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2-Connected Virtual Backbone in Wireless Networks

2-Connected Virtual Backbone in Wireless Networks. F. Wang, M. T. Thai, and D.-Z. Du, 2-Connected Virtual Backbone in Wireless Networks, IEEE Transactions on Wireless Communications , accepted with revisions, 2007.

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2-Connected Virtual Backbone in Wireless Networks

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  1. 2-Connected Virtual Backbone in Wireless Networks F. Wang, M. T. Thai, and D.-Z. Du, 2-Connected Virtual Backbone in Wireless Networks, IEEE Transactions on Wireless Communications, accepted with revisions, 2007. Presented By Donghyun KimOn February 13, 2008Mobile Computing and Wireless Networking Research Group at University of Texas at Dallas

  2. Given a UDG , find with minimum size which satisfying • Each node not in is dominated by at least nodes in • is -node-connected • : backbone nodes • This paper focus on a special case of this problem for and to construct a 2-connected 1-dominating virtual backbone. • Since finding 1-CDS is NP-hard, 2-CDS is also. Problem Definition Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  3. Present an algorithm to compute 2-CDS, namely, Connected Dominating Set Augmentation (CDSA). • The approximation ratio of CDS is 64. • Using recent results, this can be improved to 62.109. Contribution of This Paper Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  4. : a connected graph • Cut-vertex of : a vertex such that the graph is disconnected. • A block is a maximal subgraph of without cur-vertices. • A biconnected graph is a graph without cut-vertices. • A leaf block of a connected graph is a subgraph of which includes only one cut-vertex. Notations Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  5. Block Leaf Block C C Cut Vertex Notations – cont’ Observation: a leaf block include at least two adjacent blocks. Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  6. Block1 C C Block2 C Notations – cont’ Biconnected graph Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  7. Compute a CDS using any CDS construction algorithm. • Compute all the blocks in using the standard algorithm in [1], which is based on the depth first search for computing the bi-connected components. • Calculate the shortest path in the original graph that satisfies the requirements: • Repeat following two steps until is 2-connected. • The path can connect a leaf block in to other portion of . • The path does not contain any nodes in except the two endpoints. Then add all intermediate nodes in this path to . Basic Idea Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  8. Compute 1-CDS • 3 Blocks Left 2CDS construction Example Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  9. Leaf Block C 2CDS construction Example – Cont’ Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  10. 2 Blocks Left 2CDS construction Example – Cont’ Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  11. C 2CDS construction Example – cont’ Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  12. 2CDS construction Example – cont’ 1 Blocks Left Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  13. Lemma1 • At most 10 new nodes are added into the backbone at each augmenting step. • Proof of Lemma1 • Assume we have a CDS , is a leaf block of CDS, and is the cut-vertex. • Suppose we selected a shortest path between and . That is , , and all other nodes in the path are non-CDS nodes. Performance Analysis Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  14. Length of can be 1, 2, or more. • Assume • Claim • If has more than two intermediate nodes, all intermediate nodes except and must be a neighbor of the cut-vertex . • Proof of Claim • If is connected to , then is not the shortest. Performance Analysis Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  15. Four possible positions of and • Node and are both neighbors of node . • Node is a neighbor of , but node is not. • Node is a neighbor of , but node is not. • Neither node nor are neighbors of node . Performance Analysis Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  16. Intuition Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  17. At most 8 Case 1) Node u and v are both neighbors of node w Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  18. At most 8 Case 2-3) Only one of them is neighbor of w Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  19. At most 8 Case 4) Neither u nor v is the neighbor of w Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  20. Some new notations • : an optimal 2-CDS • :an optimal 1-CDS • : a 2-CDS generated by CDSA • : a CDS using the best CDS construction algorithm • Lemma2 • by [2] • Lemma3 Performance Analysis – cont’ Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  21. Performance Analysis – cont’ Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  22. Theorem 2 • In the CDSA, first a CDS is constructed, then in at most steps and each step at most 8 nodes are added, we construct a 2-connected virtual backbone. Hence, . • From Lemma 2 and 3, we have • Thus, for all . • Time complexity: Performance Analysis – cont’ Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

  23. [1] R. Tarjan, “Depth first search and linear graph algorithms,” SIAM Journal of Computing, vol. 1, no. 2, pp 146-160, 1972. • [2] M. Min, F. Wang, D.-Z. Du, and P.M. Pardalos, “A reliable virtual backbone scheme in mobile ad hoc networks,” in Proceedings of IEEE International conference on Mobile Ad Hoc and Sensor Systems (MASS), 2004. References Presented by Donghyun Kim on February 13, 2008Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

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