a pair of lightpaths, and the two pairsare of equal length.

We are only interested in the results where at least one algorithmfinds a pair, i.e., 2)–4). Let , , and denote the number ofrequests leading to 2), 3), and 4), respectively. The improvementon the number of successful requests achieved by LDLP overSAPF is then defined as . The improvement onthe quality of the lightpaths is defined as .Figs. 14 and 15 show these two improvements when .

Specifically, Fig. 14(a) and (b) illustrates the improvement onthe number of successful requests when and ,respectively. We observe that the improvement can reach upto 25% in certain conditions for both levels of partial 2-trees.The most improvement happens when the network is heavilyloaded but not severely loaded, e.g., with and

with or 50. When the network is load-freeor severely loaded, the improvement is small. The reason forthe load-free network is that SAPF is highly likely to succeedin such networks. The reason for the severely loaded networkis that there are not enough resources to have two link-disjointlightpaths even for LDLP in this case. The last observation isthat the number of nodes in the networks almost does not affect



Fig. 15. Improvement on the quality of the lightpaths with . (a). (b) .

Fig. 16. Improvement on the number of successful requests with .(a) . (b) .

Fig. 17. Improvement on the quality of the lightpaths with . (a). (b) .

Fig. 18. Scalability of LDLP.

the results. This is because the deleted links for partial 2-treegeneration and network resources reservation are both propor-tional to the network size.Fig. 15(a) and (b) shows the improvement on the quality of

lightpaths when and , respectively. We ob-serve that the improvement is more than 75% in some cases.Similar to the previous results, the most improvement happenswhen the network is moderately loaded. The reason behind thisobservation has been explained in the above. Different from theprevious results, the improvement increases with the increaseof the number of nodes. The reason is that when the number ofnodes increases, the chance that two nodes need to be connectedby a longer lightpath increases as well, which makes SAPF in-ferior to LDLP.To study the effect of the number of wavelengths on the per-

formance of LDLP, we also carried out another set of tests with. The results are shown in Figs. 16 and 17. Note that the

results are very similar to those with .

The running time of LDLP is shown in Fig. 18. We observethat the running time is proportional to for any fixed , andproportional to for any fixed , which verifies our theoret-ical analysis. We also observe that our algorithm is very fast inpractice.

VI. CONCLUSION

We have presented a polynomial-time algorithm for com-puting a pair of link-disjoint lightpaths in a WDM optical net-work whose underlying topology is a subgraph of a minimumIFI network. Our algorithm can be easily modified to computea pair of node-disjoint lightpaths in such networks.IFI networks have received a lot of attention in the litera-

ture [13], [38] because they are fault-tolerant to multiple fail-ures as long as the faults are isolated. Minimum IFI networksare attractive because they are scalable and often admit simplesolutions to many networking problems [6], [9], [33]. Besidesproviding guidelines for the design of future survivable opticalnetworks, our work can have an impact on improved decisionmaking in existing optical networks. While the backbone of ex-isting optical networks may not be subgraphs of minimum IFInetworks, we can compute a subgraph of the given network thathas the structure assumed in this paper. This can be achieved,using offline computation, by hiding a few network links so thatthe resulting network has no subgraph that is homeomorphic tothe four-vertex complete graph [34]. Our algorithm can beapplied to this resulting network, while the links that have beenhidden in the above process can be used to provide additionalprotection.

ACKNOWLEDGMENT

The authors thank the Associate Editor and the anonymousreviewers whose comments on earlier versions of this paperhave helped to significantly improve the presentation of thepaper.

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Guoliang Xue (M’96–SM’99–F’11) received theB.S. degree in mathematics and M.S. degree in oper-ations research from Qufu Normal University, Qufu,China, in 1981 and 1984, respectively, and the Ph.D.degree in computer science from the University ofMinnesota, Minneapolis, MN, USA, in 1991.He is a Professor of computer science and engi-

neering with Arizona State University, Tempe, AZ,USA. His research interests include survivability, se-curity, and privacy issues in smart grid, computer net-works, and social networks. He also applies game

theory to model selfish behavior in cooperative networks.Prof. Xue is an Associate Editor of the IEEE/ACM TRANSACTIONS ON

NETWORKING and IEEE Network. He was a Keynote Speaker at IEEE LCN2011 and a TPC Co-Chair of IEEE INFOCOM 2010.

Ravi Gottapu received the M.S. degree in computer science and engineeringfrom Arizona State University, Tempe, AZ, USA, in 2003.He is with Amazon Inc., Seattle, WA, USA. Prior to joining Amazon, he was

with Microsoft, Inc., Seattle, WA, USA. His research interests include efficientalgorithms for optimization problems in networking, with applications to QoSand survivability.

Xi Fang (S’09) received the B.S. and M.S. degreesfrom Beijing University of Posts and Telecommuni-cations, Beijing, China, and will receive the Ph.D. de-gree in computer science from Arizona State Univer-sity, Tempe, AZ, USA, in 2013.He is a Software Development Engineer with Mi-

crosoft, Inc., Redmond, WA, USA. His research in-terests include big data, cloud computing, wirelessnetworks, and smart grids.Mr. Fang received Best Paper awards at IEEE ICC

2012, IEEE MASS 2011, and IEEE ICC 2011.

Dejun Yang (S’08) received the B.S. degree in com-puter science from PekingUniversity, Beijing, China,in 2007, and is currently pursuing the Ph.D. degree incomputing, informatics, and decision systems engi-neering (CIDSE) at Arizona State University, Tempe,AZ, USA.His research interests include economic and op-

timization approaches to networks, crowdsourcing,smart grid, big data, and cloud computing.Mr. Yang has received Best Paper awards at IEEE

MASS 2011 and IEEE ICC 2011 and 2012, and wasa Best Paper Award runner-up at IEEE ICNP 2010.

Krishnaiyan Thulasiraman (M’72–SM’84–F’90)received the Bachelor’s and Master’s degrees fromthe University of Madras, India, in 1963 and 1965,respectively, and the Ph.D. degree from the IndianInstitute of Technology (IIT), Madras, India, in1968, all in electrical engineering.He is the Hitachi Chair Professor with the School

of Computer Science, University of Oklahoma,Norman, OK, USA. His research interests havebeen in graph theory, algorithms, and applicationsin electrical networks and fault tolerance in optical

networks.Dr. Thulasiraman has received several awards and honors, including being an

elected member of the European Academy of Sciences in 2002 and receiving theIEEE CAS Society Charles A. Desoer Technical Achievement Award in 2006.

Documents

IEEE/ACM TRANSACTIONS ON NETWORKING 1 A Polynomial …optimization.asu.edu/papers/XUE-JNL-2013-TON-2tree.pdfJanuary 24, 2013; accepted March 01, 2013; approved by IEEE/ACM TRANSACTIONS