1. Survey on Coverage Problems in Wireless Sensor Networks
M. T. Thai, F. Wang, H. Du, and X. Jia, "Coverage Problems in Wireless Sensor Networks: Designs and Analysis," International Journal of Sensor Networks,
special issue on Coverage Problems in Sensor Networks, vol. 3, no. 3, pp , 2008
Presented By Donghyun Kim July 10, Mobile Computing and Wireless Networking Research Group
at University of Texas at Dallas

2. Agenda Sensor Coverage Problem Area vs. Target Coverage Problem
Notations and Definitions
Maximum Lifetime Coverage Problem
Centralized Algorithms
Distributed and Localized Algorithms
Future Topics
k-Coverage Problem
Connected Coverage and Connected k-Coverage Problem
Minimum Coverage Breach
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

3. Sensing and Communication Range
Sensor Node
Sensing Range
Communication Range
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

4. Sensor Coverage Problem
Basically, it is a scheduling problem. T
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

5. Area vs. Target Coverage Problem
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

6. Target Coverage vs. Set Cover Problem
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

7. Notations We have sensors .
A sensor’s state can be transmit, receive, idle, or sleep.
The lifetime of is unit time.
is the sensing range of .
is the transmission range of .
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

8. Definitions Definition 1: Sensor Covering a Point
A sensor is said to cover a point iff the distance
Definition 2: Sensor Covering an Area
A sensor is said to cover an area iff for every point in , covers .
Definition 3: Direct Communication
Sensor and can communicate directly with each other iff and
Definition 4: Communication Graph
Given a sensor network consisting of n sensors, the communication graph of the network is an undirected graph , where is a set of sensors and is a set of edges consisting edges between any two sensors that can communicate directly with each other.
It is a Disk Graph with Bidirectional edges (DGB) – DG ignoring unidirectional edges.
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

9. Definitions Definition 5: Maximum Lifetime Coverage
Given a network consisting of sensors and an interest region, find a family of ordered pairs , where is a set of sensors that completely cover the interest region and is the time duration for to be active such that to maximize Each sensor appears in with a total time at most .
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

10. Definitions – cont’
For example, suppose four sensors and 2 targets Given , we can find the family of ordered pairs
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

11. Definitions – cont’ Definition 6: Connected Coverage Problem
Given a sensor network consisting of sensors and an interest region, find a family of ordered pairs satisfying the following two conditions.
Satisfy the conditions of the Maximum Lifetime Coverage problem
The communication graph induced by is connected
Definition 7: -Coverage Problem
Each point in is covered by at least distinct sensors in each .
Definition 8: Connected -Coverage Problem
Given a sensor network consisting of n sensors and an interest region, find a family of ordered pairs satisfying the following two conditions.
Satisfy the conditions of the -Coverage problem
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

12. Definitions – cont’
Given MAC Layer cannot handle the collision problem, the size of a subset has to be no more than some bandwidth constraint . Otherwise, coverage breach may occurs (i.e. some report can be discarded.)
Definition 9: Minimum Coverage Breach
Given a sensor network consisting of sensors and an interest region, find a family of ordered pairs such that the total coverage breach is minimum.
Theorem1
The Maximum Lifetime Coverage problem, the Connected Coverage problem, the -Coverage problem, the Connected -Coverage problem, and the Minimum Coverage Breach are NP-complete.
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

13. Maximum Lifetime Coverage Centralized Algorithms
Definition 10: Field
A field is a set of all points such that they are covered by the sane set of sensors.
Definition 11: Critical Element
A critical element is a field covered by a minimal number of sensors.
The authors in [17] modeled the area coverage problem as the maximum number of disjoint sets problem.
Divide a given interest region into a set of fields.
An heuristic algorithm, Most Constrained-Minimally Constraining Covering, is executed to compute the maximum number of disjoint sets in following ways:
Cover a high number of uncovered fields.
Cover sparsely covered fields.
Do not cover fields redundantly.
Time complexity is Assumes for all .
[17] S. Slijepcevic and M. Potkonjak, Power Efficient Organization of Wireless Sensor Networks, IEEE International Conference on Communications, June 2001.
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

14. Maximum Lifetime Coverage Centralized Algorithms – cont’
The authors in [18] modeled the coverage problem as the maximum number of disjoint dominating sets in an undirected graph , where is a set of sensors and is a set of edges such that iff and are within other’s sensing range.
A graph-coloring based two phase approximation algorithm is proposed.
Dominating set does not guarantee the complete coverage.
[18] has 1.5 to 2 times longer lifetime than [17].
The authors in [16] reduce the target coverage problem to a maximum flow problem and formulate it as a mixed integer programming.
The result of [16] is slightly better than that of [17].
[16] M. Cardei and D.-Z. Du, Improving Wireless Sensor Network Lifetime through Power Aware Organization, ACM Wireless Networks, Vol. 11, No. 3, 2005.
[18] M. Cardei, D. MacCallum, X. Cheng, M. Min, X. Jia, D. Li, and D.-Z. Du, Wireless Sensor Networks with Energy Efficient Organization, J. of Interconnection Networks, vol 3, No 3-4, pp 213~229,
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

15. Maximum Lifetime Coverage Centralized Algorithms – cont’
The authors in [13] and [15] found that organizing sensors into a maximum number of non-disjoint set covers may obtain a longer network lifetime.
Example
Disjoint set cover and Then total lifetime is 1.0 unit time.
Non-disjoint set cover – , and each set is activated for 0.5 unit time. Total lifetime is 1.5 unit time.
Find a set of set covers first and decide their activation time.
[13] P. Berman, G. Calinescu, C. Shah, and A. Zelikovsky, Power Efficient Monitoring Management in Sensor Networks, Proceedings of WCNC '04, March 2004.
[15] M. Cardei, M. T. Thai, Y. Li, and W. Wu, Energy-Efficient Target Coverage in Wireless Sensor Networks, Proceedings of the 24th conference of the IEEE Communications Society (INFOCOM),
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

16. Maximum Lifetime Coverage Centralized Algorithms – cont’
The authors in [13] used a packing Linear Programming technique.
Denote the number of set covers. Then, the coverage problem is formulated as follows.
and represent the index number of each sensor and each set cover, respectively.
For any e > 0, based on the approximation Garg-Konemann algorithm in [7], approximation algorithm is proposed.
[7] N. Garg and J. Konemann, Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems, Proc. of FOCS, 1997
[13] P. Berman, G. Calinescu, C. Shah, and A. Zelikovsky, Power Efficient Monitoring Management in Sensor Networks, Proceedings of WCNC '04, March 2004.
Coverage Condition?
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

17. Maximum Lifetime Coverage Centralized Algorithms – cont’
The authors in [15] modeled this problem as Maximal Set Covers (MSC) as follows.
is a set of all sensors that can cover a target
Assume for all sensors.
The above formulation is relaxed to an Linear Programming and solved using Ye’s algorithm in [8]. The complexity of this solution is quite high:
The authors in [15] also introduced a much faster heuristic algorithm whose complexity is , where is the number of targets and is the number of sensors that covers the most sparsely covered targets.
[8] Y. Ye, An Potential Reduction Algorithm for Linear Programming, Mathematical Programming, Vol 50, pp , 1991.
[15] M. Cardei, M. T. Thai, Y. Li, and W. Wu, Energy-Efficient Target Coverage in Wireless Sensor Networks, Proceedings of the 24th conference of the IEEE Communications Society (INFOCOM), 2005.
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

18. Maximum Lifetime Coverage Distributed and Localized Algorithms
The authors in [19] proposed a localized algorithm to solve the area coverage problem, called Node Scheduling Scheme Based On Eligibly rule (SBO).
Several sensors cover the same area.
A sensor can be off when its sensing area is covered by its neighbors, called sponsors.
A (random) backoff-based scheme was introduced to avoid blind point (i.e. two nodes expect that the other will be its sponsor).
The nodes have to be synchronized but this part is not mentioned clearly.
The author in [14] proposed two distributed and localized approximation algorithms for the target coverage problem.
Convert a target cover problem instance to the problem of computing maximum number of disjoint dominating set problem (MDS) on a bipartite graph.
To be continued…
[14] M. T. Thai, Y. Li, F. Wang, and D.-Z. Du, O(log n)-Localized Algorithms on the Coverage Problem in Heterogeneous
Sensor Networks, 26th IEEE International Performance Computing and Communications Conference (IPCCC 2007), New Orleans, LA, April 11-13, 2007.
[19] D. Tian and N. D. Georganas, A Coverage Preserving Node Scheduling Scheme for Large Wireless Sensor Networks, in Proceeding of the 1st ACM Workshop on Wireless Sensor Networks and Applications, 2002.
Presented by Donghyun Kim on July 10, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas