1. 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)

2. 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.

3. 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.

4. 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

5. 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.

6. 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.

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

8. 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.

9. 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

10. 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.

11. 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

12. 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

13. 13 Initial State

14. 14 Initial State (k=2)

15. 15 K is even (e.g. 2)

16. 16 K = 2 (cont.)

17. 17 K = 2 (cont.)

18. 18 K = 2 (cont.)

19. 19 K = 2 (cont.)

20. 20 K = 2 Total: 8 clusters

21. 21 A better State

22. 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

23. 23 Building the neighbor

24. 24 Building the neighbor

25. 25 Building the neighbor

26. 26 Building the neighbor

27. 27 Building the neighbor

28. 28 Building the neighbor

29. 29 For Odd k:

30. 30 K is odd (e.g. 3)

31. 31 K is odd (e.g. 3)

32. 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

33. 33 Experimental Evaluation Randomly Generated Graphs Grid Graphs

34. 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.

35. 35 400 nodes, k=5

36. 36 Results

37. 37 Experiments on Grids

38. 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

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

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

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

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

43. 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

44. 44 Optimal Clustering for k=4

45. 45 Optimal Clustering for k=3

46. 46

47. 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

48. 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.

49. 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