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1 K-clustering in Wireless Ad Hoc Networks using local search Rachel Ben-Eliyahu-Zohary JCE and BGU Joint work with Ran Giladi (BGU) and Stuart Sheiber and Philip Hendrix (Harvard)

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2 Cluster-based Routing Protocol The network is divided to non overlapping sub- networks (clusters) with bounded diameter. Intra-cluster routing: pro-actively maintain state information for links within the cluster. Inter-cluster routing: use a route discovery protocol for determining routes. Route requests are propagated via peripheral nodes.

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3 Cluster-based Routing Protocol +Limit the amount of routing information stored and maintained at individual hosts. +Clusters are manageable. Node mobility events are handled locally within the clusters. Hence, far-reaching effects of topological changes are minimized.

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4 Cluster-heads B A S E F H J D C G I K M N L O B A S E F H J D C G I K M N L O CH Denote Cluster-heads

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5 K-Clustering Topology: the objective is to partition the network into minimum number of sub networks (clusters) with bounded diameter, k. A more symmetric topology than cluster heads.

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6 Problem Statement Minimum k-clustering: given a graph G = (V,E) and a positive integer k, find the smallest value of ƒ such that there is a partition of V into ƒ disjoint subsets V 1,…,V ƒ and diam(G[V i ]) <= k for i = 1…ƒ. The algorithmic complexity of k-clustering is known to be NP-complete for simple undirected graphs.

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7 K-clustering K = 3 1 1 1 1 2 1 2 2 1 2 1 2 2 2 2 2

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8 System Model Two general assumptions regarding the state of the network’s communication links and topology: 1.The network may be modeled as an unit disk graph. 2.The network topology remains unchanged throughout the execution of the algorithm.

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9 Unit Disk Graph B A S E F H J D C G I K M N L O B A S E F H J D C G I K M N L O The distance between adjacent nodes <= 2 The distance between non adjacent nodes is > 2

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10 Contribution of Fernandess and Malkhi A two phase distributed asynchronous polynomial approximation for k-clustering where k > 1 that has a competitive worst case ratio of O(k): First phase – constructs a spanning tree of the network. Second phase – partitions the spanning tree into sub-trees with bounded diameter.

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11 Second Phase: K-sub-tree Given a tree T=(V,E) the algorithm finds a sub-tree whose diameter exceeds k, it then detaches the highest child of the sub-tree and repeats over on the reduced tree. k k- r detach highest sub-tree root of the sub-tree sub-tree

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12 Random Decent RANDOM_DESCENT(problem,terminate) returns solution state inputs: problem, a problem termination condition, a condition for stopping local vars: current, a solution state next, a solution state current ← Initial State (problem( while (not terminate) next ← a selected neighbor of current ∆ E← Value(next) - Value(current) if ∆ E <0 then current ←next

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13 Initial State

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14 Initial State (k=2)

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15 K is even (e.g. 2)

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16 K = 2 (cont.)

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17 K = 2 (cont.)

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18 K = 2 (cont.)

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19 K = 2 (cont.)

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20 K = 2 Total: 8 clusters

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21 A better State

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22 Random Decent RANDOM_DESCENT(problem,terminate) returns solution state inputs: problem, a problem termination condition, a condition for stopping local vars: current, a solution state next, a solution state current ← Initial State (problem( while (not terminate) next ← a selected neighbor of current ∆ E← Value(next) - Value(current) if ∆ E <0 then current ←next

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23 Building the neighbor

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24 Building the neighbor

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25 Building the neighbor

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26 Building the neighbor

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27 Building the neighbor

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28 Building the neighbor

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29 For Odd k:

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30 K is odd (e.g. 3)

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31 K is odd (e.g. 3)

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32 Random Decent RANDOM_DESCENT(problem,terminate) returns solution state inputs: problem, a problem termination condition, a condition for stopping local vars: current, a solution state next, a solution state current ← Initial State (problem( while (not terminate) next ← a selected neighbor of current ∆ E← Value(next) - Value(current) if ∆ E <0 then current ←next

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33 Experimental Evaluation Randomly Generated Graphs Grid Graphs

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34 Randomly Generated Graphs Parameters: –n – number of nodes –l – length of a unit Graph Generation: - n points are placed randomly on a 1X1 square - two vertices are connected iff the distance between them is less than l.

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35 400 nodes, k=5

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36 Results

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37 Experiments on Grids

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38 Theorem: The number of nodes in a maximal cluster in a greed: If K is even, If k is odd, e.g. 13 if k=4 e.g. 8 if k=3

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39 A maximal cluster on grid x+y=r x+y=r+k x-y=s x-y=s-k

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40 A maximal cluster on grid x+y=r x+y=r+k x-y=s x-y=s-k

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41 A maximal cluster – k is even x+y=r x+y=r+4 x-y=s x-y=s-4

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42 A maximal cluster – k is odd x+y=r x+y=r+3 x-y=s x-y=s-3

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43 In general, number of nodes in a maximal cluster: If K is even, If k is odd, e.g. 13 if k=4 e.g. 8 if k=3

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44 Optimal Clustering for k=4

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45 Optimal Clustering for k=3

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47 Related Work Local search techniques were used for network partitioning Simulated annealing and genetic algorithms Was tested on a very limited network size : 20-60 nodes. We present solid criteria for evaluating the local search

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48 Conclusions A new local search algorithm for k- clustering was introduced It outperforms existing distributed algorithm for large k and dense networks. Grids can be built using optimal clustering Clustering on grids needs improvement.

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49 Future Work Check a distributed version of local search Change the algorithm for local search Find an efficient way to fix a solution – e.g. by merging small clusters Use local search for other optimization problems in networking

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