1. 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

2. 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

3. 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.

4. 4/22 Preliminaries  Computational Geometry Voronoi Diagram Delaunay Triangulation

5. 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.

6. 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.

7. 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.

8. 8/22

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

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

11. 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)).

12. 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).

13. 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.

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

15. 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.

16. 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.

17. 17/22 Experimental Results (3/3)

18. 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.

19. 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.

20. 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.

21. 21/22 Q & A