1. Sensor network coverage Ref: 1.Seapahn Megerian et al., “Worst and best-case coverage in sensor networks” 2.Chi-Fu Huang and Yu-Chee Tseng, “The coverage problem in a wireless sensor network”

2. Since sensors may be spread in an arbitrary manner, one of the fundamental issues in a wireless sensor network is the coverage problem. In general, this reflects how well an area is monitored or tracked by sensors

3. Topics to be covered Distance to closest sensor –Worst case coverage: Maximal Breach Path –Best case coverage: Maximal Support Path K-coverage problem –Given a sensor field A and a set sensors S= {s1, s2,…,sn}, where si is located at coordinate (xi, yi), determine whether A is k-covered or not.

4. Sensor Network Coverage The Problem: –Given: Ad hoc sensor field with some number of nodes with known location Start and end positions of an agent –Want: How well can the field be observed? Example usage –Commander Weakest path: what path is the enemy likely to take? –Network manager Weakest path: where to deploy additional nodes for optimum coverage? –Soldier in the battlefield Strongest path: what path to take for maximum coverage by my command? Weakest path: how to walk through enemy sensor net or through minefield? GATEWAY MAIN SERVER CONTROL CENTER

5. Worst-case: which path can best avoid the detection of the sensors. Best-case: which path is best monitored by the sensors. The coverage problem is about how well the sensor field is monitored.

6. Worst and best coverage Sensor coverage, in general, answers the questions about the quality of service (surveillance) that can be provided by a particular sensor network. An agent is the phenomenon being detected by the sensors, for example, an enemy tank moving in the field. The worst-case coverage problem---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. (“worst-case”-the agent tries to optimally avoid the sensors)

7. The best-case coverage problem---we want to find the farthest distance to sensors that an agent traveling on any path in the sensor field must have from sensors. The agent tries to stay as close to sensors as possible. Clearly, at some points, the agent must move away from sensors in order to be able to traverse the field.

8. PRELIMINARIES(1/3) Location Discovery Techniques and Algorithms. –Global Positioning System (GPS), for position –received signal strength indicator (RSSI) of RF communication, for measuring distance That is, assume that the position of each sensor is known, and the distance between any two sensors can be calculated.

9. PRELIMINARIES(2/3) Computational Geometry: Voronoi Diagram and Delaunay Triangulation –Voronoi Diagram – 任兩點連線化平分線, – 平分線交成凸多邊形 –Bounded Voronoi Diagram –Used for solving worst-case coverage

10. PRELIMINARIES(3/3) Computational Geometry: Voronoi Diagram and Delaunay Triangulation –Delaunay Triangulation Delaunay Triangulation maximizes the smallest angle in each triangle. Neighborhood information can be extracted from the Delaunay Triangulation since sites (points) that are close together are connected. The Delaunay triangulation can be used to find the two closest sites by considering the shortest edge in the triangulation.

11. The Voronoi diagram and the Delaunay triangulation are used to limit the search space for the agent paths in the worst and best coverage problems.

12. Definition Given: A field A instrumented with 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. 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. Identify a maximal breach path P B, in A, connecting the areas I and F.

13. Theorem At least one Maximal Breach Path must lie on the line segments of the bounded Voronoi diagram formed y the locations of the sensors in S. Proof: the Voronoi diagram maximize distance from the closest sites, A Maximal breach path P B, must lie on the line segments of the Voronoi diagram. If any point p on the path P B deviates from Voronoi line segments, it must be closer to at least one sensor in S.

14. Worst-Case Coverage and Maximal Breach Path

15. Use binary search to find the maximal breach Use DFS to find a path from I to F. Weight of edge is reflect the distance from the edge to the sites (sensors) Max_weight = max edge weight in G Min_weight = min edge weight in G Search start from the middle breach_weight = min_weight + range range is the interval which is decreasing

16. Worst-Case Coverage and Maximal Breach Path

17. Prune all edge with edge_weight > breach_weight Search for any path An intermediate step where breach_weight = 40 Final solution Optimal breach_weight=57

18. If new sensors can be deployed or existing sensors moved such that this breach_weight is decreased, then the worst-case coverage is improved Breach_weight can be used as a measure of the coverage level provided by a sensor field.

19. Best-Case Coverage and Maximal Support Path Given. A field A instrumented with 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. 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. Identify Ps, the Maxmal Support Path in S, starting in I and ending in F

20. Theorem 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)

21. Best-Case Coverage and Maximal Support Path 1. The Voronoi diagram is replaced by the Delaunay triangulation as the underlying geometric structure. 2. 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. 3. 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. Complexity O(n log n)

22. EXPERIMENTAL RESULTS

25. Closest Sensor Model: Maximal Breach Path Problem: find the path between I & F with the property that for any point p on the path the distance to the closest sensor is maximized Observation: maximal breach path lies on the Voronoi Diagram Lines –by construction each line segment maximizes the distance from the nearest point 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 Search for P B : Check for existence of I  F path using BFS  Search for path with maximal, minimum edge weights Ref: based on slides by Seapahn Megerian

26. Example Result Example: Max Breach Path in a 50-node network

27. K-coverage approach Chi-Fu Huang and Yu-Chee Tseng, Department of Computer Science and Information Engineering National Chiao-Tung University

28. Problem Definition Given a set of sensors deployed in a target area, we want to determine if the area is sufficiently k-covered, in the sense that every point in the target area is covered by at least k sensors, where k is a predefined constant. The sensing range of each sensor can be a unit disk or a non-unit disk.

29. Problem Statement Given a set of sensors, S = {s1, s2,..., sn}, in a 2 dimensional area A. Each sensor si, i = 1..n, is located at coordinate (xi, yi) inside A and has a sensing range of ri, Determine if the area is sufficiently k-covered, in the sense that every point in the target area is covered by at least k sensors, where k is a predefined constant To simplify, assume that the sensing range of each sensor is a unit disk

30. DEFINITION A location in A is said to be covered by si if it is within si’s sensing range. A location in A is said to be j-covered if it is within at least j sensors’ sensing ranges. k-NC problem: Given a natural number k, the k-Non-unit-disk Coverage (k-NC) Problem is a decision problem whose goal is to determine whether all points in A are k-covered or not. k-UC problem Given a natural number k, the k-Unit-disk Coverage (k-UC) Problem is a decision problem whose goal is to determine whether all points in A are k-covered or not, subject to the constraint that r1 = r2 = · · · = rn. k-UC and k-NC

31. A naive solution Find out all sub-regions divided by the sensing regions of all n sensors, and then check if each sub-region is k-covered or not, as shown in Fig. 1.

32. Problem Statement The k-UC Problem DEFINITION 4 Consider any two sensors si and sj. A point on the perimeter of si is perimeter-covered by sj if this point is within the sensing range of sj. DEFINITION 5 Consider any sensor si. We say that si is k perimeter- covered if all points on the perimeter of si are perime- meter covered by at least k sensors other than si itself. Similarly, a segment of si’s perimeter is k-perimeter- covered if all points on the segment are perimeter-covered by at least k sensors other than si itself.

33. Consider 2 sensors si and sj located in positions (xi, yi) and (xj, yj), respectively. d(si, sj) = sqrt ( |xi − xj | 2 + |yi − yj | 2 ) Without loss of generality, let sj be resident on the west of si (i.e., yi = yj and xi > xj ). The angle α = arccos( d(si,sj ) / 2r ). So the arch of si falling in the angle [π − α, π + α] is perimeter-covered by sj. Problem Statement ( k-UC)

34. The algorithm to determine the perimeter coverage 1. For each sensor sj such that d(si, sj) ≤ 2r, determine the angle of sis arch, denoted by [αj,L, αj,R], that is perimeter-covered by sj. 2. For all neighboring sensors sj of si such that d(si,sj)< 2r, place the points αj,L and αj,R on the line segment [0, 2π] and sort all these points in an ascending order into a list L. ( 根據 αj,L ） Also, properly mark each point as a left or right boundary of a coverage range. 3. (Sketched) Traverse the line segment [0, 2π] by visiting each element in the sorted list L from the left to right and determine the perimeter-coverage of si. Problem Statement ( k-UC)

35. Determine the perimeter coverage of Si s1 s2 s7 s8 s6 s5 s4

36. LEMMA 1 Suppose that no two sensors are located in the same location. Consider any segment of a sensor si that divides 2 sub-regions in the network area A. If this segment is k-perimeter covered, the sub-region that is outside si’s sensing range is k covered and the sub-region that is inside si’s sensing range is (k+1)-covered. THEOREM 1 Suppose that no two sensors are located in the same location. The whole network area A is k-covered iff each sensor in the network is k-perimeter-covered Problem Statement ( k-UC)

37. 2 special cases When 2 sensors si and sj fall in exactly the same location, Lemma 1 will not work because for any segment of si and sj that divides 2 sub-regions in the network area, a point right inside si’s and sj ’s sensing ranges and a point right outside their sensing ranges will differ in their coverage levels by two, making Lemma 1 incorrect. Some sensors’ sensing ranges may exceed the area A In this case, we can simply assign the segments falling outside A as as∞-perimeter-covered, Problem Statement ( k-UC)

38. 2 special cases

39. Discovering Insufficiently Covered Regions For a sensor network, one basic question is whether the network area is fully covered. Our modeling of the k-UC and k-NC problems can solve the sensor coverage problem in a more general sense by determining if the network area is k-covered or not. Using a larger k, the location of the event can be reduced to a certain intersection of at least k sensors. Thus, the location of the event can be more precisely defined. This would support more fine-grained location-based services. Applications of the coverage problem

40. Discovering Insufficiently Covered Regions To determine which areas are insufficiently covered, we assume that there is a central controller in the sensor network. The central controller can broadcast the desired value of k to all sensors. Each sensor can then communicate with its neighboring sensors and then determine which segments of its perimeter are less than k-perimeter-covered. The results (i.e., segments) are then sent back to the central controller. By putting all segments together, the central controller can precisely determine which areas are less than kcovered. Applications of the coverage problem

41. Power Saving in Sensor Networks If there are more sensors than necessary, we may turn off some redundant nodes to save energy. These sensors may be turned on later on when other sensors run out of energy. Sensor nodes who can be turned off, called candidates, need to be identified. A sensor si is a candidate if all its neighbors are still k-perimeter-covered after si is removed. To do so, si can communicate with each of its neighbors and ask them to reevaluate their perimeter coverage by skipping si. Applications of the coverage problem

42. Hot Spots It is possible that some areas in the network are more important than other areas and need to be covered by more sensors. Those important regions are called hot spots. Given a hot spot, only those sensors whose perimeters are within or have crossings with the hot spot need to be checked. So the central controller can issue a request by identifying the hot spot. So a hot spot is k-covered if and only if all perimeter segments within this hot spot are k-perimeter-covered. Applications of the coverage problem

43. An example to verify if a hot spot is 2-covered