On the Coverage Problem for Myopic Sensors

1. On the Coverage Problem for Myopic Sensors Mohamed Aly, Kirk Pruhs, Taieb Znati & Brady Hunsaker University of Pittsburgh IEEE WirelessCom05

2. Sensor Networks • Challenges: • Limited power • Unattended operation • Monitoring Applications • Habitat monitoring • Need for acoustic and video sensors in such applications • Ensure the coverage of all parts of the terrain IEEE WirelessCom05

3. The Coverage Problem T1 T4 T3 T5 S2 S3 T2 S4 T7 S5 S1 S7 S8 T8 T6 S6 T11 T9 S12 T13 T12 T10 S9 S10 S11 S14 S13 IEEE WirelessCom05

4. The Coverage Problem for Myopic Sensors T1 T4 T3 T5 S2 S3 T2 S4 T7 S5 S1 S7 S8 T8 T6 S6 T11 T9 S12 T13 T12 T10 S9 S10 S11 S14 S13 IEEE WirelessCom05

5. Related Work • Mostly based on the circular coverage assumption • [Meguerdichian et al]: • Define whether each point is k-covered • Use of Voronoi diagrams to model sensor coverage areas • [Potkonjack et al]: • Partitioning sensors into mutually exclusive sets • Maximizing the lifetime of coverage by partitioning the coverage tasks among sensors covering similar areas IEEE WirelessCom05

6. Related Work • [Tian et al]: • Circular sensor coverage • Distributed heuristic for computing the coverage schedule • Polynomial in the number of sensors • [Huang et al]: • Organizing the network sensors based on the minimum number of sensors required to cover the monitored area • Checking whether every point in the area is monitored by at least k sensors instead of only one sensor • All the above schemes are based on the intersection of coverage areas IEEE WirelessCom05

7. Myopic Sensors • Unidirectional • At least in some setting, they can only be focused on one target • Video sensors • Acoustic sensors • Coverage problem definition still valid??? IEEE WirelessCom05

8. Problem Definition • S={S1…Ss} of myopic sensors • T={T1,…Tt} of targets • || S || >> || T || • Monitoring Cost Ci,j: energy/unit time • Maybe infinite • Each sensor can monitor at most one target at a time • All targets must be monitored at all times • The problem reduces to maximum cardinality matching in a complete bipartite graph with bi-partitions S and T • Monitoring Length Wi,j: • Percentage of the lifetime where sensor i is covering target j IEEE WirelessCom05

9. Problem Definition Sensors Targets W1,1 S1 T1 W1,t W1,1 Ws,t Ss Tt IEEE WirelessCom05

10. Maximize P (Network lifetime) Subject to t å £ = P · w · c e i 1 , . . . , s i,j i,j i = j 1 : Total sensor coverage does not exceed sensor' s energy t å £ = w 1 i 1 , . . . , s i,j = j 1 : Each sensor covers at most one target at a time s å ³ = w 1 j 1 , . . . , t i,j = i 1 : Each targe t is covered by at least one sensor at a time ³ w 0 i,j Mathematical Formulation IEEE WirelessCom05

11. Minimize Q (Inverse of Network lifetime) Subject to t å £ = w · c e Q i 1 , . . . , s i,j i,j i = j 1 : Total sensor coverage does not exceed sensor' s energy t å £ = w 1 i 1 , . . . , s i,j = j 1 : Each sensor covers at most one target at a time s å ³ = w 1 j 1 , . . . , t i,j = i 1 : Each targe t is covered by at least one sensor at a time ³ w 0 i,j LP Formulation IEEE WirelessCom05

12. Algorithm Steps • LP solution using SOPLEX  Coverage graph • Wi,j = weight of an edge in coverage graph • Sum of the weights of the edges incident to any vertex is at most 1 • Sum of the weights of the edges for every vertex is at most 1 • Iteratively finding matchings in the graph • A matching represents a coverage schedule IEEE WirelessCom05

13. Illustrative Example W1,1 = 0.7 T1 W2,1 = 0.3 S3 W3,3 = 1 S1 S2 T2 W2,2 = 0.7 W1,2 = 0.3 T3 S5 W5,5 = 0.4 S4 W4,5 = 0.6 T5 Time line T = 0 T = 0.3 * P IEEE WirelessCom05

14. Illustrative Example W1,1 = 0.7 T1 W2,1 = 0 S3 W3,3 = 0.7 S1 S2 T2 W2,2 = 0.7 W1,2 = 0 T3 S5 W5,5 = 0.4 S4 W4,5 = 0.3 T5 Time line T = 0 T = 0.3 * P T = 0.6 * P IEEE WirelessCom05

15. Illustrative Example W1,1 = 0.4 T1 W2,1 = 0 S3 W3,3 = 0.4 S1 S2 T2 W2,2 = 0.4 W1,2 = 0 T3 S5 W5,5 = 0.4 Dead S4 W4,5 = 0 T5 Time line T = 0 T = 0.3 * P T = P T = 0.6 * P IEEE WirelessCom05

16. Illustrative Example W1,1 = 0 T1 W2,1 = 0 Dead S3 W3,3 = 0 S1 Dead S2 T2 W2,2 = 0 W1,2 = 0 T3 S5 W5,5 = 0 Dead S4 W4,5 = 0 T5 Time line T = 0 T = 0.3 * P T = P T = 0.6 * P IEEE WirelessCom05

17. Experimental Setup • First set: • Service area of side length of 10,000 units • 10 to 1000 targets • Number of sensors = 2 * number of targets • Starting energy level of 100,000 units • Ci.j ~ square of the Euclidean distance between sensor i and target j • Uniform distribution of sensors • Uniformly distributed targets across service area • Hot spot area • Square of 500 units containing 50% of the targets • Remaining targets uniformly distributed across service area • 100 runs for each experimental setting • Increase by 10 targets up to 100 and by 50 thereafter IEEE WirelessCom05

18. Experimental Results IEEE WirelessCom05

19. Experimental Results IEEE WirelessCom05

20. Experimental Results IEEE WirelessCom05

21. Experimental Setup • Second set: • Fixed number of sensors of 500 • 20 to 500 targets • Increment by 20 • Uniform distribution of both sensors and targets • 50 runs for each setting IEEE WirelessCom05

22. Experimental Results IEEE WirelessCom05

23. Experimental Results IEEE WirelessCom05

24. Conclusion • An algorithm to solve the coverage problem for a network of myopic sensors • Experimental validation of its practicality • Possibility of generalizations: • Variable sensor energy • General cost function • Sensors number independent from targets IEEE WirelessCom05

25. Questions IEEE WirelessCom05