Constructing a m-connected k-Dominating Set in Unit Disc Graphs W. Shang, F. Yao, P. Wan, and X. Hu, On minimum m-connected k-dominating set problem in unit disc graphs, Journal of Combinatorial Optimization, Dec. 2007. Presented By Donghyun Kim June 11, 2008 Mobile Computing and Wireless Networking Research Group at University of Texas at Dallas

Virtual Backbone In simulation, DSR/AODV over virtual backbones performs better than plain DSR/AODV. Size does matter! Communication overhead can be reduced. Increase the convergence speed (in routing). Simplify the connectivity management. Support broadcasting or multicasting. Reduce the overall energy consumption. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions is a connected graph with vertex-set and edge-set . For is called the (open) neighborhood of . is called the closed neighborhood of . A cut-vertex of a connected graph is a vertex such that the graph is disconnected. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions – cont’ A block is a maximal connected subgraph having no cut- vertex. The block-cut-vertex graph of is a graph where consists of all cut-vertices of and all blocks of , with a cut-vertex adjacent to a block if is a vertex of . A leaf block of a connected graph is a block with only one cut-vertex. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions – cont’ For a graph , a Dominating Set (DS) of is a subset of such that each node in is adjacent to at least one node in . Computing an Maximal Independent Set (MIS) is the most popular way to get DS. A Connected Dominating Set (CDS) of is a dominating set of which induces a connected subgraph of . Nodes in are called as dominaters. Others are called as dominatees. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions – cont’ Coloring Technique for MIS computation Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions – cont’ The problem of computing a virtual backbone for a wireless network is modeled as one of calculating a CDS of a graph which represents the network. A -dominating set of is a set of vertices such that each vertex is either in or has at least neighbors in . Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions – cont’ -Vertex Connectivity A network is -vertex connected if it is connected and removing any 1, 2, …, nodes from will not cause partition in . -Connected -Dominating Set Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Fault Tolerant Virtual Backbone Fault-Tolerance for dominaters With -connectivity, communication may not be disrupted even when up to paths fail. Fault-Tolerance for dominatees Each dominatee node has at least neighboring dominators in a CDS. It also provides routing flexibility. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Overview How to compute ( , )-CDS? Algorithm for (1, 1)-CDS Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Relationship between an MIS and k-Dominating Set Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Relationship between an MIS and k-Dominating Set – cont’ Lemma 1 Let be a UDG and a natural number such that minimum degree . Let be a minimum -dominating set of and an MIS of . Then . Proof of Lemma 1 Let Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Relationship between an MIS and k-Dominating Set – cont’ For each , let . Then, has to be true by the property of -dominating set, and . For each , let Then, since is a UDG, for each , there are at most 5 independent vertices in its neighborhood and , and . Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (1,k)-CDS Basic Idea Compute (1,1)-CDS Sequentially produce an MIS times. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (1,k)-CDS – cont’ Theorem 1 Algorithm A is an approximation algorithm for the minimum connected -dominating set problem with performance ratios for and 7 for . Correctness of Algorithm A is (1, 1)-CDS A node has at least neighbors in . Therefore, is (1, )-CDS. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (1,k)-CDS – cont’ Proof of Theorem 1 Let for is an independent set and . By lemma1, Then, Then, and hence for | and for . Since By combining the two estimation above, the theorem holds true. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (2,k)-CDS Step 1: Apply Algorithm A to construct a 1- connected -dominating set . Step 2: Compute all the blocks in by computing the 2-connected components through the depth first search. Step 3: Produce the shortest path in the original graph such that it can connect a leaf block in with other part of but does not contain any vertices in except the two endpoints. Then add all intermediate vertices in this path to . Step 4: Repeat Step 2 and Step 3 until is 2- connected. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (2,k)-CDS – cont’ Basic Idea Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (2,k)-CDS – cont’ Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (2,k)-CDS – cont’ Lemma 2 For , at most two new vertices are added into at each augmenting step. Lemma 3 The number of cut-vertices in the connected -dominating set by Algorithm A is no bigger than the number of vertices in generated at the first step of Algorithm A. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (2,k)-CDS – cont’ Theorem 2 Algorithm B is an approximation algorithm for the minimum 2-connected -dominating set problem with performance ratio Proof of Theorem 2 At most nodes are added by Lemma 2 for 2- connectivity by Lemma 2 and 3. From Theorem1, we have and for and for . The theorem hold true by combining the results above. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (m,k)-CDS – cont’ This algorithm assumes the existence of -CDS computation algorithm and . Theorem 3. If there exists an -approximation algorithm A for the case of , then there exists an ( +6)-approximation algorithm for the case of . Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas