Randomized k -Coverage Algorithms for Dense Sensor Networks Mohamed Hefeeda, Majid Bagheri

1. Randomized k-Coverage Algorithms for Dense Sensor Networks Mohamed Hefeeda, Majid Bagheri School of Computing Science, Simon Fraser University, Canada INFOCOM 2007

2. Outlines • Introduction • Problem definition • Two approximation algorithms • Randomized k-coverage algorithm (RKC) • Distributed RKC (DRKC) • Performance evaluation • Conclusions

3. Motivations • Wireless sensor networks have been proposed for many real-life monitoring applications • Habitat monitoring, early forest fire detection, … • k-coverage is a measure of quality of monitoring • k-coverage≡every point is monitored by k+ sensors • Improves reliability and accuracy • k-coverage is essential for some applications • e.g., intrusion detection, data gathering, object tracking

4. Our k-Coverage Problem • Definition: • Given n already deployed sensors in a target area, and a desired coverage degree k ≥ 1, select aminimal subset of sensors to cover all sensorlocationssuch that every location is within thesensing range ofat leas k different sensors • Assumptions • Sensing range of each sensor is a disk with radius r • Sensor deployment can follow any distribution • Nodes do not know their locations • Point coverage approximates area coverage (dense sensor network)

5. Our k-Coverage Problem (cont’d) • k-coverage problem is NP-hard [Yang 06] • Proof: reduction to the minimum dominating set problem [4] S. Yang, F. Dai, M. Cardei, and J. Wu, “On connected multiple point coverage in wireless sensor networks,” Journal of Wireless Information Networks, May 2006.

6. Our Contributions: k-Coverage Algorithms • We propose two approximation algorithms • Randomized k-coverage algorithm (RKC) • Simple and efficient  log-factor approximation • Distributed RKC (DRKC) • Basic idea: • Modelk-coverage asa minimum hitting set problem • Finding the minimum hitting set is NP-hard, we try tofind a near-optimal hitting set. • Design an approximation algorithm for hitting set • Prove its correctness, verify using simulations

7. Set Systems and Hitting Set Set system (X,R) is composed of set X, and collection R of subsets of X H is a hitting set if it has a nonempty intersection with every element of R:

8. Minimal hitting set • First used by Raymond Reiter1987 in “A theory of diagnosis from first principles, Artificial Intelligence”. • Can be used to solve minimum set cover problem, diagnosis problem, and teachers and courses problem.

9. Minimal hitting set • Given a collection C={Si │i ∈N} of sets of elements fromsome universe U, a hitting set is a set S Usuch thatS Si   for all i. • “Minimal” means no element in S can be deleted. • Ex, C={ {M1, M2, A1}, {M1, A1, A2, M3}}. The minimal hitting sets are {M1}, {A1}, {M2, A2}, {M2, M3}.

10. Set System for k-Coverage • X:set of all sensor locations • For each point p in X, draw circle of radius r (sensing range) centred at p • All points in X which fall inside that circle constitute one set s in R • The hitting set must have at least one point in each circle • Thus all points are covered by the hitting set

11. Example: 1-Coverage r

12. Example: k-Coverage (k = 3) k-flower : a set of k sensors that all intersect at that center point, and should be activated for k-coverage. Elements of the hitting set are centers of k-flowers.

13. Centralized Algorithm (RKC) • Build an approximate hitting set • Assign weights to all points, initially 1 • Select a random set of points, referred to as ε-net • Selection biased on weights • If current ε-net covers all points, terminate • Elsedouble weight of one under-covered point, goto 2 if number of iterations is below a threshold(~log |X|) • Double size of ε-net, goto 1

14. ε-nets • N, a set and NX,is anε-net for set system (X,R) if it has nonempty intersection with every element T of R such as |T| ≥ε |X|, T N  • Let 0 < ε≦ 1 be a constant • X: the set of sensor locations • R: the collection of subsets of X created by intersecting disks ofradius r with points of X • Thus, ε-net is required to hit only large elements of R • hitting set must hit every element of R • Idea: • Find ε-nets of increasing sizes (decreasing ε) till one of them hits all points

15. ε-net Construction • ε-nets can be computed efficiently for set systems with finite VC-dimension [Bronnimann 95] • We prove that our set system has VC-dimension = 3 • Randomly selecting max {4/ε log 2/δ, 8d/ε log 8d/ε}points of X constitutes an ε-net with probability at lease1-δ, where 0<δ<1, d is the VC-dimension [7] H. Bronnimann and M. Goodrich, “Almost optimal set covers in finite VC-dimension,” Discrete and Computational Geometry, vol. 14, no. 4, April 1995.

16. Details of RKC The number of k-flowers in the ε-net.

17. Details of net-finder

18. Correctness and Complexity of RKC • Theorem 1: RKC … • ensures that very point is k-covered, • terminates in O(n2 log2 n) steps, and • returns a solution of size at most O(P log P), where P is the minimum number of sensors required for k-coverage

19. Distributed Algorithm: DRKC • RKC maintains only 2 global variables: • size of ε-net • aggregate weight of all nodes • Idea of DRKC: Emulate RKC by keeping local estimates of these 2 global variables • Nodes construct ε-net in distributed manner • Nodes double their weights with a probability • Each node verifies its own coverage

20. DRKC Message Complexity • Theorem 2: The average number of messages sent by a node in DRKC is O(1),and the maximum number isO(log n)in the worst case.

21. Performance Evaluation • Simulation • Large area of size 1000m 1000m with 30,000 sensor nodes. • Verify correctness (k-coverage is achieved) • Show efficiency (output size compared optimal ) • Compare with other algorithms • LPA (centralized linear programming) and PKA (distributed based on pruning) in [Yang 06] • CKC (centralized greedy) and DPA (distributed based on pruning) in [Zhou 04] [8] Z. Zhou, S. Das, and H. Gupta, “Connected k-coverage problem in sensor networks,” in Proc. of International Conference on Computer Communications and Networks (ICCCN’04), Chicago, IL, October 2004.

22. Correctness of RKC • RKC achieves the requested coverage degree Requested k = 1 Requested k = 8

23. Efficiency of RKC • Compare against necessary and sufficientconditions for k-coverage in [Kumar 04],k=4. The centralized algorithm produces near-optimal number of active sensors.

24. Correctness of DRKC • DRKC achieves the requested coverage degree Requested k = 1 Requested k = 8

25. Efficiency of DRKC • DRKC performs closely to RKC, especially in dense networks

26. Comparison: DRKC, PKA, DPA • DRKC consumes less energy and prolongs network lifetime

27. Conclusions • Presented a centralized k-coverage algorithm • Simple, and efficient (log-factor approximation) • Proved its correctness and complexity • Presented a fully-distributed version • low message complexity, prolongs network lifetime • Simulations verify that our algorithms are • Correct and efficient • Outperform other k-coverage algorithms

28. References [2] S. Kumar, T. H. Lai, and J. Balogh, “On k-coverage in a mostly sleeping sensor network,” in Proc. of ACM International Conference on Mobile Computing and Networking (MOBICOM’04), Philadelphia, PA, September 2004, pp. 144–158. [4] S. Yang, F. Dai, M. Cardei, and J. Wu, “On connected multiple point coverage in wireless sensor networks,” Journal of Wireless Information Networks, May 2006. [7] H. Bronnimann and M. Goodrich, “Almost optimal set covers in finite VC-dimension,” Discrete and Computational Geometry, vol. 14, no. 4, April 1995. [8] Z. Zhou, S. Das, and H. Gupta, “Connected k-coverage problem in sensor networks,” in Proc. of International Conference on Computer Communications and Networks (ICCCN’04), Chicago, IL, October 2004. [10] M. Hefeeda and M. Bagheri, “Efficient k-coverage algorithms for wireless sensor networks,” School of Computing Science, Simon Fraser University, Tech. Rep. TR 2006-22, September 2006.