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1/22 Worst and Best-Case Coverage in Sensor Networks Seapahn Meguerdichian, Farinaz Koushanfar, Miodrag Potkonjak, and Mani Srivastava IEEE TRANSACTIONS ON MOBILE COMPUTING, 2005 Presented by Cheng-Ta Lee 11/17/2009

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2/22 Outlines Introduction Preliminaries Stochastic Coverage Worst-case Coverage and Maximal Breach Path Best-case Coverage and Maximal Support Path Experimental Results Conclusion Future Works

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3/22 Introduction In general, coverage can be considered as a measure of the quality of service of a sensor network. Furthermore, coverage formulations can try to find weak points in a sensor field and suggest future deployment or reconfiguration schemes for improving the overall quality of service. By using best and worst-case coverage information as heuristics to deploy sensors to improve coverage.

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4/22 Preliminaries Computational Geometry Voronoi Diagram Delaunay Triangulation

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5/22 Stochastic Coverage In the simulation studies for this paper, authors have generally assumed uniform sensor distribution. Given: A field A. Sensors S, where for each sensor s i S, the location (x i,y i ) is known. Areas I and F corresponding to initial (I) and final (F) locations of an agent.

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6/22 Worst-case Coverage and Maximal Breach Path (maxmin) (1/6) Definition: Breach. Given a path P connecting areas I and F, breach is defined as the minimum Euclidean distance from P to any sensor in S. Problem: Maximal Breach Path. P B is defined as a path through the field A, with end- points I and F and with the property that for any point p on the path P B, the distance from p to the closest sensor is maximized, thus the P B must lie on the line segments of the Voronoi diagram. Theorem 1. At least one Maximal Breach Path must lie on the line segments of the bounded Voronoi diagram formed by the locations of the sensors in S.

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7/22 Worst-case Coverage and Maximal Breach Path (2/6) The following steps outline the algorithm for finding P B : 1. Generate Voronoi diagram D for S. 2. Apply graph theoretic abstraction by transforming D to a weighted graph. 3. Find P B using binary-search and breadth-first- search.

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9/22 Worst-case Coverage and Maximal Breach Path (4/6)

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10/22 Worst-case Coverage and Maximal Breach Path (5/6)

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11/22 Worst-case Coverage and Maximal Breach Path (6/6) The complexities of the subalgorithms For generating the Voronoi diagram, O(n log(n)), where n is the number of vertex. For BFS O(log(m)) where m is the number of edges. For binary search O(log(range)).

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12/22 Best-case Coverage and Maximal Support Path (minmax) (1/3) Definition: Support. Given a path P connecting areas I and F, support is defined as the maximum Euclidean distance from the path P to the closest sensor in S. Problem. Maximal Support Path. P S is defined as a path through the field A, with end- points I and F and with the property that for any point p on the path P S, the distance from p to the closest sensor is minimized. Theorem 2. At least one Maximal Support Path must lie on the edges of the Delaunay triangulation (with the exceptions of the start and end points connecting P S to I and F).

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13/22 Best-case Coverage and Maximal Support Path (2/3) The algorithm for finding P S is very similar to the breach algorithm above, with the following exceptions: 1. The Voronoi diagram is replaced by the Delaunay triangulation as the underlying geometric structure. 2. Each edge in graph G is assigned a weight equal to the largest distance from the corresponding line segment in the Delaunay triangulation to the closest sensor. 3. The search parameter breach_weight is replaced by the new parameter support_weight and the search is conducted in such a way that support_weight is minimized.

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14/22 Best-case Coverage and Maximal Support Path (3/3)

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15/22 Experimental Results (1/3) If new sensors can be deployed or existing sensors moved such that this breach_weight is decreased, then the worst-case coverage is improved.

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16/22 Experimental Results (2/3) If additional sensors can be deployed or existing sensors moved such that support_weight is decreased, then the best-case coverage is improved.

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17/22 Experimental Results (3/3)

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18/22 Conclusion Authors presented best and worst-case formulations for sensor coverage in wireless ad hoc sensor networks. An optimal polynomial time algorithm that uses graph theoretic and computational geometry constructs was proposed for solving for best and worst-case coverages Maximal Breach Path (worst-case coverage) Maximal Support Path (best-case coverage) Additional sensor deployment heuristics to improve coverage.

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19/22 Future Works In practice, other factors influence coverage such as Obstacles nonhomogeneous sensors Authors have introduced heuristics based on this coverage model that may perform well for single-sensor deployment, it is interesting to investigate methods of optimally deploying multiple sensors at a time.

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20/22 References SeapahnMeguerdichian, Farinaz Koushanfar, Miodrag Potkonjak, and Mani B. Srivastava, ” Coverage Problems in Wireless Ad-hoc Sensor Networks, ” IEEE INFOCOM 2001. Laura Kneckt, ” Summary of Coverage Problems in Wireless Ad-hoc Sensor Networks, ” 2005.

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21/22 Q & A

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