Survey on Coverage Problems in Wireless Sensor Networks - 2 M. T. Thai, F. Wang, H. Du, and X. Jia, "Coverage Problems in Wireless Sensor Networks: Designs and Analysis," International Journal of Sensor Networks, special issue on Coverage Problems in Sensor Networks, vol. 3, no. 3, pp. 191-200, 2008 Presented By Donghyun Kim July 31, 2008 Mobile Computing and Wireless Networking Research Group at University of Texas at Dallas

Agenda k-Coverage Problem Connected Coverage and Connected k- Coverage Problem Minimum Coverage Breach Presented by Donghyun Kim on July 31, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

K-Coverage Problem Objective Find a family of ordered pairs such that the total time is maximum and each target can be monitored by at least distinct sensors in each set . The result in [15] for 1-coverage problem can be extended as follows. Presented by Donghyun Kim on July 31, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

K-Coverage Problem - Maximum Lifetime Coverage Centralized Algorithms The authors in [15] modeled this problem as Maximal Set Covers (MSC) as follows. is a set of all sensors that can cover a target . Assume for all sensors. The above formulation is relaxed to an Linear Programming and solved using Ye’s algorithm in [8]. The complexity of this solution is quite high: The authors in [15] also introduced a much faster heuristic algorithm whose complexity is , where is the number of targets and is the number of sensors that covers the most sparsely covered targets. [8] Y. Ye, An Potential Reduction Algorithm for Linear Programming, Mathematical Programming, Vol 50, pp 239- 258, 1991. [15] M. Cardei, M. T. Thai, Y. Li, and W. Wu, Energy-Efficient Target Coverage in Wireless Sensor Networks, Proceedings of the 24th conference of the IEEE Communications Society (INFOCOM), 2005. Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

K-Coverage Problem – cont’ Modified version for K-Coverage Problem in [21] is as below. is a set of all sensors that can cover a target . Assume for all sensors. In [21], the authors tried to find a maximum number of sensors that can be off instead of global lifetime optimization. A node can be off if all the targets inside its sensing range are already k-covered. [21] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless and C. D. Gill, Integrated Coverage and Connectivity Configuration in Wireless Sensor Networks, 1st ACM Conference on Embedded Networked Sensor Systems, 2003. Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Connected Coverage and Connected k-Coverage Theorem 1 [21] If , then a set of node which completely covers whole target area is connected. Theorem 2 [11] When the number of sensors in any finite area is finite, a necessary and sufficient condition for the complete coverage of a convex region to imply connectivity is . Theorem 3 [21] Given a convex region A, a set of sensors that k-covers A forms a k-connected communication graph if . [21] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless and C. D. Gill, Integrated Coverage and Connectivity Configuration in Wireless Sensor Networks, 1st ACM Conference on Embedded Networked Sensor Systems, 2003. [11] H. Zhang and J.C. Hou, Maintaining Sensing Coverage and Connectivity in Large Sensor Networks, The Wireless Ad Hoc and Sensor Networks: An International Journal, Jan 2005. Presented by Donghyun Kim on July 31, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Connected Coverage and Connected k-Coverage – cont’ When , other algorithms to connect all nodes in a k-cover are integrated. In [22], authors proposed a greedy algorithm for the connected k-coverage problem. Approximation ratio is . [22] Z. Zhou, S. Das, and H. Gupta, Connected K-Coverage Problem in Sensor Networks, ICCCN, 2004 Presented by Donghyun Kim on July 31, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Greedy Algorithm Basic Idea is current cover. A Candidate Sensor has an intersection with any node in and target region. A Candidate Path is a set of nodes and it connects a node in to . A with the highest k-benefit is selected and all nodes in it is added to . Presented by Donghyun Kim on July 2, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas