# Distributed Clustering Method for Large-Scaled Wavelength Routed Networks

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2 Level 2 Cluster To other nodes 3 2 Level Cluster Level Cluster To other nodes Fig. 2. Network Model. BGP for the routing protocol since it meets the requirements of the inter domain routing protocol in the optical networks []. In each intermediate domain, an intra domain lightpath is set up according to a routing/signaling protocol adopted in the domain. B. Hierarchical Clustering Figure 3 has an example of hierarchical clustering. We call a set of nodes a cluster. A node whose adjacent node belongs to another cluster is referred to as a border node. An level x cluster consists of level (x ) clusters. The minimum level hierarchy is level clustering, where a level cluster includes all nodes. If the level of clustering is more than, this is called multi level clustering or a multi level hierarchy. The maximum cluster size is limited to keep the intra cluster routing table size within a reasonable size. The inter cluster routing table size can be huge when there are too many clusters. When this happens, the level of clustering is increased and higher level clusters are constructed to reduce the size of lower level inter cluster tables. Although our approach can be extended to a multi level hierarchy, we only deal with 2 level hierarchical clustering to simplify explanation. C. Conventional Clustering Problem Krishnan et. al [] formulated an optimal clustering problem for communications networks. They treated the problem as a graph partitioning problem and called it the bounded, connected, min cut problem. Bounded, connected, min cut problem Given, A undirected graph G =(V,E) with edge weights w : E Z + 0, and Upper bound on size of clusters B {,..., V }, the optimal clustering is to obtain the set of clusters V, V 2,..., V k, such that minimize w(e) () where e E, e/ E i, i {, 2,...,k} k {2,..., V }. Following restrictions need to be satisfied, Graph G i =(V i,e i) that represents the intra cluster network of cluster V i is connected, and V i B, i {, 2,...,k}. There are two characteristics the clustering problem has in communication networks. First, the clusters need to satisfy bounded, connected conditions. A bounded cluster means the maximum cluster size is bounded by B to keep the intra cluster routing table within a reasonable size. A connected cluster means any two nodes that belong to the same cluster can only reach one another via nodes in that cluster. If the connected condition is not satisfied, two nodes in the same cluster communicate through external clusters. This defeats the purpose of clustering, which is to minimize the storage and exchange of information about external clusters. The second characteristic is that each cluster does not need to be balanced. e Fig. 3. Example hierarchical clustering. This is because the construction of balanced clusters does not always result in minimized link costs between clusters. The computational complexity of the bounded, connected, min cut problem is NP complete. Therefore, Rajesh et. al [] proposed a centralized heuristic algorithm to solve this problem, which consists of three steps: () generating initial connected clusters, (2) refining clusters by trading nodes, and (3) refining clusters by merging clusters. The connected clusters in the initial step are generated through recursive bisection. Since the recursive bisection splits clusters, the heuristic algorithm requires the complete information about the entire network topology. This may cause other scalability problems with the memory having to include complete topological information. We therefore propose a clustering algorithm that is implemented in distributed fashion. Our clustering problem and algorithm will be explained in the next section. III. NODE CLUSTERING FOR HIERARCHICAL ROUTING IN LARGE SCALED WDM LIGHTPATH NETWORKS A. Node Clustering in WDM Lightpath Networks As we discussed in Section I, clustering may increase the path length. This increase is a serious problem in WDM lightpath networks because the wavelength assigned to a lightpath must be identical along the route (i.e., wavelength continuity constraint). The increased path length generally leads to increased blocking probability for lightpath requests. The routes for lightpaths in hierarchical routing depend on how the clusters are constructed. It is therefore important to construct clusters to minimize the blocking probability for lightpaths. In this section, we discuss our development of a distributed clustering algorithm that is suited to large scaled WDM lightpath networks. The requirements for this clustering algorithm are as follows. ) Keeping the size of routing tables for intra/inter cluster routing within a certain value 2) Minimizing blocking probability for lightpath requests 3) Constructing clusters in the network with a huge number of nodes We will now explain how these requirements are satisfied with our distributed algorithm. We believe that increasing the number of lightpaths available between nodes in WDM lightpath networks will lead to decreased blocking probability in lightpaths. Based on this idea, we first formulate a new clustering problem in WDM lightpath networks that maximizes the number of lightpaths available between nodes. We refer to this problem as the bounded, connected, max lightpath problem. We then propose a distributed clustering algorithm that resolves the bounded, connected, max lightpath problem and satisfies the three requirements. Bounded, connected, max lightpath problem Given: G =(V,E) that corresponds to a WDM lightpath network Upper bound on size of clusters B {,..., V } Objective function: k k maximize F ij + F il, (2) s= i,j V s s= i V s,l / V s

3 where V, V 2,..., V k are constructed clusters. F ij is the number of lightpaths available on the shortest path from node i to node j, where F ii =0, ( i =,...,N). Constrains: Graph G i =(V i,e i) that means the intra network of cluster V i is connected V i B, i {, 2,...,k} Let us try to maximize the number of lightpaths available between nodes with the above formulation. The number of lightpaths available between nodes consists of () those between nodes in the same cluster and (2) those between nodes in different clusters. The latter changes according to the construction of clusters because route with minimum cluster hop count, which changes depending on the construction of the clusters, is selected as the route of a lightpath between nodes in different clusters. Note that we do not consider wavelength conversion in this paper. If wavelength conversion is allowed, we need to weight the number of lightpaths available on a route with wavelength converters because we can set up more lightpaths on the route than that on routes without wavelength converters. Our bounded, connected, max lightpath problem is NP complete since our problem determines whether a given graph is partitioned into bounded and connected partitions, which is called as bounded, connected, partitioning problem and shown to be NP complete []. We therefore propose a heuristic algorithm, which satisfies the first and second requirements of a clustering algorithm for large scale lightpath networks. Our method satisfies the first requirement of keeping the size of routing tables for intra/inter cluster routing within a certain value because the constructed clusters are bounded and connected, which prevents intra/inter cluster routing tables from becoming too large. Our method also satisfies the second requirement of minimizing the blocking probability for lightpath requests because it maximizes the number of lightpaths available between nodes. In Sec. IV, we discuss how maximizing available lightpaths results in decreasing the blocking probability for lightpath requests. For our proposed method to satisfy the third requirement of constructing clusters in the network with a huge number of nodes, clusters need to be constructed in a distributed fashion. This is because each border node does not need to maintain all the topological information with our method. After we present information maintained by nodes with our method in Sec. III-B, we will explain our algorithm in Sec. III-C. B. Information Maintained by Nodes Figure depicts what information a node and a border node have. All nodes have () a node to cluster mapping table and (2) an intra cluster routing table. In addition, all border nodes have (3) an inter cluster routing table. The intra cluster routing table is constructed by route information exchanged among nodes that belong to the same cluster, and inter cluster routing table is constructed by route information exchanged among all the border nodes. We will next present the information in each table and when each piece of information is used. ) Node to cluster mapping table: This table includes node identifiers and cluster identifiers that include the nodes. We use the minimum node identifier in a cluster as the cluster identifier. When clusters are constructed Each node refers to this table (a) to obtain its cluster identifier, and (b) to find out whether or not it is a border node. Each node can find this out by comparing its cluster identifier with its adjacent nodes cluster identifiers. Each node refers to this table to obtain the cluster identifier for the destination node. 2) Intra cluster routing table: This table includes the shortest route from a source node to nodes in the same cluster and the minimum number of fibers Cluster Cluster Cluster Node-to-cluster mapping table Intra-cluster routing table node node 2 node 3 node node node node node node node node node 2 node 3 node Destination: node 2, 2 F: Destination: node 3, 3 F: node cluster 2 3 Cluster Cluster Inter-cluster routing table Destination: cluster (,,2), (,,-) F: Destination: cluster (,,), (,,), (,,-) F: Destination: cluster (,,), (,,-) F: Destination: cluster 2 (,,2), (,,), (2,2,-) F: Fig.. () Node to cluster mapping table, (2) inter-cluster routing table, and (3) intra-cluster routing table. on links along the route. In the intra cluster route information to node 2 in Fig.,, 2 is a list of nodes on the route and F : means the minimum number of fibers along the route, which is. When clusters are constructed Each border node refers to this table to find out the number of fibers available from it to other border nodes in the same cluster. Each node refers to this table to find out the route to nodes in the same cluster. 3) Inter cluster routing table: This table includes (a) a list of clusters on routes from the source cluster to other clusters and ingress/egress border nodes for each cluster in the list, and (b) the number of fibers available on the links along the route. In the inter cluster route information for cluster in Fig., (,, ), (,, ), (,, ) is a list of clusters on the route. Each cluster is expressed as (ingress border node identifier, cluster identifier, egress border node identifier). F : means the minimum number of fibers along the route, which is. Each border node refers to this table to obtain the route to the destination cluster that includes the destination node. The inter cluster routing table includes the ingress/egress border nodes for each cluster. This is because we distinguish the routes that pass through the same clusters but pass through different ingress/egress border nodes. We need to distinguish them because the number of fibers available on a route depends on the ingress/egress border nodes in addition to the clusters a lightpath traverses. C. Distributed Clustering Algorithm for Bounded, Connected, Max lightpath Problem Our algorithm constructs clusters by repeating a merge operation, which makes a cluster merge with one that is adjacent. Each cluster selects an adjacent cluster so that Eq. (2) is maximized. The first term in Eq. (2), which means the number of lightpaths whose source and destination belong to the same cluster, is constant despite the construction of the clusters. This is because the routes for those lightpaths are always routes with a minimum node hop count. The second term in Eq. (2), on the other hand, which means the lightpaths whose source and destination belong to different clusters, changes according to the construction of the clusters because their routes have a minimum cluster hop count. Consequently, it is important to increase F il in the second term. To achieve this, we tried to maximize

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