Jiang CD, Chen GL. Double barrier coverage in dense sensor networks. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 23(1): 154 inside back cover Jan. 2008 Double Barrier Coverage in Dense Sensor Networks Cheng-Dong Jiang ( ) and Guo-Liang Chen ( ) Department of Computer Science and Technology, University of Science and Technology of China, Hefei 230027, China E-mail: cdjiang@ustc.edu.cn; glchen@ustc.edu.cn Received December 12, 2006; revised October 19, 2007. Abstract When a sensor network is deployed to detect objects penetrating a protected region, it is not necessary to have every point in the deployment region covered by a sensor. It is enough if the penetrating objects are detected at some point in their trajectory. If a sensor network guarantees that every penetrating object will be detected by two distinct sensors at the same time somewhere in this area, we say that the network provides double barrier coverage (DBC). In this paper, we propose a new planar structure of Sparse Delaunay Triangulation (SparseDT), and prove some elaborate attributes of it. We develop theoretical foundations for double barrier coverage, and propose efficient algorithms with NS2 simulator using which one can activate the necessary sensors to guarantee double barrier coverage while the other sensors go to sleep. The upper and lower bounds of number of active nodes are determined, and we show that high-speed target will be detected efficiently with this configuration. Keywords wireless sensor network, barrier coverage, Delaunay triangulation, distributed algorithm 1 Introduction Wireless sensor network is becoming quite useful in some modern applications, such as target tracking, environmental monitoring, and intrusion detection, which require the distributed and cooperated work of separated elements. These distributed sensing applications bring forward new demands for the property of sensor networks. An important application of a sensor network is to detect penetrating objects crossing a barrier. The objective is to minimize the probability of undetected penetration through the barrier. There is not much research on it until now. The maximal breach path and the maximal support path model were introduced by Meguerdichian et al. [1] Another model, called exposure-based model, was also introduced by Meguerdichian et al. [2] There are also some other introductory work [3,4]. Kumar s prominent work [5] presents a method to determine the minimum number of sensors needed to ensure k-barrier coverage in a given belt region, and also gives the checking algorithm. This k-barrier coverage problem means that every path crossing the width of the belt region is covered by at least k distinct sensors [5]. The algorithms presented in Meguerdichian s work [1] heavily rely on geometric structures such as the Voronoi Diagram and Delaunay triangulation of the network, which was not constructed locally. The Delaunay triangulation technique is quite useful in construction and analysis of network topology [6,7]. Recently, it was used to construct a geographic routing algorithm which achieves better path dilation [8]. But the distributed algorithm of Delaunay triangulation does not work well up until now. Filipe Araújo s outstanding work [9] gives a well working distributed algorithm. But the nodes need to collect 2-hop information, and it might cause severe synchronization problem in dense sensor networks. Li proposed a sparse planar structure, namely, partial Delaunay triangulation [7], which could be constructed locally. But it is much like a subgraph which is a little denser than the Gabriel Graph. In our paper, we consider the double barrier coverage problem, which is a little different from the k- barrier coverage problem. Actually, there is not any efficient algorithm for the construction of k-covered barrier when sensors are deployed randomly. We prove in Subsection 4.3 that the 2-covered barrier has better capability in target detection than two disjoint 1-covered barriers/paths with the metric of minimum detection length. This metric denotes the exposure time that a target moving at regular speed stays in the sensing covered region. A target with bounded moving speed could be detected assuredly under the model that sensing activity occurs periodically, but it is not Regular Paper This paper is supported by the National Grand Fundamental Research 973 Program of China under Grant No. 2006CB303006.
Cheng-Dong Jiang et al.: Double Barrier Coverage in Dense Sensor Networks 155 true with disjoint 1-covered paths because the exposure time might approach zero in some certain cases. The 2-covered barrier is also constructed based on the Delaunay triangulation technique, and we present a new sparse planar network structure here, SparseDT (Sparse Delaunay Triangulation), which could be constructed locally and efficiently. In particular, it is constructed without collecting 2-hop neighbor information. It could also be quite useful in other problems of sensor networks. The rest of the paper is organized as follows. In Section 2, we define the problem formally, and propose the centralized algorithm. In Section 3, an efficient distributed algorithm to construct SparseDT structure is given, and the 2-covered barrier is formed based on it. In Section 4, we prove the correctness of our algorithm, and give the upper and lower bounds of the number of active sensors. The detection ability of the 2-covered barrier is also analyzed. In Section 5, we provide some results from simulation. Section 6 concludes the paper. 2 Preliminaries 2.1 Assumptions We assume all sensors are deployed in a two dimensional space with Poisson spatial distribution. Every node has the same sensing range R s and communication range R c with the relation of R c R s. We assume a deterministic sensing model that a sensor node s can cover any point inside its sensing circle Cs Rs which is centered at s and has a radius R s, i.e., any point with distance to s less than R s is covered by s. We define circle Cs r in the same way. Definition 2.1. Given a set of nodes V in a two dimensional space, we model a wireless sensor network with Unit Disk Graph with unit length r, G r (V, E) or G r (V ), which is comprised of all nodes in V and all edges connecting nodes of V whose distance is at most the unit length r, i.e., two nodes u and v are direct neighbors if and only if uv < r. It is generally convenient for the analysis of coverage problems that we set the unit length in this model as r = R s, and r = R c for connectivity problems. With different unit length, the edge sets could be quite different, such as G Rc (V, E 1 ) and G Rs (V, E 2 ). 2.2 Problem Formulation In this paper, we consider the Double Barrier Coverage (DBC) Problem. Given a sensor network deployed in a rectangular region A with the assumptions hereinabove, and two distant sinks sink 1 and sink 2 which are normally placed at two sides of the region, we would like to find a continuous line between sink 1 and sink 2, and the least number of active sensors that can cover every point on the line at least two times. The set V of these active nodes is called a 2-covered barrier between sink 1 and sink 2 in the region. That is, a target wanting to traverse the region A from between these two sinks must be detected by at least two sensors at the same time. 2.3 Centralized Algorithm Here, we classify the double barrier coverage problem in a series of alike problems, and present a centralized algorithm to deal with. 1) 1-Dimensional, 1-Covered Case. Given a line segment l and a set of sections S, such that every section s S covers a part of l, and the union of them covers l fully, i.e., s S s = l. The problem is to find the least number of sections to fully cover the line segment. This is in fact a continuous variety of the Set Cover Problem which is NP-hard. However, it is more tractable, and could be solved with a simple greedy algorithm. 2) 2-Dimensional, 1-Covered Case. In a two dimensional space, this problem and the DBC problem are much alike except that the barrier only needs to be 1-covered. It could be modeled as the shortest path problem in a UDG Graph, and thus settled. 3) 1-Dimensional, 2-Covered Case. This scenario could be handled just like the following 2-dimensional case, except that the intersection estimation is much easier. 4) 2-Dimensional, 2-Covered Case. This is just the problem we considered. Now we give a centralized algorithm for it. With the model of Unit Disk Graph with unit graph r, we know that an edge e E in G 2Rs (V, E) means the sensing circles of the two sensor nodes adjacent to e cross each other, and form an olive-shaped overlap (see Fig.1). We construct the Junction Graph G J (V, E ) based on G 2Rs (V, E). It represents the junction relationship of those olive-shaped overlaps. For each vertex v i V, there is a corresponding olive-shaped overlap and an edge e i E (Fig.1). We define p i as the parent node set of e i which is v i s corresponding edge in G 2Rs (V, E), such as p 1 = {s 1, s 2 } in Fig.1. Edge v 1 v 2 E if and only if the minimum bounding circle of p 1 p 2 has radius r < R s. That it, the corresponding overlaps of v 1 and v 2 are joint with each other. In the Junction graph, vertex S i denotes sink i, and it connects to those v j whose corresponding overlap
156 intersects sink i s sensing circle. That is, Si vj E 0 if and only if sink i s E for s pj. J. Comput. Sci. & Technol., Jan. 2008, Vol.23, No.1 form a 2-covered barrier with the corresponding sensor nodes of them. Conversely, there exists a path in GJ (V 0, E 0 ) for every 2-covered barrier. Fig.2. Edge in Junction graph (below) with weight 2 and the corresponding scenario in Unit disk graph (above). Fig.1. Unit disk graph G2Rs (V, E) (above) and the corresponding Junction graph GJ (V 0, E 0 )(below). Shadow area means the olive-shaped overlap of two adjacent sensing circles. We assign a weight to each edge v1 v2 in the Junction graph GJ (V 0, E 0 ), w(v1 v2 ) = 1 or 2, according to the difference of parent node sets of v1 and v2 s corresponding edges in G2Rs (V, E), i.e., w(v1 v2 ) = p1 p2. In Fig.2, v1 and v2 s corresponding edges in G2Rs (V, E) are e1 and e2 whose parent node sets are p1 = {s1, s2 } and p2 = {s3, s4 } respectively. So the weight w(v1 v2 ) = p1 p2 = {s1, s2 } {s3, s4 } = 2. In contrast, we have w(v1 v2 ) = {s1, s2 } {s1, s4 } = 1 in Fig.1, and there is no other value of edge weight in Junction graph. The edges adjacent to Si are assigned weight w(si vj ) = 1. We also define the weight of a path on Junction graph as the total weight of the edges on this path. The following theorem bounds the optimum solution to DBC Problem, that is, the least number of active sensors to construct a 2-covered barrier between two sinks. Theorem 2.2. The weight of the shortest path P between S1 and S2 in Junction Graph GJ (V 0, E 0 ) bounds the minimum number of sensors needed to construct a 2-covered barrier between sink 1 and sink 2 in this region. Proof. First, a path connecting S1 and S2 in GJ (V 0, E 0 ) depicts a chain of joint overlaps which Now we show that the minimum number of active nodes, which are needed to construct a 2-covered barrier between sink 1 and sink 2, is bounded by the weight of the shortest path P between S1 and S2. Suppose that P = {S1, v1, v2,..., vn, S2 }. We have w(p ) = 2 + n 1 X w(vi vi+1 ) = 2 + i=1 n 1 X pi+1 pi i=1 n [ n 1 [ pi. (pi+1 pi ) > >2+ i=1 (1) i=1 The rightmost quantity is the number of corresponding sensor nodes of this path. All inequalities in (1) become equalities while there is no loops in P. That is, while path P extends, all nodes of pi+1 pi are newly activated for the first time. This could be guaranteed supposing that the monitoring area is fully 2covered by the sensor network with node set V. There may be loops while it is not fully 2-covered, or the path P may not even exist while there are not enough sensors. The complexity of this centralized algorithm is O(n4 ), since V 0 = O( V 2 ) = O(n2 ), and E 0 = O( V 0 2 ) = O(n4 ). This method could be extended to 2-d, k-c scenario, except that it is relatively hard to determine overlaps and junction relationships. We construct the k-junction Graph GkJ (V 0, E 0 ) based on G2Rs (V, E). It represents the junction relationship of those k-covered overlaps. For each vertex
Cheng-Dong Jiang et al.: Double Barrier Coverage in Dense Sensor Networks 157 v i V, there is a corresponding k-covered overlap and a clique K i V with cardinality K i = k which satisfies the condition that the minimum bounding circle of K i has radius r < R s. Edge v 1 v 2 E if and only if the minimum bounding circle of K 1 K 2 has radius r < R s. That it, the corresponding overlaps of v 1 and v 2 are joint with each other. The weight of edge v i v j E is defined as w(v i v j ) = K i K j {1,..., k}. Vertex S i is defined similarly, and it could be verified in the same way that the weight of the shortest path between S 1 and S 2 in G k J (V, E ) bounds the minimum number of sensors needed to construct a k-covered barrier between sink 1 and sink 2 in this region. In a sensor network, there is no way to centralize that much information to settle, so distributed measure is more practical. In the following section, we propose a distributed algorithm to deal with the DBC problem. First we construct Delaunay triangulation on part of the nodes selected from all sensors to be active, and then construct 2-covered barrier from these nodes according to the properties of triangulation. 3 Distributed Algorithm The Delaunay triangulation (DT) of a set V of n points in 2-dimensional space possesses a host of nice and useful properties many of which are well known and understood nowadays. It has many interesting applications. We first investigate some most important properties of it: 1) Definition. The Delaunay triangulation (DT) of a node set V, represented as Del(V ), is the set of edges satisfying the empty circle property: edge uv belongs to the triangulation if and only if there is a circle containing u and v, but not containing any other node. 2) Dual Property. The planar Voronoi diagram and the Delaunay triangulation of V are duals in a graph theoretical sense. 3) Empty Circumcircle Property. The circumcircle of a triangle does not contain any other node of V. 4) Gabriel Edge. Two points u, v form an edge of DT if the minimum circle C(u, v) which contains u and v on its boundary contains no other points. These edges compose the Gabriel Graph which is a subgraph of Del(V ). There are many algorithms to construct the Delaunay triangulation. One of the most commonly used methods is Randomized Incremental Method [10] with complexity of O(n log n). 3.1 Sparse Delaunay Triangulation If R c R s, network is connected means it is also double covered. We only need to find a shortest path as the double covered barrier. Assuming that R c > R s, we could construct the sparse Delaunay triangulation only depending on local information. Using Filipe Araújo s algorithm framework [9] for reference, we propose a new distributed algorithm here for sparse Delaunay triangulation. We studied Filipe Araújo s algorithm, and found there are some Triangulate messages which are broadcasted to guarantee the rationality of local triangles. We dispose of these extra messages in our algorithm which is the crucial difference from Filipe Araújo s, so that the triangulation structure is constructed without any unnecessary communications. This nontrivial trick should save us much time and energy on communications which carry no valid sensed data. Following is the Algorithm description: 1) Neighbor Discovery Step: assuming that each sensor has been localized in advance. Localization is another important issue in the research area of sensor networks, and almost all topology control protocol need the support of localization protocol. There are many papers concentrated on it, and more detailed information is skipped here. Each sensor broadcasts periodically or randomly messages which include its position information, so that all sensors could collect the information of their neighbors, mainly the position and the status of activity. 2) Sparse Step: in random sequence, each active sensor node broadcasts a Sparse message, which would notify those sensors in a given range to sleep. This given range is called gap, which is normally set to the sensing range. It defines the extent of sparseness of the dispersed network. Those active nodes who survive this step compose the vertex set V of resultant graph G. 3) Triangulation Step: each sensor node s computes its local Delaunay triangles based on the neighbor location information. Only active neighbor nodes participate in the triangulation. Those edges incident to s whose lengths are within range of [R s, 3R s ) are kept, and others discarded. The complexity of this step is O(1) for each node as long as the number of active neighboring nodes is bounded. At the end of the algorithm, each node has the information of incident edges in a graph. It could be shown that the graph G(V, E) we created is part of the Delaunay triangulation (UDel) of the dispersed network. We call it Sparse Delaunay Triangulation (SparseDT).
158 J. Comput. Sci. & Technol., Jan. 2008, Vol.23, No.1 Fig.3. 2-covered barrier description. Fig.4. 2-covered barrier found with Algorithm DBC. 3.2 Double Barrier Coverage After the distributed Delaunay triangulation, we get Graph G(V, E). We now show that this graph could be used to construct 2-covered barrier. In Fig.3, triangles result from the above algorithm, solid circles limit the sensing range, and the dashed circles form the circumcircles of those triangles with centers at c, c 1, c 2 separately. From Lemma 4.2 which will be shown later, the center of circumcircle is within the sensing range of those sensors located at the triangle vertices, and it has coverage degree of at least 3. We could use the center as the joint of the 2-covered areas intersected by two neighboring sensors, such as c 1 -c-c 2. Barrier could then be constructed by the end-to-end connectivity of these joint segments, which is shown with the chain of thick segments in Fig.3. From another point of view, Barrier is a path between sinks in the dual Voronoi graph G of G(V, E). Algorithm DBC. Formally we should retrieve those triangles that denote the shortest path on the dual Voronoi diagram. This would give the least number of nodes that compose the 2-covered barrier. But it needs the storage of the information of dual graph, which is not preferred here. We only need to find the shortest path with any geographic or non-geographic routing protocol, and activate those sensors on this path and their neighboring sensors that stay at one side of this path (Fig.4). In the following section, we give theoretical foundation of the SparseDT planar structure, and analyze the static and dynamic performance of the 2-covered barrier. 4 Theoretical Analysis In this section, we first develop some elaborate attributes of the SparseDT planar graph, and then derive a better network dilation of it. We also analyze the detection ability of our 2-covered barrier here.
Cheng-Dong Jiang et al.: Double Barrier Coverage in Dense Sensor Networks 159 Fig.5. Delaunay triangle. Fig.6. Circumcircle of Delaunay triangle. Fig.7. Proof of Theorem 4.3. 4.1 Properties of SparseDT Under the Unit Disk Graph with unit length model we assumed, a complete Delaunay triangulation may not exist. We refer to another form of Delaunay triangulation defined as follows. Definition 4.1. Given a set of nodes V in a two dimensional space, we define Delaunay triangulation with unit length r as UDel r (V ) = Del(V ) G r (V ), which is comprised of all nodes in V and all edges in Del(V ) whose lengths are at most r. Lemma 4.2. The Delaunay triangle computed locally has a circumcircle of radius r, with (R s / 3) r < R s. Proof. According to our algorithm, every edge e in the Delaunay triangulation has the property 3R s > Length(e) R s. For arbitrary Delaunay triangle P QR (Fig.5) in the dispersed network, it must have an internal angle θ = QPR π/3. We could get some results from elementary geometry as follows. 1) Area of triangle S PQR = 1 3 2 ab sin θ 2R s 2 2 = 3R 2 s. 2) Radius of circumcircle r = c 2 sin θ R s. 3 3) r < R s. 4) S = 1 2 ah a > 1 ( 2 R 3 ) 3 s 2 R s = 4 R2 s. The results of 3) and 4) should be taken care of with some geometry illumination. Fig.6 gives all you need, and proof is omitted. We see that the set V of active nodes is a maximal independent set of Unit Disk Graph G Rs (V ), where V is the set of all deployed sensors. We call this set V of active nodes as the dispersed network. Theorem 4.3. While R c 2R s, the SparseDT graph G(V, E) we created on the dispersed network, which is a set V of active nodes, is a Delaunay triangulation with unit length r = 3R s, i.e., UDel 3R s (V ). Proof. We prove the equality of graph G(V, E) and UDel 3R s (V ) according to the definition of Delaunay triangulation and Definition 4.1. For any edge s 1 s 2 G(V, E), it must be legalized by the local Delaunay triangulation step of s 1 or s 2 in our algorithm. Without loss of generality, suppose that its s 1 (see Fig.7). If s 1 s 2 is a Gabriel edge in the local triangulation, so it is in UDel 3R s (V ). Or else, there must be a minimum circle, which legalizes edge s 1 s 2, with an active node s 3 on its circumference. It is easy to verify that s 3 C(s 1, s 2 ), which is the the minimum circle containing s 1 and s 2, so that the three edges of ABC all have lengths of less than 3R s. We have that the circumcircle of this triangle has radius r < R s according to Lemma 4.2 about the Delaunay triangle, so that there are no other nodes in V inside this circle as long as R c 2R s. We have s 1 s 2 UDel 3R s (V ). On the other hand, for any edge e UDel 3R s (V ), the minimum circle which legalizes this edge has a radius of less than R s from the reasoning given above. In this case, it could also be verified by our algorithm as long as R c 2R s. So we have e G(V, E) and prove the theorem. Theorem 4.4. While R c 3R s, the full coverage of the dispersed network implies the connectivity of the SparseDT graph G(V, E) we created. Proof. Suppose that the SparseDT graph G(V, E) is not connected, i.e., there are at least two connected components of it. Among all pairs of nodes which belong to different components, we choose one pair with the shortest distance, denoted with s 1 and s 2.
160 J. Comput. Sci. & Technol., Jan. 2008, Vol.23, No.1 There should have no active nodes inside the minimum circle C(s 1, s 2 ) containing s 1 and s 2. We know that s 1 s 2 3R s, or else the graph should have an edge s 1 s 2 E which should have been legalized by the triangulation step of our algorithm. On the other hand, we have s 1 s 2 < 2R s, or else the center of the circle C(s 1, s 2 ) is not covered by any sensor nodes. So there is an intersection point X between the sensing circles of s 1 and s 2 (see Fig.8). Fig.8. Proof of Theorem 4.4. With the knowledge of plane geometry, it is easy to verify that every point s inside the circle C Rs X has the attribute that min{ s 1 s, s 2 s } < s 1 s 2. So that there should have no active nodes inside this circle, or else s 1 and s 2 should not have the shortest distance among all pairs of disconnected nodes, which is a contradiction. It means that point X is not covered in the dispersed network, which also contradicts the presupposition of full coverage. Assume that the initial deployment of all sensors takes the Poisson distribution with density λ 0 = n πr2 s A, where A is the total area of the monitoring region. We bound the probability P a that the dispersed network fully covers the whole monitoring area, and this probability approaches one as the total number n of deployed sensors increases. The probability that there is a hole, which is not covered by the dispersed network, around any certain point X in the region, is P hole = 1 P a. It means that no sensors are active nearby when X is in a hole. We have P hole = n k=0 λ k 0/k!(1 P e λ0 DT ) k e λ0p DT, where P DT is the probability that a deployed node becomes active in the dispersed network. So that P a 1 e λ, where λ is the node density of the dispersed network. We have lim n P a = 1. In fact, if there is a hole with radius r, there must be no sensors deployed initially in the hole according to our algorithm, which has the probability P {hole is empty} = e nπr2 /A approaching zero as n. Definition 4.5. We define DT path between two nodes sink 1 and sink 2 as a chain of edge neighboring Delaunay triangles connecting them (see Fig.4). The length of a DT path is defined as the number of triangles on it. With the same analysis as the Junction graph, a DT path between two sinks means that the sensor nodes of the triangles on this path compose a 2-covered barrier. Because there is no loop in a DT path as defined, the number of triangles on a DT path, as well as the total number of sensor nodes of these triangles except two sinks, equals the weight of the corresponding path in Junction graph created based on SparseDT graph. According to the proof of Theorem 2.2, the shortest DT path depicts the optimum 2-covered barrier in dispersed network. In Algorithm DBC, we do not construct 2-covered barrier by find the shortest DT path between two sinks, because it needs the knowledge of dual Voronoi graph in each node and the path searching is relatively more complex. On the contrary, a much simpler method is used in algorithm DBC. As long as the SparseDT graph is connected, a shortest path between two sinks could be found using any multi-hop routing protocol, and by one side of this path may exist a DT path which satisfies the purpose we want. Following Theorem 4.6 guarantees the quality of the solution we found, and generally, the solution is even the optimum. Theorem 4.6. If the shortest path between two sinks in SparseDT graph is not adjacent to any nontriangular faces, the Delaunay triangles by one side of it compose a DT path between these two sinks. The length of it could be bounded as L (d max 2)L opt /4+ 1, where L opt is the length of the shortest DT path, and d max is the maximum vertex degree in SparseDT graph. On average, the length of it is within one of the optimal. Proof. Suppose that the shortest path P has length m, and the shortest DT path from sink 1 to sink 2 crosses n triangles (see Fig.4), i.e., L opt = n. We see that the total number of edges by two sides of this DT path is 3n 2(n 1) = n + 2. Those edges at either side compose a path between these two sinks, and the length of the shorter one is no more than n/2 + 1 m. Now given a shortest path P with length of m, there are at most d max (m 1) incident edges on the inner nodes of path P except the edges on path, and one side of this path has as most (d max 2)(m 1)/2 (d max 2)n/4 incident edges. As long as the shortest path is not adjacent to any non-triangular faces, these edges along with their adjacent triangles depict a DT path with length no more than (d max 2)n/4 + 1. So
Cheng-Dong Jiang et al.: Double Barrier Coverage in Dense Sensor Networks 161 we prove the theorem. We now bound the maximum vertex degree in SparseDT graph as d max 11. It is easy to verify that their minimum angle in a triangle with edge length R s e < 3R s is greater than π/6. So that d max < 12, and L 9n/4 + 1. From [11], we know that the average degree of a Delaunay triangulation is less than six, so is the average degree in SparseDT graph. On average, there are at most 4(m 1) incident edges on the inner nodes of path P except the edges on path, and one side of this path has as most 2(m 1) 2 n/2 n incident edges. The DT path constructed has length L n + 1, which is within one of the optimal, as long as the shortest path is not adjacent to any non-triangular faces. 4.2 Network Dilation and the Bound Analysis Definition 4.7. The network dilation of the network G(V, E) is defined by: D n = max u,v V τ G (u, v) uv R c where τ G (u, v) is the network length of the shortest path between nodes u and v, and uv is the Euclidean distance between them. Theorem 4.8. On average, it needs N DBC 2 D n s 1 s 2 /R c active sensors to compose a 2-covered barrier from s 1 to s 2 in a region. Proof. There is a theorem about network dilation [8]. In a sensing-covered network G(V, E), the network length of the shortest path between nodes u and v satisfies: τ G (u, v) D n uv /R c, where D n is the network dilation. According to Theorem 4.6, the DT path we constructed between s 1 and s 2 has average length of 2τ G (s 1, s 2 ). We can see that a new sensor is activated every time we cross a new triangle on the DT path. So we prove the result. Theorem 4.9. We could bound the number of active sensors as L/( 3R s ) N DBC 4L/( 3R s ), where L is the length of region. Proof. In Fig.3, the length of barrier segment (distance of circumcircle centers of two adjacent triangles) cc 1 2 Rs 2 (R s /2) 2 = 3R s according to Lemma 4.2. So that, the number of active sensors, i.e., the hops of barrier, N DBC L/( 3R s ). We get the lower bound. Now we estimate the upper bound of N DBC. First, the projection of Delaunay triangle on any line, Proj, is at least the smallest of three heights in triangle. Proj 2S /a 3/2R s (Lemma 4.2). It is trivial to verify that the projection of total DT path from s 1 to s 2 on line s 1 s 2 covers this line two times. It means that 2 s 1 s 2 = DT path(s 1,s 2) Proj N DBC 3/2R s = N DBC 4 s 1 s 2 /( 3R s ). Suppose that s 1 and s 2 are two sinks located at opposite sides of the region with length L, N DBC 4L/( 3R s ). According to Theorem 4.6, the shortest network length between any two nodes s 1, s 2 in SparseDT graph is approximately N DBC (s 1, s 2 )/2 2 s 1 s 2 /( 3R s ). So that, the asymptotic network dilation of SparseDT graph is Dn (SparseDT ) 2R c /( 3R s ). We could construct the optimum solution of double barrier coverage problem as two overlapped chains of neighboring sensors crossing the region. Opt(DBC ) = L/R s. Corollary 4.10. Assuming that R c = 2R s, the ratio of N DBC to Opt(DBC ) is D n 4/ 3. We could see that the asymptotic network dilation of our SparseDT graph is much better than the planar Delaunay triangulation presented in Filipe Araújo s work [9] and the normal sensing-covered network [8] which is 8 3π/9. 4.3 Target Detection Analysis The 2-covered barrier we presented here has better capability in target detection than two disjunct 1-covered barrier, which is constructed as a chain of connected neighboring sensors. Suppose that a target transverses the region straight downwards. According to Theorem 4.8, there Fig.9. Target detection scenario with DBC. Note that this ratio is not the approximate ratio of algorithm, but a relaxed bound.
162 J. Comput. Sci. & Technol., Jan. 2008, Vol.23, No.1 Suppose that the region has length L, and the sensing range of n randomly deployed sensors is equally R s, the isometry detection distance could be computed as d iso nπrs/l. 2 If target moves at speed v m/s, and the sensing cycle is T seconds, we can compute the average time N d that the target is detected crossing the region, N d = d iso /(vt ). It is inversely proportional to speed v while T is fixed, which could be verified later in simulation. In this model, we ignore the difference between barriers which are deployed randomly or sedulously. 5 Simulation Fig.10. Minimum 2-covered length across barrier. exists a 2-covered barrier in the shortest DT path. It also increases the detection length of 1-coverage, which offers higher detection probability than two disjunct 1-covered barrier. We now analyze the minimum detection length of 1-covered region and 2-covered region. In Fig.9, the target traces from p 1 to p 2, which delimit the 1-covered region, and the 2-covered region is indicated with thick line segment. The length of 1- covered region is p 1 p 2 > p 2 p 3 > p 3 p 4 3 3/2R s. On the other hand, the shortest length of 2-covered region could be found at line segment d 3 d 4 in Fig.10, whose prolonged line is tangent to the sensing circle of sensor C. This peculiar scenario needs very careful deployment of sensors and quite clever breach angle. It is easy to verify that d 3 d 4 0. It should be noticed that the area of short 1-covered region is just about the area of long 2-covered region, and vice versa, which is a fine complementary attribute of 2-covered barrier. In detection of high-speed target, we suppose that the target moves at speed v meters per second, and the sensor s sensing cycle is T seconds. It takes the target at least t = 3 3R s /(2v) 2.6R s /v seconds to pass the 1-covered region. As long as t T, the target must be detected. But for 1-covered barrier (even for several disjunct barriers), the target could not be assuringly detected at any speed. By the way, it needed not much more sensors to form one 2-covered barrier (Theorem 4.8) than two disjunct 1-covered barrier. As for the average performance of the barrier, we give another model for the barrier region analysis. Definition 4.11. We define the isometry detection distance d iso as the average length that a target is detected traversing the barrier region straight downwards. In this section, we present our simulation results. First, we study the average performance of our DBC algorithm. We then investigate the detection capability of the barrier it constructed. We implement the sparse Delaunay triangulation algorithm as a protocol agent and double barrier coverage algorithm as its application in NS2 simulator of version 2.30. All sensor nodes use the IEEE 802.15.4 as the MAC layer protocol. It is quite appropriate for monitoring applications with sensor networks, and its dynamic performance is better than 802.11 and some other MAC protocols aiming at sensor networks, such as s-mac and d-mac. We select two-ray ground model as the radio propagation model. Fig.11. 2-covered barrier constructed from 800 sensors deployed uniformly in an area of 50 50, with R s = 5. We set the sensing radius R s to be 5 meters, and communication radius R c to be 10 meters. Because there is no sensing model in NS2 simulator at present,
Cheng-Dong Jiang et al.: Double Barrier Coverage in Dense Sensor Networks 163 we devise a little trick to simulate the sensing event with the given sensing range and sensing cycle. Hundreds of sensors are deployed uniformly in an area of 50 50 meters to simulate a dense sensor network. Fig.11 shows sample session of constructing a 2- covered barrier. The solid lines are edges in Delaunay triangulation of those nodes selected from all sensors. The dark nodes are on the shortest path between two separate sinks at leftmost and rightmost sides of the monitored region. The 2-covered barrier is denoted by blue circles around the active nodes resulted from DBC algorithm. Next, we design a complete scheme for simulation of the barrier. Both statically and dynamically, the behavior of the barrier is analyzed, and the results are given. with the total number of sensors, and be bounded as Theorem 4.8 ensures. The ratio of ndbc to npath also matches the result of network dilation. In our simulation, it could be verified that the maximum degree nns (Fig.12) in the SparseDT graph is at most 6 on average. 5.1 Static Analysis: Number of Alive Sensors We increase the total number of nodes from 100 to 1000, with each case running tens times, and get a few in Fig.12 and Fig.13. ndt means the number statistic Fig.13. Simulation result of SparseDT algorithm. ndt means the Average number of active nodes in dispersed network. nmaxns denotes the density of deployed sensors. Fig.12. Simulation result of DBC algorithm. It displays the average number of nodes required to form a 2-covered barrier where total number varies. Data are grouped statistically with confidential interval of 0.05. of nodes in dispersed network. nmaxns denotes the density of deployed sensors. ndbc means the number of nodes needed to form a 2-covered barrier. npath means the number of nodes in the path from left to right. naperture means the number of apertures in the barrier which is explained later on. We can see that the average numbers ndt and ndbc increase In Fig.11, there is a small region which is not triangulated. This region is called hole. While the DT path passes by a hole, the barrier constructed may have an aperture in it, which means that the target may cross the aperture without being detected twice. We can see that the number of apertures increases while the total number of nodes decreases and the barrier will be degraded to be 1-covered. Because node density is small, the path should walk around many holes, and be prolonged a little. This counteracts the effect of increased hop distance due to the sparseness, and keeps the hop count of path. While number of nodes is less than 100, there is almost no path from left to right, so that the double barrier coverage is meaningless under sparse circumstance. The hole problem may have two reasons. One is that the region is not fully covered by this sensor network originally due to the random feature of deployment. The other arises from the simplicity of our sparse algorithm, because the algorithm does not consider the inequable density in different part of the area. Any methods managing it need special technique.
164 J. Comput. Sci. & Technol., Jan. 2008, Vol.23, No.1 5.2 Dynamic Analysis: Moving Target Detection We construct a 2-covered barrier for target detection analysis. It contains 14 active nodes selected from total 500 sensors deployed uniformly in region of 50m 50m. It is not perfect generally, and has an aperture in it. All sensors have the same sensing cycle of 500ms. One target crosses the region straight downwards with speed from 1m/s to 15m/s at step of 0.2m/s, and re- peats tens times at each speed. We compare the result with the 1-covered barrier which is just the shortest path spanning horizontally from left to right. We record three states: total times target is sensed by 2-covered barrier (n2-covered), times target is dually checked by two sensors at the same time in the 2-covered region (ndoublechecked), and total times target is sensed by 1-covered barrier (n1-covered). In Fig.14, we can see that the curve fits perfectly with the reversely proportional line, which proves our model mentioned in Section 4. The isometry detection distance here is d iso 7π, and the coefficient of the fitted line is 38 which approximates d iso /T with permitted error. This error attributes to the fact that there are two or three sensors which are out of the region boundary, so as to affect the computing. There is another error shown in Fig.14, that the fitted line is not approaching origin, which roots in the lack of simulation for higher speed. 6 Conclusion In this paper, we proposed double barrier coverage as an efficient scheme in sensor network deployed to detect objects penetrating a protected region. We presented a quite efficient distributed algorithm to construct the 2-covered barrier with as least as possible number of active sensors. Some fundamental theoretical results are derived according to the new sparse planar structure of SparseDT, and the performance of the 2-covered barrier is also analyzed. As for the nice property of the SparseDT, we could not bear to stop looking through it. It is quite possible to design a new topology control protocol based on it, and this work will be presented later on. References Fig.14. Simulation result of 2-covered barrier s detection capability while target moves at different speed. n2-covered denotes the average time when target is detected by sensors in 2-covered barrier. ndoublechecked denotes the average time when target is detected dually by sensors. n1-covered is just the same as n2-covered except that only 1-covered barrier is active and responsible for the detection. [1] Meguerdichian S, Koushanfar F, Potkonjak M, Srivastava M B. Coverage problems in wireless ad-hoc sensor networks. In Proc. INFOCOM, vol.3, Anchorage, Alaska, USA, 2001, pp.1380 1387. [2] Meguerdichian S, Koushanfar F, Qu G, Potkonjak M. Exposure in wireless ad-hoc sensor networks. In Proc. MOBI- COM, Rome, Italy, 2001, pp.139 150. [3] Liu B, Towsley D. On the coverage and detectability of wireless sensor networks. In Proc. WiOpt 03: Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, Sophia-Antipolis, France, 2003. From [12], we know that the necessary condition for asymptotic 1-coverage could be rephrased and approximated as n/ log(n) > 400/π = n > 300 in the region of 50m 50m with sensing range of 5m. We increase n to 500 here for one aperture hypothesis according to Fig.12. In common sense, human normally walks at speed of 5km/h 1.4m/s, and cars drive normally at speed of 54km/h = 15m/s. Considering the small sensing range here in simulation, we do not try higher speed scenario.
[4] Thai M T, Wang F, Du Ding-Zhu. Coverage problems in wireless sensor networks: Designs and analysis. International Journal of Sensor Networks, Special Issue on Coverage Problems in Sensor Networks, 2006, accepted. [5] Kumar S, Lai T H, Arora A. Barrier coverage with wireless sensors. In Proc. MOBICOM, Porta, T L, Lindemann C, Belding-Royer E M, Lu S (eds.), ACM, 2005, pp.284 298. [6] Li Xiang-Yang, Calinescu G, Wan P J. Distributed construction of planar spanner and routing for ad hoc wireless networks. In Proc. INFOCOM, New York, USA, 2002, pp.1268 1277. [7] Li Xiang-Yang, Stojmenovic I, Wang Y. Partial Delaunay triangulation and degree limited localized bluetooth scatternet formation. IEEE Trans. Parallel Distrib. Syst., 2004, 15(4): 350 361. [8] Xing Guoliang, Lu Chenyang, Pless R, Huang Qingfeng. Impact of sensing coverage on greedy geographic routing algorithms. IEEE Trans. Parallel Distrib. Syst., 2006, 17(4): 348 360. [9] Araújo F, Rodrigues L. Fast localized Delaunay triangulation. In Proc. OPODIS, Grenoble, France, LNCS 3544, Higashino T (ed.), Springer, 2004 pp.81 93. [10] Guibas L J, Knuth D E, Sharir M. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 1992, 7: 381 413, also In Proc. 17th Int. Colloq. Automata, Languages and Programming, Warwick University, England, LNCS 443, Springer-Verlag, 1990, pp.414 431. [11] Sack J, Urrutia G (eds.). Voronoi Diagrams. Handbook of Computational Geometry. Elsevier Science Publishing, 2000. [12] Kumar S, Lai T H, Balogh J. On k-coverage in a mostly sleeping sensor network. In Proc. MOBICOM, Haas Z J, Das S R, Jain R (eds.), ACM, 2004, pp.144 158. Cheng-Dong Jiang received the B.S. degree in computer science in 1996 from University of Science and Technology of China. He is now a Ph.D. candidate and faculty member in the Department of Computer Science at the same university. His research interests include topology control in wireless sensor networks, optimization techniques for NP-hard problem, and the design of VLSI. Guo-Liang Chen was born in 1938. Now he is an academician of Chinese Academy of Sciences and Ph.D. supervisor at University of Science and Technology of China, and director of the National High Performance Computing Center at Hefei, and a CCF senior member. His current research areas are parallel computing, computer architecture and combinatorial optimization.