Procedia Computer Science 63 ( 2015 ) 494 – 498
1877-0509 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the Program Chairs doi: 10.1016/j.procs.2015.08.374
ScienceDirect
The Seventh International Symposium on Applications of Ad hoc and Sensor Networks,
AASNET’15
An iterative solution for the coverage and connectivity problem in
wireless sensor network
Mahmud Mansour
a,∗, Fethi Jarray
b,caUniversity of Tripoli, Libya
bLaboratoire CEDRIC-CNAM, 75003 Paris, France cHigher institute of computer science, Medenine, Tunisia
Abstract
We study the coverage and connectivity problem in wireless sensor networks. Given an area of targets to cover by a wireless sensor network with a coverage range for each sensor, the problem consists in minimizing the number of deployed sensors in an area while ensuring the connectivity of the sensors network and the coverage of the area. We formulate the problem by a binary integer programming model to minimize the total number of used sensors. Since the problem is NP-complete, firstly we design a separation oracle to establish the feasibility of each solution. Then we provide an iterative approximation based on combinatorial relaxation.
c
2014 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Program Chairs.
Keywords: Wireless sensor network; Minimum spanning tree; coverage and connectivity, Iterative methods; Integer programming.
1. Introduction
Wireless Sensor Networks (WSNs) consists of small physical device (or nodes) having a communication, a pro-cessing, a monitoring and a sensing capability of limited ranges. In a WSN, after collecting information from the environment, sensors transmit the aggregated data to the base station (a processing unit) to ensure a global monitoring of an area. A sensor can not directly communicate with the base station because its communication range is limited. Thus the network composed by the sensors should be connected so that the information collected by each sensor can be transmitted to the base station through the network instead of a direct link.
A network is said to be fully connected if every pair of nodes can communicate with each other, either directly or via intermediate relay nodes. The connectivity of a WSN is usually modeled by a graph associated with that WSN. Stronger concepts of connectivity are sometimes considered to ensure fault tolerance networks. For example a Network is said to be k-connected if there exists at least k edge disjoint paths between any pair of nodes in that network at any given time. Equivalently, the network remains connected even when up to k-1 devices (or communication links)
∗ Corresponding author.
E-mail address:[email protected]
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
fail. Different methods have been proposed to conserve network connectivity. Especially, the graph theory based methods have been successfully employed6,9,22
Coverage in a WSN needs to guarantee that the region is monitored with the required degree of reliability. It is classified into two main categories. In area coverage, sensor nodes are deployed to completely cover a given area. In target coverage, one seeks to cover a set of given targets with predetermined coordinates or positions. A natural way to strengthen the coverage is by introducing the m-coverage where each target is covered by at least m sensor. Equivalently, the targets remain covered even when up to m-1 devices fail. Some times, we seek for m-coverage such that each sensor cover a given number of sensors to ensure a certain equilibrium between the sensors. In such case the coverage problem is equivalent to a workforce scheduling4,7or to a reconstruction problem10,11. The proposed
approaches are mostly based on metaheuristics15,5,16and heuristics for the maximum disjoint set covers17,23,8.
The coverage and connectivity problem represents a challenging problem in wireless sensor network. The objective is to minimize the number of sensors deployed in a given such that the resulting network remains connected and the targets are covered. Each sensor is assigned a communication range to communicate with its environment and a sensing or monitoring range. Coverage and connectivity has important applications in network reliability analysis and network design problems13,14,20,2.
Several approaches have been proposed to handle the coverage and connectivity problem. Shakkottai et al.21 derived necessary and sufficient conditions for the coverage of random grid network. Ke et al.12proved that the the
coverage and connectivity problem is NP-complete even when restricted to grid square area. i.e. it cannot be solved in polynomial time unless P=NP. Rebai et al.19,18proposed local search algorithms and a branch and bound approach
to approximate the problem in a rectangular grid pattern. Alam and Haas1studied the coverage and the connectivity
in 3D networks where nodes are randomly deployed in a 3D space. An excellent review on coverage and connectivity problem can be found in3.
The remainder of this paper is organized as follows. In Section 2, we introduce some definitions and notation. In Section 3, we propose an integer program to model the coverage and connectivity problem. In Section 4, we develop an iterative combinatorial relaxation. We conclude in the last section.
2. Definitions and notation
We consider a setT of mtargetsti,i ∈ {1, . . . ,m}with prefixed positions to cover by the minimum number of sensors such that all the sensors can exchange information and particularly with the sink nodeS Nwith coordinate (i0,j0). To buil the WSN model, we make the following assumptions.
Firstly, we assume that all sensors have the same transmission rangeRtand two sensors can directly communicate
if their euclidean distance is within the transmission range. Secondly, we assume that all the sensors have a predefined sensing range calledRs, and a sensor covers a target if the Euclidean distance between the sensor and the target is
smaller or equal with the predefined sensing rang.
LetPs be the set of all possible positions of sensors. For example if The coverage area is divided into a grid of
discrete points, thenPsmay be its grid points. We denote byPs(i) the set of targets covered by a sensor deployed on
positioni. Similarly we denote byCithe set of positions that cover targeti.
The coverage and connectivity problem (CCP) can be established as follows:
Instance:A setTofmtargets and a setPsof candidate positions of sensorsQuestion:Determine the minimum
number of sensors such that the sensor network graph is connected and each target is covered at least by one sensor.
3. Problem formulation
To state the problem mathematically, we introduce the binary decision variables
• xi, which gets value 1 if a sensor is placed on positioni,
• zi j, which gets value 1 two sensors are placed on positionsiandjand if a direct link betweeniandjis used to
The minimum total power spanning tree problem can be reformulated as the following integer programming model: IP ⎧⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎨ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎩ minj∈Psxi s.t. (i,j)∈Pszi j= j∈Psxi−1 (1) (i,j)∈(S,S)(zi j)≤j∈S xj−1 ∀S ⊆Ps(2) j∈Cixi≥1 ∀i (3) 2zi j≤xi+xj ∀(i,j)∈Ps (4) xi,zi j∈ {0,1}
We note that(i,j)∈(S,S)(xi j)≤ |S| −1 ∀S ⊆Ps.
The first constraint states that a tree onnvertices hasn−1 edges. The second constraint is the sub-tour elimination constraints prevents a solution with several subtours. In fact a graph has no cycle if and only if any subsetS of the vertices have less than|S|edges with both endpoints inS. The constraints (1) and (2) ensure that the deployed sensors form a tree. Unfortunately, this formulation has an exponential number of constraints. Constraint (3) ensures that eat target is covered by at least one sensor. The Fourth constraints implies that an edge (i,j) is selected only if its on its endpoints,iand j, are assigned deployed sensors. The objective function expresses the number of deployed sensors. The program can be extended to consider stronger concepts of coverage and connectivity.
4. Iterative relaxation
The iterative relaxation technique was successfully used to solve a wide range of practical and real-life optimiza-tion problems. We will apply this strategy to solve the program IP. Roughly speaking this approach firstly relaxes the second constraint and solves the remaining problem. Then it produces a violated relaxed constraint (separation pro-cess). Secondly it adds the violated constraint and resolve the problem. The procedure is iterated until no constraint is violated. Finally, this approach gives the optimal solution ofIP.
We notice that contrary to the pure minimum spanning tree problem, the linear relaxation of IP may have non integral extreme points. Thus, even though, we can separate the sub-tour elimination constraint in a polynomial time, the iterative relaxation remains exponential in the worst case.
4.1. Constrained Min-cut problem
Consider a directed graphG(V,E) with a sources, a targett, and a setS ⊆V. The constrained min-cut problem inG(V,E) consists in finding the min-cut such that there is at least one node ofS in the s-side. This problem can be solved polynomially by solving|S|min-cut subproblems and taking the best cut. In each subproblem, we add an arc with infinite capacity from the source to a node ofS to ensure that this node will be in the s-side.
4.2. Separation process
Given a solution of the relaxed IP program, a separation oracle finds a violated sub-tour elimination constraint or establish the feasibility of the solution. We establish the following result:
Proposition 1. There is a polynomial time separation oracle for the sub-tours elimination constraints in the program IP.
Proof 1. For S ∈ Ps, let E(S) = {(i,j)|i ∈ S,j ∈ S}be the set of edges with both endpoints in S . For a set of
L edges, let z(L) = e∈Lzebe the set of links of L used to ensure the connectivity. For a set F ⊆ Ps, let x(F) =
e∈Lxebe the number of sensors deployed in F. The first constraint of IP ensures that z(E(Ps))= x(Ps)−1. The
sub-tours elimination constraint is satisfied if for every subset S ⊆ Ps, z(E(S)) ≤ x(S)−1. This is equivalent to
x(S)+z(E(Ps))−z(E(S))≥x(Ps)since z(E(Ps))=x(Ps)−1. Thus the sub-tour elimination constraint is violated if
.
We will show that the subproblem Q is equivalent to a constrained min-cut problem in a bipatite graph GC(VS,VC,EC)
such that there is at least two vertices in the s-side. VS represents the candidate positions of the sensors and VC rep-resents the edges with zi j >0(see Figure 1) in the solution of the relaxed IP program. We add to GC(VS,VC,EC)
two nodes: a source s and a target t. There is an arc from s to every node VCi j ∈VC with capacity zi j. Similarly,
there is an arc from every node VSi ∈VS to t with unit capacity. There is an arc from VCi jto its endpoints,i.e. VSi
and VSjwith infinite capacity.
Suppose that there is a non empty set S that violates the sub-tour elimination constraint, i.e., z(E(S))> x(S)−1. Then we construct a cut as the following: the s-side contains all the nodes of S and E(S), i.e., all the edges with both endpoints in S . The capacity of this cut is x(S)+z(E(Ps))−z(E(S))<x(Ps)since Z(E(Ps))=x(Ps)−1. We note that
there is at least two nodes in the s-side since S is non empty.
Conversely, suppose that there is a cut in GC(VS,VC,EC)with capacity less than x(Ps)such that there is at least
two nodes in the s-side. We note S =VS∩ {s−side}and C =VC∩ {s−side}. Firstly, we note that S is non empty. Secondly, we have C =E(S )because each edge with initial vertex in C has its terminal vertex in S otherwise the cut will have an infinite capacity. So, we deduce that the capacity of the cut is x(S )+z(E(Ps))−z(E(S ))< x(Ps).
Finally, we conclude that z(E(S ))>x(S )−1and that S is a violated set.
Fig. 1. Separation oracle and the associated min-cut problem
5. Conclusion
In this paper, we have provided an iterative exact algorithm to ensure the coverage and connectivity propriety in a wireless sensor network. Our approach is based on advanced iterative techniques in combinatorial optimization and uses a min-cut flow algorithm to handle the separation oracle.
Promising future research axes include energy minimization while ensuring the coverage and connectivity.
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