1. Exposure In Wireless Ad-Hoc Sensor Networks Seapahn Meguerdichian Computer Science Department University of California, Los Angeles Farinaz Koushanfar Department of EE and CS University of California Berkeley Gang Qu Electrical and Computer Engineering Department University of Maryland Miodrag Potkonjak Computer Science Department University of California, Los Angeles Presented by John Sweeney. Slides courtesy of the author.

2. Sensor Coverage Given: Given: Field A Field A N sensors N sensors How well can the field be observed ? Closest Sensor (minimum distance) only Closest Sensor (minimum distance) only Worst Case Coverage: Maximal Breach Path Worst Case Coverage: Maximal Breach Path Best Case Coverage: Maximal Support Path Best Case Coverage: Maximal Support Path Multiple Sensors: speed and path considered Multiple Sensors: speed and path considered Minimal Exposure Path

3. Maximal Breach Path 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. Voronoi Diagram

4. 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?

5. 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.

6. Exposure - Semantics Likelihood of detection by sensors function of time interval and distance from sensors. Likelihood of detection by sensors function of time interval and distance from sensors. Minimal exposure paths indicate the worst case scenarios in a field: Minimal exposure paths indicate the worst case scenarios in a field: Can be used as a metric for coverage Can be used as a metric for coverage Sensor detection coverage Sensor detection coverage Wireless (RF) transmission coverage Wireless (RF) transmission coverage For RF transmission, exposure is a potential measure of quality of service along a specific path. For RF transmission, exposure is a potential measure of quality of service along a specific path.

7. Preliminaries: Sensing Model Sensing model S at an arbitrary point p for a sensor s : where d(s,p) is the Euclidean distance between the sensor s and the point p, and positive constants and K are technology- and environment-dependent parameters.

8. Preliminaries: Intensity Model(s) Effective sensing intensity at point p in field F : All Sensors Closest Sensor K Closest Sensors K=3 for Trilateration

9. Definition: Exposure The Exposure for an object O in the sensor field during the interval [t 1,t 2 ] along the path p(t) is:

10. Exposure – Coverage Problem Formulation Given: Field A Field A N sensors N sensors Initial and final points I and F Initial and final points I and FProblem: Find the Minimal Exposure Path P minE in A, starting in I and ending in F. P minE is the path in A, along which the exposure is the smallest among all paths from I to F.

11. Special Case – One Sensor Minimal exposure path for one sensor in a square field:

12. General Exposure Computations Analytically intractable. Analytically intractable. Need efficient and scalable methods to approximate exposure integrals and search for Minimal Exposure Paths. Need efficient and scalable methods to approximate exposure integrals and search for Minimal Exposure Paths. Use a grid-based approach and numerical methods to approximate Exposure integrals. Use a grid-based approach and numerical methods to approximate Exposure integrals. Use existing efficient graph search algorithms to find Minimal Exposure Paths. Use existing efficient graph search algorithms to find Minimal Exposure Paths.

13. Minimal Exposure Path Algorithm Use a grid to approximate path exposures. Use a grid to approximate path exposures. The exposure (weight) along each edge of the grid approximated using numerical techniques. The exposure (weight) along each edge of the grid approximated using numerical techniques. Use Dijkstra’s Single-Source Shortest Path Algorithm on the weighted graph (grid) to find the Minimal Exposure Path. Use Dijkstra’s Single-Source Shortest Path Algorithm on the weighted graph (grid) to find the Minimal Exposure Path. Can also use Floyd-Warshall All-Pairs Shortest Paths Algorithm to find P minE between arbitrary start and end points. Can also use Floyd-Warshall All-Pairs Shortest Paths Algorithm to find P minE between arbitrary start and end points.

14. Generalized Grid Generalized Grid – 1 st order, 2 nd order, 3 rd order … More movement freedom  more accurate results Approximation quality improves by increasing grid divisions with higher costs of storage and run-time.

15. Minimal Exposure Path Algorithm Complexity Single Source Shortest Path (Dijkstra) Single Source Shortest Path (Dijkstra) Each point is visited once in the worst case. Each point is visited once in the worst case. For an nxn grid with m divisions per edge: n 2 (2m-1)+2nm+1 total grid points. For an nxn grid with m divisions per edge: n 2 (2m-1)+2nm+1 total grid points. Worst case search: O(n 2 m) Worst case search: O(n 2 m) Dominated by grid construction. Dominated by grid construction. 1GHz workstation with 256MB RAM requires less than 1 minute for n=32, m=8 grids. 1GHz workstation with 256MB RAM requires less than 1 minute for n=32, m=8 grids. All-Pairs Shortest Paths (Floyd-Warshall) All-Pairs Shortest Paths (Floyd-Warshall) Has a average case complexity of O(p 3 ). Has a average case complexity of O(p 3 ). Dominated by the search: O((n 2 m) 3 ) Dominated by the search: O((n 2 m) 3 ) Requires large data structures to store paths. Requires large data structures to store paths.

16. P minE – Uniform Random Deployment Minimal exposure path for 50 randomly deployed sensors using the All-Sensor intensity model (I A ). 8x8 m=1 Exposure:0.7079 Length:1633.9 16x16 m=2 Exposure:0.6976 Length:1607.7 32x32 m=8 Exposure:0.6945 Length:1581.0

17. Exposure – Statistical Behavior Diminishing relative standard deviation in exposure for 1/d 2 and 1/d 4 sensor models.

18. P minE – Deterministic Deployment Minimal exposure path under the All-Sensor intensity model (I A ) and deterministic sensor deployment schemes. CrossSquareTriangleHexagon Exposure Level (compared to Square) 1.5x1.5x30x~120 1.5x3x6x~20HexagonTriangleCrossSensors

19. Exposure – Research Directions Localized implementations Localized implementations Performance and cost studies subject to Performance and cost studies subject to Wireless Protocols (MAC, routing, etc) Wireless Protocols (MAC, routing, etc) Errors in measurements Errors in measurements Locationing Locationing Sensing Sensing Numerical errors Numerical errors Computation based on incomplete information Computation based on incomplete information Not every node will know the exact position and information about all other nodes Not every node will know the exact position and information about all other nodes

20. Summary Exposure: Exposure: Definition Definition Efficient Algorithm Efficient Algorithm Centralized Implementation Centralized Implementation Algorithm: Algorithm: Generalized grid approximation Generalized grid approximation Application of graph search algorithms Application of graph search algorithms Ad-hoc wireless sensor networks: Ad-hoc wireless sensor networks: Coverage Coverage Quality of Service Quality of Service Research: Research: Numerous interesting open problems Numerous interesting open problems