Symmetric Connectivity in Wireless Sensor Networks with Directional Antennas

Symmetric Connectivity in Wireless Sensor Networks with Directional ntennas Tien Tran Department of Computer Science University of Texas at Dallas Richardson, TX 75080, US Email: tien.tran1@utdallas.edu Min Kyung n Department of Computer Science Sam Houston State University Huntsville, TX 7741, US Email: an@shsu.edu D. T. Huynh Department of Computer Science University of Texas at Dallas Richardson, TX 75080, US Email: huynh@utdallas.edu bstract In this paper, we study the ntenna Orientation (O) problem concerning symmetric connectivity in Directional Wireless Sensor Networks. We are given a set of nodes each of which is equipped with one directional antenna with beamwidth θ = 2/ and is initially assigned a transmission range 1 that yields a connected unit disk graph spanning all nodes. The objective of the problem is to compute an orientation of the antennas and to find a minimum transmission power range r = O(1) such that the induced symmetric communication graph is connected. We propose an algorithm that orients the antennas to yield a symmetric connected graph where the transmission power range is bounded by 6 which is currently the best result for this problem. We also study the performance of our algorithm through simulation. Index Terms wireless sensor network, directional antenna, symmetric connectivity, algorithm, complexity. I. INTRODUCTION In the literature, wireless sensor networks (WSNs) are commonly modeled as directed graphs where a directed edge exists from node U to node V if V resides in U s transmission range (broadcasting disk). Unlike the traditional WSNs where each sensor is equipped with an omnidirectional antenna whose beam-width (angle) θ = 2, recent advances have led to the development of new wireless sensor devices equipped with directional antennas whose angle is 2, so that they can collaboratively determine and orientate their antenna directions to minimize interference. In WSNs with directional antennas, the broadcasting area is represented as a sector defined by the beam-width θ < 2. s one of the main issues concerning WSNs is energy consumption due to each node s limited battery, topology control has been widely used to conserve energy of the sensor nodes in omnidirectional WSNs [1] [9] by assigning the transmission power levels to the nodes thereby establishing an energy efficient network with desired features, such as connectivity, low interference and fault tolerance [10]. Recently, researchers have started investigating the topology control in the directional WSNs as well. One interesting problem concerning topology control is the issue of establishing connectivity which is known as the ntenna Orientation (O) problem. In this problem we are given a set S of nodes, where each node V S is equipped with a directional antenna with beam-width θ [0, 2] and transmission range 1. Each node is allowed to use at most k 1 directional antennas. The objective of the problem is to find an orientation of the antennas and a minimum transmission power range r = O(1) such that the induced communication graph is a strongly connected or a symmetric connected graph. ssuming that each node has one antenna (i.e., k = 1), [11] [14] proposed algorithms that compute orientations of the antennas with a given beam-width and a power assignment as shown in Table I. In the meantime, [11] also proved that TLE I RESULTS OF THE NTENN ORIENTTION ROLEM CONCERNING STRONG CONNECTIVITY WHEN k = 1 Results in θ assigned range [11] θ < 8/5 2 sin( θ/2) [12] θ = 2/ 5 [1], [14] θ = /2 7 θ < 4/ 4/ θ < /2 2 /2 θ < 8/5 2 sin(/5) 8/5 θ 1 when the sector angle θ < 2/, the problem of determining the minimum range in order to achieve connectivity is Nhard. [15] and [16], on the other hand, studied the problem assuming that each node V can have multiple antennas (i.e., k 2) and the sum ϕ k of the angles (beam-widths) of the k antennas is bounded by θ [0, 2), and showed that there exist orientations and power assignment to obtain a strongly connected graph as shown in Table II. Later, [17] showed that for k = 2, if ϕ 2 < α, then it is N-hard to approximate the optimal range to within a factor of x, where x and α are the solutions of the equations x = 2 sin(α) = 1 + 2 cos(2α). ose et al in [18] studied the problem of replacing omnidirectional antennas with directional antennas with angle α, α < 2, to obtain strongly connected graphs. In their work, they considered not only the problem of minimizing the range of antennas but also tried to obtain a constant stretch factor with respect to the original graphs. Their results are shown in Table III. While these works concern strong connectivity, in term of symmetric connectivity, Carmi et al in [19] were the first to

TLE II RESULTS OF THE NTENN ORIENTTION ROLEM CONCERNING STRONG CONNECTIVITY WHEN 2 k 5 [15], [16] k ϕ k assigned range 2 ϕ 2 0 2 2/ ϕ 2 < 2 sin(/2 ϕ 2 /4) ϕ 2 2 sin(2/9) 6/5 ϕ 2 1 0 ϕ 4/5 ϕ 1 4 0 ϕ 4 2 2/5 ϕ 4 1 5 ϕ 5 0 1 TLE III RESULTS OF THE NTENN ORIENTTION ROLEM STRONG CONNECTIVITY ND CONSTNT STRETCH FCTOR WHEN k = 1 Results in θ assigned range stretch factor [18] θ / r = 4 2(.5κ 6) 8 log κ 1 (where κ = 2θ ) θ = / r = 6 2 10 θ > / r = 4 2( + κ) 10 and U resides in the broadcasting sector of V. The nodes are initially assigned a transmission range 1 that yields a connected unit disk graphs. Given S, the objective of the ntenna Orientation (O) problem for symmetric connectivity is to find an orientation of each antenna and the minimum transmission power range r = O(1) for every U S such that the induced communication graph is a symmetric connected graph. In the following we introduce a technique to obtain symmetric connectivity for a group of three nodes equipped with directional antennas with angle θ = 2/. The first lemma shows how we can orient the antennas of three nodes so that they form a symmetric connected graph. (See Fig. 1) Lemma 1: Given nodes each of which has a directional antenna with angle θ = 2/ and range =, there is an orientation of the antennas so that theses nodes form a symmetric connected graph. Moreover, the union of their sectors covers the entire plane. x study this problem. They showed that if θ /, k = 1 and the range is unbounded, then it is always possible to orient the antennas so that the induced symmetric graph is connected. [20] investigated the problem when θ = /2, they proved that one can orient the antennas when S = 4 and θ = /2 such that the induced symmetric graph is connected, and the union of the corresponding broadcasting sectors cover the entire plane. They also presented an algorithm that yields a symmetric connected graph based on a unit disk graph where the assigned range is 14 2 and the stretch factor is 8. In this paper, we continue the study of the ntenna Orientation (O) problem assuming that each node has one antenna (i.e., k = 1). The objective is to compute orientations of antennas with angle θ = 2/ and to find a minimum transmission power range r = O(1) such that the induced symmetric communication graph is connected. We propose an algorithm that orients the antennas to yield a symmetric connected graph where the transmission power range is bounded by 6 which is currently the best result for this problem. This paper is organized as follows. In Section II, we define the ntenna Orientation (O) problem studied in this paper and introduce techniques to obtain symmetric connectivity for groups with three nodes. In Section III, we introduce an antenna orientation algorithm for the O problem. Section IV shows some simulation, and Section V contains some concluding remarks. II. NTENN ORIENTTION ND SYMMETRIC CONNECTIVITY FOR GROUS OF THREE NODES In this paper, a Wireless Sensor Network (WSN) consists of a set S of sensor nodes deployed in the plane, and each node U S is equipped with one directional antenna with beam-width θ = 2/. symmetric (bidirectional) edge (U, V ) exists, if V resides in the broadcasting sector of U, 2 C Fig. 1. Orientation of three nodes roof: s in Fig. 1, let, and C denote the three nodes in the plane. Without loss of generality, let C be the longest edge and C be the smallest angle of C. We orient the antennas of the three nodes, and C as depicted in Fig. 1 so that x of s sector is parallel with C. ecause C is the smallest angle of C, it is smaller than /. Thus node covers node and vice versa. lso, node covers node C and vice versa. Hence, we obtain a symmetric connected graph with the three nodes, and C. In the special case when the nodes,, C are collinear and lies between and C as in Fig. 2, the same orientation of the antennas satisfies the lemma. C Fig. 2. Three nodes are collinear In the next lemma, we investigate the connectivity between two groups of nodes where each group contains three nodes

equipped with antennas with angle θ = 2/ whose range is unbounded. We show that if the antennas in each group are oriented as in Lemma 1, then these six nodes form a symmetric connected graph. Lemma 2: Let and be two groups each of which contains three nodes equipped with antennas with angle θ = 2/ and is assigned a range. If and are separated by a line, then there exist an orientation of the antennas such that the six nodes form a symmetric connected graph. (See Fig. ) 1 (b) Remove G from MST 5 (S). (c) If MST 5 (S) contains less than nodes, we are done; otherwise go back to Step (a). (Fig 4 depicts the possible groups of nodes we can obtain from MST 5 (S).) 1 2 2 lane lane Fig.. Groups of three nodes separated by a line roof: The proof of this lemma is quite technical and can be found in the full paper available from the authors. III. SYMMETRIC CONNECTIVITY FOR WSNS WITH DIRECTIONL NTENNS WITH NGLE 2/ In this section we introduce our algorithm to construct a symmetric connected graph from a UDG of a set S of nodes. The algorithm contains two main steps, and the idea is as follows. We first partition UDG(S), the UDG of S, into groups of nodes as done in [12]. ase on UDG(S), we construct a minimum spanning tree (MST) using a technique in [9] to obtain a Euclidean MST so that every node in that MST has at most 5 neighbors. Let us denote this MST by MST 5 (S). We then root MST 5 (S) at a node adjacent to at least one leaf and at most one non-leaf node as done in Lemma 5 in [9]. Finally we orient the antennas of nodes in each group separately as in Lemma 1. The algorithm is as follows. Symmetric Connectivity lgorithm 1 uild MST 5 (S) based on UDG(S). (This spanning tree is also a Euclidean MST of the set S.) 2 Choose a node which is adjacent to only one nonleaf node and at least one leaf node to be the root of MST 5 (S). (The existence of such a node has been proven by [9].) artition the tree into groups recursively as follows: (a) Starting at the farthest leaf of MST 5 (S), form a group G of nodes which satisfies the conditions below: G is as small as possible. G contains at least nodes. fter removing G, the tree MST 5 (S) remains connected. (a) (b) Fig. 4. ossible groups of sensors 4 For each group, choose, and as depicted in Fig. 4 to be the core nodes. The remaining nodes in the group are non-core nodes. * For core nodes, we orient their antennas as done in Lemma 1. * For each non-core node, we direct its antenna toward the core node in the same group that covers it. (ecause the core nodes in a group cover the entire plane, every non-core node must be covered by at least 1 core node.) 5 For any node that is not in any group, direct its antenna to the core node in an adjacent group that covers it. (Note that if the root is in a group, then every node belongs to a group. If the root is not in any group, then its adjacent leaf is also not in any group. In this case, we direct their antennas to a group adjacent to the root.) In the following we focus our attention on the performance of the algorithm, and prove that if the range of the antennas is at least 6, it is sufficient to form a symmetric connected graph spanning all nodes in S. In order to apply Lemma 2, we need the following lemma. Lemma : ny two groups of core nodes as constructed in the algorithm are separated by a line. roof: ccording to the choice of core nodes for a group, the three core nodes, say, and, together form 2 edges of the minimum spanning tree MST 5 (S). We first claim that does not contain any other node. In fact, consider the circles R 1 and R 2 with diameters and, respectively. Due to the circle property of Euclidian minimum spanning trees, there is no node inside R 1 and R 2. Since R 1 and R 2 cover entirely (R 1 and R 2 intersect at and the base H of the altitude from ), there is no node inside. (See Fig. 5.) Now, consider 2 groups of core nodes G 1 and G 2 which form two triangles. We claim that these two triangles do not intersect one another. In fact, consider the edges of the triangles of these two groups. If an edge of the triangle of (c)

G 1 intersects an edge of the triangle of G 2, there will be at least 2 edges of the triangle of G 1 that intersect 2 edges of the triangle of G 2 (because no vertex of G 1 is inside the triangle of G 2 and vice versa). Consequently, there would be at least one edge of the Euclidean MST that intersects with another edge which cannot happen. Therefore, no edge of G 1 intersects another edge of G 2 and vice versa. Thus, the triangles of G 1 and G 2 are disjoint. Hence, there is a line that separates G 1 and G 2. Group R 1 Group R2 Fig. 6. Maximum distance between 2 core nodes in 2 adjacent groups leaf H root Fig. 5. Group of core nodes We now prove our main theorem. Theorem 4: Let S be a set of nodes in the plane and UDG(S) be a unit disk graph spanning S. Futher let the nodes be equipped with directional antennas with angle θ = 2/ and range r 6. Then there is an orientation of the antennas that yields a symmetric connected graph. roof: We first prove the following Fact. The distance between 2 core nodes in two adjacent groups is bounded by 6. Consider 2 adjacent groups and. Let group be parent of group in MST 5 (S). This means that a node of is adjacent to some node of. We can see that the maximum distance between a core node of group and any node of group which is adjacent to node of group is. (See Fig. 4(c).) nd the maximum distance between node of and another core node of is 2. (See Fig. 4(a).) Hence the maximum distance between a core node of and any core node of is 6. (See Fig. 6 and note the distance from node in group to node in group.) So far, we have proven that range 6 is sufficient for any adjacent groups to connect with each other to form a symmetric connected graph. We still need to connect nodes that are not in any group. To this end, consider the root of MST 5 (S) and its adjacent leaves which do not belong to any group as depicted in Fig. 7. We simply direct their antennas to the core nodes in the group adjacent to the root that cover them. Note that range 6 is sufficient. IV. ERFORMNCE ND SIMULTION In this section, we will analyze the stretch factor of neighbor nodes in the unit disk graph UDG(S) that spans the set S of nodes. The stretch factor of two neighbor nodes in UDG(S) is the number of hops in the shortest path that connects them in the symmetric connected graph obtained by our algorithm. We show that in the worst case the stretch factor is Ω(n), where Fig. 7. Root and its adjacent leaves n is the number of nodes in the network. We then perform some simulations to obtain the practical performance of our algorithm concerning the average stretch factor as well as the average range that is sufficient to create a symmetric connected graph for the network. The results show that our algorithm performs well and yields networks with small average ranges and stretch factors. Finally, we analyze the time complexity of our algorithm. Claim. There exist networks of which the stretch factor is Ω(n), where n is number of nodes in the networks. roof: Consider a network depicted in Fig. 8. Let S be a set of n nodes, where X and X are neighbors in UDG(S). Let X and X belong to group X and group X, respectively. ll members of group X and group X are placed on a line and the distance between any two adjacent nodes in group X and group X is slightly less than 1 unit as depicted in Fig 8. The other nodes in the network form a circle that connect group X and group X so that a group in the circle can only connect to its adjacent groups on the circle. lso, the distance between two adjacent nodes on the circle is slightly less than 1 unit. ased on how we orient the antennas of group X and group X according to the algorithm, the node will cover the nodes, and X of group X, and the node will cover the sensor, and X of group X. However, the distance between node and node is set to be slightly less than 7 but larger than 6, which is the range defined for the directional antennas. Thus, in order to connect node X and node X in the

symmetric connected graph, we cannot go directly from group X to group X but have to go through every intermediate group on the circle. s there are (n 8) remaining nodes on the circle, we have at least (n 8) 2 groups (in this case, the root and its adjacent leaf are not in any group). To connect node X to node X, we have to go through every group, and for each group we need to orient the antennas of two nodes in the group towards its adjacent groups on the left and right sides. Thus, the number of hops on the connection between X and X is at least Ω(n) follows. leaf root (n 8) 2 2. Hence, the lower bound X X group X group X Fig. 8. Worst case spanner factor of two neighbors in UDG(S) In our simulation, we randomly deploy nodes on an area of dimension 1000 1000. We then choose a uniform range for every node in such a way that we can obtain different types of networks: dense networks, spare networks and random networks from the UDG. In the case of dense networks, the number of edges in the UDG is n 2 /4, where n is the number of nodes. In contrast to this, the number of edges in the spare networks is 4n. For random networks, there is no constraint concerning the number of edges in the network. We implement our algorithm and estimate the spanner factor for 1000 different cases for each type of networks and vary the number of nodes from 25 to 100. The result is showed in Fig. 9. 2.25 2.2 2.15 2.1 2.05 2 1.95 1.9 25 50 100 Spare Network Fig. 9. verage spanner factor of two neighbors in UDGs Random Network Densed Network We also try to minimize the range of nodes to obtain energyefficient networks. To this end, once we obtain a symmetric connected graph after running our algorithm, we compute an MST and set the range to be the length of the longest edge of the MST. We then average over the ranges of all networks generated. The simulation results in Fig. 10 show that on average the minimum range to get a symmetric connected graph is actually not as large as indicated in the theorem. Moreover, the range tends to decrease as the networks become denser. 2.5 2 1.5 1 0.5 0 25 50 100 Spare Network Random Network Densed Network Fig. 10. verage of the minimum ranges for symmetric connectivity It is interesting to note that for dense networks the average range is even smaller than 1 as seen in Fig. 10. This is actually not unreasonable because in dense networks the unit range is large in order to obtain n 2 /4 edges in the UDGs. nd the denser a network becomes, the larger the gap between the unit range and minimum range to obtain a symmetric connected graph. Finally, we analyze the complexity of our algorithm in Section 2. One can easily see that the complexity of Steps 2, 4 and 5 is O(n). Step is a recursive process, and at each iteration, we have to find the farthest leaf and form a group. We can achieve this by using breadth-first search to visit the MST and store nodes by levels. Then for each iteration, we just get a node from the farthest level from the root and form a group, and then remove all nodes in that group. Thus, the complexity of Step is O(n). s the complexity of the algorithm is dominated by finding the MST which is O(n log n), the overall running time of the algorithm is O(n log n). V. CONCLUSION In this paper, we studied the ntenna Orientation (O) problem concerning symmetric connectivity in wireless sensor networks whose nodes are equipped with directional antennas with a beam-width θ = 2/. We showed that if the range of the antennas is set to be 6, they can be oriented to yield a symmetric connected communication graph. To the best of our knowledge, the assignment of a range of 6 is the best upper bound for the problem. We also did some simulations and showed that our algorithm performs very well. Future research will focus on how to improve on the upper bound for the stretch factor, perhaps at the expense of a larger range. REFERENCES [1] L. M. Kirousis, E. Kranakis, D. Krizanc, and. elc, ower Consumption in acket Radio Networks, in roceedings of the 14th nnual Symposium on Theoretical spects of Computer Science (STCS), 1997, pp. 6 74. [2] E. L. Li, J. Y. Halpern,. ahl, Y.-M. Wang, and R. Wattenhofer, nalysis of a Cone-based Distributed Topology Control lgorithm for Wireless Multi-hop Networks, in ODC, 2001, pp. 264 27. [] G. Călinescu, S. Kapoor,. Olshevsky, and. Zelikovsky, Network Lifetime and ower ssignment in d Hoc Wireless Networks, in ES, 200, pp. 114 126. [4] D. Ye and H. Zhang, The Range ssignment roblem in Static d-hoc Networks on Metric Spaces, in SIROCCO, 2004, pp. 291 02. [5] E. L. Lloyd, R. Liu, M. V. Marathe, R. Ramanathan, and S. S. Ravi, lgorithmic spects of Topology Control roblems for d Hoc Networks, in MONET, 2005, pp. 19 4.

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