Coverage Problems in Wireless Ad-hoc Sensor Networks

1. Coverage Problems in Wireless Ad-hoc Sensor Networks Seapahn Meguerdichian, Farinaz Koushanfar, Miodrag Potkonjak and Mani Srivastava INFOCOMM – 2001 Presetation : suh-mahn hur

2. Contents Introduce Worst Case Coverage-Maximal Breach Path Best Case Coverage-Maximal Support Path Result Conclusoin Network Security Technology Lab.

3. Introduction • Some issues arising in ad-hoc wireless sensor networks are: • location calculation, deployment, tracking and coverage • Coverage: Measure of the quality of service of a sensor network • how well can network observe an area? • Weak points can suggest future deployment or reconfiguration schemes • what is the probability of an event being detected within a time frame?

4. Problem definition • Different viewpoints of coverage • Worst-case coverage: Quantify the QoS by finding areas of lower observability from sensor nodes and detecting breach regions • Best Case Coverage: Detect areas of high observability from sensors, and regions of best support • Goal: Given a sensor network deployment, efficiently find the maximal breach and supporting paths

5. Solutions • Based on computation geometry ideas • Voronoi diagram • Delaunay Triangulation

6. Voronoi Diagrams • The Voronoi diagram of a set of points partitions the plane into convex polygons such that all points inside a polygon are closest to point inside the polygon

7. Voronoi Diagram Construction • Construct a bisector between one site and all others. • A Voronoi cell is the intersection of all half-planes defined by the bisectors.

8. Delaunay Triangulation • Related to the Voronoi diagram (dual of each other) • Connects the sites(points) in the Voronoi diagram whose polygons share a common edge • Ensures that sites that are close together are connected Site Delaunay edge Voronoi Edge Voronoi vertex

9. Worst Case Coverage • Given: Field A instrumented with sensors; areas I and F. • Problem: Identify PB, the maximal breach path in S, starting in I and ending in F. • PB is defined as a path with the property that for any point p on the path PB, the distance from p to the closest sensor is maximized. • Intuition: • Find a path such that any point on path always is at least breach_width distance away • Maximum value of breach_width leads to worst case coverage and minimizes observability along path

10. 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. Formulation • Given: Voronoi diagram D with vertex set V and line segment set L and sensors S • Construct graph G(N,E): • Each vertex viV corresponds to a node niN • Each line segment li L corresponds to an edge eiE • Each edge eiE, Weight(ei) = Distance of li from closest sensor skS • Formulation: Is there a path from I to F which uses no edge of weight less than K?

12. Algorithm • Generate Voronoi Diagram • Apply Graph-Theoretic Abstraction (generate graph from diagram) • Search for PB • Check existence of path I --> F using Breadth First Search and Binary Search • Perform a binary search between the smallest and largest edge weights in the graph • During each step of the Binary Search, check to see if a path exists using only edges with weights larger than the specified search criteria (breach_weight) • PB is maximal breach path • Every edge in the breach path has weight larger than or equal to the breach_weight, and at least one edge will have a weight equal to the breach_weight

13. Best Case Coverage • Given: Field A instrumented with sensors; areas I and F • Problem: Identify Ps, the maximal support path in S, starting in I and ending in F. • Ps is defined as a path with the property that for any point p on the path Ps, the distance from p to the closest sensor is minimized. • Intuition: • Find a path such that any point on path always is at most support_width distance away • Minimum value of support_width leads to worst case coverage and maximizes observability along path

14. Maximal Support Path Use the Delaunay Triangulation Property: Triangles formed have minimum edge lengths Ps has to lie on these edges

15. Algorithm • The algorithm used is exactly the same as for Maximal breach path, with the following changes: • The Voronoi diagram is replaced by the Delaunay triangulation as the underlying geometric structure; • The edges in graph G are assigned weights equal to the length of the corresponding line segments in the Delaunay triangulation; • The search parameter breach_weight is replaced by the new parameter support_weight. • Support_weight is now an upper bound on all the edge weights that lie on the maximal support path, and there must exist at least one edge with weight equal to support weight

16. Complexity • Generation of Voronoi Diagram: O(n log n) • Graph conversion and weight assignment: O(n) • BFS search: O(m), • where m is the number of edges • O(n) for sparse networks, and O(n2) in the worst case • Binary Search: O(log range) Total: O(n log n) (for sparse networks), or O(n2log n) in the worst case

17. Results The paths for a simulation of 30 sensors randomly deployed

18. Results Voronoi Diagram and Delaunay Triangulation of the 30 node network

19. Maximal Breach Path Example (50 nodes)

20. Maximal Breach Path Example (200 nodes)

21. Deployment Heuristics Adding sensor along breach and support weight edges to improve breach coverage and support coverage

22. Asymptotic Behavior • Average over 1000 random deployments of 100 nodes • Support = 1 – support_weight • Certain levels of coverage can be expected even if the sensor deployment is random, given that a sufficient number of sensors are deployed

23. Conclusions • Problem formulation to determine worst-case and best-case coverage as a QoS metric • Used computation geometry constructs and properties • Heuristics can help future deployments • Using breach_weight and support_weight edges