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K-clustering in Wireless Ad Hoc Networks

K-clustering in Wireless Ad Hoc Networks . Fernandess and Malkhi Hebrew University of Jerusalem Presented by: Ashish Deopura. Outline of the presentation. Motivation for clustering in Mobile Ad hoc networks Problem Statement Algorithm Description Conclusions / Summary.

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K-clustering in Wireless Ad Hoc Networks

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  1. K-clustering in Wireless Ad Hoc Networks Fernandess and Malkhi Hebrew University of Jerusalem Presented by: Ashish Deopura

  2. Outline of the presentation • Motivation for clustering in Mobile Ad hoc networks • Problem Statement • Algorithm Description • Conclusions / Summary

  3. Wireless Ad-hoc networks • Dynamic topology • Power and bandwidth limitations • Broadcast network • Routing • How to determine path from source to destination

  4. 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.

  5. Cluster-based Routing Protocol (Cntd.) • 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. • Overcome mobility by adjusting cluster size (diameter) according to network stability.

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

  7. 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 V1,…,Vƒ and diam(G[Vi])<= kfor i = 1…ƒ. • The algorithmic complexity of k-clustering is known to be NP-complete for simple undirected graphs.

  8. Example K-clustering (for K = 3) 1 2 1 2 2 1 1 2 1 2 1 2 2 1 2 2

  9. Algorithm Description • A two phase distributed algorithm for k-clustering where k > 1 that has a competitive worst case ratio of O(k) • First phase: construct a MCDS tree of the network • Second phase: partition the spanning tree into sub-trees with bounded diameter.

  10. System Model Two general assumptions regarding the state of the network’s communication links and topology: • The network may be modeled as an unit disk graph (represents effective broadcast range). • The network topology remains unchanged throughout the execution of the algorithm.

  11. The distance between adjacent nodes = 1 S S M M E E L L F F B B C C J J A A G G H H D D K K I I N N O O The distance between non adjacent nodes is >= 2 Unit Disk Graph

  12. Preliminaries I Given an undirected graph G = (V,E) consider the following general definitions regarding k-clustering: • Diameter: • Dominating Set (DS): • Connected Dominating Set (CDS): The induced sub graph G[D]is connected.

  13. Preliminaries II • Independent Set (IS): • Maximal Independent Set (MIS): An independent set S where no proper superset of Sis also an IS. • A MIS is also DS.

  14. Maximal Independent Set (MIS) S M E L F B C J A G H D K I N O CH Denote MIS & DS nodes

  15. First Phase: MCDS Tree Construction Given an unit disk graph G = (V,E)the algorithm executes as follows: • Step 1: Construct a spanning tree T. • partitions the nodes into disjoints sets Si • Siis a set of nodes at level equal to i • Every node knows its neighbors • A rank associated with every node

  16. Spanning tree (Cntd.) • Maximal Independent Set Construction A A C C B B D D E E F F G G

  17. INV INV Spanning tree (Cntd.) • Connected Dominating Set, parent child pointers A A C C B B D D E E F F G G

  18. 2 5 2 3 5 3 6 4 6 4 1 1 3 5 root root 4 2 2 6 5 5 3 3 4 6 4 6 Denote MIS nodes Denote NS nodes Connected Dominating Set Denote spanning tree edge. 3 5 4 6

  19. Second Phase: K-sub tree • Partition the spanning tree into sub-trees • Bounded Diameter • Each node maintains • Height • Highest child • Detach child if • H+ Height + 1 > k • Where H is the height received from a child

  20. The tree rooted at this node exceeds k  detach the highest child K-sub-tree Converge-cast (K=4) 2 5 3 leaf 6 4 1 3 5 root 4 2 6 5 3 4 6 Denote MCDS spanning tree edge.

  21. The tree rooted at this node exceeds k  detach the highest child. K-sub-tree Converge-cast (K=4) 2 5 3 leaf 6 4 1 3 5 root 2 6 Denote MCDS spanning tree edge.

  22. K-sub-tree Converge-cast (K=4) 2 1 3 root 2 Denote MCDS spanning tree edge.

  23. Summary • A distributed k-clustering algorithm • Competitive worst case ratio of O(k) • Building-block – essential for cluster-based routing protocols. • Flexible - cluster diameter is a part of the algorithm parameter.

  24. Thanks

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