1 On Multiple Point Coverage in Wireless Sensor Networks Shuhui Yangy, Fei Daiz, Mihaela Cardeiy, and Jie Wuy Department of Computer Science and Engineering Florida Atlantic University Boca Raton, FL IEEE 2nd Intl. Conf. on Mobile Ad-hoc and Sensor Systems (MASS'05), Nov

2 References M. Cardei, M. Thai, Y. Li and W. Wu, Energy- Efficient Target Coverage in Wireless Sensor Networks, IEEE INFOCOM 2005, Mar. 2005, Miami, USA.Energy- Efficient Target Coverage in Wireless Sensor Networks M. Cardei and D.-Z. Du, Improving Wireless Sensor Network Lifetime through Power Aware Organization, ACM Wireless Networks, Vol. 11, No. 3, pp , May 2005.Improving Wireless Sensor Network Lifetime through Power Aware Organization

3 Outlines Introduction Problem definitions A global solution for the k-CS problem Non-global solutions for the k-CS/k-CCS problem Simulation Conclusion

4 Introduction Coverage  The way to select active nodes Area coverage  Maintaining full coverage of the monitoring area.  The sensor network is to cover (monitor) an area (region).  Each point of the area is monitored by at least one sensor.

5 Introduction Point coverage  A limited number of points (targets) with known locations need to be monitored.  A large number of sensors are dispersed randomly in close proximity to the targets.  Send the monitored information to a central processing node.  Every target must be monitored at all times by at least one sensor, assuming that every sensor is able to monitor all targets within its sensing range. Reference :  M. Cardei and J. Wu, Energy-Efficient Coverage Problems in Wireless Ad Hoc Sensor Networks, Computer Communications Journal (Elsevier), Vol. 29, No. 4, pp , Feb Energy-Efficient Coverage Problems in Wireless Ad Hoc Sensor Networks

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7 Introduction 3 coverage models  1) Targets form a contiguous region and the objective is to select a subset of sensors to cover the region[18]. Typical solutions involve geometry properties based on the positions of sensor nodes. [18] D. Tian and N. Georganas. A coverage-preserving node scheduling scheme for large wireless sensor networks. In Proc. of the 1st ACM Workshop on Wireless Sensor Networks and Applications,  2) Targets form a contiguous region and the objective is to select a subset of sensors to cover the rest of sensors [4]. This model assumes the network is sufficiently dense so that point coverage can approximate area coverage. Typical solutions involve constructing dominating sets or connected dominating sets based on traditional graph theory. [4] J. Carle and D. Simplot-Ryl. Energy efficient area monitoring by sensor networks. IEEE Computer, 37(2):40–46, 2004.

8 Introduction  3) Targets are discrete points and the objective is to select a subset of sensors to cover all of the targets. Typical solutions use the traditional set coverage or bipartite graph models. When R >= 2r, a sensor network that achieves k-coverage is k- connected.[19]  Communication range (R) that is at least twice the sensing range (r).[26]  [19] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless, and C. Gill. Integrated coverage and connectivity configuration in wireless sensor networks. In Proc. of the 1st ACM Conference on Embedded Networked Sensor Systems,  [26] H. Zhang and J. Hou. Maintaining sensing coverage and connectivity in large sensor networks. Technical Report UIUC. UIUCDCS-R , 2003.

9 Contributions 1) Define and formalize the k-(Connected) Coverage Set problems (k-CCS/k-CS). 2) Develop a global algorithm for the k-CS problem using linear programming. 3) Design two non-global solutions for k- CS/k-CCS.

10 Problem definitions Sensor network consisting of n sensor nodes : s1, s2,….., sn. An edge exists between two nodes if the two corresponding sensors are each within the other’s communication range. We assume that the network is sufficiently dense, such that the network is connected, and each node has at least k neighbors for a given constant k. Objective :  Select a minimum subset of sensors with the property that each sensor is monitored by at least k sensors in the selected subset. Assumption :  The network is sufficiently dense, such that the network is connected, and each node has at least k neighbors for a given constant k.  In this paper, we propose to maintain 1-connectivity rather than k- connectivity, to reduce the size of the k-coverage set.

11 Problem definitions k-Coverage Set (k-CS) Problem: Given a constant k >0 and an undirected graph G = (V,E) find a subset of nodes C V such that  (1) each node in V is dominated (covered) by at least k different nodes in C, and  (2) the number of nodes in C is minimized.

12 Problem definitions k-Connected Coverage Set (k-CCS) Problem: Given a constant k > 0 and an undirected graph G = (V,E) find a subset of nodes C V such that  (1) each node in V is dominated (covered) by at least k different nodes in C,  (2) the number of nodes in C is minimized, and  (3) the nodes in C are connected.

13 Problem definitions k-CS and k-CCS are extensions of the Dominating Set (DS) and Connected Dominated Set (CDS) problems. DS if every node in the network is either in the set or a neighbor of a node in the set. CDS : any two nodes in the DS can be connected through intermediate nodes from the DS. When k = 1, k-CS (k-CCS) problem reduces to the DS (CDS) problem. Therefore, for k = 1, both k-CS and k-CCS are NP-complete.

14 A global solution for the k-CS problem

15 A global solution for the k-CS problem Guarantees : each node in V is covered by at least k nodes in C. : the maximum node degree in G. Design a ρ-approximation algorithm,  ρ = + 1 Since IP is NP-hard, we first relax the IP to Linear Programming (LP), solve the LP in linear time,  Then round the solution in order to get a feasible solution for the IP.

16 Best performance : O(n^3), n : the number of variables.

17 A global solution for the k-CS problem

18 Non-global solutions for the k-CS/k-CCS problem A Cluster-based Solution  we propose a scheme for the k-CS/k-CCS problems, Sequentially apply a traditional clustering algorithm k times, whereby the clusterheads selected each time are marked and removed immediately from the network. Find gateways to connect the first set of the clusterheads and also mark them. For each marked node (clusterhead or gateway), if it does not have k marked neighbors, it designates some unmarked neighbors to be marked.

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20 Non-global solutions for the k-CS/k-CCS problem Any two clusterheads are not neighbors, and the clusterhead set is a maximum independent set (MIS) of the network in addition to a DS. Initially, all the nodes are unmarked. When the algorithm terminates, all the marked nodes (clusterheads or gateways) form the k-CCS/k-CS. Theorem 2: All the clusterheads (and gateways) marked in CKA form a k-CS (k-CCS) of the networks. This algorithm can be easily extended to achieve k-coverage in O(k log3 n) time with high probability.  A randomized clustering algorithm has been proposed to achieve1-coverage in O(log3 n) time with high probability.  A traditional algorithm takes O(n) rounds in the worst case, in a network with n nodes.

21 Non-global solutions for the k-CS/k-CCS problem A Local Solution  A node u is “k-covered” by a subset of C of its neighbors if and only if three conditions hold: The subset C is connected by nodes with higher priorities than u. Any neighbor of u is a neighbor of at least k nodes from C. Each node in C has a higher priority than u.  Each node determines its status (marked or unmarked) based on its 2-hop neighborhood information.  Initially, it is assumed that all nodes are marked. After the algorithm terminates, all the marked nodes form the k- CCS/k-CS.  Theorem 3: The marked nodes from PKA form a k-CS/k- CCS of the network.

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23 Fig. 1. A small scale example (n = 15, k = 2, r = 40). Examples All the marked nodes are connected, and every node in the network is covered at least twice by the marked nodes.

24 Fig. 1. A small scale example (n = 15, k = 2, r = 40). Generally speaking, the size of resultant k-CS by CKA is smaller than that of k-CCS.

25 Fig. 1. A small scale example (n = 15, k = 2, r = 40).

26 Simulation Simulator  Linear programming is implemented using Matlab.  All other approaches are implemented on a custom simulator. n nodes are randomly placed in a restricted 100 X100 area.  Assume all nodes have the same transmission range. Number of nodes n : 100 ~ 1000 The transmission range r : 20 and 40 The coverage parameter k : 2, 3, and 4

27 Fig. 3. Comparison of k-CCS & k-CS by different algorithms (k = 2).

28 Fig. 3. Comparison of k-CCS & k-CS by different algorithms (k = 2). CKA has better performance than PKA because CKA is quasi- local while PKA is local algorithm. More information leads to a more precise process.

29 Fig. 3. Comparison of k-CCS & k-CS by different algorithms (k = 2).

30 Fig. 3. Comparison of k-CCS & k-CS by different algorithms (k = 2). LPA has worse performance than CKA and PKA, especially when the network is dense. This is because a dense network increases the maximum node degree, and thus the LPA’s performance ratio.

31 Fig. 4. Algorithms with and without connectivity (k = 2).

32 Fig. 4. Algorithms with and without connectivity (k = 2). k-CS by CKA has the smallest size, and next is the k-CCS by CKA. k-CS by PKA has almost the same size as k-CCS by PKA.

33 Fig. 5. k-CCS & k-CS by PKA & CKA with k = 2, 3, 4 (r = 20).

34 Fig. 5. k-CCS & k-CS by PKA & CKA with k = 2, 3, 4 (r = 20). With larger k, the size of k-CCS or k-CS from CKA or PKA is larger. But when the number of node is great, this increase is less significant

35 Simulation Result (1) CKA has better performance than PKA, especially in generating k-CS. (2) CKA and PKA have better performance than LPA, especially when network is relatively dense. (3) Greater k leads to larger sized k-CS/k-CCS. (4) CKA and PKA have better scalability than LPA, especially when the network is relatively dense. (5) LPA performs better in sparse topologies.

36 Conclusion Define and formalize the k-(Connected) Coverage Set problems (k-CCS/k-CS). Develop a global algorithm for the k-CS problem using linear programming. Design two non-global solutions for k-CS/k- CCS.

37 References [3] M. Cardei, J. Wu, M. Lu, and M. O. Pervaiz. Maximum network lifetime in wireless sensor networks with adjustable sensing ranges. In Proc. of IEEE International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob), [4] J. Carle and D. Simplot-Ryl. Energy efficient area monitoring by sensor networks. IEEE Computer, 37(2):40–46, [6] F. Dai and J. Wu. On constructing k-connected k-dominating set in wireless networks. In Proc. of IPDPS, Apr [19] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless, and C. Gill. Integrated coverage and connectivity configuration in wireless sensor networks. In Proc. of the 1st ACM Conference on Embedded Networked Sensor Systems, [20] J. Wu and H. Li. On calculating connected dominating set for efficient routing in ad hoc wireless networks. In Proc. of the 3rd Int’l Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications (Dial M), [26] H. Zhang and J. Hou. Maintaining sensing coverage and connectivity in large sensor networks. Technical Report UIUC. UIUCDCS-R , 2003.