at University of Texas at Dallas A Localized Algorithms for Coverage Problems in Heterogeneous Sensor Networks My T. Thai, Yingshu Li, Feng Wang, and Ding-Zhu Du, "O(log n)-Localized Algorithms on the Coverage Problem in Heterogeneous Sensor Networks," 26th IEEE International Performance Computing and Communications Conference (IPCCC 2007), New Orleans, LA, April 11-13, 2007. Presented By Donghyun Kim July 23, 2008 Mobile Computing and Wireless Networking Research Group at University of Texas at Dallas

Sensing and Communication Range Sensor Node Sensing Range Communication Range Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Sensor Coverage Problem Basically, it is a scheduling problem. T Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Area vs. Target Coverage Problem Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Target Coverage vs. Set Cover Problem Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations We have sensors . A sensor’s state can be transmit, receive, idle, or sleep. The lifetime of is unit time. is the sensing range of . is the transmission range of . Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Definitions Definition 1: Maximum Lifetime Coverage Given a network consisting of sensors and an interest region, find a family of ordered pairs , where is a set of sensors that completely cover the interest region and is the time duration for to be active such that to maximize . Each sensor appears in with a total time at most . Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Definitions – cont’ For example, suppose four sensors and 2 targets . Given , we can find the family of ordered pairs . Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Definitions – cont’ Some people found that organizing sensors into a maximum number of non- disjoint set covers may obtain a longer network lifetime. Example Disjoint set cover - and . Then total lifetime is 1.0 unit time. Non-disjoint set cover – , and each set is activated for 0.5 unit time. Total lifetime is 1.5 unit time. Find a set of set covers first and decide their activation time. Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Definitions Definition 2: Domatic Number The domatic number of a directed graph is the maximum number of disjoint dominating sets of graph . Maximum lifetime Dominating Sets Problem (MDS) Given a directed graph , where each node has a maximum lifetime , find a family of ordered pairs , where is a dominating set and is the time duration for to be active, such that the total active time of all dominating sets is maximized. Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Overall Idea Maximum Lifetime Target Coverage Problem Maximum Disjoint Set Cover Problem in Bipartite Graph Maximum Lifetime Dominating Sets Problem in Directed Graph Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Graph Transformation Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Graph Transformation – cont’ Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Correctness Lemma 1. A dominating set of is a cover set of . Lemma 2. A cover set in can be transformed to a dominating set of . Theorem 1. A feasible solution to the MDS problem is a feasible solution of the MTC problem. Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Lower and Upper Bounds of Domatic Number in Directed Graphs Theorem 2. Given a directed graph , the domatic number is bounded as where and the term “ ” goes to zero as goes to , is the minimum indegree. Proof Upper bound: for any node , it can be dominated by at most nodes. Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Lower and Upper Bounds of Domatic Number in Directed Graphs Lower bound For each vertex in the graph, randomly assign a color in the range . Let be the event that there is no vertex of color in , which is incoming neighbors of . Let denote the probability of the event . Let be the event that vertex does not have all color in . Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Lower and Upper Bounds of Domatic Number in Directed Graphs Then, the expected number of bad event is : And the expected number of colors that forms dominating sets is at least: Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Lower and Upper Bounds of Domatic Number in Directed Graphs Hence, When Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Two localized approximation algorithms Case 1: for all and Case 2: Otherwise Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas