1. 1 Worst and Best-Case Coverage in Sensor Networks Seapahn Meguerdichian, Farinaz Koushanfar, Miodrag Potkonjak, Mani Srivastava IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL.4, NO. 1, JANUARY-FEBRUARY 2005 IEEE Infocom 2001IEEE Infocom 2001, Vol. 3, pp. 1380-1387, April 2001.

2. 2 Outlines Introduction Sensing models and assumptions Coverage formulations Maximal Breach Maximal Support Experimental Conclusion

3. 3 Coverage Coverage can be considered as a measure of the quality of service of a sensor network. Coverage formulations can  try to find weak points in a sensor field  suggest future deployment or reconfiguration schemes for improving the overall quality of service.

4. 4 Coverage Problem Given:  Field A  S sensors, specified by coordinates  Initial(I) and final(F) locations of an agent (I, F) How well can the field be observed ? nWorst Case Coverage: Find a maximal breach path for an agent moving in A. nBest Case Coverage: Find a maximal support path for an agent moving in A.

5. 5 Worst Case Coverage We want to find the closest distance to sensors that an agent traveling on any path in the sensor field must encounter at least once. We determine the closest distance to sensors even if the agent tries to optimally avoid the sensors.

6. 6 Best Case Coverage We want to find the farthest distance to sensors that an agent traveling on any path in the sensor field must have from sensors, even if it tries to stay as close to sensors as possible. At some points, the agent must move away from sensors in order to be able to traverse the field.

7. 7 Key Highlight Transform the difficult to represent coverage problems to discrete-domain optimization using computational geometry( 計算幾何 ) and graph theory constructs: Voronoi Diagram Delaunay Triangulation

8. 8 Sensing Model We express the general sensing model S at an arbitrary point p for a sensor s as: where d(s,p) is the Euclidean distance between the sensor s and the point p, and positive constants and K are sensor technology dependent parameters

9. 9 Assumption Sensing effectiveness diminishes as distance increases  Homogeneous sensor nodes  Sensor node locations are known  Non-directional sensing technology  Centralized computation model

10. 10 Coverage Formulation How well can the field be observed ? nWorst Case Coverage: Maximal Breach Path nBest Case Coverage: Maximal Support Path The “ paths ” are generally not unique. They quantify the best and worst case observability (coverage) in the sensor field.

11. 11 Maximal Breach Path Given: Field A instrumented with sensors S; areas I and F. Breach: the minimum Euclidean distance from P to any sensor in S. Problem: Identify P B, the Maximal Breach Path in S, starting in I and ending in F. P B is defined as a path with the property that for any point p on the path P B, the distance from p to the closest sensor is maximized.

12. 12 Enabling Step: Voronoi Diagram By construction, each line-segment maximizes distance from the nearest point (sensor). Consequence: Path of Maximal Breach of Surveillance in the sensor field lies on the Voronoi diagram lines.

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15. 15 Graph-Theoretic Formulation Given : Voronoi diagram D with vertex set V and line segment set L and sensors S Construct graph G(N,E): Each vertex v i  V corresponds to a node n i  N Each line segment l i  L corresponds to an edge e i  E Each edge e i  E, Weight(e i ) = Distance of l i from closest sensor s k  S Formulation : Is there a path from I to F which uses no edge of weight less than K?

16. 16 Finding Maximal Breach Path Algorithm 1.Generate Voronoi Diagram 2.Apply Graph-Theoretic Abstraction 3.Search for P B Check existence of path I --> F using BFS Search for path with maximal, minimum edge weights This is a Maximal Breach Path, P B, and it is not unique.

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18. 18 Critical Regions 30 sensors are deployed at random.

19. 19 Bounded Voronoi Diagram Sensor field with Voronoi Diagram and a Maximal Breach Path.

20. 20 Maximal Support Path Given: Field A instrumented with sensors S; areas I and F. Support : the maximum Euclidean distance from the path P to the closest sensor in S.. Problem: Identify Ps, the Maximal Support Path in S, starting in I and ending in F. Only requirement: the distance from the farthest point on Ps to the closest sensor is minimized.

21. 21 Maximal Support Path Given : Delaunay Triangulation of the sensor nodes Construct graph G(N,E): The graph is dual to the Voronoi graph previously described Formulation : what is the path from which the agent can best be observed while moving from I to F? (The path is embedded in the Delaunay graph of the sensors) Solution: Similar to the max breach algorithm, use BFS and Binary Search to find the shortest path on the Delaunay graph. Sensor field with Delaunay triangulation and a Maximal Support Path (Ps)

22. 22 Maximal Breach Path Example (50 nodes)

23. 23 Maximal Breach Path Example (200 nodes)

24. 24 Maximal Breach Path – Sensor Deployment Even after deploying 100 sensors, breach coverage can be improved by about 10 percent by deploying just one more sensor.

25. 25 Maximal Support Path – Sensor Deployment

26. 26 Asymptotic Behavior On average, after deploying about 100 sensors, additional random sensors do not improve coverage very significantly.

27. 27 Conclusions Best and Worst case coverage formulations Efficient optimal algorithms using computational geometry and graph theory  Maximal Breach Path (worst-case coverage)  Maximal Support Path (best-case coverage) Applications in:  Deployment  Asymptotic analysis