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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 33431
IEEE 2nd Intl. Conf. on Mobile Ad-hoc and Sensor Systems
(MASS'05), Nov. 2005.
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References
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M. Cardei, M. Thai, Y. Li and W. Wu, EnergyEfficient Target Coverage in Wireless Sensor
Networks, IEEE INFOCOM 2005, Mar. 2005,
Miami, USA.
M. Cardei and D.-Z. Du, Improving Wireless
Sensor Network Lifetime through Power
Aware Organization, ACM Wireless Networks,
Vol. 11, No. 3, pp. 333-340, May 2005.
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Outlines
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Introduction
Problem definitions
A global solution for the k-CS problem
Non-global solutions for the k-CS/k-CCS
problem
Simulation
Conclusion
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Introduction
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Coverage
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The way to select active nodes
Area coverage
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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.
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Introduction
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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. 413-420, Feb. 2006.
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Introduction
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3 coverage models
 1) Targets form a contiguous region and the objective is to select
a subset of sensors to cover the region[18].
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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,
2002.
2) Targets form a contiguous region and the objective is to select
a subset of sensors to cover the rest of sensors [4].
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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.
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Introduction
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3) Targets are discrete points and the objective is to select a
subset of sensors to cover all of the targets.
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Typical solutions use the traditional set coverage or bipartite graph
models.
When R >= 2r, a sensor network that achieves k-coverage is kconnected.[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, 2003.
 [26] H. Zhang and J. Hou. Maintaining sensing coverage and
connectivity in large sensor networks. Technical Report UIUC.
UIUCDCS-R-2003-2351, 2003.
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Contributions
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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 kCS/k-CCS.
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Problem definitions
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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 kconnectivity, to reduce the size of the k-coverage set.
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Problem definitions
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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
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(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.
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Problem definitions
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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
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(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.
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Problem definitions
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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.
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A global solution for the k-CS problem
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A global solution for the k-CS problem
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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,
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ρ=
+1
Since IP is NP-hard, we first relax the IP to
Linear Programming (LP), solve the LP in
linear time,
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Then round the solution in order to get a feasible
solution for the IP.
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Best performance : O(n^3), n : the number of variables.
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A global solution for the k-CS problem
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Non-global solutions for the k-CS/k-CCS
problem
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A Cluster-based Solution
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we propose a scheme for the k-CS/k-CCS
problems,
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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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Non-global solutions for the k-CS/k-CCS
problem
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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.
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Non-global solutions for the k-CS/k-CCS
problem
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A Local Solution
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A node u is “k-covered” by a subset of C of its neighbors if
and only if three conditions hold:
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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 kCCS/k-CS.
Theorem 3: The marked nodes from PKA form a k-CS/kCCS of the network.
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Examples
All the marked nodes
are connected, and
every node in the
network is covered at
least twice by the
marked nodes.
Fig. 1. A small scale example (n = 15, k = 2, r = 40).
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Generally speaking, the size of resultant k-CS by CKA is smaller than that of k-CCS.
Fig. 1. A small scale example (n = 15, k = 2, r = 40).
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Fig. 1. A small scale example (n = 15, k = 2, r = 40).
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Simulation
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Simulator
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n nodes are randomly placed in a restricted 100
X100 area.
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Linear programming is implemented using Matlab.
All other approaches are implemented on a custom
simulator.
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
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Fig. 3. Comparison of k-CCS & k-CS by different algorithms (k = 2).
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CKA has better
performance than PKA
because CKA is quasilocal while PKA is local
algorithm.
More information leads
to a more precise
process.
Fig. 3. Comparison of k-CCS & k-CS by different algorithms (k = 2).
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Fig. 3. Comparison of k-CCS & k-CS by different algorithms (k = 2).
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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.
Fig. 3. Comparison of k-CCS & k-CS by different algorithms (k = 2).
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Fig. 4. Algorithms with and without connectivity (k = 2).
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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.
Fig. 4. Algorithms with and without connectivity (k = 2).
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Fig. 5. k-CCS & k-CS by PKA & CKA with k = 2, 3, 4 (r = 20).
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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
Fig. 5. k-CCS & k-CS by PKA & CKA with k = 2, 3, 4 (r = 20).
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Simulation Result
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(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.
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Conclusion


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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/kCCS.
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References
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[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), 2005.
[4] J. Carle and D. Simplot-Ryl. Energy efficient area monitoring by
sensor networks. IEEE Computer, 37(2):40–46, 2004.
[6] F. Dai and J. Wu. On constructing k-connected k-dominating set in
wireless networks. In Proc. of IPDPS, Apr. 2005.
[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, 2003.
[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), 1999.
[26] H. Zhang and J. Hou. Maintaining sensing coverage and
connectivity in large sensor networks. Technical Report UIUC.
UIUCDCS-R-2003-2351, 2003.
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