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502 J. OPT. COMMUN. NETW./VOL. 2, NO. 8 /AUGUST 2010 Chen et al.

Clustering Methods for HierarchicalTraffic Grooming in Large-Scale Mesh WDM Networks

Bensong Chen, George N. Rouskas, and Rudra Dutta

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Abstract—We consider a hierarchical approach fortraffic grooming in large multiwavelength networksof a general topology. Inspired by similar concepts inthe airline industry, we decompose the network intoclusters, and select a hub node in each cluster togroom traffic originating and terminating locally. Atthe second level of the hierarchy, the hub nodes forma virtual cluster for the purpose of grooming intra-cluster traffic. Clustering and hierarchical groomingenables us to cope with large network sizes and facili-tates the control and management of traffic and net-work resources. Yet, determining the size and compo-sition of clusters so as to yield good groomingsolutions is a challenging task. We identify thegrooming-specific factors affecting the selection ofclusters, and we develop a parameterized clusteringalgorithm that can achieve a desired trade-off amongvarious goals. We also obtain lower bounds on two im-portant objectives in traffic grooming: the number oflightpaths and wavelengths needed to carry the sub-wavelength traffic. We demonstrate the effectivenessof clustering and hierarchical grooming by present-ing the results of experiments on two network topolo-gies that are substantially larger than those consid-ered in previous traffic grooming studies.

Index Terms—Optical networking; Trafficgrooming; Network design; Resource provisioning;Hierarchical grooming; Routing; Control planealgorithms; Large networks.

I. INTRODUCTION

O ngoing advances in optical network and commu-nication technologies continue to expand the ca-

pacity of individual wavelengths and increase the

Manuscript received January 21, 2010; accepted April 2, 2010;published July 9, 2010 �Doc. ID 123151�.

Bensong Chen ([email protected]) is with Google, Inc.,Mountain View, California 94043, USA.

George N. Rouskas ([email protected]) and Rudra Dutta([email protected]) are with the Department of Computer Science,North Carolina State University, Raleigh, North Carolina 27695-8206, USA.

Digital Object Identifier 10.1364/JOCN.2.000502

1943-0620/10/080502-13/$15.00 ©

vailability of wavelength channels for direct opticalonnections. Traffic grooming, the area of researchoncerned with efficient and cost-effective transport ofubwavelength traffic over multigranular networks,as emerged as an important field of study in recentears. In static grooming [1], the objective is to provi-ion the network to carry a set of long-term traffic de-ands while minimizing the overall network cost; the

atter is typically taken as the number of electronicorts needed to originate and terminate the set ofightpaths in the logical topology. In dynamic groom-ng [2], on the other hand, the goal is to develop onlinelgorithms for efficiently grooming and routing of con-ections that arrive in real time. The reader is re-

erred to [3] for a comprehensive survey and classifi-ation of research on traffic grooming.

Early work on traffic grooming focused on the ringopology [1,4,5], reflecting the technological push inesponse to the industry’s effort to upgrade the de-loyed SONET infrastructure to WDM technology.ore recently, several studies have begun to address

rooming issues in networks with a general topology6–10]. Nevertheless, most studies regard the networks a flat entity for the purposes of lightpath routing,avelength assignment, and traffic grooming. In gen-ral, such approaches do not scale well to networks ofealistic size for two reasons: first, the running-timeomplexity of traffic grooming algorithms increasesapidly with the size of the network, and second, theperation, management, and control of multigranularetworks becomes a challenging issue in large, un-tructured topologies.

We have recently proposed a scalable hierarchicalramework for traffic grooming that can be applied toetworks of practical size covering a national or inter-ational geographical area [11]. In our model, the net-ork is organized in a hierarchical manner to facili-

ate the control and management of resources (e.g.,rooming ports and wavelengths) and to ensure thecalability of grooming algorithms and functionality.he model borrows ideas from the hub-and-spokearadigm used within the airline industry. The net-ork is partitioned into clusters, and one node within

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Chen et al. VOL. 2, NO. 8 /AUGUST 2010/J. OPT. COMMUN. NETW. 503

each cluster is selected as the hub. Nonhub nodesroute all their traffic to the hub, where it is groomedbefore it is forwarded to the destination cluster; as aresult, the hub is the only node in a cluster respon-sible for grooming traffic not originating or terminat-ing locally. At the second level of the hierarchy, thefirst-level hubs form another cluster for grooming androuting intercluster traffic. This hierarchical ap-proach is quite scalable, can be used directly for net-works with tens or even hundreds of nodes, and is ap-plicable to both static and dynamic grooming contexts.

One important yet challenging issue in the hierar-chical grooming approach is the selection of clustersand hub nodes. Although clustering techniques areused in a wide range of network design problems,there is little work related to traffic grooming; wepresent a survey in Section IV. Therefore, we developa new parameterized clustering algorithm appropri-ate for traffic grooming. The algorithm is flexible andallows the network designer to achieve a desired bal-ance among a number of conflicting goals. We also de-velop lower bounds on two metrics of importance, thenumber of lightpaths and wavelengths needed for thestatic grooming problem. To demonstrate the effec-tiveness of the clustering and hierarchical groomingalgorithms, we apply them to two large networks, in-cluding a 128-node, 321-link topology correspondingto a worldwide backbone network; the latter is ap-proximately an order of magnitude larger than net-works that have been considered in previous groomingstudies.

Following the introduction, we describe the hierar-chical grooming approach in Sections II and III. InSection IV we discuss clustering methods in general,and in Section V we present and analyze our cluster-ing algorithm for hierarchical grooming in general to-pologies. We obtain lower bounds in Section VI, andpresent numerical results on various network topolo-gies and traffic patterns in Section VII. We concludethe paper in Section VIII.

II. HIERARCHICAL GROOMING IN MESH NETWORKS

We consider a network of general topology with Nnodes. Physical links are bidirectional and support Wwavelengths per direction. The capacity C of eachwavelength channel is an integer multiple of a basictransmission unit (e.g., OC-3); C is also known as thegrooming factor. The demands placed on the networkare provided in a traffic demand matrix, T= �t�sd��,where integer t�sd� denotes the amount of (forecast)long-term traffic to be carried from node s to node d.Although the traffic demands may change over time,we assume that such changes take place over longertime scales; hence, for the rest of the paper we assume

hat the traffic matrix is fixed. We also allow the traf-c demand between any pair of nodes to exceed the ca-acity of a wavelength.

The objective of the traffic grooming problem is toonfigure the network (i.e., determine the lightpathso be set up) to carry the entire traffic matrix T whileinimizing the total number of electronic ports re-

uired at the network nodes. Since each lightpath re-uires exactly two electronic ports (one at the node atach end of the lightpath), this objective is equivalento minimizing the number of lightpaths in the result-ng logical topology. For a more formal definition ofhe problem and a general-purpose integer linear pro-ramming (ILP) formulation, the reader is referredo [3].

The traffic grooming problem in general topologyetworks has long been known to be NP-hard, since itontains as a subproblem, the lightpath routing andavelength assignment (RWA) problem, which is it-

elf NP-hard [12]. Furthermore, our earlier work13,14] has shown that traffic grooming remains in-ractable even in simple network topologies, such asaths and stars, for which the RWA subproblem cane solved in polynomial time. Consequently, for WDMetworks with more than a few nodes, it is importanto develop heuristic algorithms that are scalable andan be used to obtain provably good solutions in poly-omial time.

Our framework for hierarchical traffic groomingas developed for large-scale mesh networks consist-

ng of several tens or even hundreds of nodes, forhich existing grooming algorithms, especially thoseased on ILP formulations, are not practical. Thisramework was inspired by the hub-and-spoke para-igm that is widely used by the airline industry. Inur approach, a large network is partitioned into aumber of clusters, each consisting of a contiguousubset of nodes. The clusters may correspond to inde-endent administrative entities or may be createdolely for the purpose of simplifying resource manage-ent and control functions.

In the traffic grooming context, we view each clusters a virtual star, and we designate one node as theub of the cluster. We refer to each cluster as a virtualtar because, even though the physical topology of theluster may take any form (and in fact may be quiteifferent than a physical star topology), the hub is thenly node responsible for grooming intracluster andntercluster traffic. Consequently, hub nodes are ex-ected to be provisioned with more resources (e.g.,arger number of electronic ports and higher switch-ng capacity for grooming traffic) than nonhub nodes.eturning to the airline analogy, a hub node is similar

n function to airports that serve as major hubs; theseirports are typically larger than nonhub airports, in

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terms of both the number of gates (“electronic ports”)and physical space (for “switching” passengers be-tween gates).

Our hierarchical framework consists of threephases:

1) Clustering of network nodes. In this phase,the network is partitioned into m clusters, andone node in each cluster is designated as thehub. The clustering phase is crucial to the qual-ity of the grooming solution. In particular, wehave found that the selection of cluster size andhub nodes, as well as the existence of critical cut-ting edges in the network topology, has a pro-found effect on the number of lightpaths andwavelengths required for the final solution. Wediscuss related work on clustering techniques inSection IV, and we describe in detail our cluster-ing algorithm for traffic grooming in Section V.

2) Hierarchical logical topology design andtraffic routing. The outcome of this phase is aset R of lightpaths for carrying the traffic de-mand matrix T, and a routing of individual traf-fic components t�sd� over these lightpaths. Thisphase is further subdivided into three parts:a) setup of direct lightpaths for large traffic de-

mands,b) intracluster traffic grooming, andc) intercluster traffic grooming.This hierarchical approach is discussed in Sec-tion III.

3� Lightpath routing and wavelength assign-ment (RWA). The goal of the RWA phase is toroute the lightpaths in R over the physical topol-ogy and color them by using the minimum num-ber of wavelengths. The RWA problem on arbi-trary network topologies has been studiedextensively in the literature [8,9,12,15,16]. Inthis work, we adopt the LFAP (longest first alter-nate path) algorithm [15], which is fast, concep-tually simple, and has been shown to use a num-ber of wavelengths that is close to the lowerbound for a wide range of problem instances.

The outcome of the clustering phase is a partition ofthe network nodes into some number m of clusters,denoted B1 , . . . ,Bm, and the selection of one node, de-noted hi, to serve as the hub for cluster Bi. Consider,for instance, the 32-node network in Fig. 1. The figureshows a partition of the network into eight clusters,B1 , . . . ,B8, each cluster consisting of four nodes. Theseclusters represent the first level of the hierarchy.Within each cluster, one node is the hub; for instance,node 2 is the hub for cluster B1. At the second level ofthe hierarchy, we view the hub nodes of the eight first-level clusters as forming another cluster, and we se-lect one of these nodes as the hub node of this second-level cluster. We emphasize that, while we view each

luster as a virtual star, the actual physical topologyf the cluster is determined by the physical topology ofhe part of the original network where the clusterodes lie; for example, the four nodes of cluster B8

orm a ring. Since the RWA algorithm is performed onhe underlying physical topology after the logical to-ology has been determined, the lightpaths will followhe most efficient paths in the network and are notonstrained to go through the hub, as is the case for ahysical star. Consider, for example, cluster B8 withode 32 as its hub. Suppose that the logical topologyn the corresponding virtual star with node 32 as theub includes the one-hop lightpath (28, 32) and thewo-hop lightpath (31, 28). After running the RWA al-orithm, the one-hop lightpath may be routed over theath 28–30–32 (since node 28 is not directly connectedo the hub node 32 of the virtual star), while the two-op lightpath may in fact be routed over the direct

ink 31–28, completely bypassing the hub node 32 (un-ike a physical star, where a two-hop lightpath is op-ically switched at the hub). Similar observations ap-ly to all clusters at both levels of the hierarchy.

III. HIERARCHICAL LOGICAL TOPOLOGY DESIGN

Suppose that the network has been partitioned intoclusters Bi with corresponding hub nodes hi, i

1, . . . ,m; we will describe such a clustering algo-ithm shortly. Our objective is to determine the logicalopology and associated routing of the traffic demands�sd� that minimizes the number of lightpaths; a sec-ndary but important objective is to keep the numberf required wavelengths low. The hierarchical logicalopology algorithm we outline in this section reflectshe two-level cluster hierarchy we described in therevious section and sets up lightpaths in a sequencef three steps, each step dealing with a different typef traffic demand: large demands, intracluster traffic,nd intercluster traffic.

First−level clusters with hubs forming a second level

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A. Setup of Direct Lightpaths for Large TrafficDemands

Some elements of the traffic demand matrix may belarge enough to utilize the whole capacity of one ormore wavelengths. For such demands t�sd�, we simplyassign as many direct lightpaths as necessary to carrythem even if nodes s and d belong to different clusters.Given our goal of minimizing the total number oflightpaths in the logical topology, setting up directlightpaths for such traffic is preferable to carrying itover a series of lightpaths with intermediate stops athubs. We call this operation the reduction of the origi-nal matrix, after which all elements of the matrix willbe smaller than the wavelength capacity C.

In this step, we also apply a “direct to the destina-tion hub” rule to set up lightpaths between somesource node s and a remote hub h, if the total amountof traffic from s to destination nodes d in h’s cluster�dt�sd��p�C, where p� �0.5,1� is a parameter deter-mined by the network designer; in our work, we letp=0.8. Setting up such lightpaths for large demandsto bypass the local hub node (i.e., the hub in the clus-ter of node s) has several benefits: the number of light-paths in the logical topology is reduced, the number ofelectronic ports and switching capacity required athub nodes is reduced (leading to higher scalability),and the RWA algorithm may require fewer wave-lengths (since hubs will be less of a bottleneck).

Let Rinit be the set of direct lightpaths created inthis step. Let Tr= �tr

�sd�� denote the matrix of residualtraffic demands (i.e., excluding those carried by thelightpaths in Rinit) that need to be groomed. Obviously,tr

�sd��C for all s, d.

B. Intracluster Traffic Grooming

Consider the ith cluster Bi with ni nodes and nodehi as its hub. We view cluster Bi as a virtual star witha ni�ni traffic matrix Ti= �ti

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�sd� includes not only the intracluster traffic compo-nent tr

�sd�, but also the aggregate intercluster trafficoriginating at node s (terminating at node d). Thisdefinition of ti

�sd�, when either s or d is the hub node,implements the hierarchical grooming of traffic: all in-tercluster traffic, other than that carried by direct

ightpaths set up earlier, is first carried to the localub, groomed there with intercluster traffic fromther local nodes, carried on lightpaths to the destina-ion hub (as we discuss shortly), groomed there withther local and nonlocal traffic, and finally carried tohe destination node.

Given traffic matrix Ti= �ti�sd��, we view cluster Bi as

virtual star with hub hi and ni−1 nonhub nodes. Wepply the StarTopology algorithm described in [14] tobtain the set of lightpaths Ri for carrying the de-ands �ti

�sd�. The lightpaths in Ri are either one hopi.e., from a nonhub node to the hub, or vice versa), orwo hop (i.e., from one nonhub node to another).ence, the routing of the traffic components ti

�sd� is im-licit in the logical topology Ri.

We emphasize that, at this stage, we only identifyhe lightpaths to be created; the routing of these light-aths over the physical topology is performed duringhe third (RWA) phase of our approach. Depending onhe actual topology of the cluster Bi, which may beuite different than that of a physical star, onceouted, the lightpaths in Ri may follow paths that doot resemble at all the paths of a physical star. For in-tance, a one-hop lightpath from a nonhub node of theluster to the hub hi is routed on the unique link fromhe node to the hub in a physical star; in our case,owever, the path followed by the lightpaths may con-ist of several links, depending on the physical topol-gy of the network. Similarly, a two-hop lightpath islways switched optically at the hub of a physicaltar; in a virtual star cluster, on the other hand, a two-op lightpath will be routed by the RWA algorithm onhe actual underlying topology, and its path may notven pass through the hub hi at all, if not doing so isore efficient in terms of resource usage (e.g., if the

wo nonhub nodes are connected by a direct link).

We perform intracluster grooming in this manner,y applying the StarTopology algorithm to each clus-er Bi , . . . ,Bm, in isolation. As a result, at the end ofhis step, we identify a set of lightpaths RintraR1�R2� ¯ �Rm for carrying all intracluster traffic.

. Intercluster Traffic Grooming

At the end of intracluster grooming, all traffic (otherhan that carried by the initial direct lightpaths) fromhe nodes of a cluster Bi with destination outside theluster, is carried to the hub hi for grooming andransport to the destination hub. In order to groomhis traffic, we consider a new cluster B that forms theecond-level hierarchy in our approach. Cluster B con-ists of the m hub nodes h1 , . . . ,hm, of the first-levellusters. Let h� �h1 , . . . ,hm be the node designated ashe second-level hub. We view cluster B as a virtualtar with an m�m traffic matrix T = �t�hihj�� repre-
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We now apply the StarTopology algorithm (see [14])to the virtual star B with hub h, and we obtain the setof lightpaths Rinter to carry the traffic demands �tinter

�hihj�.Again, we emphasize that the routing of these light-paths is performed on the underlying physical topol-ogy; thus, the same observations regarding the rout-ing of the intracluster lightpaths above also apply tothe lightpaths in Rinter.

Figure 2 provides a pseudocode description of thehierarchical logical topology (MeshTopology) algo-rithm. The time complexity of the algorithm is deter-mined by the application of the StarTopology algo-rithm for intracluster and intercluster grooming inSteps 5–8 and 13, respectively. Since the complexity ofthe StarTopology algorithm for a star with n nodes isO�n2�, it is straightforward to see that the complexityof the MeshTopology algorithm for a network of Nnodes is O�N2�.

The outcome of the logical topology design phase isa set of lightpaths R=Rinit�Rintra�Rinter and an im-plicit routing of the original traffic components t�sd�

over these lightpaths. This set R of lightpaths is theinput to the RWA phase of our framework, which, as

Fig. 2. Hierarchical logical topology algorithm for mesh networks.

e discussed in the previous section, uses the LFAPlgorithm [15] to route and color the lightpaths.

Finally, we note that we considered only two levelsf clusters in our grooming algorithm. However, foretworks of very large size, our approach can be ex-ended to three or more levels of hierarchy in atraightforward manner.

IV. CLUSTERING ALGORITHMS IN NETWORK DESIGN

Clustering is a function that arises frequently inroblems related to network design and organization.classic book [17] defines clustering as “grouping of

imilar objects” and discusses many mainstream clus-ering algorithms. The algorithms are classified as ei-her minimum cut or spanning tree, depending on thenderlying methodology. The input to the algorithmsenerally consist of a set of nodes (objects) and edgeeights (node relationships), while the output is aartition of the nodes that optimizes a given objectiveunction. In our case, the goal is to find a clusteringhat will minimize the number of lightpaths after ap-lying the hierarchical grooming (logical design) ap-roach, a fact that adds significant complexity to theroblem. Specifically, the input to our problem con-ists of a traffic demand matrix and several con-traints, in addition to the physical network topology;urthermore, unlike typical objective functions consid-red in the literature (e.g., the physical cut size or themount of intercluster traffic), ours cannot be easilyxpressed as a function of the resulting clusters.herefore, most of the existing clustering techniquesre not directly applicable to the problem at hand.

Some clustering studies consider only the communi-ation (traffic) pattern between nodes. For instance,n algorithm that can group a nearly completely de-omposable matrix into blocks, so that the weightedrcs between blocks have values not exceeding a givenhreshold, was introduced in [18]. The algorithm,alled TPABLO, can be used to group the states ofarge Markov chains. A similar objective exists in theraffic grooming context, as it is desirable for trafficemands within a cluster to be denser than interclus-er traffic. However, the TPABLO algorithm does notake into account the physical topology; hence it mayroup together nodes that are far apart. Such clustersre inappropriate for the hierarchical logical topologye consider, since the long lightpaths created for in-

racluster traffic may significantly increase the num-er of wavelengths required in the whole network.

Other work has focused on the physical topologynly. Typically, the goal is to partition the nodes intoontiguous clusters containing roughly equal numbersf nodes, and at the same time minimize the overallut size. An example is the work in [19] on multiobjec-

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tive graph partitioning, which was implemented inthe METIS software package. These algorithms weredesigned for VLSI (very-large-scale integration) de-sign, a very different problem, where equality in sizeand a minimum of cross-layer connections are essen-tial for each module, and are not applicable to trafficgrooming. First, there is no requirement that all clus-ters be equal in size; more important, a small physicalcut size may result in bottlenecks for intercluster traf-fic, which in turn may increase the wavelength re-quirements during the RWA phase.

Another family of clustering problems concernedwith the physical network topology includes the well-known k-center, k-clustering, k-median, and facilitylocation problems [20–23]. Unlike the applicationstargeted by METIS, they do not require clusters to beof equal size. Of all the variants, the k-center problemis of most interest to us. The goal of the k-center prob-lem is to find a set S of K nodes (centers) in the net-work, so as to minimize the maximum distance fromany network node to the nearest center. Thus, the setS implicitly defines K clusters with corresponding hubnodes in S.

A solution to the k-center problem may be useful forhierarchical traffic grooming, since it is likely to leadto short lightpaths within a cluster, thus lowering thewavelength requirements. Also, this type of clusteringtends to avoid creating pathlike physical topologies foreach cluster; pathlike topologies are not a good matchfor the StarTopology algorithm (described in [14]),which treats each cluster as a virtual star.

The k-center problem is NP-complete, and the bestapproximation ratio that can be obtained in polyno-mial time is 2 [24,25]. We implemented the2-approximation algorithm in [24] for the k-centerproblem, and we compare it with our own clusteringmethod in Section VII. For completeness, the steps ofthe algorithm are listed below:

1) Create a single cluster, B1= �v1 , . . . ,vn, with hubnode h1=v1. Calculate the all-pair shortestpaths, record the distances in matrix dist, andlet x←1.

2) Let x be the number of clusters and d be themaximum distance between any node and itshub, i.e., d=max�dist�vi ,hj� ,vi�Bj. Let v be anode such that the distance between v and itshub is d.

3) Create a new cluster Bx+1 with hx+1=v as theonly node. Then for each node v�, if v� is closer tov than to its current hub, move v� from its cur-rent cluster to the new cluster Bx+1. Let x←x+1.

4) Repeat Steps 2 and 3 K−1 times, adding onecluster at each iteration, for a total of K clusters.

More recently, some studies have explored cluster-ng techniques in the context of traffic grooming: a hi-rarchical design for interconnecting SONET ringsith multirate wavelength channels was proposed in

26], and in [27], the blocking island paradigm is usedo abstract network resources and find groups of band-idth hierarchies for a restricted version of the trafficrooming problem. Our work, which we present in theext section, is more comprehensive, and it is appli-able to many variants of the grooming problem.

V. CLUSTERING FOR HIERARCHICAL GROOMING

We now describe a clustering algorithm tailored tohe hierarchical grooming framework we propose. Thebjective of the algorithm is twofold: to partition theetwork into some number m of clusters, denoted1, . . . ,Bm, and to select one node in each cluster to

erve as the hub where grooming of intracluster andntercluster traffic is performed. As we discussed inection III, the clusters and hubs are the input to theubsequent logical topology design and RWA phases ofhe framework. Consequently, the quality of the clus-ering phase is an important factor in the quality ofhe overall design. Next, we discuss the trade-offs in-olved in selecting the clusters, which set the designrinciples for our clustering algorithm.

. Important Considerations

To obtain a good clustering, the number of clusters,heir composition, and the corresponding hubs muste selected in a way that helps achieve our goal ofinimizing the number of lightpaths and wave-

engths required to carry the traffic demands. There-ore, the selection of clusters and hubs is a complexnd difficult task, as it depends on both the physicalopology of the network and the traffic matrix T. To il-ustrate this point, consider the trade-offs involved inetermining the number m of clusters. If m is small,he amount of intercluster traffic will likely be large.ence, the m hubs may become bottlenecks, resulting

n a large number of electronic ports at each hub andossibly a large number of wavelengths (since manyightpaths may have to be carried over the fixed num-er of links to and from each hub).

On the other hand, a large value for m implies amall number of nodes within each cluster. In thisase, the amount of intracluster traffic will be small,esulting in inefficient grooming (i.e., a large numberf lightpaths); similarly, at the second-level cluster,�m2� lightpaths will have to be set up to carry smallmounts of intercluster traffic. Therefore, the networkesigner must select the number and size of clusterso strike a balance between capacity utilization and

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number of lightpaths for both intracluster and inter-cluster traffic.

Now consider the composition of each cluster. If theaverage traffic demand between nodes within a clus-ter is higher than the average intercluster demand,there will tend to be fewer intercluster lightpaths,which are typically longer than local lightpaths.Therefore, it is desirable to cluster together nodeswith denser traffic between each other: doing so re-duces the number of longer lightpaths, alleviates hubcongestion, and provides more flexibility to the RWAalgorithm (since long lightpaths are more likely to col-lide during the routing and wavelength assignmentphase).

On the physical topology side, we also need to con-sider the cut links that connect different clusters.Each cluster has a number of fibers that link to nodesoutside the cluster, and all traffic between a node out-side the cluster and one within must traverse thesecut links. Since the cut links must have sufficient ca-pacity to carry the intercluster traffic, it is importantto select clusters such that their cut size is not toosmall, in order to keep the wavelength requirementslow.

Another important consideration arises in physicaltopologies for which there exists a critical small cutset that partitions the network into two parts. In sucha topology, all traffic between the two sides of the bi-section will have to go through the cut. In this case,creating clusters that consist of nodes on differentsides of the cut may be undesirable, because it maygenerate unnecessary traffic that goes back and forththrough the cut. Consider a cluster with nodes i , j, onone side of the bisection and the hub h on the other.Due to the nature of the hierarchical grooming ap-proach, traffic between i and j may need to be sent tothe hub first, creating additional traffic across the cutlinks, with a corresponding increase in the number ofrequired wavelengths. This additional traffic can beeliminated by forcing nodes on different sides of thebisection to be in different clusters. We describeshortly a precutting technique that can be useful insuch situations.

The physical shape of each cluster may also affectthe wavelength requirements. In particular, it is im-portant to avoid the creation of clusters whose topol-ogy resembles that of a path, since in such topologiesthe links near the hub can become congested. Sincewe use a virtual star approach for logical topology de-sign within each cluster, topologies with relativelyshort diameter are more attractive in terms of RWA.

In the next subsection, we describe an algorithmthat takes into account all the above factors in parti-tioning the network into clusters to yield good groom-ing solutions.

. MeshClustering Algorithm

Figure 3 provides a pseudocode description of oureshClustering algorithm, which we use to partitionnetwork of general topology in order to apply our hi-

rarchical traffic grooming framework. The algorithmncludes several user-defined parameters that can besed to control the size and composition of clusters, ei-her directly or indirectly. Parameters MinCS andaxCS represent the minimum and maximum cluster

ize, respectively. Our algorithm treats these param-ters as an indication of the desirable range of clusterizes, rather than as hard thresholds that cannot beiolated. Although the algorithm attempts to keep theize of each cluster between the values of these twoarameters (inclusive), it has the freedom, based onhe values of the other parameters, to determine whatt thinks may be the best clustering. Consequently,he final result may contain clusters larger than

Fig. 3. Clustering algorithm for mesh networks.

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MaxCS (see also the discussion below regarding Step26 of the algorithm).

The parameter � (0.5��0.8, default value �=0.8) is used to test whether there is sufficient capac-ity at the hub node, as well as the edges connectingthe cluster to the rest of the network, to groom andcarry the traffic demands. Specifically, we require thatthe intercluster traffic originating from or terminatingat a given cluster not exceed a fraction � of the hubcapacity (this is the HUBTEST in Step 9 of the algo-rithm); similarly, this intracluster traffic must not ex-ceed a fraction � of the capacity of the links connect-ing the cluster to the rest of the network (theCUTTEST in Step 10 of the algorithm). The algorithmwill consider a node to add to a cluster only if doing sowill not violate these two constraints.

The parameter � is meant to control the ratio of thediameter of a cluster to the number of nodes it con-tains. In order to avoid cluster topologies that re-semble long paths, we require that 0��0.75. Weused the value �=0.75 in our experiments; this valuecorresponds to a four-node path, hence restricting thelongest path within a cluster to no more than threelinks. Finally, the parameter �, 0.8��1.25, specifiesthe acceptable range for the ratio of intracluster-to-intercluster traffic for a given cluster. As we discussedearlier, it is desirable to cluster together nodes thatexchange a substantial amount of traffic among them-selves relative to traffic they exchange with the rest ofthe network; therefore, we let �=1.25 in our imple-mentation.

The MeshClustering algorithm in Fig. 3 generatesone cluster during each iteration of the main whileloop between Steps 1 and 25. Initially, in Steps 2–4,the hub of a new cluster B is selected as the node withthe maximum remaining capacity among those notyet assigned to a cluster; by “remaining capacity,” wemean the capacity remaining on its incident links af-ter subtracting the bandwidth taken up by any directlightpaths set up as discussed in Section III.A. Follow-ing the hub selection, we grow the cluster by addingone node during each iteration of the innermost whileloop between Steps 5 and 24. At each iteration, the setQ of candidate nodes for inclusion in cluster B consistsof all nodes, not yet assigned to another cluster, thatare adjacent to nodes in B. For each node q�Q, wefirst check whether including q in B would result in acluster that passes both the HUBTEST (Step 9) andCUTTEST (Step 10); if not, node q is removed for con-sideration for inclusion into cluster B (Step 17). Forall nodes q that pass both tests, we compute thediameter-to-nodes ratio �q and intracluster-to-intercluster traffic ratio �q, assuming that q is addedto cluster B (Steps 11–16). Let q0 be a node thatpasses both tests and has the largest �q value amongthe candidates; if there are multiple such nodes, we

elect the one with the smallest �q value. We include0 in cluster B (Steps 21–23), and the process is re-eated as long as the size of B is less than MaxCS.

Once all nodes have been assigned to clusters, it isossible for one or more of the clusters to have fewerhan MinCS nodes. In this case, at Step 26, the algo-ithm removes these clusters and includes their nodesnto adjacent clusters. As a result, at the end of the al-orithm some clusters may contain more than MaxCSodes.

The running time complexity of the algorithm is de-ermined by Steps 1–25. The main while loop ex-cutes a number of times equal to the number m oflusters; for each cluster, the innermost while loopxecutes a number of times equal to the size of theluster. Therefore, Steps 5–24 of the algorithm are re-eated N times, where N is the number of nodes in theetwork. Each time, the foreach loop executes atost N times. Steps 9, 10, 12, and 13 each take noore than O�N2� time in the worst case, while all

ther steps take constant time. Therefore, thesymptotic complexity of the algorithm is O�N4�. How-ver, this bound is quite loose; in practice, we haveound that the algorithm takes only a few seconds forhe 128-node, 321-link network we consider in SectionII.

. Precutting for Imbalanced Topologies

As we mentioned in Subsection V.A, when the topol-gy has a bisection of small cut size, the cut links areikely to become congested, as they have to carry allraffic between the two parts of the network on eitheride of the bisection. To reduce congestion, and hencehe number of required wavelengths, in such topolo-ies, it may be necessary to disallow nodes on differ-nt sides of the bisection from being in the same clus-er. However, identifying such a critical bisection in aarge, imbalanced topology, is a difficult task. In Sec-ion VI.B, we describe a method we developed for thisurpose and which we also use to obtain a loweround on the number of wavelengths.

Once we identify a critical bisection, we apply theollowing approach. First, we use the MeshClusteringlgorithm to determine a clustering that does not takehe bisection into consideration. Then, we partitionhe network into two parts along the bisection, and wepply the MeshClustering algorithm on each parteparately; this ensures that no cluster contains nodesrom both sides of the bisection. We then select thelustering that requires the fewest lightpaths afterhe logical topology and RWA phases, unless it re-uires a significantly larger number (e.g., 10% orore) of wavelengths; in this way, we achieve balance

etween the lightpath objective and the wavelengthequirements.

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VI. LOWER BOUNDS

We now obtain lower bounds on both the number oflightpaths and the number of wavelengths required tocarry the traffic matrix T. These bounds are obtainedindependently of the manner (e.g., hierarchical or oth-erwise) in which traffic grooming is performed; hencethey are useful in characterizing the effectiveness ofour algorithm.

A. Integer Linear Programming Lower Bound onNumber of Lightpaths

A simple lower bound Fl on the total number oflightpaths (our main objective) can be obtained as

Fl = max�s

��d t�sd�

C�,�

d��s

t�sd�

C��. �3�

This bound is based on the observation that each nodemust source and terminate a sufficient number oflightpaths to carry the traffic demands from and tothis node, respectively. This bound can be determineddirectly from the traffic matrix T.

It is possible to obtain a better lower bound by usinga standard ILP relaxation technique. Starting fromthe ILP formulation (e.g., see [3]) of the traffic groom-ing problem, we remove some variables and con-straints, so that we can obtain a relaxed solution thatserves as a lower bound for the original objective. Inthe relaxed ILP formulation, let bsd denote the num-ber of direct lightpaths set up from s to d. Since alltraffic originating at source node s must be carried onsome lightpath also originating at s, the following con-straints must be observed:

�d

bsdC � �d

t�sd� ∀ s. �4�

Similarly, for each destination d we have that

�s

bsdC � �s

t�sd� ∀ d. �5�

Since our goal is to minimize the overall number oflightpaths, the resulting relaxed ILP formulation is

Minimize: �s,dbsdSubject to: Constraints (4) and (5).

We emphasize that this ILP will not necessarilyyield a meaningful solution to the original groomingproblem, only a lower bound. By configuring CPLEXto use dual steepest-edge pricing, we are able to com-pute this bound within a few seconds even for the 128-node topology that we consider in the next section. Al-though this bound is better than the simple bound Fl

above, we believe that it is somewhat loose. However,

e have found that introducing additional constraintsn traffic flow and/or routing into the relaxed ILP inrder to improve the lower bound tends to increasehe running time of CPLEX substantially, to the pointhat it becomes impractical for the large networks weonsider in this work. Therefore, we use the loweround obtained from the above simple ILP in our ex-erimental study.

. Lower Bound on Number of Wavelengths

Consider a bisection cut of the network, and let t behe maximum amount of traffic that needs to be car-ied on either direction of the links in the cut set. If xs the number of links in the cut set, then the quantityt /xC� is a lower bound on the number of wavelengthsor carrying the given traffic matrix. Computing thisound does not require any information regarding theogical topology or the routing and wavelength assign-

ent of lightpaths.

The main challenge, then, is to find a bisection ofhe network that yields a good lower bound. To thisnd, we make the observation that there is a trade-offetween the cut size and the relative sizes of the nodeets at each side of the bisection. On the one hand, us-ng software such as METIS [19] to divide the networknto two equal parts may not yield a good bound if,ue to the irregular nature of the topology, the cut sizeurns out to be large. On the other hand, a small cutize may not be effective either, if one node set is sig-ificantly larger than the other, resulting in little traf-c across the cut links.

To reconcile these conflicting objectives, we used theHACO software [28], which implements the parti-

ioning algorithm in [29]. The software uses the pa-ameter KL-IMBALANCE to control the relative sizesf the node sets on either size of the bisection. To de-ermine a cut that achieves a good balance betweenhe two objectives, we apply CHACO several times,arying the KL-IMBALANCE parameter, and obtaineveral different bisections of the physical topology.e then select the bisection that corresponds to the

est (highest) lower bound, under the assumption ofniform traffic. We use this bisection to calculateavelength bounds for all problem instances on this

opology, even though the actual traffic matrix is gen-rated based on a different pattern (discussedhortly).

VII. NUMERICAL RESULTS

In this section, we present experimental results toemonstrate the performance of our clustering and hi-rarchical grooming algorithms. The traffic matrix T�t�sd�� of each problem instance we consider is gener-ted by drawing N�N−1� random numbers (rounded

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FrhioIrooroitkhtte

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to the nearest integer) from a Gaussian distributionwith a given mean t and standard deviation � that de-pend on the traffic pattern. We consider three trafficpatterns here:

1) Random pattern. Random patterns are oftenchallenging, since the matrix does not have anyparticular structure that can be exploited by agrooming algorithm. To generate a traffic ma-trix, we let the standard deviation � be 150% ofthe mean t. Consequently, the traffic elementst�sd� take values in a wide range around themean, and the link loads also vary widely. If therandom number generator returns a negativevalue for some element, we set the correspond-ing t�sd� value to zero.

2) Falling pattern. This pattern is designed to cap-ture the locality property that has been observedin some networks. Specifically, if the mean of thedistribution for node pairs that have shortestdistance 1 is t, then the mean for node pairs withshortest distance 2 (3) is set to 0.8t (0.6t); for allother pairs, the mean is set to 0.2t. We also letthe standard deviation � be 20% of the mean.

3) Rising pattern. This is the opposite of the fallingpattern. We use t as the mean for traffic de-mands between node pairs with the longestpath. Node pairs with the second and third long-est paths have mean values 0.8t and 0.6t, respec-tively. All other node pairs have mean 0.2t. Thestandard deviation is also set to 20% of themean.

For a given network topology and traffic pattern, wegenerate 30 problem instances, and we compare ourMeshClustering algorithm (shown in Fig. 3) to thek-center algorithm [24] we described in Section IV. Weconsider two performance metrics in our study: thenormalized lightpath count and the normalized wave-length count. The former is the ratio of the number oflightpaths required for hierarchical traffic grooming,when clustering is performed by one of the two algo-rithms above, to the lightpath lower bound obtainedusing the relaxed ILP in Subsection VI.A; the latter isthe ratio of the number of wavelengths required to thewavelength lower bound computed as explained inSubsection VI.B. The normalized metrics filter out theeffect of the traffic matrix, allowing us to compare re-sults among problem instances created by very differ-ent traffic patterns; obviously, a smaller value of thetwo metrics implies a better solution. We also considertwo network topologies, one in each of the followingtwo subsections. We emphasize that the size of thesecond topology is about an order of magnitude largerthan the typical topology considered in previousgrooming studies, a fact that demonstrates the scal-ability of our hierarchical grooming approach.

. 47-Node Balanced Topology Network

We first consider a 47-node, 96-link network topol-gy that appeared in a historical paper on network de-ign [30]. The node degree of the network is relativelyigh, and the topology is balanced, in the sense thathere is no bisection with a small cut size that can bebottleneck in traffic grooming.

Figures 4 and 5 plot the normalized lightpath andavelength count, respectively, for each of 30 problem

nstances whose traffic matrix was generated accord-ng to the falling pattern. For each problem instance,our values are shown, corresponding to four differentlusterings. The first two are from the k-center algo-ithm, with the number of clusters K equal to 4 and 6,espectively. The other two are from our MeshCluster-ng algorithm. Recall that our algorithm does not takehe number of clusters as input; rather, it tries to op-imize it. Consequently, the algorithm may produceifferent clusters for two different problem instances,ven if they are defined on the same topology andheir matrices are drawn from the same distribution.o make the comparison against the k-center algo-ithm as fair as possible, we selected two sets of val-es for the user-defined parameters of MeshCluster-

ng (refer to Fig. 3) so that the average number oflusters over all 30 instances are 3.52 and 5.45, re-pectively.

Let us first consider the normalized lightpath count.rom Fig. 4, we observe that the number of lightpathsequired for hierarchical grooming is about 40%igher than the lower bound, regardless of the cluster-

ng algorithm used. Recall that the lower bounds werebtained by relaxing most of the constraints in theLP formulation; hence we believe that they areather loose. Therefore, these results (as well as thenes to be discussed shortly) serve as a validation ofur approach, as they demonstrate that the lightpathequirements for hierarchical grooming are close toptimal. We also observe that, except for a couple ofnstances, the curves corresponding to the MeshClus-ering algorithm lie below those corresponding to the-center algorithm. Although the difference is notigh, we would like to point out that a 1% reduction inhe number of lightpaths in this network with rela-ively dense demands would result in about 40 fewerlectronic ports, a substantial savings in cost.

Let us now turn our attention to the normalizedavelength count. As Fig. 5 demonstrates, the Mesh-lustering algorithm requires significantly feweravelengths than the k-center algorithm. This result

s due to the fact that unlike the k-center algorithm,urs is designed to take the wavelength requirementsnto account. In absolute terms, the difference in theumber of wavelengths for these problem instances isf the order of 10–12. Put another way, given a con-

lao

as

B

wapwbwpttt

Fn

Fn


straint on the number of available wavelengths, ouralgorithm is more likely to generate clusters that ad-mit a feasible grooming solution. We also note that thelarge values of the normalized wavelength count aredue to the fact that the lower bound is loose in thiscase. Recall that a good bound depends on finding anetwork cut of small size and large crosscut traffic.However, this particular 47-node network was de-signed to avoid such a bottleneck cut. Furthermore,the falling traffic pattern makes it unlikely that alarge amount of traffic will cross any network cut; aswe shall see in a moment, the rising pattern yieldsbetter bounds.

Figures 6 and 7 are similar to Figs. 4 and 5, respec-tively, but show results for the rising traffic pattern.Note that, due to the nature of this pattern, relativelylarge amounts of traffic will cross any network cut, re-sulting in the much tighter wavelength bounds in Fig.7. Again, except for a few instances, our clustering al-gorithm outperforms the k-center algorithm. Regard-

1.36

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Problem Instance

K-Center, 4 clustersK-Center, 6 clusters

MeshClustering, 3.52 clustersMeshClustering, 5.45 clusters

Fig. 4. (Color online) Lightpath comparison, falling pattern, 47-node network.

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Fig. 5. (Color online) Wavelength comparison, falling pattern, 47-node network.

ess of the clustering algorithm, we observe that hier-rchical grooming is also close to optimal, confirmingur earlier observations.

We have obtained similar results for the randomnd other traffic patterns, which we omit because ofpace constraints; these results can be found in [14].

. 128-Node Imbalanced Topology Network

We now consider a 128-node, 321-link network,hich corresponds to the worldwide backbone oper-ted by a large service provider; we obtained the to-ology information from data documented on CAIDA’seb site (http://www.caida.org). This topology is im-alanced, in the sense that there exists a bisectionith a small cut size of 5 links that divides it into twoarts of 114 and 14 nodes, respectively. We identifyhis critical cut with the method discussed in Subsec-ion VI.B and use it to calculate the lower bound onhe number of wavelengths. We also steer our algo-

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ig. 6. (Color online) Lightpath comparison, rising pattern, 47-ode network.

1.2

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ig. 7. (Color online) Wavelength comparison, rising pattern, 47-ode network.

http://www.caida.org

sro

F1

Fn

F1


rithm towards creating clusters that contain nodes onone side of the cut only, as we explained in SubsectionV.C.

Figures 8 and 9 plot the normalized lightpath andwavelength count, respectively, for 30 instances gen-erated according to the random traffic pattern; Figs.10 and 11 show similar plots but for instances gener-ated according to the rising pattern. Additional re-sults for other traffic patterns may be found in [14].For the k-center algorithm, we let the number K ofclusters be either 9 or 10, and we selected the param-eters of the MeshClustering algorithm so that it alsoproduces either 9 or 10 clusters (the average over allinstances is 9.33 and 9.03 for the random and risingpatterns, respectively). As we can see, our clusteringalgorithm slightly outperforms k-center in terms ofthe number of lightpaths, and both algorithms arerelatively close to the (loose) lower bound. However, interms of the number of wavelengths, our algorithmproduces results that are within 5% of the lowerbound, whereas k-center requires more than twice thenumber of wavelengths of our algorithm.

VIII. CONCLUDING REMARKS

Hierarchical traffic grooming is a new approach forefficient and cost-effective design of large-scale opticalnetworks with multigranular traffic demands. Wehave demonstrated that the hierarchical groomingframework must be coupled with clustering tech-niques that follow grooming-specific design principles;we have presented such a clustering algorithm that isflexible in balancing various conflicting goals via user-defined parameters. We have also developed practicalmethods for computing lower bounds on metrics of in-terest. We have applied our algorithms to networks ofrealistic size. Overall, our experimental results dem-onstrate that (1) our hierarchical grooming approach

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K-Center, 9 clustersK-Center., 10 clusters

MeshClustering, 9.33 clusters

Fig. 8. (Color online) Lightpath comparison, random pattern, 128-node network.

cales to very large networks; (2) our clustering algo-ithm outperforms algorithms that were not devel-ped with traffic grooming in mind; and (3) hierarchi-

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ig. 9. (Color online) Wavelength comparison, random pattern,28-node network.

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MeshCluster, 9.03 clusters

ig. 10. (Color online) Lightpath comparison, rising pattern, 128-ode network.

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ig. 11. (Color online) Wavelength comparison, rising pattern,28-node network.


cal grooming combined with specially designedclustering techniques produce logical topologies thatperform well in terms of both lightpath and wave-length requirements across a variety of traffic pat-terns.

ACKNOWLEDGMENT

This work was supported by National Science Foun-dation (NSF) grant ANI-0322107.

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