1. Coverage Problems in Wireless Ad-hoc Sensor Networks Seapahn Meguerdichian 1 Farinaz Koushanfar 2 Miodrag Potkonjak 1 Mani Srivastava 2 University of California, Los Angeles

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

3. Sensor Network Architecture GATEWAY MAIN SERVER CONTROL CENTER

4. Key Highlight Transform the difficult to represent coverage problems to discrete-domain optimization using computational geometry and graph theory constructs: Voronoi Diagram Voronoi Diagram Delaunay Triangulation Delaunay Triangulation

5. Outline Sensing models and assumptions Sensing models and assumptions Coverage formulations Coverage formulations Maximal Breach Maximal Breach Maximal Support Maximal Support Future directions Future directions Conclusion Conclusion

6. 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

7. Assumption Sensing effectiveness diminishes as distance increases (monotonic) Sensing effectiveness diminishes as distance increases (monotonic)  Homogeneous sensor nodes  Non-directional sensing technology  Centralized computation model

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

9. Maximal Breach Given: Field A instrumented with sensors; areas I and F. 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.

10. 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.

11. 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?

12. Finding Maximal Breach Path Algorithm 1. 1.Generate Voronoi Diagram 2. 2.Apply Graph-Theoretic Abstraction 3. 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

13. Bounded Voronoi Diagram

14. Maximal Support 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.

15. Maximal Breach Path Example (50 nodes)

16. Maximal Breach Path Example (200 nodes)

17. Future Directions Distributed Schemes Distributed Schemes Path planning Path planning Multi-sensor deployment optimization Multi-sensor deployment optimization Average-case coverage calculations Average-case coverage calculations Exposure Exposure

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