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Randomized k-Coverage Algorithms for Dense Sensor Networks

Mohamed Hefeeda, Majid BagheriSchool of Computing Science, Simon Fraser University, Canada

INFOCOM 2007

OutlinesOutlines

• Introduction• Problem definition• Two approximation algorithms

– Randomized k-coverage algorithm (RKC)– Distributed RKC (DRKC)

• Performance evaluation• Conclusions

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MotivationsMotivations

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

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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)

Our Our kk-Coverage Problem-Coverage Problem

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k-coverage problem is NP-hard [Yang 06]- Proof: reduction to the minimum dominating set

problem

Our Our kk-Coverage Problem (cont’d)-Coverage Problem (cont’d)

[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.

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Our Contributions:Our Contributions: k k-Coverage Algorithms-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

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:

sHRs

XH

,

Minimal hitting setMinimal 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.

Minimal hitting setMinimal 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}.

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Set System for Set System for kk-Coverage-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

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Example: 1-CoverageExample: 1-Coverage

r

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Example: Example: kk-Coverage (-Coverage (k = 3k = 3))

Elements of the hitting set are centers of k-flowers.

k-flower : a set of k sensors that all intersect at that center point, and should be activated for k-coverage.

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Centralized Algorithm (RKC)Centralized Algorithm (RKC)

Build an approximate hitting set1. Assign weights to all points, initially 12. Select a random set of points, referred to as ε-net

• Selection biased on weights

3. If current ε-net covers all points, terminate 4. Else double weight of one under-covered point, goto 2

if number of iterations is below a threshold(~log |X|)

5. Double size of ε-net, goto 1

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εε-nets-nets

N, a set and NX, 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

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ε-net Constructionε-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.

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Details of RKCDetails of RKC

The number of k-flowers in the ε-net.

Details ofDetails of net-finder net-finder

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Correctness and Complexity of RKCCorrectness 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

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Distributed Algorithm: DRKCDistributed 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

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DRKC Message ComplexityDRKC 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.

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Performance EvaluationPerformance 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.

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Correctness of RKCCorrectness of RKC

RKC achieves the requested coverage degree

Requested k = 1 Requested k = 8

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Efficiency of RKCEfficiency of RKC

Compare against necessary and sufficient conditions for k-coverage in [Kumar 04], k=4.

The centralized algorithmproduces near-optimal number of active sensors.

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Correctness of DRKCCorrectness of DRKC

DRKC achieves the requested coverage degree

Requested k = 1 Requested k = 8

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Efficiency of DRKCEfficiency of DRKC

DRKC performs closely to RKC, especially in dense networks

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Comparison: DRKC, PKA, DPAComparison: DRKC, PKA, DPA

DRKC consumes less energy and prolongs network lifetime

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ConclusionsConclusions

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

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.