Worst and Best-Case Coverage in Sensor Networks

1. Worst and Best-Case Coveragein 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. 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. 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. Computational Geometry Voronoi Diagram Delaunay Triangulation Preliminaries

5. 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 siS, the location (xi,yi) is known. • Areas I and F corresponding to initial (I) and final (F) locations of an agent.

6. 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. • PB 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 PB, the distance from p to the closest sensor is maximized, thus the PB 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. Worst-case Coverage and Maximal Breach Path (2/6) • The following steps outline the algorithm for finding PB: • Generate Voronoi diagram D for S. • Apply graph theoretic abstraction by transforming D to a weighted graph. • Find PB using binary-search and breadth-first-search.

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

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

11. 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 ofedges. • Forbinary search O(log(range)).

12. 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 . • PS 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 PS, 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 PS to I and F).

13. Best-case Coverage and Maximal Support Path (2/3) • The algorithm for finding PS is very similar to the breach algorithm above, with the following exceptions: • The Voronoi diagram is replaced by the Delaunay triangulation as the underlying geometric structure. • 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. • 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. Best-case Coverage and Maximal Support Path (3/3)

15. 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. 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. Experimental Results (3/3)

18. 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. 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. 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. Q &A