Strong Barrier Coverage with Directional Sensors

Strong Barrier Coverage with Directional Sensors Li Zhang, Jian Tang Department of Computer Science Montana State University Bozeman, MT 59717-3880. Email: {zhang,tang}@cs.montana.edu

Weiyi Zhang Department of Computer Science North Dakota State University Fargo, ND 58105. Email:[email protected]

Abstract—The barrier coverage model was proposed for applications in which sensors are deployed for intrusion detection. In this paper, we study a strong barrier coverage problem in wireless sensor networks with directional sensors. First, we introduce the directional coverage graph to model barrier coverage with directional sensors. Based on this graph model, we present an integer linear programming formulation for the barrier coverage problem, which can be used to provide optimal solutions. Moreover, we present efficient centralized algorithms and a distributed algorithm to solve the problem. It has been shown by simulation results that the proposed algorithms provide close-to-optimal solutions and consistently outperform a simple greedy algorithm. Keywords: Wireless sensor networks, directional sensor, barrier coverage.

(a) Weak Barrier Coverage

A barrier

I. I NTRODUCTION A Wireless Sensor Network (WSN) usually consists of a large number of small sensor nodes, which are deployed in the field to perform certain sensing tasks [3]. Even though WSNs have been extensively studied, previous research has mostly concentrated on omni-directional sensors with disk-like sensing coverage regions. However, some emerging WSNs, such as wireless multimedia sensor networks [4], contain directional sensors with directional Field Of Views (FOVs), i.e., they can only sense events that happen in a certain direction at one time. The widely used directional sensors include video sensors, image sensors, ultrasonic sensors and infrared sensors [5]. Coverage with directional sensors has not been well studied before. Intrusion detection is one of major applications of WSNs. For example, infrared sensors are usually placed in a region or along a roadway to detect the crossing of wild animals [16]. In order to detect the crossing of a moving object, there is no need to fully cover the whole region. The barrier coverage model [10] was proposed for intrusion detection. According to this model, sensors are only needed to be deployed and activated to cover any possible crossing path, which is a path that crosses the complete width of a target region from one side to the other side. There are two kinds of barrier coverage: weak and strong [12], which are illustrated by Fig. 1. Weak barrier coverage can only guarantee to detect intruders moving along congruent paths, indicated by dash lines. A path indicated by the solid line in Fig. 1(a) cannot be covered by any sensors. However, strong barrier coverage (illustrated by Fig. 1(b)) guarantees to detect intruders no matter what crossing paths they take, which is the focus of this paper.

(b) Strong Barrier Coverage

Fig. 1.

Strong and weak barrier coverage with directional sensors

In order to provide strong barrier coverage with omnidirectional sensors, we only need to find a subset of sensors such that their sensing disks form one or multiple barriers. However, providing strong barrier coverage with directional sensors is much harder since in order to form barriers (see Fig. 1(b)), we not only need to select a subset of sensors but also need to determine their orientations. The algorithms proposed for barrier coverage with omni-directional sensors [6], [10], [12] cannot be directly applied here. To our best knowledge, we are the first to study strong barrier coverage in wireless sensor networks with directional sensors. We summarize our contributions in the following: 1) We introduce the directional coverage graph to model barrier coverage with directional sensors. Based on this graph model, we present an Integer Linear Programming (ILP) formulation for a barrier coverage problem to provide optimal solutions, which can be used as benchmarks for performance evaluation. 2) We present two simple centralized and a simple distributed algorithm to solve the problem. 3) Extensive simulation results have been presented to show that the proposed algorithms provide close-to-optimal solutions.

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The rest of this paper is organized as follows. We discuss related work in Section II. The system model and problem definition are described in Section III. An ILP formulation, centralized algorithms and distributed algorithms are presented in Section IV. We present numerical results in Section V and conclude the paper in Section VI. II. R ELATED W ORK Directional sensor networks have attracted research attention recently. In [14], the authors analyzed how an algorithm designed for omni-directional WSNs, which solves the coverage and routing problems, behaves in video WSNs. Their results showed that because of the unique way that cameras capture data, previous algorithms do not offer expected results in terms of coverage preservation. An optimal polynomial time algorithm for computing the worst-case breach coverage in directional WSNs was presented in [1]. In [5], Cai et al. considered the problem of covering a set of static target points using a minimum number of directional sensors, and the related scheduling problem. They showed that the coverage problem is NP-complete and presented heuristic algorithms to solve the related problems. In [2], Ai et al. studied the Maximum Coverage with Minimum Sensors (MCMS) problem. They presented an ILP formulation, as well as a distributed greedy algorithm and protocol for the MCMS. In [15], Tezcan and Wang proposed an algorithm to determine a node’s coverage and find the sensor orientation that minimizes the negative effect of occlusions and overlapping regions for multimedia WSNs with directional sensors. How to efficiently deploy directional sensors to form a connected network and cover a set of static target points and a target region was investigated in [9]. Our work is different from these works proposed for WSNs with directional sensors since none of them consider barrier coverage. Barrier coverage with omni-directional sensors has been studied by several recent papers. In [10], Kumar et al. defined the notion of k-barrier coverage of a belt region using wireless sensors. They proposed efficient algorithms to determine whether a region is k-barrier covered or not after deployment. Then they established the optimal deployment pattern for achieving k-barrier coverage. Moreover, they derived critical conditions for weak k-barrier coverage in a given belt region, which can be used to compute the minimum number of sensors needed to provide weak k-barrier coverage with high probability. In [11], optimal centralized algorithms were proposed to solve the sleep-wakeup problems for strong barrier coverage in WSNs. In [6], the authors introduced the concept of local barrier coverage and showed that for thin belt regions, local barrier coverage almost always provides global barrier coverage. They also developed a novel sleep-wakeup algorithm for maximizing network lifetime. In [12], the authors showed that in a rectangular area with a width of w and a length of l, if w = Ω(log l) and the sensor density reaches a certain value, then there exist multiple strong disjoint sensor barriers with high probability; on the other hand, if w = o(log l), then there is a crossing path not covered by any sensor with

high probability regardless of sensor density. Based on these results, they devised an efficient distributed algorithm to find multiple disjoint barriers. In [7], Chen et al. proposed a metric, theorems and a distributed algorithm to measure the quality of k-barrier coverage and identify regions that need to be repaired. Our work is also different from these works since none of them consider barrier coverage in the context of directional sensors. III. P ROBLEM D EFINITION In this paper, we consider a WSN with n stationary directional sensor nodes, each of which is aware of its location. Our sensing model is similar to that in [2], [5]. Each node s has a directional sensor whose sensing region is a sector of the disk centered at s with a sensing range of r and a FOV of θ. In the following, we will use sensor and sensor node interchangeably. The orientation of a sensor is defined as the direction of the central axis of its sensing sector. The orientation of a directional sensor can be tuned to any direction in a given set Ω. An example is given in Fig. 2 to illustrate our sensing model. In this example, θ = π2 , 5π 7π Ω = { π4 , 3π 4 , 4 , 4 }. This directional sensor has 4 possible sensing sectors. Note that two possible sensing sectors may overlap with each other. A tuple (i, j) is used to denote the jth sensing sector of sensor si . The set of all possible sectors is denoted as Π = {(i, j) : i ∈ {1, 2, · · · n}, j ∈ {1, 2, · · · qi }}, where qi is the total number of possible sectors of sensor si .

θ=π/2 Sensor si

Fig. 2.

ω=π/4

The possible sensing sectors of a directional sensor

Similar as in [10], [12], a directional WSN is assumed to be placed in a belt region (see Fig. 1). Sensor node placement is out of scope of this paper and will be addressed by our future research. In order to provide barrier coverage for the target region, we need to find a subset of sensors and their orientations (i.e., a set of sensing sectors) to form multiple disjoint barriers such that any crossing path from top to bottom is covered by sensing sectors of multiple distinct sensors. We formally define the Directional Sensor Barrier (DSB) and the corresponding coverage problem as follows. Definition 1: A Directional Sensor Barrier (DSB) is a minimal set of sensing sectors such that 1) the leftmost and


rightmost sectors overlap with the left and right boundaries of the target region respectively; 2) any two neighboring sectors overlap; and 3) it includes at most one sector from each sensor node. The Maximum Directional Sensor Barrier Problem (MDSBP) is to find the maximum number of disjoint DSBs. Once the MDSBP is solved, the optimal algorithms proposed in [11] can be used to find sleep-wakeup schedules with the maximum lifetimes. IV. P ROPOSED A LGORITHMS In this section, we present different methods to solve the MDSBP. First of all, we introduce the directional coverage graph G(V, E) to model barrier coverage with directional sensors. In G, each vertex corresponds to a possible sensing sector in Π. There is a directed edge from vertex v to vertex v if the two corresponding sectors overlap and belong to two different sensor nodes. Note that there are no edges connecting vertices whose corresponding sectors belong to the same sensor. In addition, there are a virtual source z and a virtual sink t connected with the vertices whose corresponding sectors cover left and right boundaries respectively. The capacity of every edge is set to 1. A directional coverage graph has O(nq) vertices and O(n2 q 2 ) edges, where q = max{qi : i ∈ {1, 2, · · · , n}}. The importance of the directional coverage graph lies in the fact that each simple z −t path corresponds to a DSB. However, a pair of disjoint paths P and P in G may not correspond to two disjoint DSBs because two vertices that are on P and P respectively may belong to the same sensor. Therefore, finding the maximum number of disjoint DSBs is equivalent to finding the maximum number of sensor-disjoint paths in G. A. ILP Formulation Based on the directional coverage graph G, we present an ILP formulation for the MDSBP. In this formulation, we have flow variables fe : fe = 1 if edge e is selected on one of sensor-disjoint paths; fe = 0, otherwise. ILP-MDSBP max

fe

(1)

∀v ∈ V \ {z, t};

(2)

∀i ∈ {1, 2, · · · , n};

(3)

∀e ∈ E.

(4)

e∈Ezout

subject to: e∈Evout

fe =

fe ,

e∈Evin

fe ≤ 1,

e∈Esout i

fe = {0, 1},

In this formulation, Evin and Evout denote the set of incomdenotes ing and outgoing edges at vertex v respectively. Esout i the set of outgoing edges of vertices (sectors) associated with sensor si . Constraint (2) is a general flow conservation constraint, which makes sure that multiple paths can be found.

Constraint (3) guarantees the multiple paths computed by solving the ILP are sensor-disjoint, i.e., no two paths are associated with a common sensor. With the objective function (1), we can maximize the number of sensor-disjoint paths. B. Centralized Algorithms In this section, we present two centralized heuristic algorithms for the MDSBP, which are formally presented as follows. Algorithm 1 The Disjoint Path Algorithm 1 Step 1 P = ∅; B = ∅; Step 2 Construct a directional coverage graph G(V, E); Step 3 Compute the maximum number of vertex-disjoint paths from source z to sink t in G and store them in P; Step 4 Select a path P with the minimum hop count from P and insert it into B; Step 5 Remove path P and all the paths conflicting with P from P; Step 6 if (P = ∅) goto Step 4; else output B; endif As mentioned before, a pair of vertex-disjoint paths in the directional coverage graph G may not be sensor-disjoint. However, we can find a subset of paths from a set of vertex-disjoint paths to form multiple disjoint barriers. The basic idea of this algorithm is to find the maximum number of vertex-disjoint paths in G and then identify as many as possible sensordisjoint paths from them. A flow-based algorithm can be used to find the maximum number of vertex-disjoint paths in a graph. In this algorithm, a new auxiliary graph is constructed to assist computation, in which every vertex v (except z and t) in G is split to two vertices v in and v out and there is a directed edge from v in to v out with a capacity of 1. Then an augmenting path based maximum flow algorithm such as the Ford-Fulkerson algorithm [8] is used to find a maximum flow from z to t. Based on the computed flow allocation, we can identify the maximum number of vertex-disjoint z − t paths in G. In addition, a step-by-step heuristic method is used to find a subset of sensor-disjoint paths. In each step, a shortest path (minim hop count) is selected and all the conflicting paths are removed from P. Note that a path P is said to conflict with path P if they share a common sensor, i.e., a vertex (sector) v ∈ P and a vertex (sector) v ∈ P belong to the same sensor. In this way, hopefully as few as possible paths will be removed in each step, which will give the algorithm better chances to find more feasible paths later. The running time of Step 2 is O(n2 q 2 ) since the numbers of vertices and edges in G are O(nq) and O(n2 q 2 ) respectively. Step 3 can be done in O(n2 q 2 N ) assuming that the FordFulkerson algorithm [8] is used to find a maximum flow, where N is the maximum number of vertex-disjoint paths that is


usually much smaller than n (See Figs. 3-4). Steps 4 and 5 can be done within O(nN ), which may be repeated O(N ) times. Therefore, the worst-case time complexity of Algorithm 1 is O(n2 q 2 N ). Algorithm 2 The Disjoint Path Algorithm 2 Step 1 P = ∅; B = ∅; Step 2 Construct a directional coverage graph G(V, E); Step 3 Compute the maximum number of vertex-disjoint paths from source z to sink t in G and store them in P; Step 4 Select a path P conflicting with the fewest number of other paths in P and insert it into B; Step 5 Remove path P and all the paths conflicting with P from P; Step 6 if (P = ∅) goto Step 4; else output B; endif The basic idea of Algorithm 2 is similar to that of Algorithm 1. However, the path selection procedure (Step 4) is different. Every time, a path that conflicts with the fewest number of other paths in P is selected. In this way, it is likely that more feasible paths can be found by the algorithm later, which has been verified by simulation results. The worst-case time complexity of Algorithm 2 is same as that of Algorithm 1. However, we observed it took a little longer than Algorithm 1 in practice due to its path selection procedure. C. Distributed Algorithm In this section, we present a simple distributed algorithm to solve the MDSBP problem, which is formally presented as Algorithm 3. A sensor node is said to be another node’s sensing neighbor if their sensing regions potentially overlap. The transmission range (R) of each sensor node is assumed to be no less than 2 times its sensing range, i.e., R ≥ 2r. That is to say, a sensor node can directly communicate with all of its sensing neighbors. In practice, this condition can usually be satisfied [5], [13]. In the algorithm, the function ID(·) returns the index of a given cluster header and x(·) gives the x coordinate of a sensor node. A cluster header h constructs its local directional (Vhlocal , Ehlocal ) as follows: each vertex coverage graph Glocal h in Vhlocal corresponds to a possible sensing sector belonging to a sensor in Sh . There is a directed edge from vertex v to vertex v if the two corresponding sectors overlap and belong to two different sensor nodes . In addition, there are a virtual source zh and a virtual sink th . First of all, the cluster header h needs to find out the neighboring cluster headers on the left and right of h with the maximum weight values given by function w(·). We call these two cluster headers , there is a directed edge from zh hlef t and hright . In Glocal h to a vertex whose corresponding sector belongs to a sensor node in Sh Shlef t . Similarly, there is a directed edge from

Algorithm 3 The Distributed Algorithm Step 1 Each sensor node si constructs a set Si of its sensing neighbors (including itself) by periodically exchanging HELLO messages; It broadcasts |Si | to all nodes in Si . Step 2 Every sensor node si calculates imax := max{j : sj ∈ Si and |Sj | = Kmax }, where Kmax = max{|Sj | : sj ∈ Si }; It compares its index with imax ; if (i = imax ) si claims itself as a cluster header by broadcasting a HEADER message to all nodes in Si ; endif Each sensor node receiving a HEADER message then forwards it to its sensing neighbors. Step 3 Each cluster header h constructs a set Hh of neighboring cluster headers, and calculates hidmax := max{ID(h ) : h ∈ Hh and |Sh | = Hmax } h ∩Sh | and w(h ) := α × max{|Sh|S∩S + β × |: h ∈Hh } h

|x(h)−x(h )| max{|x(h)−x(h )|: h ∈Hh } , ∀h ∈ Hh , where Hmax := max{|Sh | : h ∈ Hh }, Sh is the set of sensing

neighbors of h and α + β = 1; Step 4 Each cluster header h compares its index with hidmax ; if (ID(h) = hidmax ) goto Step 5; else Waits for 2 seconds; goto Step 3; endif Step 5 The cluster header h constructs a directional coverage (Vhlocal , Ehlocal ); (Details are given in graph Glocal h Section IV-C.) Step 6 The cluster header h computes the local disjoint barriers by computing the zh −th sensor-disjoint paths using the method in Algorithm 2. in Glocal h Step 7 Let Bh denote the set of sectors that have been selected to form a barrier in Step 6; The cluster header h broadcasts a BARRIER message with Bh to all nodes in Sh and all cluster headers in Hh ; When receiving the message, each cluster head h removes the corresponding sensors from Sh .

a vertex whose corresponding sector belongs to a sensor node in Sh Shright to th . For a special case, i.e., if there are no other cluster headers on the left of h, then zh is connected with the vertices whose corresponding sectors overlap with the left boundary. Similarly, if there are no other cluster headers on the right of h, then zh is connected with the vertices whose corresponding sectors overlap with the right boundary. After extensive simulations, we found out that by setting the two parameters α = 0.4 and β = 0.6 in w(·), the algorithm can achieve the best performance. The capacity of every edge is set to 1. According to the algorithm, the sensor nodes with more


The Number of Barriers

30 25 20 15 10

Optimal Disjoint Path 1 Disjoint Path 2 Distributed Greedy

5 0 100

150

200

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300

Fig. 3.

400

Scenario 1: r = 30m and q = 4

45 40 35 30 25 20 15


10 5

V. N UMERICAL R ESULTS

0 100

150

200

250

300

350

400

Network Size (n)

Fig. 4.

Scenario 2: r = 30m and q = 6

35


We evaluated the performance of the proposed algorithms via simulations. In the simulation, directional sensor nodes are randomly placed in a rectangle region with a size of 200 × 150m2 . R = 2r, where R and r are the transmission and sensing ranges of each sensor node respectively. Similar as in [2], [5], a disk with a radius of r was divided into q = 2π θ disjoint sensing sectors with equal sizes, where θ is the FOV. The number of disjoint barriers is used as the performance metric. Intuitively, the following parameters may have a significant impact on the performance: the number of sensor nodes (network size) n, the sensing range r, the number of possible sensing sectors q and FOV θ. We evaluated their impacts by setting them to different values in different scenarios. In scenario 1, r = 30m, q = 4 and the network size was changed from 100 to 400 with 50 as the step size. The settings in scenario 2 are the same as those in scenario 1 except q = 6. In scenario 3, n = 300, q = 4 and the sensing range was changed from 15m to 40m with 5m as the step size. In scenario 4, n = 300, r = 30m and q was changed from 1 to 6. The results are presented in Figs. 3– 6. In these figures, “Optimal”, “Disjoint Path 1”, “Disjoint Path 2” and “Distributed” refer to solutions given by solving ILP-MDSBP, Algorithm 1, Algorithm 2 and Algorithm 3 respectively. “Greedy” refers to a simple centralized greedy algorithm which constructs barriers sequentially from left to right one by one until no more disjoint barriers can be found. We make the following observations from the results:

350

Network Size (n)


sensing neighbors than all its neighbors claims itself as a cluster header. It is possible that in a common neighborhood, multiple nodes have the same number of neighbors. The tie is broken using the indices of sensor nodes and the one with the maximum index wins. In this way, the whole network will be divided into multiple clusters. Basically, each cluster header constructs local disjoint barriers using its neighbors. Hopefully, these local disjoint barriers can be connected together to form multiple global disjoint barriers. Basically, those cluster headers with a relatively large number of neighbors will start computation first because it is likely that they could find a large number of local disjoint barriers. Note that if each cluster header constructs its local disjoint barriers independently without considering its left and right neighboring clusters, those local disjoint barriers may fail to form global disjoint disjoint barriers. According to our weight function w(·) and the way of constructing the local coverage graph, a cluster header h tries to find some local disjoint barriers which are likely to be connected with other local disjoint barriers belonging to neighboring clusters that are far away in terms of distance in the x-axis (so that the algorithm can make a good coverage progress towards both boundaries.) and share a large number of common sensors with the cluster h. In this way, the algorithm is likely to find a large number of disjoint barriers by using a small number of sensor nodes.


30 25 20 15 10 5 0 15

20

25

30

35

40

Sensing Range (r)

Fig. 5.

Scenario 3: n = 300 and q = 4

1) The optimal solutions given by solving the ILP problem are used as the baseline for comparisons since we are the first to address the barrier coverage in directional WSNs. From Figs. 3 and 4, we can see that the numbers of disjoint barriers found by our Disjoint Path Algorithm 2 are almost the same as the corresponding optimal values when the network sizes are relatively small. On average, the numbers of disjoint barriers


R EFERENCES

45 Optimal Disjoint Path 1 Disjoint Path 2 Distributed Greedy


40 35 30 25 20 15 10 5 0 1

2

3

4

5

6

The Number of Possible Sectors (q)

Fig. 6.

Scenario 4: n = 300 and r = 30m

found by the two centralized algorithms are 75.8% and 88.4% of the optimal values, respectively. As expected, our distributed algorithm does not perform as well as the centralized algorithms in most of cases due to lack of global information. However, the average number of disjoint barriers given by our distributed algorithm is still 71.2% of the optimal value. In addition, both our centralized and distributed algorithms consistently outperform the simple greedy algorithm. 2) No matter which algorithm is employed, the number of disjoint barriers increases monotonically with the network size and with the sensing range. This is because the more sensors a network has or the larger the sensing range is, the more sensing neighbors a sensor node will have, which will lead to more disjoint barriers. 3) From Fig. 6, we find out that the number of disjoint barriers does not monotonically increase or decrease with the number of possible sensing sectors. One one hand, increasing the number of possible sensing sectors introduces more possible orientations and better flexibility, which leads to more disjoint barriers. On the other hand, the more possible sensing sectors a sensor has, the smaller the FOV becomes, which is not favorable for finding disjoint barriers and may counteract the improvement brought by introducing more possible sensing sectors in a sensor. As expected, all the algorithms perform best in networks with omni-directional sensors (i.e., q = 1).

[1] J. Adriaens, S. Megerian and M. Potkonjak, Optimal worst-case coverage of directional field-of-view sensor networks, Proceedings of IEEE Secon’2006, pp. 336–345. [2] J. Ai and A. A. Abouzeid, Coverage by directional sensors, Proceedings of IEEE International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, 2006, pp. 1–10. [3] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam and E. Cayirci, Wireless sensor networks: a survey, Computer Networks Journal, Vol. 38, No. 4, 2002, pp. 393–422. [4] I. F. Akyildiz, T. Melodia and K. R. Chowdhury, A survey on wireless multimedia sensor networks, Computer Networks Journal, Vol. 51, No. 4, 2007, pp. 921–960. [5] Y. Cai, W. Lou, M. Li and X-Y Li, Target-oriented scheduling in directional sensor networks, Proceedings of IEEE Infocom’2007, pp. 1550– 1558. [6] A. Chen, S. Kumar and T. H. Lai, Designing localized algorithms for barrier coverage, Proceedings of ACM MobiCom’2007, pp. 63–74. [7] A. Chen, T. H. Lai and D. Xuan, Measuring and guaranteeing quality of barrier-coverage in wireless sensor networks, Proceedings of ACM MobiHoc’2008, pp. 421–430. [8] T. H. Cormen et al. , Introduction to algorithms, 2nd edition, The MIT Press, 2001. [9] X. Han, X. Cao, E. L. Lloyd and C-C. Shen, Deploying directional sensor networks with guaranteed and coverage and connectivity, Proceedings of IEEE Secon’08, pp.153–160. [10] S. Kumar, T. H. Lai and A. Arora, Barrier coverage with wireless sensors, Proceedings of ACM MobiCom’2005, pp. 284–298. [11] S. Kumar, T. H. Lai, M. E. Posner and P. Sinha, Optimal sleepwakeup algorithms for barriers of wireless sensors, Proceedings of IEEE Broadnets’2007, pp. 327-336. [12] B. Liu, O. Dousse, J. Wang and A. Saipulla, Strong barrier coverage of wireless sensor networks, Proceedings of ACM MobiHoc’2008, pp. 411– 419. [13] Mica2 data sheet, http://www.xbow.com. [14] S. Soro and W. B. Heinzelman, On the coverage problem in videobased wireless sensor networks, Proceedings of IEEE Broadnet’2005, pp. 932–939. [15] N. Tezcan and W. Wang, Self-orienting wireless multimedia sensor networks for occlusion-free viewpoints, Computer Networks Journal, Vol. 52, 2008, pp. 2558–2567. [16] Road Ecology Program in Western Transportation Institute, http://www.coe.montana.edu/wti/road ecology/home.php.

VI. C ONCLUSIONS In this paper, we studied a strong barrier coverage problem in directional WSNs. First, we introduced the directional coverage graph to model barrier coverage with directional sensors. Based on this graph model, we presented an ILP formulation for the barrier coverage problem. Moreover, we presented efficient centralized algorithms and a distributed algorithm to solve the problem. It has been shown by simulation results that the proposed algorithms provide close-to-optimal solutions and consistently outperform a simple greedy algorithm.
