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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006 147

On the Complexity of and Algorithms for Findingthe Shortest Path With a Disjoint Counterpart

Dahai Xu, Member, IEEE, Yang Chen, Student Member, IEEE, Yizhi Xiong, Chunming Qiao, Member, IEEE, andXin He, Member, IEEE

Abstract—Finding a disjoint path pair is an important compo-nent in survivable networks. Since the traffic is carried on the ac-tive (working) path most of the time, it is useful to find a disjointpath pair such that the length of the shorter path (to be used asthe active path) is minimized. In this paper, we first address such aMin-Min problem. We prove that this problem is NP-complete ineither single link cost (e.g., dedicated backup bandwidth) or duallink cost (e.g., shared backup bandwidth) networks. In addition,it is NP-hard to obtain a -approximation to the optimal solu-tion for any 1. Our proof is extended to another open ques-tion regarding the computational complexity of a restricted versionof the Min-Sum problem in an undirected network with ordereddual cost links (called the MSOD problem). To solve the Min-Minproblem efficiently, we introduce a novel concept called conflictinglink set which provides insights into the so-called trap problem, anddevelop a divide-and-conquer strategy. The result is an effectiveheuristic for the Min-Min problem called COnflicting Link Exclu-sion (COLE), which can outperform other approaches in terms ofboth the optimality and running time. We also apply COLE to theMSOD problem to efficiently provide shared path protection andconduct comprehensive performance evaluation as well as compar-ison of various schemes for shared path protection. We show thatCOLE not only processes connection requests much faster than ex-isting integer linear programming (ILP)-based approaches but alsoachieves a good balance among the active path length, bandwidthefficiency, and recovery time.

Index Terms—Bandwidth sharing, dynamic provisioning, graphtheory, optimization, protection, survivability.

I. INTRODUCTION

SURVIVABILITY is a critical design problem forhigh-speed networks. To protect a mission-critical connec-

tion from a single link (node) failure, a common solution is tofind a link (node) disjoint pair of paths from a source (ingress)node to a destination (egress) node. In such path protectionapplication, traffic is routed along one path, called the active(working) path (AP) unless it is affected by a link (node)

Manuscript received December 9, 2003; revised September 13, 2004, March20, 2005, and April 22, 2005; approved by IEEE/ACM TRANSACTIONS ON

NETWORKING Editor J. Yates. This work was supported in part by the NationalScience Foundation (NSF) under Contracts ANIR 0208331 and CCR-0309953.A preliminary version of this paper entitled “On Finding Disjoint Paths inSingle and Dual Link Cost Networks” was presented at the IEEE INFOCOM2004, Hong Kong.

D. Xu, C. Qiao, and X. He are with the Department of Computer Science andEngineering, State University of New York at Buffalo, Buffalo, NY 14260 USA(e-mail: [email protected]; [email protected]; [email protected]).

Y. Chen is with the College of Computing, Georgia Institute of Technology,Atlanta, GA 30332 USA (e-mail: [email protected]).

Y. Xiong is with Cedars-Sinai Medical Center, Los Angeles, CA 90048 USA(e-mail: [email protected]).

Digital Object Identifier 10.1109/TNET.2005.863451

failure, upon which, the traffic is re-routed along the other path,called the backup path (BP).

In this paper, we focus on the problem of finding a pair of link(node) disjoint AP and BP in an on-line setting. More specifi-cally, we assume that not all requests for survivable connectionsarrive at the same time, and a decision as to how to satisfy a re-quest (if possible at all) has to be made without knowing whichrequests will arrive in the future, and for the sake of guaranteedquality of service (QoS) (e.g., in terms of both delay and loss),without being able to rearrange the way existing connectionsare established, including the existing BPs. The latter restric-tion is placed to avoid the overhead and delay involved in co-ordinating the rearrangement of existing connections, and moreimportantly, avoid data loss due to a link (or node) failure onan AP while its corresponding BP is being rearranged, for ex-ample. Techniques for and potential benefits of rearranging theexisting connections (either both APs and BPs or just BPs) as apart of network maintenance have been described in [1]–[3].

In the above on-line path protection application, one of thechallenges is how to allocate minimal amount of network re-source (e.g., bandwidth) for a connection requesting, say,units of guaranteed bandwidth. In the case of so-called dedi-cated protection, additional units of bandwidth are to be allo-cated along (each link of) both the chosen AP and BP. Hence, theproblem of minimizing the sum of the additional units of band-width to be allocated along the AP and BP, hereafter called theMin-Sum problem, can be easily solved using, e.g., the ShortestPair of Path (SPP) algorithm [4], [5] (by treating as the costof using each link), which has a polynomial time complexity.However, in the case of so-called shared path protection, nosuch polynomial time algorithm exists today which can solve theMin-Sum problem optimally. Intuitively, this is because while

additional units of bandwidth has to be allocated on a chosenAP, the exact number of additional units of bandwidth to be al-located on a link along BP (called backup bandwidth) dependson how the corresponding AP as well as the existing APs andBPs are established. More specifically, denote by the ad-ditional units of backup bandwidth needed on a link, we have

because of the possible sharing of the backupbandwidth among this BP and other existing BPs using the samelink. In fact, in spite of extensive research [6]–[9], one of theopen questions is whether the Min-Sum problem is NP-com-plete (NPC) in undirected network, where each link has two or-dered “costs” and the cost incurred by a BP is always a fractionof the cost incurred by an AP. In addition, given the fact thatthe AP is used most of the time while BP is used only when APfails, it is natural to ask the question of how to minimize the AP.

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148 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006

In this paper, as one of our contributions, we answer this openquestion by proving this instance of the Min-Sum problem (tobe called MS problem with ordered dual costs, or MSOD) to beNPC (and hence the general Min-Sum problem is also NPC).We also address the second open problem, which is to find apair of link (node) disjoint paths in (directed or undirected) net-works such that the length of the shorter path (to be used as anAP), in terms of the number of hops or geographical distance ithas, is minimized. We call it a Min-Min problem, as opposedto the Min-Sum problem above. The Min-Min problem is alsodifferent from the so-called Min-Max problem studied in [10]and [11] where the objective was to minimize the length of thelonger one of the two paths (to be used as a BP).

Specifically, as another major contribution, we prove that theMin-Min problem above is also NPC, whether each link in adirected or undirected network has only one cost (as in the caseof dedicated protection) or dual costs (as in the case of sharedpath protection).1 Furthermore, we prove that even the problemof obtaining a -approximation solution to the Min-Minproblem is NPC for any , which means there is andwill be no polynomial time heuristic algorithm that can achieveguaranteed near-optimal performance as long as .

In addition to the above theoretical results, this paper providesa novel and efficient heuristic algorithm to obtain practicallynear-optimal solutions to the above two problems that we proveto be NPC. More specifically, we propose a heuristic calledCOnflicting Link Exclusion (COLE) for the Min-Min problem,and extend it for the Min-Sum problem in shared path protec-tion. Our comprehensive performance evaluation and compar-ison studies reveal that COLE can achieve better results thanother heuristic algorithms for the Min-Min problem. Further-more, a simple application of COLE to the MSOD problemcan achieve near-optimal performance for shared path protec-tion when compared to its counterpart that solves integer linearprogramming (ILP) formulations using the branch and boundtechniques, whose time complexity can be prohibitively highfor large networks.

The rest of this paper is organized as follows. Section IIdescribes related work, and provides the motivations for thiswork, especially the Min-Min problem. In Section III, wefirst prove that the Min-Min problem is NPC, and so is its

-approximation problem, as well as the MSOD problemin undirected networks. Section IV describes the proposedCOLE heuristic for the Min-Min problem. In Section V, wepresent numerical results from the comparison between theproposed heuristics and other existing heuristics as well as ILPbased approaches. Finally, Section VI concludes this paper.Frequently used acronyms are listed at the end of the paper foreasy reference.

II. PRELIMINARY

In this section, we first describe related work on finding a link(node) disjoint path pair and then provide the motivation for ourwork.

1Since having a single cost network is a special case of having a dual-costnetwork, as long as we can prove that the Min-Min problem in a single-costnetwork is NPC, the Min-Min problem in a dual-cost network is also NPC.

A. Related Work

As mentioned earlier, in solving the problem of finding a linkor node disjoint pair of AP and BP, one can model a network aseither a single (link) cost graph or a dual (link) cost graph. Inthe former, the cost of using a link by AP, denoted by , is thesame as that by BP, denoted by . In the latter, could bedifferent from, and often higher than (due to backup band-width sharing as mentioned earlier). In addition, we will con-sider the most general dual-cost network model where each linkis associated with a vector of two costs, ( , ), such thatand can have an arbitrary relationship (e.g., it is possible that

). Accordingly, the single-cost model is a special caseof this dual-cost model. In the following discussion, we will usethe terms “network” and “graph” interchangeably.

Computational Complexity of Various Problems: Manyproblems related to finding a pair of link or node disjoint pathsin single cost networks have been studied [4], [5], [10]–[12].For instance, the Min-Sum problem mentioned earlier, in whichthe objective is to minimize the sum of the costs of the twopaths can be solved using a polynomial time algorithm calledSPP [4], [5]. Unlike Min-Sum, the Min-Max problem, whoseobjective is to minimize the length of the longer one of thetwo paths was proved to be NPC [10], [11]. However, as faras we know, no existing work has addressed what we call theMin-Min problem, in which the objective is to minimize thelength of the shorter one of the two paths. In later subsections,we will provide the motivations for the Min-Min problem aswell as its NP completeness proof.

While it is clear that the Min-Max problem in a more generaldual-cost network is NPC, it is not straightforward to show thatthe Min-Sum problem in a dual-cost network is also NPC (giventhat a polynomial time algorithm exists in a single-cost net-work). In [9], it was shown that in a network where the relation-ship between and on each link is arbitrary, the Min-Sumproblem is NPC. A more restricted version of the problem wasalso studied, where for each and every link, . Thisversion of the Min-Sum problem, referred to as the MS with“ordered” dual cost (or MSOD) problem, is also NPC [7]. Re-cently, an even more restricted version of the MSOD problemwas studied, where for each and every link, , and

is a constant for all the links. It has been shown thatsuch a MS with uniformly ordered dual link costs (or MSOD-U)is NPC but only for a directed network [8]. It was left as anopen question as to whether the MSOD-U problem is NPC foran undirected network.2

Later, we will prove the MSOD-U problem is NPC for anundirected network. Our proof can be applied to a directed net-work, yielding a different proof from that in [8]. Note that,the MSOD-U problem is more restricted than the previouslymentioned two versions, namely, “ordered” but only “nonuni-formly,” and “arbitrary” (or “unordered”). Hence, our proof canalso be used to show that these versions are NPC. In partic-ular, we note that the proof of NP completeness of the Min-Sumproblem in [7] only applies to the “nonuniformly ordered” dual-

2Because the proof in [8] used a known NPC problem that is only applicableto a directed network in its reduction process, the proof cannot be extended toan undirected network.

XU et al.: ON THE COMPLEXITY OF AND ALGORITHMS FOR FINDING THE SHORTEST PATH WITH A DISJOINT COUNTERPART 149

TABLE ICOMPUTATIONAL COMPLEXITY OF VARIOUS PROBLEMS OF FINDING

TWO DISJOINT PATHS

cost case, where of a link to be used by a BP depends onhow its corresponding AP and other existing APs and BPs areestablished. However, our proof will also lead to the proof ofthe NP-completeness of a more general version of the Min-Sumproblem in the “nonuniformly ordered” case, where of a linkis independent of the other factors.

Table I summarizes the main results on the computationalcomplexity of various problems related to finding a link or nodedisjoint pair of paths, where our major contributions are high-lighted.

Algorithms: In this work, we focus on the algorithms usefulfor path protection. Note that, although SPP, which is an al-gorithm developed for single-cost networks, can be applied toshared path protection, the paths found by SPP are not optimalfor shared path protection in that the sum of the costs (band-width) of AP and BP is not minimum. Accordingly, several al-gorithms have been developed for shared path protection, whichessentially are solutions to the MSOD problem.

One typical class of algorithms develop ILP formulationswhose objective is to jointly optimize the selection of bothAP and BP for the MSOD problem, and then solve the ILPformulations using the branch-and-bound search techniques toachieve optimal results. Two representative schemes belongingto this class, which have previously undergone extensive quan-titative performance evaluation, are Sharing with CompleteInformation (SCI) [13] and Distributed Partial InformationManagement (DPIM) [14].

In addition to the above mentioned ILP formulation basedalgorithms, whose time complexity could be prohibitively highfor a large network, another class of algorithms for the MSODproblem use the active-path-first (or APF) heuristic [2], [15],[16]. In these APF-based heuristics, an AP is found first by usingthe Dijkstra algorithm (or any other shortest path algorithms)without considering the need to find a corresponding BP forthe time being, and the BP is found (again using the Dijkstraalgorithm for example) after removing the links or nodes alongthe AP.

Previous studies have shown that APF-based heuristics canachieve near-optimal solutions to the MSOD problem comparedto their ILP-based counterparts [2], [15], [17]. The major chal-lenge in using the APF heuristic, however, is that once an AP isfound, one may not be able to find a disjoint BP (even thougha pair of disjoint paths do exist using a different AP). This isthe so-called trap problem. The trap problem can happen evenif the network is highly connected [8], and certainly cannot beignored in a sparsely-connected network. In fact, when APF isused, the trap problem can occur with a probability of up to 13%in a typical national-scale transport network.

An effective APF-based algorithm which can solve the trapproblem is the K Shortest Path algorithm (KSP). It finds the first

shortest paths between the source and destination as candi-date APs, and then test them one by one in the increasing orderof their costs to see if it has a corresponding (disjoint) BP, untilsuch a BP is found. As long as one sets to be large enough, theKSP algorithm can avoid traps, that is, either a correspondingBP can be found, or there does not exist any pair of disjointpaths.

For the Min-Max problem, an ILP formulation was proposedto obtain an optimal solution in a single-cost network [11]. Ithas also shown that the SPP is a 2-approximation algorithm forthe Min-Max problem. As to the Min-Min problem, so far thereexist no specific algorithms.

B. Motivations

In this subsection, we provide the motivations for this work,and in particular, our study of the Min-Min problem.

Motivation for the Min-Min Problem: First, the Min-Minproblem itself has its own intrinsic theoretical values and prac-tical applications. For example, for safety or security reasons,two types of commodities (e.g., encrypted data and decryptionkey) need to be transported separately along two disjoint pathsin a network, and finding the shortest path for one commodity(e.g., encrypted data) may be the major design objective be-cause transporting one commodity is much more expensivethan transporting the other commodity. Second, solving theMin-Min problem can lead to near-optimal solutions to theMSOD problem as it has been shown that the APF-basedheuristics, which tend to select a shortest possible AP (whichis the objective of the Min-Min problem), can achieve near-op-timal solutions to the MSOD problem. Later, we will showthat our proposed heuristic algorithm can achieve near-optimalsolutions to not only the Min-Min problem but also the MSODproblem.

Intuitively, the reason for the above correlation between theMin-Min problem and the Min-Sum problem within the contextof shared path protection is that, in a typical network, after sat-isfying a large number of pairs of disjoint paths in an on-linefashion using an optimal approach developed for the MSODproblem such as SCI [13], the ratio of the average amount ofbackup bandwidth allocated per link along a BP to the averageamount of active bandwidth allocated per link along an AP isonly 0.2 because of backup bandwidth sharing.3 This observa-tion motivates us to give a higher priority to minimizing thelength of AP (and hence the active bandwidth). Another mo-tivation for solving the Min-Min problem is that, in most of thetime, traffic is carried on the AP while the backup bandwidthallocated to a BP can be used to carry low-priority and preempt-able traffic so that any “excessive” backup bandwidth allocatedwill not be totally wasted. Finally, taking units of bandwidthaway from a link for use by an AP and taking units of band-width away from the same link for use by a BP result in different“real-world” costs. The latter results in a lower cost because of

3Although the ratio of the total backup bandwidth to the total active bandwidthmay be different due to the fact that BPs are usually longer than APs.


Fig. 1. Inefficiency of KSP.

potential use (sharing) of the bandwidth by other BPs (to be es-tablished later).

In short, it is wise from the bandwidth utilization point ofview to minimize the length of AP (and accordingly the totalamount of active bandwidth allocated to the AP). In fact, as tobe shown later, extending the heuristic algorithm (COLE) for theMin-Min algorithm COLE to the MSOD problem can achieveinteresting trade-offs between total bandwidth consumption andrecovery time when compared to the existing solutions devel-oped for the MSOD problem and the Min-Max problem.

Motivation for Our Heuristic Algorithms: The major de-ficiency of most of the existing APF-based heuristics, whenbeing used to solve either the MSOD problem or the Min-Minproblem, is that they will run into the trap problem, especiallyin sparsely connected networks, as mentioned in the precedingsubsection. Even the KSP algorithm, which is one of the mosteffective APF-based heuristics dealing with the trap problem,has a serious weakness. More specifically, a major problemof KSP is that after the current candidate AP fails the test(that is, it does not have a corresponding disjoint BP), the nextcandidate AP to be tested is selected solely based on the pathlength, without considering which link (or links) along thecurrent candidate AP has caused the failure in finding a disjointBP. As a result, usually a large number of paths need to betested in order to find a disjoint path pair (if it exists between apair of nodes), or declare the path pair does not exist.

Fig. 1 illustrates an example in which the KSP is extremelyinefficient. In this example, any path from to consists of twosegments (or subpaths) from to and from to , respec-tively. Assume that the length of the links in domain is muchshorter than that in domain , and that the shortest segmentfrom to consists of links , and . Then, the firstshortest paths from to will contain this shortest segment from

to , and hence, even though could be as high as an ex-ponential function of the number of nodes in domain , noneof those shortest paths will have a link disjoint BP (becausethere is no segment in domain that is disjoint with the seg-ment consisting of links , , and ).

The above example also shows that if is set to be too small,the KSP will run into the trap problem. On the other hand, if onealways sets to be very large, the time complexity of the KSPalgorithm can be high, In fact, the best implementation of theKSP algorithm, has a time complexity of [18],where is the number of nodes in the network. In this classicalimplementation of KSP, after the th shortest path is selected

Fig. 2. Undirected lobe for x .

among a path pool, whose length (in hops) is, say, , up tonew paths should be computed and added into a path pool

before the th shortest path is selected from the pool [18].This means that the Dijkstra algorithm (or any other shortestpath algorithm) will be invoked many times.

Finally, although the SPP algorithm may be used to solve theMin-Min problem (without running into the trap problem), it isobvious that SPP can only obtain a suboptimal solution to theMin-Min problem. Hence, efficient algorithms for the Min-Minproblems are needed.

We note that for Shared Risk Link Group (SRLG) networkswhere an SRLG is a group of network links that share a commonphysical resource such as a cable conduit whose failure willcause the failure of all links of the group [19], [20], finding anSRLG-disjoint pair of paths is NPC [12], [21], and many al-gorithms have been proposed to find an SRLG-disjoint pair ofpaths [19], [20]. However, none of these algorithms can be ex-tended to solving the Min-Min problem effectively because theirobjective is to simply find the shortest pair of SRLG-disjointpaths rather than minimize the AP.

III. NP-COMPLETE PROOF

The Min-Sum problem in an arbitrary dual-cost network hasbeen proved to be NPC [9]. But the network used in [9] for itsNPC proof is not for MSOD (with ordered dual-cost links). Inthis section, we use a well-known NPC problem, called 3-Satis-fiability (3SAT) to show that Min-Min problem in a single costnetwork is NPC and the MSOD problem is also NPC in an undi-rected network by extending the graph construction in [9] and[22].

An instance of 3SAT is a Boolean formula that is the ANDof clauses . A clause is the OR of threeliterals, each of which is an occurrence of variable

or its negation. A truth assignment is a function. is satisfied by if it contains a literal

with value. The question is to determine whether there isa truth assignment that simultaneously satisfies all clauses.

Theorem 1: The problem of finding two node/link-disjointpaths between a pair of source and destination nodes in adirected/undirected network with minimum cost for the shorterone is NPC.

Proof: We first prove that the undirected link-disjoint ver-sion of the problem is NPC by reducing 3SAT to its decisionform: Given an undirected network with nonnegative link costand a nonnegative value , does it contain two link dis-joint paths with the cost of the shorter one at most ?

Given an instance of 3SAT, a lobe is constructed for eachvariable as shown in Fig. 2, where is the number of its


Fig. 3. 3SAT reduction in an undirected network.

occurrences as literal (either or ) in all the clauses. Thesolid link has a cost of 1 while the dashed one has a cost of 0.The lobes are connected in series from (which is also thesource ) to destination .

For each clause , two vertices , are added togetherwith dashed (0 cost) links , ( ),and . We connect those vertices representing clauses tovariables with dashed links as follows:

• and , if the th occurrence of the th vari-able is literal in clause .

• and , if the th occurrence of the th vari-able is literal in clause .

For example, network G corresponding to instanceis depicted

in Fig. 3.Since G can be constructed in polynomial time, it suffices to

show that there exists a truth assignment which satisfies allclauses if and only if G has two link-disjoint paths from sourceto destination and one of paths has a cost of 0.

Firstly, suppose there exists a truth assignment that simul-taneously satisfies all clauses. Each satisfied clause con-tains either a literal such that or a literalsuch that . A subpath or

is constructed accordingly. These sub-paths together with links and form the pathhaving total cost of 0. There is another path from to thatpasses through the upper portion of the lobe if orthe lower portion otherwise. and are obviously link-dis-jointed.

Conversely, suppose there exist two link-disjoint simplepaths between and and one of them has cost 0. Then, the0-cost path , i.e., shorter path, must contain links and

. Since contains no loop, it must contain a subpathor . Similarly,

passes through all node pairs and in the order of increasing. On the other hand, path passes through either upper or

lower portion of each lobes. Therefore, for , weset if passes through lower portion of the thlobe or if passes through upper portion of theth lobe. According to this truth assignment, all clauses are

simultaneously satisfied. This complete the proof that link-dis-joint version of Min-Min problem in an undirected single costnetwork is NPC.

To show the directed version of the Min-Min problem is alsoNPC, we modify the undirected lobes and the network shown

in Figs. 2 and 3 as follows (note that all the links’ costs are notchanged):

For each lobe:

1) Replace and with directed linksand .

2) Replace and withdirected links and .

3) Apply the same changes above to the lower portion of thelobe.

For the entire network:

1) Replace and with directed linksand .

2) Replace all and pairs with directed pairsand .

3) Replace all and pairs with directed pairsand .

4) Replace all with directed links .Similar to the proof above for an undirected network, if we

are given a truth assignment which simultaneously satisfies allclauses of the 3SAT instance, a link-disjoint path pair with

shorter one’s length being at most 0 can be found and vice versain this directed network.

Finally, it is obvious that and are not only link disjointbut also node disjoint in all the previous arguments. Therefore,both link-disjoint and node-disjoint versions of the Min-Minproblem in either a directed or undirected network are NPC.

The following theorem states that it does not make sense totry to develop an approximation algorithm with polynomial timecomplexity for the Min-Min problem.

Theorem 2: For any fixed constant , it is NP-hard toapproximate the length of AP in the optimal solution denotedby to Min-Min problem within a factor of . In otherwords, assuming , no polynomial time algorithmcan find a solution denoted by and

.Proof: It is enough to show that a polynomial-time ap-

proximation algorithm for Min-Min problem with approxi-mation factor will lead to a polynomial-time algorithm for3SAT.

For a given 3SAT instance , we run the following algorithm:

1) Construct a network G as shown in Fig. 3 with the fol-lowing modifications: we assign cost of to all solidlinks and modify only one dashed link ’s cost to 1.

2) Run the algorithm on G, and answer “ is satisfiable”iff finds a solution with cost .

Since is polynomial time, is clearly a polynomial timealgorithm. To see that correctly solves 3SAT, we note that(from the proof of Theorem 1):

• If there is a truth assignment that satisfies all clauses of, then there exist two edge-disjoint to paths in G such

that the cost of the shorter path is 1. In this case, sinceand is a -approximation algorithm, we

must have . Thus, will correctly answer“ is satisfiable.”

• If there is no truth assignment that satisfies all clausesof , then, for any pair of edge-disjoint to paths in G,


the cost of the shorter path is at least . Hence, wehave . So will correctlyanswer “ is not satisfiable.”

As mentioned in Section II-A and Table I, the MSOD-Uproblem of finding disjoint paths in a uniformly ordered dualcost network where and , is provedto be NPC used in the directed networks [8]. But it was leftopen the question whether the MSOD-U problem is NPC in anundirected network because the original NPC problem in thereducibility process used in [8] is only applicable to a directednetwork. We will use similar technique in the Min-Min problemproof to prove the following theorem.

Theorem 3: The MSOD-U problem is NPC in both directedand undirected networks.

Proof: We start with the link disjoint version of theproblem in an undirected network. As before, we reduce 3SATto this problem. For a given instance , let .Choose a value of so that .

We construct the lobes and the network G as shown in Figs. 2and 3. However, the links represented by solid line now havea cost of 1 or unit when it is on the AP or BP, respectively.(Namely, when a link is on BP, its cost will be time of costwhen it is on AP). Dash link still has a cost of 0 on both AP andBP. We ask the decision question in the undirected uniformlyordered dual cost network G: are there two link-disjoint pathsfrom to in G with total cost at most ?

If there exists a truth assignment that simultaneously sat-isfies all clauses of a given 3SAT instance , we can findtwo link-disjoint paths with sum of cost not greater than fol-lowing the same argument in the previous Min-Min problem’sNPC proof.

Suppose there exist two link-disjoint paths with total cost lessthan , we must have AP pass through and pairs inan increasing order of and contain and becausethe 1 unit cost of the solid link forces the AP to use only dashedlinks. BP will contain either upper or lower portion of all thelobes and have a total cost bounded by . A truth assignmentto the 3SAT instance can be constructed accordingly.

Similarly, we can prove that finding a link or node disjointpath pair in the MSOD-U problem is NPC in directed networks,so is the problem of finding a node-disjoint path pair in bothdirected and undirected MSOD-U networks.

IV. A HEURISTIC SOLUTION TO MIN-MIN PROBLEM

In this section, we describe the proposed COnflicting LinkExclusion (COLE) heuristic for the Min-Min problem, whichcan also be used for the MSOD problem to provide shared pathprotection as will be discussed in Section V. Like other APF-based heuristics, COLE tries to find the shortest path first forus as an AP. If a corresponding link or node disjoint BP canbe found, the algorithm terminates with the optimal solution.Otherwise, it may have run into a possible trap so it will try adifferent AP for which a BP may exist. COLE differs from otherAPF-based heuristics, including KSP, in how they decide whichother AP to try next.

Fig. 4. Demonstration of conflicting link set.

A. Basic Idea

We will first explain the basic idea of COLE before describingthe algorithm in detail. Since the problem of finding a node-dis-joint path can be transformed to that of finding a link-disjointpath by splitting each node into two halves with a “virtual”zero-cost directed link in between, and an undirected link canbe represented as two directed links, we will use the example offinding two link-disjoint paths in a directed network shown inFig. 4, to illustrate the intuition behind our COLE heuristic. InFig. 4, the shortest AP found so far is shown in bold for whichno link-disjoint BP exists.

For a given AP, we define its Conflicting Link Set to be asubset of the links on the AP such that any AP using all these“problematic” links cannot find a link-disjoint BP. Although to-gether all the links along the AP form such a conflicting link set,we are interested in a set having as few links as possible. Forthe example in Fig. 4, can be ,4 i.e., .Later, we will describe an algorithm to find and prove that ifany AP uses all the links in this , no link-disjoint BP can befound. Note that, it is possible that an AP using all but one linkin can still have a BP.

B. Divide-and-Conquer

Now, we will make use of the conflicting link set to di-vide-and-conquer the Min-Min problem. More specifically, let(the inclusion set) and (the exclusion set) be two disjoint sub-sets of . Denoted by the subproblem (of the Min-Minproblem) of finding a pair of AP and BP, where the AP is theshortest among all possible APs that must use the links in butnot the links in . Then, the (original) Min-Min problem canbe represented by .

Note that, for any given link, the solution to , if existsat all, will use an AP that either contains the link or not. Accord-ingly, for the example in Fig. 4, can be divided into twosubproblems: and . Similarly,can be further divided into two subproblems and

. The concept of the conflicting link set comeshandy because there is no need to consider whichwe know has no solution.

We will try to find an optimal solution to each of the twosubproblems and , and then select thebetter one (i.e., the path pair with the shorter AP) to be the final(optimal) solution to the original problem . If there is nosolution to any of the two subproblems, we can guarantee thatthere is no solution to the original problem, simply because ofthe way in which the problem was divided. Note that, each sub-problem can be solved in a similar divide-and-conquer way. Inaddition, in a subproblem, such as , even though wehave excluded a conflicting link from AP, it is possible that the

4 is not unique. For example, can also be f2! 3; 5! 6g, etc.


new (shortest) AP found still does not have a link-disjoint BP. Insuch a case, we will find a new conflicting link set for this newAP, based on which the subproblem can be further divided intoseveral sub-subproblems, until either a solution is found for asub-subproblem, or no AP can be found for this sub-subproblemdue to a large number of excluded conflicting links from thissub-subproblem.

In general, given a conflicting link set, the original Min-Min problem can

be first divided into the following subproblems, ,. In other

words, the number of subproblems to be solved is equal to, (This is why we want to have a small set of ). Speaking

of complexity, it should take us less time to solve eachsubproblem than the original problem itself as at least one link(which is a conflicting link) will be removed from any furtherpath computation for an AP, which also ensures that a differentAP will be found and tested for the existence of a link-disjointBP. In practice, such a divide-and-conquer approach based onthe exclusion of conflicting links can yield an optimal solutionfast, and in fact, faster than any existing approaches such asKSP, as will be shown later.

C. Finding Conflicting Link Set

In this section, we describe how to find a conflicting link setfor a given AP for which there is no link-disjoint BP in a directednetwork G. Let be the length of AP in hops, be the setof links on AP and be the set of reverse links of .

Finding a link-disjoint path pair between nodes and in agiven graph is the same as finding two unit-flowsin a flow network where each link’s capacity is set to 1 unit.We assume that the network is symmetrical, i.e., even for link

whose reverse link does not exist, we add a reverse linkwith 0 capacity to the network. In Fig. 4, only the links with1 unit capacity are shown as arrowed solid lines. We will usethe technique of finding a min-cut to obtain the conflicting linkset.

A cut of a directed network is a pairwhere and is a partition of[23]. We call the capacitated links (links with nonzero capacity)from to positive links and the capacitated links from to

negative links. The (positive) capacity of the cut is the sumof capacity of the positive links and the negative capacity of thecut is the sum of capacity of the negative links. A min-cut isa s-d cut whose positive capacity is minimal. The value of anyflow from s to d in G is bounded by the positive capacity of themin-cut of G [23].

To find the conflicting link set for the chosen AP, we constructa new graph as follows.

1) uses the same and of G.2) The capacity of the links in is set to 1.3) The capacity of the links in is set to 0.4) The capacity of all other links with nonzero capacity in G

(except those in or ) is set to (or a largervalue).

Let be a min-cut of , and represent the setof positive links of . We will prove that the set of negative

links on AP of is , i.e., a conflicting link set according toour definition in Section IV-A. For the example in Fig. 4 whose

does not have the capacitated links and ,can partition into (the nodes shown in circles)and (the nodes shown in squares), whosepositive links are and

.Lemma 1: Any path from to in must pass through at

least one link in .Proof: Otherwise, the AP would have a link-disjoint BP

in G.Lemma 2: The value of any max flow of is at most .

Proof: A max flow of with value can be partitionedinto unit-flows from to in . According to Lemma 1, eachof these unit-flows must pass through at least one link in andthey cannot share the same link in because the capacity ofsuch link is 1 unit. Therefore, there can be at most suchunit flows.

Lemma 3: All positive links of the min-cut of belongto , i.e., .

Proof: According to the max-flow min-cut theorem [24],the positive capacity of should be equal to the max-flow,which is at most by Lemma 2. Note that the capacity oflinks in is 0, and the capacity of the links that are neither in

nor in is . Thus, none of them can be a positivelink of . So all the positive links of must belong to .

Hereafter, we define to be the cut in G which has the samenode partition as in , i.e., . We call anAP-constrained cut.

Lemma 4: The number of links in the set isand the positive capacity of cut of G is at most .

Proof: Call all the nodes in “labeled,” and other nodes in“un-labeled.” From Lemma 3, , and thus the path AP

must start with one or more labeled nodes, called labeled nodegroup, (include ) and end with one or more un-labeled nodes,called un-labeled node group, (include ). Since is the set oflinks going from a labeled group to a un-labeled group, there are

labeled groups, interlaced with un-labeled groups alongthe AP. For example, in Fig. 4, AP consists of labeled node ,un-labeled nodes {1, 2}, labeled node {3}, un-labeled nodes {4,5, 6}, labeled node {7} and un-labeled node . Since is theset of links in from an un-labeled node to a labeled node,while is the set of links in from an labeled node to aun-labeled node, .

To prove the second part of the lemma, define to be the setof reverse of links of , (a subset of the links in ), and hence

.Note that all the positive links of the min-cut of are also

positive links of cut of G. In addition, unlike in , the links inin G may have nonzero capacity, and thus become additional

positive links of the cut of G. In particular, the capacitatedlinks in going from a labeled node to an un-labeled nodewill become the new positive links. It is obvious that they are asubset of , which contains the reverse links of . Besides thepositive links in , the total positive capacity of cut of G is atmost , which is .

Theorem 4: If a unit-flow of G uses all the links in in G,then no more flow exists.


Fig. 5. Example showing that using a simple min-cut of G is inefficient.

Fig. 6. Demonstration of COLE heuristic.

Proof: If a unit-flow uses all the links in , then no flowcan use links in any more, even though there may be somepositive links in on of G. The rest positive links of cut ofG, , or positive capacity will be used up by this flow topass the cut to reach destination , thus no more flow can passcut .

Hence, if a path in G uses all the links in , it has no otherlink-disjoint path in G. So, is a conflicting link set forthe given AP.

D. Reason for Not Using an Ordinary Min-Cut

Although it appears to be attractive to use the links in amin-cut of G, or an AP-constrained cut described aboveto divide and conquer , such an approach might not helpreducing the computational complexity. For example, in Fig. 5,

(dashed) is the shortest path for which no link-disjoint BPexists and (dotted) is the shortest path with a link-disjointBP. The min-cut here is the partition where , whosepositive links are and . The original Min-Min problem can bedivided into and becausewill not have feasible solution. However, solving the firstsubproblem will only lead to a nonoptimal solution and tryingto solve the second subproblem will again yield . Usingall the positive links in will similarly result in inefficiency.On the other hand, for this example, we can use the methoddescribed earlier to yield and consequentlywe can solve th following two-subproblems and

with a reduced computational complexity.

E. COnflicting Link Exclusion (COLE)

The proposed COLE heuristic (as demonstrated in Fig. 6) forfinding a link disjoint path pair for the Min-Min problem worksas follows. The shortest AP in the network is found first. If wecan find a link-disjoint BP, the Min-Min problem is solved. Oth-erwise, we identify the conflicting link set using the methodin Section IV-C, and divide the problem into subproblems inthe form of based on . The subproblem in which nolink-disjoint path pairs can be found is further divided accordingto the new conflicting link set identified until no AP or BP canbe found. The link-disjoint path pairs found in each subproblemare compared, and the one with the shortest AP will be used asthe solution to the original Min-Min problem.

Fig. 7. Example showing that COLE is a heuristic.

Instead of pursuing an optimal solution to each NP-Hardsubproblem , we will use a heuristic method. For ex-ample, assume the conflicting links in set arein the order of their appearance along the , which is theshortest path found for . To solve the subproblem

where , let the endingnode for link be , and be the subpath offrom the source to . We then remove all the nodes (except

) along and link from the network before tryingto find a shortest path from to the destination inthe residual network. If does not exist, no solution willbe returned for this subproblem. Otherwise, (i.e., ifexists), the concatenation of and , denoted by

, will be used as a possible candidate solution for sub-problem . In other words,the algorithm will try to find a link disjoint BP for and ifsuch a BP exists, will be the solution to the subproblem.If not, a conflicting link set for will be determined,and the subproblem can be solved recursively in a similardivide-and-conquer way. Note that, since the algorithm doesnot consider all possible ways to include inthe new other than using the , it is only a heuristic.For example, in Fig. 7 where is shown as a straight linefrom to and the conflicting link set , COLEwill fail to evaluate all possible solutions to the subproblem

thus may produce a suboptimal result.More specifically, after one removes and nodes 1 through 4along (which is a subpath of and which goes fromnode to node 5), COLE will find to be .Accordingly, isfor which the BP is . However, anothersolution to the above subproblem which is not considered byCOLE is choosing AP to be ,for which a disjoint BP is . This APmay be shorter than the considered by COLE in terms ofhop number or distance. Nevertheless, as to be shown, in mostcases, COLE can also find optimal solutions.

V. PERFORMANCE EVALUATION

In this section, we investigate the performance of variousschemes for finding a link or node disjoint path pair with oneof two optimization objectives, namely Min-Min or Min-Sum,through simulation. First, our COLE algorithm is comparedwith SPP and KSP, two existing approaches based on a shortestpath algorithm (such as Dijkstra) that can also avoid trapswhenever possible as the proposed COLE, in terms of theiroptimality and time complexity when being used to solve theMin-Min problem in a single-cost network. Then, we com-pare COLE with two ILP-based approaches [hereafter calledMin-Sum (ILP) and Min-Max(ILP)] in terms of their bandwidth


TABLE IITOPOLOGIES USED IN SIMULATION

TABLE IIIOPTIMALITY OF HEURISTICS IN SINGLE-COST NETWORK

efficiency and running time when being used to provide sharedpath protection (or in other words, solve the Min-Sum problemin an ordered dual-cost network or the MSOD problem).

A. Topologies Used

In our simulation, we use three sample networks whose im-portant parameters are listed in Table II. Network 1 is the USnetused in [25] and shown as domain in Fig. 1. While Network 2is taken from [8], Network 3 is a typical carrier-like transportbackbone used by researchers working for a major U.S. carrier.For the sake of completeness, two cases where the path lengthis measured in terms of the hop number and (geographical) dis-tance are considered, respectively.

The table also provides the APF failure ratio, which is thepercentage of failed attempts of an APF heuristic to find two link(node) disjoint paths over all node pairs. Note that this ratio canbe as high as 13.5%, and thus is not negligible. As mentionedearlier, this is the motivation of using SPP, KSP, the proposedCOLE, and other APF-based heuristics. Hereafter, we will focuson the problem of finding a node-disjoint path pair, for whichtraps occur more frequently than for the problem of finding link-disjoint paths.

B. Heuristics for the Min-Min Problem

For any node pairs between which the ordinary APF heuristicfailed to find a pair of node-disjoint paths, the proposed COLEas well as SPP and KSP can guarantee to succeed if such a node-disjoint path pair exists. However, not all these algorithms canfind a shortest AP (as a part of its solution) all the time exceptfor KSP when is large enough.

We define the optimality of a heuristic for the Min-Minproblem to be the percentage of times that a node-disjoint pathpair is found (whenever it exists) whose AP is the shortest.Table III shows the optimality of SPP, KSP, and COLE wheneither the hop number or distance is used as the path length.Since SPP does not intend to minimize AP’s length, as thesize of network increases, its optimality decreases (to as low as

TABLE IVAVERAGE (MAXIMAL) TIMES OF DIJKSTRA INVOCATION

23.6% in Network 2), and thus is not suitable for the Min-Minproblem. When distance instead of hop number is used asthe path length metric, SPP performs even worse because thevariance in the lengths of all possible paths can be much larger.On the other hand, both KSP and COLE can achieve 100%optimality.

The major computational complexities of KSP and COLEcome from their repeated invocations of the Dijkstra algorithm(to find the shortest path) as a subroutine. Table IV shows theaverage and maximal number of times the Dijkstra subroutineis called by these two heuristics to find the node disjoint pathpair for each node pair in the three networks. In the worst case,KSP invokes 2622 times of the Dijkstra subroutine, which isaround ten times as much as that in COLE (which invokes thesubroutine only 221 times). On average, KSP also calls the sub-routine significantly more times than COLE, especially for largenetworks. This is expected as the former searches all APs in abrute force way while the latter modifies AP to avoid using atleast one “problematic” link at a time.

C. Schemes for Shared Path Protection

In this subsection, we evaluate the performance of our COLEwhen being extended to provide shared path protection (an in-stance of MSOD problem, and compare it with two existingschemes.

In our simulation, we inject a large number (e.g., a few hun-dreds) of randomly generated connection requests (one after an-other in an on-line fashion) into the three networks. We usethe incremental traffic model in which an established connec-tion lasts sufficiently long, and each link in the networks has asufficiently large capacity as in [2], [13], and [14]. We assumethe availability of complete aggregated information on how theexisting APs and BPs are established, or equivalently, how thebandwidth on each and every link is allocated to the APs andBPs traversing it as in [2], [13], [14], and [16].

COLE can be applied to shared path protection as follows.Upon receiving a new request for a survivable connection ofbandwidth units, a shortest AP (in terms of either hop numberor distance) is found. Then, we determine the additional backupbandwidth needed on each and every remaining link using thetechniques described in [13] for example. Using the additionalbackup bandwidth needed as the cost metric of each link, we usethe Dijkstra algorithm to try to find a node-disjoint BP. If sucha BP is found, the algorithm terminates successfully with the“cheapest” BP for the shortest AP. The AP is also the “cheapest”in terms of bandwidth (as the amount of additional bandwidth


TABLE VAVERAGE RUNNING TIME PER CONNECTION (SECONDS)

needed for each link along an AP is units5). If such a BPcannot be found, then we attempt to escape from the trap bydividing the problem into subproblems and solving each sub-problem recursively as described in Section IV-B. Note that, al-though other APF-based heuristics, such as KSP, which achieves100% optimality for the Min-Min problem, can also be extendedto provide shared path protection as above, they will run muchslower as discussed in previous subsection.

In our implementation, in order to avoid using too many zero-cost links (which are possible due to backup bandwidth sharing)as a part of BP as in [26] and [27], which would make BP toolong, a base cost is added to each and every link (ex-cept the links along the AP) before using the Dijkstra algorithmto find a node-disjoint BP. Note that, this technique can be ap-plied to any APF-based heuristic for shared path protection.

We also compare the performance of COLE with Min-Sum(ILP), which is an ILP-based approach similar to the one pro-posed in [13], and which can find a pair of AP and BP such thatthe sum of the additional bandwidth needed for them is mini-mized.

COLE is also compared with Min-Max(ILP), which is anILP-based approach similar to the one proposed in [11], andwhich can find a pair of AP and BP such that the length (inhop number or distance) of BP is minimized (with the objec-tive being to minimize the length of BP). Even though Min-Max(ILP) is not designed to minimize the sum of the addi-tional bandwidth needed for AP and BP, we can augment itby using the minimal bandwidth allocation method describedin [14] for example to allocate the minimal amount of backupbandwidth on BP. In our simulations, we use the augmentedMin-Max(ILP) to provide shared path protection to make thecomparison between different approaches as fair as possible. Forboth Min-Sum (ILP) and the augmented Min-Max(ILP), we useCPLEX 7.1 to resolve the ILP formulations.

The average running times (on a Pentium IV 1.7G PC) re-quired by the three algorithms to satisfy a connection in thethree networks are shown in Table V. Note that the average run-ning time of each of the three algorithms increases with the sizeof the network as expected. The reason that Min-Max(ILP) ismuch faster than Min-Sum(ILP) is that in the latter, the costof a link on BP depends on the AP being selected in a joint-optimization process. Compared to Min-Max(ILP), COLE ob-tains even better performance in terms of the running time be-cause of its use of an intelligent AP searching algorithm. Thereduction in the average running time by using COLE is signif-icant, especially for large networks. For example, in Network2, COLE only takes 0.27 seconds to obtain a solution, which is500 times faster than Min-Sum(ILP) [and also ten times faster

5Note that, the sum of the additional bandwidth needed by AP and BP maynot be the lowest possible.

Fig. 8. Performance in shared path protection on Network 2.

than Min-Max(ILP)]. More importantly, we note that for a largeproblem size (e.g., one in which a network has a couple of hun-dreds of nodes, and a large number of requests have to be pro-cessed), Min-Sum(ILP) will become infeasible. A primal-dual-based heuristic for the Min-Sum problem was proposed in [13].However, to satisfy a connection request on a 70-node network,several hundred invocations of the Dijkstra subroutine are re-quired for every subproblem and the number of subproblemscan be over a hundred. Accordingly, it is also slower than COLE(not to mention that it is less bandwidth-efficient than COLE aswell).

In addition to the average running time, we compare the per-formance of the three algorithms in terms of the following twometrics:6

1) Bandwidth Overhead: Since various schemes we evaluatemay choose different APs for the same connection request, wefirst define the service bandwidth as the minimum amountof active bandwidth (also called working capacity) required tosatisfy a given connection request along a shortest path withoutconsidering the need for protection, which is thus independentof protection schemes. Then, Bandwidth Overhead is definedas the percentage increase in the total bandwidth (active +backup) required by a protection scheme over the above servicebandwidth to satisfy all requests, i.e., .

2) Recovery Time: The AP length and BP length can serveas a good indication of the failure notification time, and BP sig-naling and set-up time, respectively. Their sum is also a goodindication of the overall recovery time (other components of therecovery time such as failure detection time are almost indepen-dent of the protection schemes used).

The average bandwidth overhead in Network 2 during theprocess of satisfying 500 connections is given in Fig. 8(a). Band-width overhead decreases as the number of existing connec-tions increases because more backup bandwidth sharing can beachieved. Also as expected, Min-Sum(ILP) achieves the lowestbandwidth overhead as it optimizes the total bandwidth (active+ backup) taking bandwidth sharing into consideration, whileCOLE and Min-Max(ILP) minimizes the length of AP and BP,

6Note that KSP has almost the same performance as COLE in terms of thesetwo metrics since they select and use almost the same AP and BP. Accordingly,we will not include the results from KSP in the following study.


respectively, as shown in Fig. 8(b) and (c). However, as men-tioned earlier as a motivation for using a solution to the Min-Minproblem for shared path protection, the cost of a BP link in termsof additional bandwidth needed is always no greater than thecost of an AP link due to possible backup bandwidth sharing.Therefore, although COLE uses a BP which is a few hops longerthan that of Min-Max, the AP used by COLE is a few hopsshorter, and this is why its bandwidth overhead is lower thanMin-Max which has the highest bandwidth overhead. In ad-dition, the results in Fig. 8(b) and (c) also show that COLEprovides a satisfactory performance in terms of recovery timeas well. More specifically, COLE uses a BP that is only aboutthree hops longer than Min-Max(ILP) and one hop longer thanMin-Sum(ILP), and in addition, there is a little difference amongthe three schemes in terms of the average sum of the length ofAP and BP.

In short, COLE results in the fastest running time, a shortestAP length (which is used to carry working traffic most ofthe time anyway), a comparable bandwidth overhead toMin-Sum(ILP) and a satisfactory recovery time, and hence is aviable choice even for the shared path protection scheme.

VI. CONCLUSION

In this paper, we have proved the Min-Min problem as wellas its -approximation in single and dual cost networks to beNP-complete. In addition, we have proved that a restricted ver-sion of the Min-Sum problem in undirected ordered dual costnetworks is also NP-complete, which was left as an open ques-tion in [8]. In order to solve the Min-Min problem efficientlyand optimally, we have developed a novel heuristic algorithmcalled COLE, which divides and conquers the Min-Min problembased on the concept of conflicting link set. To illustrate the effi-ciency of our heuristic, we have conducted comprehensive per-formance evaluation and comparison. We have found that COLEobtains 100% optimality in solving the Min-Min problem with amuch shorter running time than other schemes such as KSP. Fur-thermore, when used to provide shared path protection, COLEhas the fastest running time, up to 500 times less than an ex-isting ILP-based approach, the shortest AP length, a comparablebandwidth overhead, as well as a satisfactory recovery time, andhence is a viable choice.

ACRONYMS

AP Active path.APF Active path first.BP Backup path.COLE COnflicting Link Exclusion.ILP Integer linear programmingKSP K Shortest Path algorithm.MSOD Min-Sum problem in ordered dual-cost networkNPC NP-complete.SPP Shortest Pair of Path algorithm.

ACKNOWLEDGMENT

The authors would like to thank P. Laborczi, Budapest Uni-versity of Technology and Economics, Hungary, for providingthe network topology used in the simulation study.

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Dahai Xu (S’01–M’05) received the B.Eng. degreein applied electronics in 1996 and the M.Eng. degreein computer science and engineering in 1999 fromShanghai Jiao Tong University, China, and the Ph.D.degree in computer science and engineering from theState University of New York at Buffalo in 2005

Currently, he is a Postdoctoral Research Asso-ciate in the Department of Electrical Engineering,Princeton University, Princeton, NJ. His research in-terests include network survivability and restorationin IP/MPLS, optical networks, network design and

protocol development for next-generation Internet, and performance evaluation(modeling, simulation, and measurements).

Yang Chen (S’01) received the B.Eng. degree in in-formation engineering from Xi’an Jiaotong Univer-sity, China, in 1999, and the M.S. degree in computerscience and engineering from the State University ofNew York at Buffalo in 2003. He is currently workingtoward the Ph.D. degree in the College of Computing,Georgia Institute of Technology, Atlanta.

His current research interests include protocol de-velopment, performance analysis, and resource man-agement in mobile ad hoc networks.

Yizhi Xiong received the B.S. and M.S. degrees inoptical engineering from Nanjing University of Sci-ence and Technology, China, in 1994 and 1997, re-spectively, and the Ph.D. degree in communicationsand information systems from Shanghai Jiao TongUniversity, China, in 2000.

He was a Postdoctoral Associate in the ComputerScience and Engineering Department, State Univer-sity of New York at Buffalo. His research interests in-clude network survivability, architecture design, real-time provisioning in IP/WDM, and optical network

management. Currently, he is a researcher with Cedars-Sinai Medical Center,Los Angeles, CA.

Chunming Qiao (S’89–M’92) is a full Professor atthe State University of New York at Buffalo where hedirects the Lab for Advanced Network Design, Anal-ysis, and Research (LANDER), which conducts cut-ting-edge research work on optical networks, wire-less networks, survivable networks, and TCP/IP tech-nologies. He has over 12 years of academic and in-dustrial experience in optical networks. He has pub-lished more than 100 papers in leading technical jour-nals and conference proceedings, and is recognizedfor his pioneering research on optical internet and,

in particular, the optical burst switching (OBS) paradigm. His work on inte-grated cellular and ad hoc networking systems (iCAR) is also internationallyacclaimed.

Dr. Qiao was the IEEE Communication Society’s Editor-at-Large for opticalnetworking and computing until recently, when he became an Area Editor ofIEEE Communications Magazine for the new Optical Communications Supple-ment. He is also an editor of several other journals and magazines, includingIEEE/ACM TRANSACTIONS ON NETWORKING, and has been a guest editor forthe IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS and other pub-lications. He has chaired and co-chaired many conferences and workshops onoptical communications and networking, including the Opticomm 2002. He isalso the founder and chair of the Technical Group on Optical Networks spon-sored by SPIE, and a Vice Chair of the IEEE Technical Committee on GigabitNetworking.

Xin He (M’92) received the M.S. degree in mathe-matics in 1981, the M.S. degree in computer and in-formation sciences in 1984, and the Ph.D. degree incomputer and information sciences in 1987, all fromThe Ohio State University, Columbus.

Since 1987, he has been with the Department ofComputer Science and Engineering, State Universityof New York at Buffalo, where he is currently a fullProfessor. His research interests include algorithmdesign and graph theory.