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Energy efficient k-barrier coverage in limited mobile wireless sensor networks
Computer Communications 35 (2012) 1749–1758
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Energy efficient k-barrier coverage in limited mobile wireless sensor networks Huan Ma, Deying Li ⇑, Wenping Chen, Qinghua Zhu, Huiqiang Yang School of Information, Renmin University of China, Beijing 100872, PR China
a r t i c l e
i n f o
Article history: Available online 4 May 2012 Keywords: k-barrier coverage Sensing power level Energy efficient Mobility Wireless sensor network
a b s t r a c t Energy cost and reliability are two main concerns in barrier coverage for wireless sensor networks. In this paper, we take the energy cost and reliability as objectives respectively to study two problems of k-barrier coverage: the minimum energy cost k-barrier coverage problem in static wireless sensor networks and the maximum k-barrier coverage problem in limited mobile wireless sensor networks. For the minimum energy cost k-barrier coverage problem, all sensors are stationary, and each sensor has l + 1 sensing power levels in the network, the objective of the problem is to find a sensing level assignment to form k-barrier coverage such that the total power consumed by the k-barrier is minimized. We firstly transform it into a minimum cost flow problem with side constraints and use Lagrangian relaxation technique to solve the minimum cost flow problem. Then, we also propose a heuristic algorithm. For the maximum k-barrier coverage problem, each sensor can move within the limited range, the objective of the problem is to form more barriers while some sensors can move within limited range. We formulate the problem into an integer linear programming (ILP), then propose two heuristic algorithms based on the linear programming (LP) relaxation. The simulation results demonstrate our algorithms are efficient. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Barrier coverage is one of the most important issues in wireless sensor networks [1], which was originally motivated by military applications. It is an efficient way to guard boundaries of critical infrastructures or assets, such as battlefield and country borders [2]. In wireless sensor networks, a barrier is formed by a set of sensors such that any intruder crossing this region can be detected by at least one sensor in the barrier. Compared with full coverage, barrier coverage requires significantly fewer sensors and less energy. The authors of [3] proved that if the width of the deployment region is three times the sensing range, full coverage requires more than twice the sensor density of barrier coverage. Hence barrier coverage is more cost effective and attractive for intrusion detection and border surveillance applications. In order to increase the reliability of barriers, a certain degree of redundancy is required. k-barrier coverage is proposed to address this issue. The k-barrier coverage makes that the intruders can be detected by at least k sensors while they move along any crossing paths from one boundary to another. If one barrier fails, the other ðk 1Þ barriers can detect the intruders. However, whether k-barrier coverage can be formed depends on a number of parameters, such as the sensor density, sensor deployment and so on. In many ⇑ Corresponding author. Tel.: +86 010 62519223; fax: +86 010 62511184. E-mail addresses: [email protected] (H. Ma), [email protected] (D. Li), [email protected] (W. Chen), [email protected] (Q. Zhu), yhq@ruc. edu.cn (H. Yang). 0140-3664/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comcom.2012.04.021
existing works on k-barrier coverage, the sensing power of each sensor is supposed to be fixed no matter the distance between sensors is long or short. It may result in the waste of energy. Hence, constructing energy efficient k-barrier from randomly deployed sensors in a region is a challenging problem. Most studies on k-barrier coverage mainly focused on how to construct barriers in static sensor networks where each sensor is assumed to be stationary. However, when the sensor density is not high enough, the number of formed barrier coverage may be too few to achieve application requirements. Compared with static wireless sensor network, sensors with limited mobility can move to some desired locations to form stronger barriers at the cost of only a little energy cost. How to move the mobile sensors to form more barriers is also a challenging problem. In this paper, we study the minimum energy cost k-barrier coverage problem (P1) in static wireless networks and the maximum kbarrier coverage problem (P2) in limited mobile wireless sensor networks. For the problem P1, each sensor has several sensing power levels and each sensing power level corresponds to a sensing range. We need to assign the sensing power level to each sensor such that the sensor network can form a k-barrier coverage. Our goal is finding such an assignment that the total sensing power is minimized. To solve this problem, we transform it into the minimum cost network flow problem with side constraints. By using the Lagrangian relaxation technique [4], we get a lower bound of the problem. We also design a heuristic algorithm. For the problem P2, there is a given set of mobile sensors in which each sensor has limited moving range, our goal is finding k-barrier coverage such that k is
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maximized. To solve the problem P2, we first transform the problem into an integer linear programming. Then, two heuristic algorithms based on the relaxation linear programming are proposed to find disjoint barriers in a target region. The main contributions of this paper are as follows. We formulate the minimum energy k-barrier coverage problem (P1) as the minimum cost network flow problem with side constraints and find a lower bound of the total energy cost by the classical Lagrangian algorithm. We propose a heuristic algorithm for P1 whose total energy cost is at most two times of the lower bound of the optimal solution. We formulate the maximum k-barrier coverage problem with limited moving range (P2) as an integer linear programming and find an upper bound of the problem. We employ the relaxation rounding technique to design two heuristic algorithms for P2. The rest of this paper is organized as follows. Section 2 reviews the related work. Section 3 addresses the minimum energy k-barrier coverage problem. In Section 4 we study the maximum k-barrier coverage problem and present two heuristic algorithms. The simulation results are given in Section 5. Finally, Section 6 concludes the paper.
2. Related work The traditional coverage problems, including point coverage [5] which requires every target point to be covered by at least one or k sensors, and area coverage [6] which requires every point in the area to be covered by at least one or k sensors, have been studied intensively. The concept of barrier coverage was first introduced by Gage in robotic sensors [7]. Chen et al. [3] proposed a localized barrier coverage protocol to detect all intruders whose movement are confined to a slice of the original strip region. The localized algorithm, however, only provides barrier coverage for slices of bounded length. It does not protect the network against intruders which can move beyond the range of a slice. Saipulla et al. [8] studied the barrier coverage for line-based deployments and established a tight lower-bound. Liu et al. [2] showed that in a rectangular area with width w and length l for w ¼ Xðlog lÞ, there exist, with high probability, multiple disjoint barriers across the entire length of the area, and proposed an efficient distributed algorithm to construct multiple disjoint barriers in a long boundary area of an irregular shape. Kumar et al. [9] studied k-barrier coverage problem in which sensors were deployed deterministically. Later, Kumar et al. [10] introduced weak coverage and proved equivalence conditions that deduce efficient algorithms for determining whether a given belt region is k-barrier covered. Moreover, they derived the critical conditions for weak barrier coverage with high probability in a belt region. Ssu et al. [11] studied the k-barrier coverage problem in directional sensing model and developed a distributed algorithm to construct k-barrier coverage in randomly deployed directional sensor fields. Since sensors are battery powered and it is difficult to replace the batteries, sensors may be designed to adjust sensing range adaptively to save energy consumption. However, all the above works assumed that the sensing powers of all sensors were fixed. In this paper, one of problems we study is minimizing the total energy cost of k-barrier coverage, where sensors are stationary and have multiple sensing power levels. Some results about this problem are shown in [12]. There are also much effort on investigating how to move sensors to form more barriers. Saipulla et al. [13] presented a sensor
mobility scheme that constructs the maximum number of barriers with minimum sensor moving distance. Shen et al. [14] proposed a virtual force based on heuristic algorithm to relocate mobile sensors to form barriers. Barrier coverage is affected by the deployment region of sensors. Chellappan et al. [15] studied the deployment problem which schedule the movement of the sensors to minimize the number of moving sensors and simultaneously minimize the sensor movements. Ban et al. [16] focused on the problem how to relocate mobile sensors to construct k sensor barriers with minimum total moving distance. The authors modeled the problem as integer linear programming (ILP) and proposed an approximation algorithm AHGB to construct one energy-efficient sensor barrier. Based on AHGB, a Divide-and-Conquer algorithm was proposed to achieve k-barrier coverage for large sensor networks. Most existing works for barrier coverage are based on regions with long belt shape. In this paper, to adapt more practical application, we study how to move the mobile sensors to form maximum number of barriers under that each mobile sensor only can move within a limited range on arbitrary region. 3. Minimum energy cost k-barrier coverage In this section, we address the minimum energy cost k-barrier coverage problem in wireless sensor networks. We assume that a set S of sensors are deployed randomly in a two-dimensional rectangular area where sensors do not move after initial deployment. Each sensor si has its coordinates si ðx; yÞ, which may be detected through GPS or obtained through a localization mechanism. We use si :x to represent x-coordinate of si . Suppose each sensor node has l þ 1 sensing power levels, 0 6 p1 6 p2 6 pl , i.e., each sensor has l þ 1 different corresponding sensing ranges f0; r 1 ; r 2 ; . . . ; rl g. Every active sensor is assigned with one sensing power level PLðsi Þ 2 f0; p1 ; p2 ; . . . ; pl g. We define the total energy P cost of the network as si 2S PLðsi Þ. We give some definitions as follows: Definition 1 (Crossing path (or Incursive path)). A path is said to be a crossing (or incursive) path in a belt region if it crosses the complete width of the region from one side to the other side. An example is illustrated in Fig. 1. The two red sketch paths are incursive paths while the two blue paths are not crossing paths. Definition 2. A crossing path P is said to be k-covered if P intersects with at least k active sensor’s sensing disks. Definition 3 (k-barrier coverage). A belt region is said to be k-barrier covered by a sensor network deployed over it if and only if all the crossing paths through the belt region are k-covered by this sensor network.
Fig. 1. An example of k-barrier coverage.
H. Ma et al. / Computer Communications 35 (2012) 1749–1758
Fig. 1 shows an example of 3-barrier coverage. Once intruders cross the areas, they will be detected at least 3 times. Definition 4 (Minimum energy k-barrier coverage problem). Given a sensor network over the objective belt region and each sensor has l þ 1 sensing power levels, find a sensing power level assignment such that the given belt region is k-barrier covered and the total energy cost is minimized. The goal of the problem is to find a sensing power level assignment for all sensors, i.e., a set of sensing power levels fPLðs1 Þ; PLðs2 Þ; . . . ; PLðsn Þg such that the sensor network can form a k-barrier coverage and the total energy cost is minimum among all the assignments. 3.1. Lagrangian method for minimum energy cost k-barrier coverage problem Given a set of sensors S ¼ fs1 ; s2 ; . . . ; sn g, each sensor node si has l þ 1 sensing power levels, 0 6 p1 6 p2 6 6 pl . We consider the minimum energy cost k-barrier coverage problem in a long belt, where sensors are deployed randomly over it. Before solving the minimum energy cost k-barrier coverage problem, we first restate the method to determine whether a belt region is k-barrier covered [10]. Corresponding to a sensor network deployed in a belt region, we derive a coverage graph CG ¼ ðV; EÞ, where V is the set of all sensor nodes plus two virtual vertices s and t (see Fig. 2). The set of edges E is derived as follows: each pair of sensors whose sensing disks overlap is connected by an edge. Additionally, the sensors whose sensing disks intersect with the left boundary are connected to node s and the sensors whose sensing disks intersect with the right boundary are connected to node t. An example of constructed coverage graph is shown in Fig. 2. Lemma 1 [10]. An open belt region is k-barrier covered if and only if s and t are k-connected in the corresponding coverage graph, CG. In order to solve the minimum energy cost k-barrier coverage problem, we construct an auxiliary weighted directed graph Ga ¼ ðV a ; Ea ; C a ; U a Þ, where V a denotes the set of nodes, Ea denotes the set of edges, C a denotes the set of edges’ cost, and U a denotes the set of edges’ capacity. First, we construct l node pairs fqti;1 ; qhi;1 g, fqti;2 ; qhi;2 g; . . . ; fqti;l ; qhi;l g for each sensor si in S. For each node pair fqti;k ; qhi;k g, we add a directional edge from qti;k to qhi;k with cost pk . We add a directed edge ðqhi;k ; qtj;m Þ with cost 0 from qhi;k to qtj;m if and only if si ’s sensing range by sensing power level pk intersects with sj ’s sensing range by sensing power level pm . In addition, we add a new virtual node s at the left side of the belt region and a new virtual node t at right side of the belt region. There is a directed edge from s to qti;k with cost 0 if and only if si ’s sensing range by sensing power level pk intersects with the left border of the belt region. Last, there is a directed edge from qhi;k to t with cost 0 if and only if si ’s sensing range by sensing power level pk intersects with the right border of the belt region. Each edge in Ea is associated with capacity 1.
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To formulate our problem rigorously, we use a binary variable xi;j to represent if there is a flow through an edge from i to j in Ga ¼ ðV a ; Ea ; C a ; U a Þ. For the simplicity of the mathematical representation of the problem, we use eðsi Þ to denote the set of edges fðqti;1 ; qhi;1 Þ; ðqti;2 ; qhi;2 Þ; . . . ; ðqti;l ; qhi;l Þg for each sensor si in S. Let X denote the column vector of variables corresponding to each edge (i.e., X ¼ fxi;j jði; jÞ 2 Ea g). Let C 0 denote the row vector of the cost of edges in the same order with X (i.e., C ¼ fci;j jði; jÞ 2 Ea g). Let C be the transpose of vector C 0 . Our problem can be formulated by an integer programming problem as follows:
min C X
ð1Þ
subject to
X
xi;v
ði;v Þ2Ea
X
xu;i ¼ 0;
ðu;iÞ2Ea
for each node i 2 V a fs; tg X xs;v ¼ k
ð2Þ ð3Þ
ðs;v Þ2Ea
X
xu;t ¼ k
ð4Þ
ðu;tÞ2Ea
X
xu;v ¼ 1;
ðu;v Þ2eðsi Þ
for each edge set eðsi Þ xi;j ¼ 0; 1
ð5Þ ð6Þ
From the formulation, it is easy to know that if there is not constraint (5), the problem is a classical minimum cost flow problem. Adding constraint (5), the problem becomes the minimum cost network flow problem with side constraint (5). From Lemma 1 and the structure of the auxiliary graph, we have the following result. Theorem 1. The minimum energy cost k-barrier coverage problem is equivalently transformed to the minimum cost k-flow problem with side constraints. We express the above formulation in matrix form as follows:
min C X
ð7Þ
subject to
AX ¼ b DX þ Ir S ¼ d 0 6 X 6 1; S P 0
ð8Þ ð9Þ ð10Þ
where constraint (8) corresponds to constraints (2)–(4), and A is the ðjV a j þ 2; jV a j þ 2Þ coefficient matrix which is composed by the left coefficient of the three constraints. If ði; v Þ 2 Ea , the coefficient element is 1; if ðu; iÞ 2 Ea , the coefficient element is 1; if ðs; v Þ 2 Ea , the coefficient element is 1; if ðu; tÞ 2 Ea , the coefficient element is 1, else in the other situations, the coefficient element is 0. Constraint (9) corresponds to constraint (5), and D is the ðjV a j; jV a jÞ coefficient matrix which can be got from the left coefficient of (5). If
Fig. 2. A simple example of coverage graph.
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ðu; v Þ 2 eðsi Þ, the coefficient element is 1, else 0; Ir is ðr; rÞ-identity matrix and r equal to jSj. We can use the Lagrangian relaxation algorithm in [4] to obtain a lower bound of our problem. Lagrangian relaxation has been used in various approaches to solve the constrained capacitated network flow problems [7,17,18,20]. The basic idea is to relax the side constraints so that only a capacity network flow problem remains to be solved. Let X ¼ fxjAx ¼ b; 0 6 x 6 lg be the set of feasible solutions for the pure network part of Minimum Flow Structures (MFS). Then the associated Lagrangian subproblem is
LðlÞ ¼ minfCX þ lðDX dÞg where l 2 Rr is a row vector of Lagrange multipliers. For every l P 0, the value of the Lagrangian function LðlÞ is a lower bound for the objective function value of problem (MFS). In order to find an optimal solution of MFS, the Lagrangian dual L ¼ maxlP0 LðlÞ must be solved. As LðlÞ is a concave, non-differentiable function of l, a subordinate search method is employed to find L , see [17] for details. Algorithm 1 represents the basic steps about the Lagrangian relaxation as follows: Algorithm 1. The basic algorithm for the Lagrangian relaxation
the sensor si ’s sensing range intersects with the left border of the belt region when si working at the maximum sensing power level pl . And there is a directed edge from qhi to t with cost 0, if and only if si ’s sensing range intersects with the right border of the belt region when si working at the maximum sensing power level pl . Each edge in Ec is associated with capacity 1. Then we use the minimum cost flow algorithm on Gc to find k node-disjoint paths from s to t. And we adjust the sensing power level of each sensor node in each path to make this path form 1-barrier coverage. According to Lemma 1, we know that the found paths form a k-barrier coverage in the sensor network. Note that we use Rðpi Þ to denote the sensing range corresponding to the sensing power level pi , and let LEN represent the length of the belt region. The heuristic algorithm is formally represented as follows: Algorithm 2. Heuristic algorithm for problem P1 1: Construct an auxiliary graph Gc ¼ ðV c ; Ec ; C c Þ. 2: Using minimum cost flow algorithm in [4] to find kminimum cost flow from s to t. n o 3: For each found path P i ¼ s; qti1 ; qhi1 ; . . . ; qtim ; qhim ; t , assign a power level for each sensor node as follows: PLðsi1 Þ ¼ minfpk jr k P maxðsi1 :x; distðsi1 ; si2 Þ r l Þg; PLðsij Þ ¼ minfpk jr k P maxðdistðsij1 ; sij Þ RðPLsi
j1
1: Initialization: Set iteration count k ¼ 0; Lagrange multipliers l1 ¼ 0; lower bound LB ¼ 1; tolerance e; determine an upper bound UB for MFS. 2: Subproblem: Solve Lðlk Þ ¼ minfCX þ lk ðDX dÞg, let xk be the optimal solution corresponding to Lðlk Þ, then ðDxk dÞ is a subgradient of L at lk . 3: Optimality check: If Dxk 6 d and lk ðDxk dÞ ¼ 0 Stop: xk is an optimal solution of MFS. If Dxk 6 d Set UB ¼ minfUB; cxk g; set LB ¼ maxfLB; Lðlk Þg. If UB=LB 6 1 þ e Terminate. 4: Subgradient: Set k ¼ k þ 1; calculate lk ¼ ½lk1 þ hk ðDxk1 dÞþ , where ½lk1 þ hk ðDxk1 dÞþ is the positive part of ½lk1 þ hk ðDxk1 dÞ, and hk is the step size determined by the standard heuristic hk ¼ kk ðUB Lðlk1 ÞÞ=jjDxk1 djj2 and kk is a scalar between 0 and 2; go to 2.
3.2. Heuristic algorithm for minimum energy cost k-barrier coverage problem In this subsection, we propose a heuristic algorithm (Heuristic). In order to design this Heuristic, an auxiliary weighted directed graph Gc ¼ ðV c ; Ec ; C c Þ where V c denotes the set of nodes, Ec denotes the set of edges, and C c denotes the set of edges’ cost, is constructed as follows: we construct one node pair fqti ; qhi g for each sensor si originally in S. For each node pair fqti ; qhi g, we add a directed edge ðqti ; qhi Þ from qti to qhi with cost pl . We add a directed edge (qhi ; qtj ) with cost 0 from qhi to qtj if and only if the Euclidian distance between si and sj (i.e., distðsi ; sj Þ) is less than 2 r l , where r l is the largest sensing range. We also add a new node s at the left side of the belt region and a new node t at right side of the belt region. There is a directed edge from s to node qti with cost 0, if and only if
Þ; distðsij ; sijþ1 Þ r l Þg,
for2 6 j 6 m 1; PLðsim Þ ¼ minfpk jr k P maxðdistðsim1 ; sim Þ RðPLsim1 Þ; LEN sim :xÞg. In Heuristic, the time complexity of step 1 is OðnÞ. In step 2, using the minimum cost flow algorithm has the time complexity of Oðn3 Þ. Step 3 takes times of Oðn kÞ. Because k is always less than n, step 3 takes times at most Oðn2 Þ. Thus, the whole algorithm has time complexity of Oðn3 Þ. w4. The maximum k-barrier coverage problem In this section, we address problem P2, which is the maximum k-barrier coverage problem in limited mobile wireless sensor networks. This section is organized as follows: We first formulate the problem into an integer linear programming (ILP) problem, then propose two round-up algorithms based on LP relaxation. In the initial deployment, we assume that a set S of n mobile sensors are distributed uniformly at random in a large region, and the sensing range of each sensor is r. Without loss of generality, we assume that the region is two-dimensional belt area with length l. We also assume that each sensor knows its own location denoted by the coordinates (x; y). The location information can be known through a localization mechanism or GPS. Suppose there are k0 grids in the network model. There are lq grid points in the qth grid. The grid points are candidate positions of mobile sensors. Due to power constraint, the mobile sensors can move at most distance R after initial deployment. Therefore, the mobile sensors can move to a grid point as long as the distance between them is not more than R. One grid point only can deploy at most one sensor, and one sensor only can move to at most one grid point. After the movement, if each grid point in the qth grid is occupied by a sensor, these sensors can connect with each other to form a barrier. Our goal is how to move these sensors such that the number of formed barrier coverage is maximized. Definition 5 (Maximum k-barrier coverage problem). Given a sensor network over the objective region and each sensor can move at most R, find a moving strategy of these sensors such that the given region is k-barrier covered and k is maximized.
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Definition 6 ([19] Grid length). When n grid points are deployed in an objective region, the grid length is generally considered to be the minimum distance between two border regions through the grid points. Before designing our algorithms, we give the following theorem. Theorem 2. When n mobile sensors are deployed in an objective range with the minimum grid length l and each sensor covers a disk area of radius r, the number of sensors for any barrier coverage is at least dl=ð2 rÞe, and the number of formed barriers is at most bn=dl=ð2 rÞec, where r is the sensing radius. Proof. No matter what the shape of the objective range is, any barrier with the minimum number of sensors should have property that the sensing area of any two adjacent sensors are tangent with each other in the grid. That is, if the minimum grid length is l and each sensor covers a disk area of radius r, the minimum number of sensors in a barrier coverage is at least dl=ð2 rÞe, therefore, when there are n mobile sensors deployed in an objective range, the maximum number of barriers which can be formed is at most bn=dl=ð2 rÞec. h
For given bipartition graph G ¼ ðX; Y; EÞ; X represents sensor nodes and Y represents grid points. We add two new nodes S and D. For any x 2 X, there is an edge between x and S. For any y 2 Y, there is an edge between y and D. We define binary variables:
xij ¼
1; if sensor xi occupy yj ; 0; otherwise
ð11Þ
xSi ¼ 1;
8i ¼ 1; . . . ; n
ð12Þ
xjD ¼ 1;
1 6 j 6 l1 þ l2 þ þ lk0
ð13Þ
yj ¼
zq ¼
1; if there is a sensor in yj ; 0;
otherwise
ð14Þ
1; if each grid point on the qth grid is filled with a sensor; 0; otherwise ð15Þ
Then we have an ILP formulation as follows:
max
k0 X zq
ð16Þ
q¼1
Suppose there are k0 grids in which if each point of some grid is occupied by a sensor, then it can form a barrier. Once the positions of grids are identified, the network can be represented as a bipartition graph G ¼ ðX; Y; EÞ, while X represents a set of n mobile sensors, Y represents a set of grid points, where Y={Y 1 ; . . . ; Y i ; . . . ; Y k0 }, Y i ¼ fli1 þ 1; li1 þ 2; . . . li1 þ li g while l0 ¼ 0. For any sensor x and any grid point y, there is an edge between them if and only if the distance between them is at most R. The problem can be composed of the following two subproblems: Subproblem (a). Grid matching problem: the problem is to match mobile sensor set X with grid points set Y. Each mobile sensor can only occupy a grid point and each grid point can be filled by only one mobile sensor. If each grid point in grid Y i is matched with a mobile sensor in X; Y i is called as a barrier; Subproblem (b). Maximizing barriers problem: the problem is how to find a grid matching such that the number of formed barriers is maximized. Fig. 3(a) shows an example, the red nodes x1 ; . . . ; x6 represent the mobile sensors, and the black nodes y1 ; . . . ; y6 represent the grid points. The shaded areas are the coverage areas for sensors and the gray circles are the moving ranges of the mobile sensors. In this case, x1 can move to grid point y1 . x2 can move to grid point y1 ; y2 and y3 . x3 can move to grid point y3 and y4 . x4 can move to grid point y4 and y5 . x5 can move to grid point y5 and y6 . x6 can move to grid point y6 . (1) If x3 moves to y4 , the other mobile sensors: x1 moves to y1 ; x2 moves to y2 ; x4 moves to y5 ; x6 moves to y6 , we can get a barrier Y 2 by fx3 ; x4 ; x6 g, i.e., if x3 move to y4 , no matter how to move the other mobile sensors, we can get only one barrier coverage at most. (2) If x3 moves to y3 , the other mobile sensors: x1 moves to y1 ; x2 moves to y2 ; x4 moves to y4 ; x5 moves to y5 ; x6 moves to y6 , we can get two barriers Y 1 and Y 2 by fx1 ; x2 ; x3 g and fx4 ; x6 ; x8 g. Therefore, if we want to maximize the number of barriers, x3 have to move to y3 although the Euclidean distance between x3 and y4 is less than that between x3 and y3 . 4.1. ILP formulation For the convenience of discussion, we reintroduce our notations as follows: i: index of sensors, where 1 6 i 6 n. j: index of grid points, where 1 6 j 6 l1 þ l2 þ þ lk0 . q: index of barriers, where 1 6 q 6 k0 . lq : the number of grid points on the qth barrier.
subject to:
X
xSi ¼
81 6 i 6 m
ð17Þ
81 6 j 6 l 1 þ l 2 þ þ l k 0
ð18Þ
xij ;
ði;jÞ2E
xjD ¼
n X xij ; i¼1
2yj 6
n X xij þ xjD ;
81 6 j 6 l 1 þ l 2 þ þ l k 0
ð19Þ
i¼1
lq zq 6
l1 þþl Xq
yj ;
81 6 q 6 k 0
ð20Þ
j¼l1 þþlq1 þ1
xSi ; xij ; xjD ; yj ; zq 2 f0; 1g;
81 6 i 6 n; 1 6 j 6 l1 þ l2 þ þ lk0 ;
1 6 q 6 k0
ð21Þ
In the formulation, constraint (17) ensures the flow conservation for all sensors; Constraint (18) ensures the flow conservation for all grid nodes; Constraint (19) evaluates whether each grid node is occupied by a sensor; Constraint (20) ensures that the qth grid forms a barrier coverage if and only if each grid point on the qth grid is occupied by one sensor. Fig. 3(b) shows the matching graph based on Fig. 3(a). Nodes x1 ; . . . ; x6 in (b) are corresponding to the mobile sensors in (a). Nodes y1 ; . . . ; y6 in (b) are corresponding to the grid points in (a). There is an edge ðxi ; yj Þ in (b) if the distance between xi and yj is at most R. S and D are virtual nodes, 8xi 2 X, add an edge ðS; xi Þ to E; 8yj 2 Y, add an edge ðyj ; DÞ to E. Then we get the bipartition matching graph (b). 4.2. Simple round-up algorithm based on linear programming relaxation In this subsection, we will design a heuristic for P2. The heuristic includes two steps. Firstly, we relax the ILP to a IP as follows: removing the integral constraints in (21), i.e., (21) is replaced by
0 6 xSi 6 1;
0 6 xij 6 1; 0 6 xjD 6 1; 0 6 zq 6 1;
81 6 i 6 n; 1 6 j 6 l1 þ l2 þ þ lk0 ; 1 6 q 6 k0
ð22Þ
Then solve the relaxation LP to get an optimal solution for the relaxation LP. Secondly, a greedy round-up algorithm is employed
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to find an integral feasible solution for ILP based on the optimal solution of the relaxation LP. We use xLij to represent the optimal solution of the relaxation LP, where some xLij may not be integers, and use xGij to represent the solution obtained by rounding xLij . In our greedy round-up algorithm, we first sort xLij in the order of non-incremental in algorithm. Then, we remove the edges which have the common vertex, and set xGij ¼ 1; yGj ¼ 1 for the edges that have not common vertex. Finally, set zGq ¼ 1 if each yGj of the barrier is 1, i.e., the q-th barrier coverage has been formed when zGq ¼ 1. The basic steps about the simple round-up algorithm is represented formally as Algorithm 3. Algorithm 3. Simple round-up algorithm for problem P2 1: Solve the LP relaxation Eq. (16)–(20) and (22) to obtain a fractional solution {xLij ; yLj ; zLq }. 2: Set the xLij by order of non-incremental. (a) for each i 2 f1; . . . ; ng, each j 2 f1; . . . ; ng set x0ij ¼ 1; y0j ¼ 1 for the edges that have not common vertex, and (b) set other x0ij ¼ 0; y0j ¼ 0. 3: For each j, each q (a) set xGij ¼ 1; zGq ¼ 1 if each y0j in Y j is 1, and (b) set other zGq ¼ 0.
4.3. Improved round-up algorithm based on linear programming relaxation In this subsection, an improved round-up algorithm is proposed. Note that in Step 2 of the simple round-up algorithm, we select the maximum value of xLij with no common vertex to be rounded up to 1. However, it only considered the maximum xLij in the barrier. The simple round-up algorithm can be further improved. Since a barrier coverage needs that every grid point in the barrier is occupied by a sensor, we can select a grid with maxP imum nij2E;i¼1 xij , it may be closer to the optimal solution. After fixing the grid, we will match the sensors with the grid nodes. In this matching, we will first match grid with the smaller degree which has not been used, but for the same degree of the grid points, we will firstly select the sensor whose flow is the maximum value from the grid and set x0ij ¼ 1, and match the grid point one by one. If all x0ij ¼ 1 for the grid points of some barrier, it means that a barrier coverage has been formed, set xGij ¼ 1. Otherwise, set x0ij ¼ 0; xGij ¼ 0. Algorithm 4 represents the basic steps about the improved round-up algorithm. Algorithm 4. Improved round-up algorithm for problem P2 1: Solve the Linear relaxation problem Eq. (16)–(20), 22 and obtain a fractional optimal solution fxLij ; yLj ; zLq g.
In the simple round-up algorithm, after ordering the optimal solution of the relaxation LP non-incrementally, the algorithm checks whether form barriers, if there is no barrier it will continue working until forming barriers. Thus, we can get at least one barrier coverage.
2: For each q 2 f1; . . . ; kg Select the largest W Y q ¼
Pl1 þþlq
j¼l1 þþlq1
Proof. We prove it by contradiction. If not, i.e., there is an xGij ¼ 1, but xLij P k xGij . Since xGij ¼ 1, the ðl1 þ l2 þ þ lk Þ largest edges have not common vertex xLij , there are at most ðl1 þ l2 þ þ lk Þ 1 edges such that xLij 6 k xGij . Consequently, there are at least ðn ðl1 þ l2 þ þ lk Þ þ 1Þ sensors such that xLij P k xGij . So, þþlk n l1 þl2X X i¼1
xLij > l1 þ l2 þ þ lk 1 þ k ðn ðl1 þ l2 þ þ lk0 Þ þ 1Þ
ij2E;i¼1 xij
For each t 2 fl1 þ þ lq1 ; . . . ; l1 þ þ lq g (a) set x0ij ¼ 1 for the minimum degree node in the t-th grid, and
Lemma 2. For any zGq ¼ 1, we must have xLij 6 k xGij , where k is the maximum number of barrier coverage
Pn
(b) set x0ij ¼ 1 for the maximum
Pl1 þþlq
x j¼l1 þþlq1 ij
for the
same degree of grid point. 3: (a) set xGij ¼ 1; zGq ¼ 1 for each x0ij ¼ 1 in the grid t, and (b) zGq ¼ 0.
5. Simulation results In this section, we demonstrate the effectiveness of proposed algorithms through simulations.
j¼1 l1 þl2 þþlk
P l1 þ l2 þ þ lk0 ¼
X
0
yj
ð23Þ
j¼1
It contradicts with that fxLij g is feasible to the LP relaxation. h Theorem 3. The simple round-up algorithm has an approximation ratio k , where k is the maximum number of barrier coverage. Proof. Suppose that aG is the objective function value obtained by the simple round-up algorithm, aL is the optimal objective function value for the LP relaxation, and a is the optimal objective function value. By Lemma 1, when xGij ¼ 1, we have xLij 6 k 6 k xGij , and while G G L L xij ¼ 0, we have xij 6 0 6 k xij . Therefore, we have xij 6 k xGij , for all i and j. Then
a ¼
l1 þþl k k X Xq 1X zq 6 aL ¼ lq q¼1 j¼l þþl q¼1 1
6
l1 þþl k Xq 1X lq q¼1 j¼l þþl 1
q1
yLj 6
q1 þ1
l1 þþl k Xq 1X lq q¼1 j¼l þþl 1
n X xLij
q1 þ1
i¼1
k n X k X k xGij ¼ lq zG ¼ k aG lq q¼1 q þ1 i¼1
ð24Þ
5.1. Performance of problem P1 We develop a custom simulator and use it to evaluate the performance of our algorithm. In the simulation, we randomly deploy sensors in a rectangular barrier with 100 units long and 50 units wide. Each sensor has 4 different sensing power levels f0; 16:0; 36:0; 64:0g. Without loss of generality, the sensing ranges are the square root of the power levels in the simulation, i.e., each sensor has 4 different sensing ranges f0; 4:0; 6:0; 8:0g. In Lagrangian Relaxation algorithm, we set the ¼ 0:0001. Since time cost for converging to a feasible solution is very high, we terminate the algorithm after 100 iterations and get a lower bound of an instance. In experiments, we will consider the effect of factors such as belt length, sensor density and the number of required barrier coverage. By measuring the total energy cost of our algorithm and Lagrangian Relaxation algorithm, we evaluate the performance of our constructing k-barrier coverage schemes. 5.1.1. Varying number of sensors. In this simulation, we set the number of sensors from 100 to 220 with an increment of 20. Fig. 4 shows the total energy cost of the algorithms with different number of sensors. We can observe that the total energy cost of Heuristic algorithm is at most two times of the lower bound of
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Heuristic Lagrangian
1600 1400 1200 1000 800 600 400 200 0
Total Energy Cost
Total Energy Cost
Fig. 3. An example of grid matching problem.
100
120
140
160
180
200
Heuristic Lagrangian
4000 3500 3000 2500 2000 1500 1000 500 0 100
120
140
160
180
200
Number of Sensors
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Fig. 4. Total energy cost vs. number of sensors.
the problem got by Lagrangian algorithm. This shows the proposed heuristic algorithm is very efficient. 5.1.2. Varying length of belt region. To study the effect of the length of belt region, we vary the length of belt region from 100 to 200 with an increment of 20. The number of sensors is set to 250. As shown in Fig. 5, the total energy cost increases when the length of belt region increases. This is because every path has to cover longer region. For different length of belt region, the total energy cost consumed by Heuristic algorithm is at most two times of the lower bound of the optimal solution. Above simulation results show that the proposed heuristic for P1 is efficient. 5.2. Performance of problem P2 We compare our algorithms with the upper bound which is the optimal solution of linear programming relaxation to evaluate our algorithms’ performance. We study how the number of formed barrier is affected by the four parameters: the width of the region, the number of sensors, the coverage radius and the mobile range. In the simulation, we randomly deploy sensors in a rectangular region with 200 130, we set r0 ¼ dl=ð2 rÞe and k0 ¼ bn=r0 c, and arrange k0 r0 grid points, in which there are k0 rows, each row has r 0 points. We select ðr; 3r; . . . ; ð2r0 1ÞrÞ as x-coordinates of points in each row, and ð1; . . . ; 2r0 ), (r 0 ,. . .,3r 0 ),. . ., (ðk0 1Þr0 ,. . ., n) as y-coordinates of the r0 columns respectively. Fig. 6–9 illustrate the experiment results, each of results is the average of 100 separate runs. 5.2.1. Effect of number of sensors. We first study the effect of number of sensors on barrier coverage. We randomly deploy n (n 6 250) mobile sensors in four rectangle regions with different size: 200 60, 200 80, 200 100 and 200 130, respectively.
The sensing range of each sensor is set to 10 and the moving range is set to 20. We compare the number of the formed barriers by Algorithm 4 (denoted as Greedy) with the upper bound, which is the optimal solution of LP relaxation. As shown in Fig. 6, the number of barriers computed by our greedy algorithm and LP OPT increase with the number of nodes increasing when fixing sensing range and moving range. It is obvious that our algorithm is close to the upper bound.
5.2.2. Effect of coverage radius. In this experiment, we randomly deploy 140 mobile sensors in two different rectangle regions: 200 130 and 200 150, respectively. We vary the sensing range from 1 to 25. The mobile range is set to 20. Fig. 7 shows the relationship between the barrier coverage and the sensing range of mobile sensors. It can be observed that the barrier coverage is 0 and the optimal solution of linear programming relaxation is 0.33 when the sensing range is equal to 1. As we increase the sensing range to some value, a barrier can be formed successfully, and if we continue to increase the sensing range, the quantity of barriers rises up rapidly. It is because when the sensing range becomes greater, more area will be covered by a sensor, and the sensing region of adjacent sensors may be overlapped, the probability of successfully forming a barrier will be greater. For fixed number of sensors, the setting with larger sensing range always can yield more barriers.
5.2.3. Effect of region width. In this experiment, we randomly deploy 80, 140, and 220 mobile sensors in a rectangle, respectively. The width of the region varies from 30 to 130 and the length of the region is 200. The sensing range is set as 10 and the mobile range is set as 20. It can be observed from Fig. 8 that the setting with smaller region width always forms more barriers when the number of
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Heuristic Lagrangian
350
Total Energy Cost
Total Energy Cost
400 300 250 200 150 100 50 0 100
120
140
160
180
200
900 800 700 600 500 400 300 200 100 0
Heuristic Lagrangian
100
120
Length of Belt Region
140
160
180
200
Length of Belt Region
Fig. 5. Total energy cost vs. length of belt region.
15 10 5
15
95 11 5 13 5 15 5 17 5 19 5 20 5 22 5 24 5
75
55
35
15
0
LP_OPT Greedy
95 11 5 13 5 15 5 17 5 19 5 20 5 22 5 24 5
20
20 18 16 14 12 10 8 6 4 2 0
55 75
LP_OPT Greedy
35
Barrier Coverage
Barrier Coverage
25
Number of sensors
14
18 16 14 12 10 8 6 4 2 0
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LP_OPT Greedy
12
LP_OPT Greedy
10 8 6 4 2
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95 11 5 13 5 15 5 17 5 19 5 20 5 22 5 24 5
55 75
15 35
95 11 5 13 5 15 5 17 5 19 5 20 5 22 5 24 5
75
55
0 35
15
Barrier Coverage
Number of sensors
Number of sensors
Fig. 6. Effect of number of sensors.
Fig. 7. Effect of coverage radius.
sensors is fixed. With the width becoming larger gradually, the quantity of barriers decrease.
5.2.4. Effect of mobile range. In this experiment, the mobile range of sensors varies from 0 to 35. The coverage range is set as 20. Without loss of generality, we test two different settings described as follows:
1. 100 mobile sensors are randomly deployed in a 200 130 rectangular area. 2. 200 mobile sensors are randomly deployed in a 200 60 rectangular area. As the Fig. 9 illustrated, when mobile range is equal to 0, the barrier coverage is 0 and the optimal solution of linear programming relaxation is close to 0, i.e., when the sensors are stationary,
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9 8 7 6 5 4 3 2 1 0
LP_OPT
Barrier Coverage
Barrier Coverage
H. Ma et al. / Computer Communications 35 (2012) 1749–1758
Greedy
16 14 12 10 8 6 4
LP_OPT Greedy
2 30
40
50
60
70
80
0
90 100 110 120 130
30
40
Region Width
50
60
70
80
90 100 110 120 130
Region Width
Barrier Coverage
25 LP_OPT Greedy
20 15 10 5 0
30
40
50
60
70
80
90 100 110 120 130
Region Width
Fig. 8. Effect of region width.
Fig. 9. Effect of mobile range.
barrier coverage cannot be achieved with the two settings. With the mobile range of sensors increasing, the number of barriers increases rapidly and get close to the optimal solution of linear programming relaxation eventually. Above simulation results show that our greedy algorithm is very close to the upper bound of the problem. So our greedy algorithm is efficient. 5.3. Mobility vs. stationary We also study how the number of formed barriers is affected by the mobility. When all the sensors are deployed in the region and keep stationary, we can get maximum number of formed barriers based on the maximum flow corresponding this deployment. We use MCF to represent the maximum number of formed barriers in the corresponding stationary deployment. When the sensors are allowed with limited mobility, we use Algorithm 4 to find the number of formed barriers, which is denoted as Greedy. To compare the above two states: mobility and stationary, we study how the number of formed barrier is affected by the three parameters: the mobile range, the width of the region, the number of sensors. In the following experiments, we set the same grid points as that mentioned in Section 5.2. 5.3.1. Effect of mobile range. In this experiment, we randomly deploy 200 mobile sensors in a rectangle region: 200 100. The cov-
erage range of the sensors is set as 10, and the mobile range varies from 11 to 25. As the Fig. 10(a) illustrated, the number of formed barrier (MCF) is 3. While by Greedy, with the increasing of the mobile range of sensors, the number of barriers increase rapidly, which is much larger than MCF in stationary state.
5.3.2. Effect of region width. In the experiment, we randomly deploy 200 mobile sensors in a rectangle. The length of the region is 200 and the width of the region varies from 40 to 100. The sensing range is set as 10 and the mobile range is set as 15. It can be observed from Fig. 10(b) that the setting with smaller region width always forms more barriers when the number of sensors is fixed, and the sensors with limited mobility can form more barriers. When the width becoming larger gradually, the quantity of barriers decrease.
5.3.3. Effect of number of sensors. In this experiment, we randomly deploy n (n 6 200) mobile sensors in a rectangle region: 200 100. The sensing range of each sensor is set to 10 and the moving range is set to 15. As shown in Fig. 10(c), with the increasing of the number of sensors, the number of formed barriers increases rapidly. Compared with stationary senors, senors with limited mobility could form more barriers.
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MCF
12
Barrier Coverage
Barrier Coverage
14
Greedy
10 8 6 4 2 0
16 14 12 10 8 6 4
MCF Greedy
2 11
13
15
17
19
21
23
0
25
40
60
Barrier Coverage
Mobile Range
10 9 8 7 6 5 4 3 2 1 0
80
100
Region Width
MCF Greedy
100
120
140
160
180
200
220
Number of sensors
Fig. 10. MCF vs. greedy.
From the simulations in Section 5.3, we know that if the sensors are allowed with limited mobility, the number of formed barriers can increase greatly. The above simulation results demonstrate our algorithms are efficient. 6. Conclusion In this paper, we study k-barrier coverage problem in wireless sensor networks. Firstly, we study the minimum energy cost k-barrier coverage problem P1 and formulate P1 into a minimum cost flow problem with side constraints. We use the classical Lagrangian algorithm to find a lower bound of the total energy cost, and we propose a heuristic algorithm for this problem. Simulation results demonstrate that the total energy cost consumed by the heuristic algorithm is at most two times of the lower bound of the problem. Then we study the maximum k-barrier coverage problem P2, where some sensors can move within limited range. We formulate the problem into an integer linear programming (ILP) and relax it into linear programming (LP). We propose two algorithms based on the LP relaxation. Simulation experiments show that our algorithms are efficient. Acknowledgments This research was jointly supported in part by National Natural Science Foundation of China under Grants 61070191 and 91124001, the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China under Grant 10XNJ032. And Research Fund for the Doctoral Program of Higher Education of China under Grant 20100004110001. References [1] I. Akyildiz, W. Su, Y. Sankarasubramaniam, E. Cayirci, A survey on sensor networks, IEEE Communications Magazine 40 (8) (2002) 102–114. [2] B. Liu, O. Dousse, J. Wang, A. Saipulla, Strong barrier coverage of wireless sensor networks, in: Proceedings of the ACM MobiHoc’08, 2008, pp. 411–420.
[3] A. Chen, S. Kumar, T.H. Lai, Designing localized algorithms for barrier coverage, in: Proceedings of ACM MobiHoc’07, 2007, pp. 63–74. [4] R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows: Theory, Algorithms, and Applications, Prentice-Hall, 1993. [5] M. Cardei, M.T. Thai, Y. Li, W. Wu, Energy-efficient target coverage in wireless sensor networks, in: Proceedings of IEEE INFOCOM’05, 2005, pp. 1976–1984. [6] Z. Zhou, S. Das, H. Gupta, Connected k-coverage problem in sensor networks, in: Proceedings of the 13th Annual Conference of the IEEE International Conference on Computer Communications and Networks(ICCCN), 2004. [7] D. Gage, Command control for many-robot systems. in: Proceedings of the 19th Annual AUVS Technical Symposium (AUVS-92), 1992. [8] A. Saipulla, C. Westphal, B. Liu, J. Wang, Barrier coverage of line-based deployed wireless sensor networks, in: Proceedings of the IEEE INFOCOM’09, 2009, pp. 127–135. [9] S. Kumar, T.H. Lai, J. Balogh, On k-coverage in a mostly sleeping sensor network, in: Proceedings of the ACM MobiCom’04, 2004, pp. 144–158. [10] S. Kumar, T.H. Lai, A. Arora, Barrier coverage with wireless sensors, in: Proceedings of the ACM MobiCom’05, 2005, pp. 284–298. [11] K. Ssu, W. Wang, F. Wu, T. Wu, K-barrier coverage with a directional sensing model, in: Proceedings of the International Journal on Smart Sensing and Intelligent Systems, 2009, pp. 75–83. [12] H. Yang, D. Li, Q. Zhu, W. Chen, and Y. Hong, Minimum energy cost k-barrier coverage in wireless Sensor networks, In WASA2010, pp. 80–89. [13] A. Saipulla, B. Liu, G. Xing, X. Fu, J. Wang, Barrier coverage with sensors of limited mobility, in: Proceedings of ACM MobiHoc’10, 2010, pp. 201–210. [14] C. Shen, W. Cheng, X. Liao, S. Peng, Barrier coverage with mobile sensors, in: Proceedings of the IEEE The International Symposium on Parallel Architectures, Algorithms, and Networks (ISPAN), 2008, pp. 99–104. [15] S. Chellappan, W. Gu, X. Bai, D. Xuan, B. Ma, K. Zhang, Deploying wireless sensor networks under limited mobility constraints, in: Proceedings of the IEEE Transactions on Mobile Computing (TMC), vol. 6 (10), 2007, pp. 1142– 1157. [16] D. Ban, W. Yang, J. Jiang, J. Wen, W. Dou, Energy-efficient algorithms for kbarrier coverage in mobile sensor networks, International Journal of Computers Communications and Control, 1841-9836 5 (5) (2010) 616–624. [17] M. Fisher, The Lagrangian relaxation method for solving integer programming problems, Management Science 27 (1) (1981) 1–18. [18] M. Held, P. Wolfe, H. Crowder, Validation of subgradient optimization, Mathematical Programming 6 (1) (1974) 62–88. [19] Ralph E. Huschke, Glossary of Meteorology, American Meteorological Society, Boston, 1959. [20] S. Mathies, P. Mevert, A hybrid algorithm for solving network flow problems with side constrains, Computers and Operations Research 25 (9) (1998) 745– 756.
Contents lists available at SciVerse ScienceDirect
Computer Communications journal homepage: www.elsevier.com/locate/comcom
Energy efficient k-barrier coverage in limited mobile wireless sensor networks Huan Ma, Deying Li ⇑, Wenping Chen, Qinghua Zhu, Huiqiang Yang School of Information, Renmin University of China, Beijing 100872, PR China
a r t i c l e
i n f o
Article history: Available online 4 May 2012 Keywords: k-barrier coverage Sensing power level Energy efficient Mobility Wireless sensor network
a b s t r a c t Energy cost and reliability are two main concerns in barrier coverage for wireless sensor networks. In this paper, we take the energy cost and reliability as objectives respectively to study two problems of k-barrier coverage: the minimum energy cost k-barrier coverage problem in static wireless sensor networks and the maximum k-barrier coverage problem in limited mobile wireless sensor networks. For the minimum energy cost k-barrier coverage problem, all sensors are stationary, and each sensor has l + 1 sensing power levels in the network, the objective of the problem is to find a sensing level assignment to form k-barrier coverage such that the total power consumed by the k-barrier is minimized. We firstly transform it into a minimum cost flow problem with side constraints and use Lagrangian relaxation technique to solve the minimum cost flow problem. Then, we also propose a heuristic algorithm. For the maximum k-barrier coverage problem, each sensor can move within the limited range, the objective of the problem is to form more barriers while some sensors can move within limited range. We formulate the problem into an integer linear programming (ILP), then propose two heuristic algorithms based on the linear programming (LP) relaxation. The simulation results demonstrate our algorithms are efficient. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Barrier coverage is one of the most important issues in wireless sensor networks [1], which was originally motivated by military applications. It is an efficient way to guard boundaries of critical infrastructures or assets, such as battlefield and country borders [2]. In wireless sensor networks, a barrier is formed by a set of sensors such that any intruder crossing this region can be detected by at least one sensor in the barrier. Compared with full coverage, barrier coverage requires significantly fewer sensors and less energy. The authors of [3] proved that if the width of the deployment region is three times the sensing range, full coverage requires more than twice the sensor density of barrier coverage. Hence barrier coverage is more cost effective and attractive for intrusion detection and border surveillance applications. In order to increase the reliability of barriers, a certain degree of redundancy is required. k-barrier coverage is proposed to address this issue. The k-barrier coverage makes that the intruders can be detected by at least k sensors while they move along any crossing paths from one boundary to another. If one barrier fails, the other ðk 1Þ barriers can detect the intruders. However, whether k-barrier coverage can be formed depends on a number of parameters, such as the sensor density, sensor deployment and so on. In many ⇑ Corresponding author. Tel.: +86 010 62519223; fax: +86 010 62511184. E-mail addresses: [email protected] (H. Ma), [email protected] (D. Li), [email protected] (W. Chen), [email protected] (Q. Zhu), yhq@ruc. edu.cn (H. Yang). 0140-3664/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comcom.2012.04.021
existing works on k-barrier coverage, the sensing power of each sensor is supposed to be fixed no matter the distance between sensors is long or short. It may result in the waste of energy. Hence, constructing energy efficient k-barrier from randomly deployed sensors in a region is a challenging problem. Most studies on k-barrier coverage mainly focused on how to construct barriers in static sensor networks where each sensor is assumed to be stationary. However, when the sensor density is not high enough, the number of formed barrier coverage may be too few to achieve application requirements. Compared with static wireless sensor network, sensors with limited mobility can move to some desired locations to form stronger barriers at the cost of only a little energy cost. How to move the mobile sensors to form more barriers is also a challenging problem. In this paper, we study the minimum energy cost k-barrier coverage problem (P1) in static wireless networks and the maximum kbarrier coverage problem (P2) in limited mobile wireless sensor networks. For the problem P1, each sensor has several sensing power levels and each sensing power level corresponds to a sensing range. We need to assign the sensing power level to each sensor such that the sensor network can form a k-barrier coverage. Our goal is finding such an assignment that the total sensing power is minimized. To solve this problem, we transform it into the minimum cost network flow problem with side constraints. By using the Lagrangian relaxation technique [4], we get a lower bound of the problem. We also design a heuristic algorithm. For the problem P2, there is a given set of mobile sensors in which each sensor has limited moving range, our goal is finding k-barrier coverage such that k is
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maximized. To solve the problem P2, we first transform the problem into an integer linear programming. Then, two heuristic algorithms based on the relaxation linear programming are proposed to find disjoint barriers in a target region. The main contributions of this paper are as follows. We formulate the minimum energy k-barrier coverage problem (P1) as the minimum cost network flow problem with side constraints and find a lower bound of the total energy cost by the classical Lagrangian algorithm. We propose a heuristic algorithm for P1 whose total energy cost is at most two times of the lower bound of the optimal solution. We formulate the maximum k-barrier coverage problem with limited moving range (P2) as an integer linear programming and find an upper bound of the problem. We employ the relaxation rounding technique to design two heuristic algorithms for P2. The rest of this paper is organized as follows. Section 2 reviews the related work. Section 3 addresses the minimum energy k-barrier coverage problem. In Section 4 we study the maximum k-barrier coverage problem and present two heuristic algorithms. The simulation results are given in Section 5. Finally, Section 6 concludes the paper.
2. Related work The traditional coverage problems, including point coverage [5] which requires every target point to be covered by at least one or k sensors, and area coverage [6] which requires every point in the area to be covered by at least one or k sensors, have been studied intensively. The concept of barrier coverage was first introduced by Gage in robotic sensors [7]. Chen et al. [3] proposed a localized barrier coverage protocol to detect all intruders whose movement are confined to a slice of the original strip region. The localized algorithm, however, only provides barrier coverage for slices of bounded length. It does not protect the network against intruders which can move beyond the range of a slice. Saipulla et al. [8] studied the barrier coverage for line-based deployments and established a tight lower-bound. Liu et al. [2] showed that in a rectangular area with width w and length l for w ¼ Xðlog lÞ, there exist, with high probability, multiple disjoint barriers across the entire length of the area, and proposed an efficient distributed algorithm to construct multiple disjoint barriers in a long boundary area of an irregular shape. Kumar et al. [9] studied k-barrier coverage problem in which sensors were deployed deterministically. Later, Kumar et al. [10] introduced weak coverage and proved equivalence conditions that deduce efficient algorithms for determining whether a given belt region is k-barrier covered. Moreover, they derived the critical conditions for weak barrier coverage with high probability in a belt region. Ssu et al. [11] studied the k-barrier coverage problem in directional sensing model and developed a distributed algorithm to construct k-barrier coverage in randomly deployed directional sensor fields. Since sensors are battery powered and it is difficult to replace the batteries, sensors may be designed to adjust sensing range adaptively to save energy consumption. However, all the above works assumed that the sensing powers of all sensors were fixed. In this paper, one of problems we study is minimizing the total energy cost of k-barrier coverage, where sensors are stationary and have multiple sensing power levels. Some results about this problem are shown in [12]. There are also much effort on investigating how to move sensors to form more barriers. Saipulla et al. [13] presented a sensor
mobility scheme that constructs the maximum number of barriers with minimum sensor moving distance. Shen et al. [14] proposed a virtual force based on heuristic algorithm to relocate mobile sensors to form barriers. Barrier coverage is affected by the deployment region of sensors. Chellappan et al. [15] studied the deployment problem which schedule the movement of the sensors to minimize the number of moving sensors and simultaneously minimize the sensor movements. Ban et al. [16] focused on the problem how to relocate mobile sensors to construct k sensor barriers with minimum total moving distance. The authors modeled the problem as integer linear programming (ILP) and proposed an approximation algorithm AHGB to construct one energy-efficient sensor barrier. Based on AHGB, a Divide-and-Conquer algorithm was proposed to achieve k-barrier coverage for large sensor networks. Most existing works for barrier coverage are based on regions with long belt shape. In this paper, to adapt more practical application, we study how to move the mobile sensors to form maximum number of barriers under that each mobile sensor only can move within a limited range on arbitrary region. 3. Minimum energy cost k-barrier coverage In this section, we address the minimum energy cost k-barrier coverage problem in wireless sensor networks. We assume that a set S of sensors are deployed randomly in a two-dimensional rectangular area where sensors do not move after initial deployment. Each sensor si has its coordinates si ðx; yÞ, which may be detected through GPS or obtained through a localization mechanism. We use si :x to represent x-coordinate of si . Suppose each sensor node has l þ 1 sensing power levels, 0 6 p1 6 p2 6 pl , i.e., each sensor has l þ 1 different corresponding sensing ranges f0; r 1 ; r 2 ; . . . ; rl g. Every active sensor is assigned with one sensing power level PLðsi Þ 2 f0; p1 ; p2 ; . . . ; pl g. We define the total energy P cost of the network as si 2S PLðsi Þ. We give some definitions as follows: Definition 1 (Crossing path (or Incursive path)). A path is said to be a crossing (or incursive) path in a belt region if it crosses the complete width of the region from one side to the other side. An example is illustrated in Fig. 1. The two red sketch paths are incursive paths while the two blue paths are not crossing paths. Definition 2. A crossing path P is said to be k-covered if P intersects with at least k active sensor’s sensing disks. Definition 3 (k-barrier coverage). A belt region is said to be k-barrier covered by a sensor network deployed over it if and only if all the crossing paths through the belt region are k-covered by this sensor network.
Fig. 1. An example of k-barrier coverage.
H. Ma et al. / Computer Communications 35 (2012) 1749–1758
Fig. 1 shows an example of 3-barrier coverage. Once intruders cross the areas, they will be detected at least 3 times. Definition 4 (Minimum energy k-barrier coverage problem). Given a sensor network over the objective belt region and each sensor has l þ 1 sensing power levels, find a sensing power level assignment such that the given belt region is k-barrier covered and the total energy cost is minimized. The goal of the problem is to find a sensing power level assignment for all sensors, i.e., a set of sensing power levels fPLðs1 Þ; PLðs2 Þ; . . . ; PLðsn Þg such that the sensor network can form a k-barrier coverage and the total energy cost is minimum among all the assignments. 3.1. Lagrangian method for minimum energy cost k-barrier coverage problem Given a set of sensors S ¼ fs1 ; s2 ; . . . ; sn g, each sensor node si has l þ 1 sensing power levels, 0 6 p1 6 p2 6 6 pl . We consider the minimum energy cost k-barrier coverage problem in a long belt, where sensors are deployed randomly over it. Before solving the minimum energy cost k-barrier coverage problem, we first restate the method to determine whether a belt region is k-barrier covered [10]. Corresponding to a sensor network deployed in a belt region, we derive a coverage graph CG ¼ ðV; EÞ, where V is the set of all sensor nodes plus two virtual vertices s and t (see Fig. 2). The set of edges E is derived as follows: each pair of sensors whose sensing disks overlap is connected by an edge. Additionally, the sensors whose sensing disks intersect with the left boundary are connected to node s and the sensors whose sensing disks intersect with the right boundary are connected to node t. An example of constructed coverage graph is shown in Fig. 2. Lemma 1 [10]. An open belt region is k-barrier covered if and only if s and t are k-connected in the corresponding coverage graph, CG. In order to solve the minimum energy cost k-barrier coverage problem, we construct an auxiliary weighted directed graph Ga ¼ ðV a ; Ea ; C a ; U a Þ, where V a denotes the set of nodes, Ea denotes the set of edges, C a denotes the set of edges’ cost, and U a denotes the set of edges’ capacity. First, we construct l node pairs fqti;1 ; qhi;1 g, fqti;2 ; qhi;2 g; . . . ; fqti;l ; qhi;l g for each sensor si in S. For each node pair fqti;k ; qhi;k g, we add a directional edge from qti;k to qhi;k with cost pk . We add a directed edge ðqhi;k ; qtj;m Þ with cost 0 from qhi;k to qtj;m if and only if si ’s sensing range by sensing power level pk intersects with sj ’s sensing range by sensing power level pm . In addition, we add a new virtual node s at the left side of the belt region and a new virtual node t at right side of the belt region. There is a directed edge from s to qti;k with cost 0 if and only if si ’s sensing range by sensing power level pk intersects with the left border of the belt region. Last, there is a directed edge from qhi;k to t with cost 0 if and only if si ’s sensing range by sensing power level pk intersects with the right border of the belt region. Each edge in Ea is associated with capacity 1.
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To formulate our problem rigorously, we use a binary variable xi;j to represent if there is a flow through an edge from i to j in Ga ¼ ðV a ; Ea ; C a ; U a Þ. For the simplicity of the mathematical representation of the problem, we use eðsi Þ to denote the set of edges fðqti;1 ; qhi;1 Þ; ðqti;2 ; qhi;2 Þ; . . . ; ðqti;l ; qhi;l Þg for each sensor si in S. Let X denote the column vector of variables corresponding to each edge (i.e., X ¼ fxi;j jði; jÞ 2 Ea g). Let C 0 denote the row vector of the cost of edges in the same order with X (i.e., C ¼ fci;j jði; jÞ 2 Ea g). Let C be the transpose of vector C 0 . Our problem can be formulated by an integer programming problem as follows:
min C X
ð1Þ
subject to
X
xi;v
ði;v Þ2Ea
X
xu;i ¼ 0;
ðu;iÞ2Ea
for each node i 2 V a fs; tg X xs;v ¼ k
ð2Þ ð3Þ
ðs;v Þ2Ea
X
xu;t ¼ k
ð4Þ
ðu;tÞ2Ea
X
xu;v ¼ 1;
ðu;v Þ2eðsi Þ
for each edge set eðsi Þ xi;j ¼ 0; 1
ð5Þ ð6Þ
From the formulation, it is easy to know that if there is not constraint (5), the problem is a classical minimum cost flow problem. Adding constraint (5), the problem becomes the minimum cost network flow problem with side constraint (5). From Lemma 1 and the structure of the auxiliary graph, we have the following result. Theorem 1. The minimum energy cost k-barrier coverage problem is equivalently transformed to the minimum cost k-flow problem with side constraints. We express the above formulation in matrix form as follows:
min C X
ð7Þ
subject to
AX ¼ b DX þ Ir S ¼ d 0 6 X 6 1; S P 0
ð8Þ ð9Þ ð10Þ
where constraint (8) corresponds to constraints (2)–(4), and A is the ðjV a j þ 2; jV a j þ 2Þ coefficient matrix which is composed by the left coefficient of the three constraints. If ði; v Þ 2 Ea , the coefficient element is 1; if ðu; iÞ 2 Ea , the coefficient element is 1; if ðs; v Þ 2 Ea , the coefficient element is 1; if ðu; tÞ 2 Ea , the coefficient element is 1, else in the other situations, the coefficient element is 0. Constraint (9) corresponds to constraint (5), and D is the ðjV a j; jV a jÞ coefficient matrix which can be got from the left coefficient of (5). If
Fig. 2. A simple example of coverage graph.
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ðu; v Þ 2 eðsi Þ, the coefficient element is 1, else 0; Ir is ðr; rÞ-identity matrix and r equal to jSj. We can use the Lagrangian relaxation algorithm in [4] to obtain a lower bound of our problem. Lagrangian relaxation has been used in various approaches to solve the constrained capacitated network flow problems [7,17,18,20]. The basic idea is to relax the side constraints so that only a capacity network flow problem remains to be solved. Let X ¼ fxjAx ¼ b; 0 6 x 6 lg be the set of feasible solutions for the pure network part of Minimum Flow Structures (MFS). Then the associated Lagrangian subproblem is
LðlÞ ¼ minfCX þ lðDX dÞg where l 2 Rr is a row vector of Lagrange multipliers. For every l P 0, the value of the Lagrangian function LðlÞ is a lower bound for the objective function value of problem (MFS). In order to find an optimal solution of MFS, the Lagrangian dual L ¼ maxlP0 LðlÞ must be solved. As LðlÞ is a concave, non-differentiable function of l, a subordinate search method is employed to find L , see [17] for details. Algorithm 1 represents the basic steps about the Lagrangian relaxation as follows: Algorithm 1. The basic algorithm for the Lagrangian relaxation
the sensor si ’s sensing range intersects with the left border of the belt region when si working at the maximum sensing power level pl . And there is a directed edge from qhi to t with cost 0, if and only if si ’s sensing range intersects with the right border of the belt region when si working at the maximum sensing power level pl . Each edge in Ec is associated with capacity 1. Then we use the minimum cost flow algorithm on Gc to find k node-disjoint paths from s to t. And we adjust the sensing power level of each sensor node in each path to make this path form 1-barrier coverage. According to Lemma 1, we know that the found paths form a k-barrier coverage in the sensor network. Note that we use Rðpi Þ to denote the sensing range corresponding to the sensing power level pi , and let LEN represent the length of the belt region. The heuristic algorithm is formally represented as follows: Algorithm 2. Heuristic algorithm for problem P1 1: Construct an auxiliary graph Gc ¼ ðV c ; Ec ; C c Þ. 2: Using minimum cost flow algorithm in [4] to find kminimum cost flow from s to t. n o 3: For each found path P i ¼ s; qti1 ; qhi1 ; . . . ; qtim ; qhim ; t , assign a power level for each sensor node as follows: PLðsi1 Þ ¼ minfpk jr k P maxðsi1 :x; distðsi1 ; si2 Þ r l Þg; PLðsij Þ ¼ minfpk jr k P maxðdistðsij1 ; sij Þ RðPLsi
j1
1: Initialization: Set iteration count k ¼ 0; Lagrange multipliers l1 ¼ 0; lower bound LB ¼ 1; tolerance e; determine an upper bound UB for MFS. 2: Subproblem: Solve Lðlk Þ ¼ minfCX þ lk ðDX dÞg, let xk be the optimal solution corresponding to Lðlk Þ, then ðDxk dÞ is a subgradient of L at lk . 3: Optimality check: If Dxk 6 d and lk ðDxk dÞ ¼ 0 Stop: xk is an optimal solution of MFS. If Dxk 6 d Set UB ¼ minfUB; cxk g; set LB ¼ maxfLB; Lðlk Þg. If UB=LB 6 1 þ e Terminate. 4: Subgradient: Set k ¼ k þ 1; calculate lk ¼ ½lk1 þ hk ðDxk1 dÞþ , where ½lk1 þ hk ðDxk1 dÞþ is the positive part of ½lk1 þ hk ðDxk1 dÞ, and hk is the step size determined by the standard heuristic hk ¼ kk ðUB Lðlk1 ÞÞ=jjDxk1 djj2 and kk is a scalar between 0 and 2; go to 2.
3.2. Heuristic algorithm for minimum energy cost k-barrier coverage problem In this subsection, we propose a heuristic algorithm (Heuristic). In order to design this Heuristic, an auxiliary weighted directed graph Gc ¼ ðV c ; Ec ; C c Þ where V c denotes the set of nodes, Ec denotes the set of edges, and C c denotes the set of edges’ cost, is constructed as follows: we construct one node pair fqti ; qhi g for each sensor si originally in S. For each node pair fqti ; qhi g, we add a directed edge ðqti ; qhi Þ from qti to qhi with cost pl . We add a directed edge (qhi ; qtj ) with cost 0 from qhi to qtj if and only if the Euclidian distance between si and sj (i.e., distðsi ; sj Þ) is less than 2 r l , where r l is the largest sensing range. We also add a new node s at the left side of the belt region and a new node t at right side of the belt region. There is a directed edge from s to node qti with cost 0, if and only if
Þ; distðsij ; sijþ1 Þ r l Þg,
for2 6 j 6 m 1; PLðsim Þ ¼ minfpk jr k P maxðdistðsim1 ; sim Þ RðPLsim1 Þ; LEN sim :xÞg. In Heuristic, the time complexity of step 1 is OðnÞ. In step 2, using the minimum cost flow algorithm has the time complexity of Oðn3 Þ. Step 3 takes times of Oðn kÞ. Because k is always less than n, step 3 takes times at most Oðn2 Þ. Thus, the whole algorithm has time complexity of Oðn3 Þ. w4. The maximum k-barrier coverage problem In this section, we address problem P2, which is the maximum k-barrier coverage problem in limited mobile wireless sensor networks. This section is organized as follows: We first formulate the problem into an integer linear programming (ILP) problem, then propose two round-up algorithms based on LP relaxation. In the initial deployment, we assume that a set S of n mobile sensors are distributed uniformly at random in a large region, and the sensing range of each sensor is r. Without loss of generality, we assume that the region is two-dimensional belt area with length l. We also assume that each sensor knows its own location denoted by the coordinates (x; y). The location information can be known through a localization mechanism or GPS. Suppose there are k0 grids in the network model. There are lq grid points in the qth grid. The grid points are candidate positions of mobile sensors. Due to power constraint, the mobile sensors can move at most distance R after initial deployment. Therefore, the mobile sensors can move to a grid point as long as the distance between them is not more than R. One grid point only can deploy at most one sensor, and one sensor only can move to at most one grid point. After the movement, if each grid point in the qth grid is occupied by a sensor, these sensors can connect with each other to form a barrier. Our goal is how to move these sensors such that the number of formed barrier coverage is maximized. Definition 5 (Maximum k-barrier coverage problem). Given a sensor network over the objective region and each sensor can move at most R, find a moving strategy of these sensors such that the given region is k-barrier covered and k is maximized.
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Definition 6 ([19] Grid length). When n grid points are deployed in an objective region, the grid length is generally considered to be the minimum distance between two border regions through the grid points. Before designing our algorithms, we give the following theorem. Theorem 2. When n mobile sensors are deployed in an objective range with the minimum grid length l and each sensor covers a disk area of radius r, the number of sensors for any barrier coverage is at least dl=ð2 rÞe, and the number of formed barriers is at most bn=dl=ð2 rÞec, where r is the sensing radius. Proof. No matter what the shape of the objective range is, any barrier with the minimum number of sensors should have property that the sensing area of any two adjacent sensors are tangent with each other in the grid. That is, if the minimum grid length is l and each sensor covers a disk area of radius r, the minimum number of sensors in a barrier coverage is at least dl=ð2 rÞe, therefore, when there are n mobile sensors deployed in an objective range, the maximum number of barriers which can be formed is at most bn=dl=ð2 rÞec. h
For given bipartition graph G ¼ ðX; Y; EÞ; X represents sensor nodes and Y represents grid points. We add two new nodes S and D. For any x 2 X, there is an edge between x and S. For any y 2 Y, there is an edge between y and D. We define binary variables:
xij ¼
1; if sensor xi occupy yj ; 0; otherwise
ð11Þ
xSi ¼ 1;
8i ¼ 1; . . . ; n
ð12Þ
xjD ¼ 1;
1 6 j 6 l1 þ l2 þ þ lk0
ð13Þ
yj ¼
zq ¼
1; if there is a sensor in yj ; 0;
otherwise
ð14Þ
1; if each grid point on the qth grid is filled with a sensor; 0; otherwise ð15Þ
Then we have an ILP formulation as follows:
max
k0 X zq
ð16Þ
q¼1
Suppose there are k0 grids in which if each point of some grid is occupied by a sensor, then it can form a barrier. Once the positions of grids are identified, the network can be represented as a bipartition graph G ¼ ðX; Y; EÞ, while X represents a set of n mobile sensors, Y represents a set of grid points, where Y={Y 1 ; . . . ; Y i ; . . . ; Y k0 }, Y i ¼ fli1 þ 1; li1 þ 2; . . . li1 þ li g while l0 ¼ 0. For any sensor x and any grid point y, there is an edge between them if and only if the distance between them is at most R. The problem can be composed of the following two subproblems: Subproblem (a). Grid matching problem: the problem is to match mobile sensor set X with grid points set Y. Each mobile sensor can only occupy a grid point and each grid point can be filled by only one mobile sensor. If each grid point in grid Y i is matched with a mobile sensor in X; Y i is called as a barrier; Subproblem (b). Maximizing barriers problem: the problem is how to find a grid matching such that the number of formed barriers is maximized. Fig. 3(a) shows an example, the red nodes x1 ; . . . ; x6 represent the mobile sensors, and the black nodes y1 ; . . . ; y6 represent the grid points. The shaded areas are the coverage areas for sensors and the gray circles are the moving ranges of the mobile sensors. In this case, x1 can move to grid point y1 . x2 can move to grid point y1 ; y2 and y3 . x3 can move to grid point y3 and y4 . x4 can move to grid point y4 and y5 . x5 can move to grid point y5 and y6 . x6 can move to grid point y6 . (1) If x3 moves to y4 , the other mobile sensors: x1 moves to y1 ; x2 moves to y2 ; x4 moves to y5 ; x6 moves to y6 , we can get a barrier Y 2 by fx3 ; x4 ; x6 g, i.e., if x3 move to y4 , no matter how to move the other mobile sensors, we can get only one barrier coverage at most. (2) If x3 moves to y3 , the other mobile sensors: x1 moves to y1 ; x2 moves to y2 ; x4 moves to y4 ; x5 moves to y5 ; x6 moves to y6 , we can get two barriers Y 1 and Y 2 by fx1 ; x2 ; x3 g and fx4 ; x6 ; x8 g. Therefore, if we want to maximize the number of barriers, x3 have to move to y3 although the Euclidean distance between x3 and y4 is less than that between x3 and y3 . 4.1. ILP formulation For the convenience of discussion, we reintroduce our notations as follows: i: index of sensors, where 1 6 i 6 n. j: index of grid points, where 1 6 j 6 l1 þ l2 þ þ lk0 . q: index of barriers, where 1 6 q 6 k0 . lq : the number of grid points on the qth barrier.
subject to:
X
xSi ¼
81 6 i 6 m
ð17Þ
81 6 j 6 l 1 þ l 2 þ þ l k 0
ð18Þ
xij ;
ði;jÞ2E
xjD ¼
n X xij ; i¼1
2yj 6
n X xij þ xjD ;
81 6 j 6 l 1 þ l 2 þ þ l k 0
ð19Þ
i¼1
lq zq 6
l1 þþl Xq
yj ;
81 6 q 6 k 0
ð20Þ
j¼l1 þþlq1 þ1
xSi ; xij ; xjD ; yj ; zq 2 f0; 1g;
81 6 i 6 n; 1 6 j 6 l1 þ l2 þ þ lk0 ;
1 6 q 6 k0
ð21Þ
In the formulation, constraint (17) ensures the flow conservation for all sensors; Constraint (18) ensures the flow conservation for all grid nodes; Constraint (19) evaluates whether each grid node is occupied by a sensor; Constraint (20) ensures that the qth grid forms a barrier coverage if and only if each grid point on the qth grid is occupied by one sensor. Fig. 3(b) shows the matching graph based on Fig. 3(a). Nodes x1 ; . . . ; x6 in (b) are corresponding to the mobile sensors in (a). Nodes y1 ; . . . ; y6 in (b) are corresponding to the grid points in (a). There is an edge ðxi ; yj Þ in (b) if the distance between xi and yj is at most R. S and D are virtual nodes, 8xi 2 X, add an edge ðS; xi Þ to E; 8yj 2 Y, add an edge ðyj ; DÞ to E. Then we get the bipartition matching graph (b). 4.2. Simple round-up algorithm based on linear programming relaxation In this subsection, we will design a heuristic for P2. The heuristic includes two steps. Firstly, we relax the ILP to a IP as follows: removing the integral constraints in (21), i.e., (21) is replaced by
0 6 xSi 6 1;
0 6 xij 6 1; 0 6 xjD 6 1; 0 6 zq 6 1;
81 6 i 6 n; 1 6 j 6 l1 þ l2 þ þ lk0 ; 1 6 q 6 k0
ð22Þ
Then solve the relaxation LP to get an optimal solution for the relaxation LP. Secondly, a greedy round-up algorithm is employed
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to find an integral feasible solution for ILP based on the optimal solution of the relaxation LP. We use xLij to represent the optimal solution of the relaxation LP, where some xLij may not be integers, and use xGij to represent the solution obtained by rounding xLij . In our greedy round-up algorithm, we first sort xLij in the order of non-incremental in algorithm. Then, we remove the edges which have the common vertex, and set xGij ¼ 1; yGj ¼ 1 for the edges that have not common vertex. Finally, set zGq ¼ 1 if each yGj of the barrier is 1, i.e., the q-th barrier coverage has been formed when zGq ¼ 1. The basic steps about the simple round-up algorithm is represented formally as Algorithm 3. Algorithm 3. Simple round-up algorithm for problem P2 1: Solve the LP relaxation Eq. (16)–(20) and (22) to obtain a fractional solution {xLij ; yLj ; zLq }. 2: Set the xLij by order of non-incremental. (a) for each i 2 f1; . . . ; ng, each j 2 f1; . . . ; ng set x0ij ¼ 1; y0j ¼ 1 for the edges that have not common vertex, and (b) set other x0ij ¼ 0; y0j ¼ 0. 3: For each j, each q (a) set xGij ¼ 1; zGq ¼ 1 if each y0j in Y j is 1, and (b) set other zGq ¼ 0.
4.3. Improved round-up algorithm based on linear programming relaxation In this subsection, an improved round-up algorithm is proposed. Note that in Step 2 of the simple round-up algorithm, we select the maximum value of xLij with no common vertex to be rounded up to 1. However, it only considered the maximum xLij in the barrier. The simple round-up algorithm can be further improved. Since a barrier coverage needs that every grid point in the barrier is occupied by a sensor, we can select a grid with maxP imum nij2E;i¼1 xij , it may be closer to the optimal solution. After fixing the grid, we will match the sensors with the grid nodes. In this matching, we will first match grid with the smaller degree which has not been used, but for the same degree of the grid points, we will firstly select the sensor whose flow is the maximum value from the grid and set x0ij ¼ 1, and match the grid point one by one. If all x0ij ¼ 1 for the grid points of some barrier, it means that a barrier coverage has been formed, set xGij ¼ 1. Otherwise, set x0ij ¼ 0; xGij ¼ 0. Algorithm 4 represents the basic steps about the improved round-up algorithm. Algorithm 4. Improved round-up algorithm for problem P2 1: Solve the Linear relaxation problem Eq. (16)–(20), 22 and obtain a fractional optimal solution fxLij ; yLj ; zLq g.
In the simple round-up algorithm, after ordering the optimal solution of the relaxation LP non-incrementally, the algorithm checks whether form barriers, if there is no barrier it will continue working until forming barriers. Thus, we can get at least one barrier coverage.
2: For each q 2 f1; . . . ; kg Select the largest W Y q ¼
Pl1 þþlq
j¼l1 þþlq1
Proof. We prove it by contradiction. If not, i.e., there is an xGij ¼ 1, but xLij P k xGij . Since xGij ¼ 1, the ðl1 þ l2 þ þ lk Þ largest edges have not common vertex xLij , there are at most ðl1 þ l2 þ þ lk Þ 1 edges such that xLij 6 k xGij . Consequently, there are at least ðn ðl1 þ l2 þ þ lk Þ þ 1Þ sensors such that xLij P k xGij . So, þþlk n l1 þl2X X i¼1
xLij > l1 þ l2 þ þ lk 1 þ k ðn ðl1 þ l2 þ þ lk0 Þ þ 1Þ
ij2E;i¼1 xij
For each t 2 fl1 þ þ lq1 ; . . . ; l1 þ þ lq g (a) set x0ij ¼ 1 for the minimum degree node in the t-th grid, and
Lemma 2. For any zGq ¼ 1, we must have xLij 6 k xGij , where k is the maximum number of barrier coverage
Pn
(b) set x0ij ¼ 1 for the maximum
Pl1 þþlq
x j¼l1 þþlq1 ij
for the
same degree of grid point. 3: (a) set xGij ¼ 1; zGq ¼ 1 for each x0ij ¼ 1 in the grid t, and (b) zGq ¼ 0.
5. Simulation results In this section, we demonstrate the effectiveness of proposed algorithms through simulations.
j¼1 l1 þl2 þþlk
P l1 þ l2 þ þ lk0 ¼
X
0
yj
ð23Þ
j¼1
It contradicts with that fxLij g is feasible to the LP relaxation. h Theorem 3. The simple round-up algorithm has an approximation ratio k , where k is the maximum number of barrier coverage. Proof. Suppose that aG is the objective function value obtained by the simple round-up algorithm, aL is the optimal objective function value for the LP relaxation, and a is the optimal objective function value. By Lemma 1, when xGij ¼ 1, we have xLij 6 k 6 k xGij , and while G G L L xij ¼ 0, we have xij 6 0 6 k xij . Therefore, we have xij 6 k xGij , for all i and j. Then
a ¼
l1 þþl k k X Xq 1X zq 6 aL ¼ lq q¼1 j¼l þþl q¼1 1
6
l1 þþl k Xq 1X lq q¼1 j¼l þþl 1
q1
yLj 6
q1 þ1
l1 þþl k Xq 1X lq q¼1 j¼l þþl 1
n X xLij
q1 þ1
i¼1
k n X k X k xGij ¼ lq zG ¼ k aG lq q¼1 q þ1 i¼1
ð24Þ
5.1. Performance of problem P1 We develop a custom simulator and use it to evaluate the performance of our algorithm. In the simulation, we randomly deploy sensors in a rectangular barrier with 100 units long and 50 units wide. Each sensor has 4 different sensing power levels f0; 16:0; 36:0; 64:0g. Without loss of generality, the sensing ranges are the square root of the power levels in the simulation, i.e., each sensor has 4 different sensing ranges f0; 4:0; 6:0; 8:0g. In Lagrangian Relaxation algorithm, we set the ¼ 0:0001. Since time cost for converging to a feasible solution is very high, we terminate the algorithm after 100 iterations and get a lower bound of an instance. In experiments, we will consider the effect of factors such as belt length, sensor density and the number of required barrier coverage. By measuring the total energy cost of our algorithm and Lagrangian Relaxation algorithm, we evaluate the performance of our constructing k-barrier coverage schemes. 5.1.1. Varying number of sensors. In this simulation, we set the number of sensors from 100 to 220 with an increment of 20. Fig. 4 shows the total energy cost of the algorithms with different number of sensors. We can observe that the total energy cost of Heuristic algorithm is at most two times of the lower bound of
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Heuristic Lagrangian
1600 1400 1200 1000 800 600 400 200 0
Total Energy Cost
Total Energy Cost
Fig. 3. An example of grid matching problem.
100
120
140
160
180
200
Heuristic Lagrangian
4000 3500 3000 2500 2000 1500 1000 500 0 100
120
140
160
180
200
Number of Sensors
Number of Sensors
Fig. 4. Total energy cost vs. number of sensors.
the problem got by Lagrangian algorithm. This shows the proposed heuristic algorithm is very efficient. 5.1.2. Varying length of belt region. To study the effect of the length of belt region, we vary the length of belt region from 100 to 200 with an increment of 20. The number of sensors is set to 250. As shown in Fig. 5, the total energy cost increases when the length of belt region increases. This is because every path has to cover longer region. For different length of belt region, the total energy cost consumed by Heuristic algorithm is at most two times of the lower bound of the optimal solution. Above simulation results show that the proposed heuristic for P1 is efficient. 5.2. Performance of problem P2 We compare our algorithms with the upper bound which is the optimal solution of linear programming relaxation to evaluate our algorithms’ performance. We study how the number of formed barrier is affected by the four parameters: the width of the region, the number of sensors, the coverage radius and the mobile range. In the simulation, we randomly deploy sensors in a rectangular region with 200 130, we set r0 ¼ dl=ð2 rÞe and k0 ¼ bn=r0 c, and arrange k0 r0 grid points, in which there are k0 rows, each row has r 0 points. We select ðr; 3r; . . . ; ð2r0 1ÞrÞ as x-coordinates of points in each row, and ð1; . . . ; 2r0 ), (r 0 ,. . .,3r 0 ),. . ., (ðk0 1Þr0 ,. . ., n) as y-coordinates of the r0 columns respectively. Fig. 6–9 illustrate the experiment results, each of results is the average of 100 separate runs. 5.2.1. Effect of number of sensors. We first study the effect of number of sensors on barrier coverage. We randomly deploy n (n 6 250) mobile sensors in four rectangle regions with different size: 200 60, 200 80, 200 100 and 200 130, respectively.
The sensing range of each sensor is set to 10 and the moving range is set to 20. We compare the number of the formed barriers by Algorithm 4 (denoted as Greedy) with the upper bound, which is the optimal solution of LP relaxation. As shown in Fig. 6, the number of barriers computed by our greedy algorithm and LP OPT increase with the number of nodes increasing when fixing sensing range and moving range. It is obvious that our algorithm is close to the upper bound.
5.2.2. Effect of coverage radius. In this experiment, we randomly deploy 140 mobile sensors in two different rectangle regions: 200 130 and 200 150, respectively. We vary the sensing range from 1 to 25. The mobile range is set to 20. Fig. 7 shows the relationship between the barrier coverage and the sensing range of mobile sensors. It can be observed that the barrier coverage is 0 and the optimal solution of linear programming relaxation is 0.33 when the sensing range is equal to 1. As we increase the sensing range to some value, a barrier can be formed successfully, and if we continue to increase the sensing range, the quantity of barriers rises up rapidly. It is because when the sensing range becomes greater, more area will be covered by a sensor, and the sensing region of adjacent sensors may be overlapped, the probability of successfully forming a barrier will be greater. For fixed number of sensors, the setting with larger sensing range always can yield more barriers.
5.2.3. Effect of region width. In this experiment, we randomly deploy 80, 140, and 220 mobile sensors in a rectangle, respectively. The width of the region varies from 30 to 130 and the length of the region is 200. The sensing range is set as 10 and the mobile range is set as 20. It can be observed from Fig. 8 that the setting with smaller region width always forms more barriers when the number of
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Heuristic Lagrangian
350
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Fig. 7. Effect of coverage radius.
sensors is fixed. With the width becoming larger gradually, the quantity of barriers decrease.
5.2.4. Effect of mobile range. In this experiment, the mobile range of sensors varies from 0 to 35. The coverage range is set as 20. Without loss of generality, we test two different settings described as follows:
1. 100 mobile sensors are randomly deployed in a 200 130 rectangular area. 2. 200 mobile sensors are randomly deployed in a 200 60 rectangular area. As the Fig. 9 illustrated, when mobile range is equal to 0, the barrier coverage is 0 and the optimal solution of linear programming relaxation is close to 0, i.e., when the sensors are stationary,
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H. Ma et al. / Computer Communications 35 (2012) 1749–1758
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Fig. 8. Effect of region width.
Fig. 9. Effect of mobile range.
barrier coverage cannot be achieved with the two settings. With the mobile range of sensors increasing, the number of barriers increases rapidly and get close to the optimal solution of linear programming relaxation eventually. Above simulation results show that our greedy algorithm is very close to the upper bound of the problem. So our greedy algorithm is efficient. 5.3. Mobility vs. stationary We also study how the number of formed barriers is affected by the mobility. When all the sensors are deployed in the region and keep stationary, we can get maximum number of formed barriers based on the maximum flow corresponding this deployment. We use MCF to represent the maximum number of formed barriers in the corresponding stationary deployment. When the sensors are allowed with limited mobility, we use Algorithm 4 to find the number of formed barriers, which is denoted as Greedy. To compare the above two states: mobility and stationary, we study how the number of formed barrier is affected by the three parameters: the mobile range, the width of the region, the number of sensors. In the following experiments, we set the same grid points as that mentioned in Section 5.2. 5.3.1. Effect of mobile range. In this experiment, we randomly deploy 200 mobile sensors in a rectangle region: 200 100. The cov-
erage range of the sensors is set as 10, and the mobile range varies from 11 to 25. As the Fig. 10(a) illustrated, the number of formed barrier (MCF) is 3. While by Greedy, with the increasing of the mobile range of sensors, the number of barriers increase rapidly, which is much larger than MCF in stationary state.
5.3.2. Effect of region width. In the experiment, we randomly deploy 200 mobile sensors in a rectangle. The length of the region is 200 and the width of the region varies from 40 to 100. The sensing range is set as 10 and the mobile range is set as 15. It can be observed from Fig. 10(b) that the setting with smaller region width always forms more barriers when the number of sensors is fixed, and the sensors with limited mobility can form more barriers. When the width becoming larger gradually, the quantity of barriers decrease.
5.3.3. Effect of number of sensors. In this experiment, we randomly deploy n (n 6 200) mobile sensors in a rectangle region: 200 100. The sensing range of each sensor is set to 10 and the moving range is set to 15. As shown in Fig. 10(c), with the increasing of the number of sensors, the number of formed barriers increases rapidly. Compared with stationary senors, senors with limited mobility could form more barriers.
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Fig. 10. MCF vs. greedy.
From the simulations in Section 5.3, we know that if the sensors are allowed with limited mobility, the number of formed barriers can increase greatly. The above simulation results demonstrate our algorithms are efficient. 6. Conclusion In this paper, we study k-barrier coverage problem in wireless sensor networks. Firstly, we study the minimum energy cost k-barrier coverage problem P1 and formulate P1 into a minimum cost flow problem with side constraints. We use the classical Lagrangian algorithm to find a lower bound of the total energy cost, and we propose a heuristic algorithm for this problem. Simulation results demonstrate that the total energy cost consumed by the heuristic algorithm is at most two times of the lower bound of the problem. Then we study the maximum k-barrier coverage problem P2, where some sensors can move within limited range. We formulate the problem into an integer linear programming (ILP) and relax it into linear programming (LP). We propose two algorithms based on the LP relaxation. Simulation experiments show that our algorithms are efficient. Acknowledgments This research was jointly supported in part by National Natural Science Foundation of China under Grants 61070191 and 91124001, the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China under Grant 10XNJ032. And Research Fund for the Doctoral Program of Higher Education of China under Grant 20100004110001. References [1] I. Akyildiz, W. Su, Y. Sankarasubramaniam, E. Cayirci, A survey on sensor networks, IEEE Communications Magazine 40 (8) (2002) 102–114. [2] B. Liu, O. Dousse, J. Wang, A. Saipulla, Strong barrier coverage of wireless sensor networks, in: Proceedings of the ACM MobiHoc’08, 2008, pp. 411–420.
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