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 a minimal subset of sensors to cover all sensor locations such that every location is within the sensing range of at 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:
Model k-coverage as a minimum hitting set problem
Finding the minimum hitting set is NP-hard, we try to find 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 Reiter 1987 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 from some universe U, a hitting set is a set S  U such that S  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
Else double 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 an ε-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 of radius 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 lease 1-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 is O(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 sufficient conditions 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 , September 2006.