1. Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois Institute of Technology, Chicago, IL IEEE Transactions on Parallel and Distributed Systems 2009

2. Outline Introduction Problem Formulation The k-th Nearest Point Voronoi Diagram Best case coverage: Minimum k-Support Path Distributed Algorithm for Compute the Minimum k-Support Path Worst case coverage: Maximum k-Breach Path Simulation Conclusion

3. Introduction Coverage is a measure of quality of service (QoS) of a sensor network to some extend In many wireless sensor network applications, we are often required to find a path from a source point to a destination point such that the found path is the optimum one under a certain quality measurement For example, when some emergency happens, the sensor network should provide safe path(s) which can guide the users leaving from the working place to some safe exit(s) In this scenario, the path should be close to some sensor(s) such that the situation along the path can be monitored well

4. Goal (1) Finding a path connecting a source point S and a destination point D inside the given area, which maximizes the smallest observability of all points along the path. This is called best coverage problem (2) Finding a path connecting a source point S and a destination point D inside the given area, which minimizes the largest observability of all points on the path. This is called worst coverage problem

5. Problem Formulation we assume all sensor nodes have enough sensing range such that it can sense any point in wireless sensor network However, the sensing ability(observability) of a sensor node for a point depends on the Euclidean distance between them We use Euclidean distance as the measurement of QoS.

6. Problem Formulation Definition 1: Given a point p in the field Ω and the set of sensors U, the k-th distance of p, with respect to U, denoted as, is defined as the Euclidian distance from p to its k-th nearest sensor node in U. p

7. Problem Formulation Definition 2: Given a path P connecting a source point S and a destination point D, the k-support of P, denoted by S k (P), is defined as the maximum k-th distance of all points on P. In other words, where p is a point on path P S D P

8. Problem Formulation Problem 1: Optimal k-support Path (Best Case Coverage) Problem: Given a source point S and destination point D, find a path P in the field to connect S and D such that S k (P) is minimized Problem 2: Optimal k-breach Path (Worst Case Coverage) Problem: Given a source point S and destination point D, find a path P in the field to connect S and D such that B k (P) is maximized. S D P S D P

9. The k-th Nearest Point Voronoi Diagram we call each independent polygon k-th nearest-point Voronoi cell of node u i and use C k (u i ) to denote it we simply call u i is the owner of C k (u i ) C 2 (u 3 ) and its owner is u 3 KNP Voronoi edge KNP Voronoi vertex

10. Compute The kNP Voronoi Diagram (1) Compute the order-k Voronoi diagram of given sensor nodes set U using the algorithm given in [7] (2) Compute the farthest Voronoi diagram of its corresponding k sensor nodes [14] ▫It is a partition of the plane into polygons such that points in a polygon have the same farthest sensor node in U.

11. Compute The kNP Voronoi Diagram (3) For each sensor node u i, we merge the partial cells computed above into one KNP Voronoi cell if they share one edge.

12. Best Case Coverage Problem 1: Optimal k-support Path (Best Case Coverage) Problem: Given a source point S and destination point D, find a path P in the field to connect S and D such that S k (P) is minimized S D P S D P

13. Best case coverage: Optimal k-Support Path -Preliminaries Theorem 2: Based on any given path P 1 connecting source node S and destination node D, we can always construct another (maybe same) path P 2 composed by only a finite number of line segments such that kNP Voronoi Diagram

14. (1) S k (p ab ) line segment ab is entirely contained in a disk centered at u i with radius (2) S k (ab) (3) S k (ab) S k (p ab )

15. Best case coverage: Optimal k-Support Path -Preliminaries Theorem 3: Based on any given path P 1 connecting source node S and destination node D, we can construct another path P 3 consisting of only line segments whose end points are perfect support location of the KNP Voronoi edges such that Definition 4 (Perfect Support Location): The perfect support location of a KNP Voronoi edge is defined as the point (on this edge) which has the minimum Euclidean distance to its owner (k-th nearest sensor node)

16. perfect support location We use to denote this part of path P1

17. Best case coverage: Optimal k-Support Path -Preliminaries Theorem 4: There is one optimal k-support path consisting of only line segments whose end points are located at the perfect support locations of the KNP Voronoi edges. This theorem is straightforward from Theorem 2 and 3.

18. Compute the Minimum k-Support Path After getting the KNP Voronoi diagram G with respect to U by Algorithm 1, we present our algorithm to compute the optimal k-support path based on Theorem 4 As shown in Theorem 4, there must exist one minimum k-support path consisting of only line segments and all of these line segments’ end points are located on the perfect support location of some KNP Voronoi edges. Clearly we only need to consider all the paths that using only line segments connecting the perfect support locations of the KNP Voronoi edges

19. Compute the Minimum k-Support Path First, we construct a new graph G’ based on KNP Voronoi diagram G as follows: S’ D’ w(v’) is equal to the k-th distance of the perfect support location of edge perfect support location

20. Compute the Minimum k-Support Path First, we construct a new graph G’ based on KNP Voronoi diagram G as follows: S’ D’ 1 2 3 4 5 6

21. [15] Shaojie Tang, Xufei Mao, and Xiang-Yang Li. Optimal k-support coverage paths in wireless sensor networks. In IQ2S Workshop of PerCom 2009, 2009. S’ D’ 1 2 3 4 5 6 adding an edge between any two nodes u’ and v’ in G’ if and only if their corresponding KNP-Voronoi edges belong to the same KNP-Voronoi cell in G Compare

22. Compute the Minimum k-Support Path Next, we use Algorithm 2, which originates from Dijkstra’s shortest path algorithm, to find a minimum weight path P’ in G’ to connect S’ and D’

23. 5/5 3/5 6/6 minimum weight path

24. Distributed Algorithm for Compute the Optimal k-Support Path we present our distributed algorithm to compute the optimal k- support path after getting the KNP Voronoi diagram with respect to U by Algorithm 1 First, we construct a new graph G’ based on KNP Voronoi diagram G in a distributed manner we let each sensor node record its owned KNP-Voronoi cells Next, we present our distributed algorithm to compute the optimal k- support path based on Theorem 4

27. Worst Case Coverage Problem 2: Optimal k-breach Path (Worst Case Coverage) Problem: Given a source point S and destination point D, find a path P in the field to connect S and D such that B k (P) is maximized Definition 3: Given a path P which is connecting source point S and destination point D, the k-breach of P, denoted by B k (P), is defined as the minimum k-th distance of all points on P, S D P S D P

28. Worst case coverage: Optimal k-Breach Path -Preliminaries Theorem 10: Based on any given path P 1 connecting source node S and destination node D, we can always construct another (maybe same) path P 4 which only use KNP Voronoi edges such that

29. B k (p ab ) has upper bound B k (p’) B k (p ab )

30. compute the Maximum k-breach path Theorem 11: There is one maximum k-breach path which lies along the KNP Voronoi edges (except the first edge or last edge when S or D is not on some Voronoi edge) (1) Use Algorithm 1 to generate KNP Voronoi Diagram G of U (2) Each KNP Voronoi vertex v G is assigned a weight w(v) (3) We add an edge between S (resp. D) and a (4) We let the weight of (u, v) be equal to the minimum k-th distance among all points on (u, v) uiui a s

31. maximum weight path

32. Simulation In our simulation, a set of n wireless sensors is randomly and uniformly deployed in the target square region with size 500 * 500 meter 2

33. Simulation

34. This result can be used to estimate the coverage quality if the number of sensors and required coverage degree are given.

35. Conclusion In this paper, we proposed polynomial time algorithms (both centralized and distributed) for two k-coverage problems in wireless sensor networks An interesting future work, we would like to design algorithms that can address the coverage problem when the sensing abilities of sensors are heterogeneous

36. Thank You!