CLUSTERING IN WIRELESS SENSOR NETWORKS: A GRAPH THEORY PERSPECTIVE

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if (state== ClusterHead ){ return 1; //DONE } if (state== Clustered ){ //wait for cluster-head to terminate waitClusterHead(); //pick one as cluster-head pickNewClusterHead(); return 1; //DONE } if (state== Unclustered ){ //pick a random clustered node to act as proxy //after it terminates pickRandomProxy(); //wait for random proxy to terminate waitRandomProxy(); return 1; //DONE } }else{ if (state== Unclustered ){ numLoyalFollowers = countLoyalFollowers(); if (numLoyalFollowers >= fmin(time)){ clusterID = generateNewRandomID(); locallyBroadcast( Recruit ,ID, clusterID); } }else{ if (state== ClusterHead ){ bestLeader = ID; bestFollowerCount = numLoyalFollowers; //all nodes where the node is a potential new cluster-head ArrayList potCH = definePotentialClusterHeads(); Iterator iter = potCH.iterator(); ACENode potential=null; while (iter.hasNext()){ potential = (ACENode) iter.next(); followerCount = PollForNumLoyalFollowers (potential,clusterID); if (followerCount > bestFollowerCount){ bestLeader=potential.ID; bestFollowerCount= followerCount; } } if (bestLeader!= ID) { send(bestLeader, Promote ,clusterID); //waits for bestLeader to broadcast

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GRAPH-BASED APPROACHES FOR CLUSTERING IN WSN

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it s Recruit //message while(!recruitMessageReceived()); locallyBroadcast( Abdicate , ID, clusterID); } } } } } }

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Because of being an strictly localized algorithm, ACE uses a xed amount of time O(d) for its execution, regardless of the total amount of nodes in the network. The estimated average degree d of a node in the network is calculated prior to the network deployment.

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7.3.3 Distributed Dominating Set-Based Algorithms One of the problems that arise in a number of distributed network applications is how to locate a small number of centers in the network such that every node is close to at least one center. Clearly, this is the same objective targeted by clustering, and one of the main objectives when doing this is to de ne the cluster concentration points while creating the minimum number of clusters in the network and guaranteeing the total coverage of the network. Graph Theory approaches this problem by calculating a dominating set DSG in the graph. Computing a DSG is also a way of increasing the network s lifetime. The idea is to build a network backbone (the DS) consisting of nodes with higher energy. In reference 21, the nodes in the dominating set (called coordinators) change from time to time in order to maximize their lifetime. An approach similar to the ones presented in reference 14 is proposed in reference 22 using both the node ID and the node degree of the vertices to select the CH. After that, they use clustering algorithms designed based on a Depth First Search (DFS) traversal of nodes in the network. These algorithms are initialized by any node and are fully distributed. They create and maintain k-clusters, in which any node is at distance at most k hops from the CH. In the clustering process, each node is either a CH, a covered nodes or an undecided node. In the basic k-cluster algorithm, each undecided visited node in DFS declares itself a CH and covers all its k-hop neighbors. In the highest connectivity k-cluster algorithms, each undecided visited node checks all its undecided k-hop neighbors and chooses one with the largest connectivity to be its CH. Chen and Stojmenovic consider the mobility of nodes and describe algorithms for modifying cluster structure in the presence of topological changes. The ef ciency of the clustering algorithms is tested by measuring the average number of created clusters, the number of border nodes, and the cluster size in random unit graphs. Basagni [23], also uses the IDSG approximation but generalizes the idea of using weight, such that any meaningful parameter can be used in order to best exploit the network properties.

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CLUSTERING IN WIRELESS SENSOR NETWORKS: A GRAPH THEORY PERSPECTIVE

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The fast distributed approximation algorithms proposed in reference 24 use a synchronous model of computation using a variant of DSG called the k-dominating set where every node is within a distance k of some server or directory copy in this case, of some CH. The proposed algorithm is called Local Randomized Greedy Algorithm (LRG) and is similar to the greedy algorithm for Minimum Set Covering Computation presented in reference 25, but instead of only considering the span of the nodes, LRG considers how the nodes covered by an individual node are also covered by other nodes. The algorithm creates the clusters in three steps. First each node v calculates its span d(v); this is how many of the current adjacent uncovered nodes it will cover if it becomes a CH. This span d(v) plus the node ID is sent to all nodes within a distance equal to 2 around u. Each neighbor computes the maximum and forwards it one more step. With this information, every node can determine whether it is a candidate or not. In the third step, called the selection step, a node u calculates its support s(u), which is the number of candidates that cover u. The node u sends its s(u) to the candidates in its N(u). Each node v adds itself to the dominating set lexicographically higher than that of any node within its 2-hop neighbors. One drawback of this approach for WSN is that once a node and it neighbors are covered, the node does not participate further in the cluster formation process, which will cause a higher rate of energy consumption in the node if it is a CH, leading it to its death because of battery depletion. 7.3.4 Combined and Additional Approaches In this section we present some other clustering mechanisms that combine some of the previously explained approaches and therefore cannot be classi ed in only one of the categories. We include the Zonal Algorithm, the Max-Min D-Cluster formation, and the Topology Adaptive Spatial Clustering algorithm (TASC). The Zonal Algorithm proposed in reference 26 aims to nd weakly connected dominating sets in the network. The algorithm consist of three phases: (1) An input graph representing the network is partitioned into regions of approximately size x. (2) The distributed algorithm for weakly connected sets is run in each region. (3) Some additional border vertices are added. In phase 1, graph partitioning using minimum spanning forests, a sender v of degree d(v) can broadcast to its neighbors in constant time with O(d(v)) messages. A vertex can also send a message to any adjacent vertex in constant time. For nding a Minimum Spanning Tree (MST), all edge weights are distinct, breaking ties using the vertex IDs of the endpoints. The MST is unique for a given graph with distinct edge weights. The algorithm maintains a spanning forest. Initially, the spanning forest is a collection of trees of single vertices. At each step the algorithm merges two trees by including an edge in the spanning forest. During the process of the algorithm, an edge can be in any of the three states: tree edge, rejected edge, or candidate edge. All edges are candidate edges at the beginning of the algorithm. When an edge is included in the spanning forest, it becomes a tree edge. If the addition of a particular edge would create a cycle in the spanning forest, the edge is called a rejected edge and will not be considered further. In each iteration, the algorithm looks for the

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