1. 1 On Constructing k- Connected k-Dominating Set in Wireless Networks Department of Computer Science and Information Engineering National Cheng Kung University, Taiwan R.O.C. Authors: Fei Dai and Jie Wu Publisher: Parallel and Distributed Processing Symposium, 2005. Proceedings. 19th IEEE International Present: Min-Yuan Tsai ( 蔡旻原 ) Date: March, 11, 2008

2. 2 Outline 1. Introduction 2. Proposed Scheme K-Gossip K-Coverage Condition Color Based K-CDS 3. Simulation 4. Conclusion

3. 3 Introduction Why k-CDS? For fault tolerance and routing flexibility. Dominating Set (DS) : All nodes in the network are either in this set or have a neighbor in this set. k-DS : Every node is either in the set or has k neighbors in the set. Connected Dominating Set (CDS) : The subgraph induced from this dominating set is connected. k-CDS : Its induced subgraph is k-vertex connected. A graph is k-vertex connected if removing any k-1 nodes from it does not cause a partition. Induced subgraph: a subgraph obtained by deleting a set of vertices.

4. 4 Outline 1. Introduction 2. Proposed Scheme k-Gossip k-Coverage Condition Color Based k-CDS 3. Simulation 4. Conclusion

5. 5 Related work Localized algorithm can achieve fast convergence with low maintenance. Localized CDS algorithms are either probabilistic or deterministic. Probabilistic : Gossip. Each node v has a backbone status with probability p. Deterministic : By some rules, but all they are special case of Coverage Condition. Node v has a non-backbone status if for any two neighbors u and w, a replacement path exists that connects u and w via several intermediate nodes (if any) with higher priorities than v.

6. 6 k-Gossip Original Gossip : Each node v has a backbone status with probability p. k-Gossip : Each node v has a backbone status with probability p k. p k depends on k, network size, deploying area size, and transmission range – global information. Pros : Very low overhead at each node, requires no information exchange among neighbors, and very low computation cost. Cons : Need global information, and produce large size backbone. Expected size of backbone is n* p k.

7. 7 k-Coverage Condition Original Coverage Condition : Node v has a non-backbone status if for any two neighbors u and w, a replacement path exists that connects u and w via several intermediate nodes (if any) with higher priorities than v. k-Coverage Condition : Node v has a non-backbone status if for any two neighbors u and w, k node disjoint replacement paths exist that connect u and w via several intermediate nodes (if any) with higher priorities than v. That is, node v has a non-backbone status if all its neighbors are k-connected with each other via higher priority nodes.

8. 8 k-Coverage Condition (contd.) Pros : Depend on local information only (just use hello to collect the k-hop information). Cons : More complex. (compute the k node disjoint replacement paths) Complexity Each node has to test whether there are k-disjoint paths between every pair of its neighbors If 2-hop information is collected, O(k  4 ), where  means the maximal node degree

9. 9 Color Based k-CDS Construction A hybrid algorithm. Construction Step : Each node v selects a random color c v (c v is between 1 and k). Thus the network is divided into k-disjoint subsets. (probabilistic) For each sub-network with color c, construct a CDS that covers the original network. (deterministic) k-CDS is the union of all k CDSs.

10. 10 Color Based k-CDS Construction (cont.) Pros : Use smaller “Hello” messages to collect 2- hop information, lower computation cost, and construct a smaller backbone than probabilistic. Cons : The scheme works well only when the network is sufficient dense.

11. 11 Outline 1. Introduction 2. Proposed Scheme K-Gossip K-Coverage Condition Color Based K-CDS 3. Simulation 4. Conclusion

12. 12 Simulation The simulation focus on success ratio and backbone size. Success ratio :

13. 13 Simulation (contd) The simulation focus on success ratio and backbone size. Backbone size :

14. 14 Outline 1. Introduction 2. Proposed Scheme K-Gossip K-Coverage Condition Color Based K-CDS 3. Simulation 4. Conclusion

15. 15 Conclusion Future work: Conduct extensive simulation study on the performance in k-CDS routing. Try to find a probabilistic approximation ratio of the k-convergence condition.

16. 16 Backup Material CBCC-I : Covers all neighbors regardless of their colors; that is, any two neighbors of a non-backbone node must be connected via a replacement place. Node 3 is a backbone node, because it has two neighbors 2 and 7 that are not connected via a gray replacement path. CBCC-II : Only neighbors with the same color are considered. Node 3 becomes a non-backbone node because it has only one visible neighbor 7. However, the resultant gray backbone {5, 7} is not a CDS of the entire network, and union of all backbone nodes {5, 7, 8} is not 2-dominating. The failure of node 8 will leave node 2 uncovered.