Hierarchical Agglomerative Aggregation Scheduling in Directional Wireless Sensor Networks

Transcription

1 Hierarchical Agglomerative Aggregation Scheduling in Directional Wireless Sensor Networks Min Kyung An Department of Computer Science Sam Houston State University Huntsville, Texas 77341, USA Hyuk Cho Department of Computer Science Sam Houston State University Huntsville, Texas 77341, USA Lei Chen Department of Information Technology Georgia Southern University Statesboro, Georgia 30458, USA Abstract In this paper, we study the Minimum Latency Aggregation Scheduling (MLAS) problem in Wireless Sensor Networks (WSNs). The MLAS problem targets to attain data aggregation schedules that satisfy the two desirable properties: minimum latency and no collisions. Most existing works explored the problem under the uniform power model with no power control in omnidirectional WSNs. However, we investigate it under a more realistic non-uniform power model with power control in directional WSNs. To the best of our knowledge, addressing the MLAS problem in directional WSNs under non-uniform power model with power control is unprecedented. Unlike existing works that schedule nodes based on trees, our proposed scheduling algorithm does not create trees. Specifically, our algorithm employs multilevel divide-and-conquer steps, where a whole network is repeatedly partitioned into smaller networks and the smaller networks are systematically agglomerated to achieve the two desirable properties. We assess the performance of the proposed algorithm in terms of latency and power level for simulated networks. I. INTRODUCTION Wireless Sensor Networks (WSNs) consist of a number of small-sized sensor nodes deployed in a plane. Each node is equipped with sending and receiving antennas. Sending antennas can be either omnidirectional with a beam-width θ = 2π or directional with θ (0, 2π), whereas receiving antennas are omnidirectional. Nodes communicate among others using radio signals, and nodes with directional sending antennas can collaboratively determine and orientate their antenna directions [1]. WSNs consisting of nodes with omnidirectional sending antennas are called omnidirectional WSNs, and similarly those with directional sending antennas are called directional WSNs. One crucial task of WSNs is to monitor nearby environmental conditions periodically, and aggregate the sensed data from all nodes to a designated destination called a sink node. This type of application is commonly known as data aggregation in the literature. However, upon aggregating data, collisions can occur if data transmitted by a node is interfered with signals concurrently sent by other neighboring nodes, and then the data should be re-transmitted. As sensor nodes have limited energy resources, it is desirable to reduce such unnecessary retransmissions in order to reduce energy consumption as it can prolong the network s lifetime. An interesting existing approach is to assign timeslots to nodes in order obtain a schedule, through which all data can be aggregated without any collisions on their way to a sink node. Since the data aggregation occurs periodically with limited energies, reducing the latency of the schedule, i.e., constructing a schedule with a minimum number of timeslots, has been a critical issue. The problem, whose objective is to construct a collision-free aggregation schedule with the minimum latency, is called the Minimum Latency Aggregation Scheduling (MLAS) problem. The MLAS problem has been studied over the last several years in omnidirectional WSNs. Under uniform power model, where every node is initially assigned a uniform transmission power level, Chen et al. [2] proved its NP-hardness in the grid topologies using a reduction from restricted planar 3-SAT problem that is NP-complete [3]. They also introduced a ( 1)-approximation algorithm named Shorted Data Aggregation (SDA), where is the maximum node degree, which proceeds by incrementally constructing smaller and smaller shortest path trees (SPTs) rooted at a sink node. Du et al. [4] introduced Latency Bounded Data Aggregation Tree (LBDAT) algorithm, which is developed from Light Approximation Shortest-path Tree (LAST) algorithm [5] that aims at balancing minimum spanning tree (MST) and SPT. They proved that LBDAT produces schedules with latencies bounded by D min{, n}, where D is the network diameter and n is the number of nodes in a network. Later, Huang et al. [6], Yu et al. [7], and Xu et al. [8] proposed algorithms which build aggregate data trees based on Connected 2-hop Dominating Set (CDS), and schedule nodes using subprocedures, named first-fit algorithm [6] and distributed aggregation scheduling algorithm (SCHDL for short) [7]. The authors of [6] [8] proved that their algorithms produce schedules whose latencies are bounded by 23R + 18, 24D , and 16R + 14, respectively, where R is the network radius. Also, Wan et al. [9] used Sequential Aggregation Scheduling (SAS), Piplelined Aggregation Scheduling (PAS), and Enhanced Pipelined Aggregation Scheduling (E-PAS) algorithms as subprocedures to schedule nodes based on trees. They proved that the latencies produced by by SAS, PAS, and E-PAS are 15R + 4, 2R + O(log R) +, and Ä 1 + O Ä log R/ 3 R ää +, respectively. Later, Xu et al. [10] proposed an algorithm named Improved data Aggregation Scheduling (IAS) whose latency is 16R An et al.

2 [11] also proposed a constant factor approximation algorithm named Cell Coloring whose latency is bounded by O( +R). Yousefi et al. [12] proposed an algorithm that builds a tree with a balanced Connected 3-hop Dominating Sets (C3DS)- based structure, which is different from the commonly used CDS-based approaches, and simultaneously schedules nodes. Their latency bound is 12R+ 2. Recently, Feng et al. [13] proposed two algorithms: a dynamic programming scheduling algorithm in the given SPT and a greedy scheduling algorithm that simultaneously constructs the aggregation tree and then schedules nodes. Unlike uniform power model, non-uniform power model allows each node to have various transmission power levels. Therefore, determining the right power levels to be assigned (known as power control) is also an important task. Assuming that the maximum transmission power of a node is unbounded, Kesselman et al. [14] proposed a simple randomized distributed convergecast algorithm, called DC, whose latency is bounded by O(log n). The DC, however, schedules nodes with a constant probability p of successful transmission. Later, An et al. [11] showed the first result of Ω(log n) approximation lower bound for the MLAS problem in the metric model by constructing a polynomial-time approximation-preserving reduction from Set Cover problem which is known to be hard to approximate [15], [16], i.e., there is no approximation algorithm with an approximation ratio better than Ω(log n) unless N P DT IM E(n log log n). They also designed a heuristic algorithm called Minimum Data Aggregation on Backbone tree (MDAB) based on the SDA [2] that aggregates data along the shortest path towards a sink node. In directional WSNs, along with the assignment of timeslots, determining and orientating antenna directions (also known as antenna orientation problem) is also one of the objectives of the MLAS problem. Under uniform power model, Liu et al. [17] proposed a nearly constant factor approximation algorithm with 0 (0, 2π]. It first builds a CDS-based tree, and then assigns timeslots to nodes and orient their antennas based on the tree. Under non-uniform power model, there have been no studies done, to the best of our knowledge. Therefore, in this paper we study the MLAS problem under the non-uniform power model with power control in the directional WSNs. We first propose an algorithm that does not create trees on which most existing works have been based. Specifically, our algorithm employs hierarchical agglomerative steps, where a whole network is repeatedly partitioned into smaller networks and the smaller networks are systematically agglomerated to achieve the two desirable properties. We believe that it is the first work that addresses the MLAS problem in directional WSNs under non-uniform power model with power control. Table I summaries the works for the MLAS problem in different antenna and power models. Note that all algorithms [2], [4], [6] [13], [17] are tree-based. The rest of this paper is organized as follows. In Section II, we introduce the technique to equip directional antennas TABLE I: Existing works for the MLAS problem Uniform Power Non-uniform Power Omnidirectional WSNs [2], [4], [6] [10], [11], [14] [12], [13] Directional WSNs [17] This paper in WSNs, describe our network model, and then formally define the MLAS problem. In Section III, we describe our aggregation scheduling algorithm, and show its correctness. In Section IV, we evaluate the performance of our algorithm with simulated networks and compare with an existing work by Liu et al. [17] as it considers directional WSNs. Finally, we conclude with some remarks in Section V. II. PRELIMINARIES In this section, we briefly discuss the background so as to equip directional antennas in WSNs. Then, we describe our network model, and define the MLAS problem. Refer to Table II for notations. TABLE II: Notations Symbol Definition V Set of nodes in a network V Set of sender nodes n Number of nodes in a network R Network (graph) radius D Network (graph) diameter The maximum node degree of a network (u, v) Directional edge from nodes u to v d(u, v) Euclidean distance between nodes u and v p(u) Transmission power level of a node u p max Maximum transmission power level r(u) Transmission range (radius of broadcasting sector) of node u K Number of sectors of a node sec k (u) Broadcasting sector k of a node u (1 k K) S(u) Set of K sectors of a node u ω(u) Antenna orientation of a node u t Timeslot t(u) Timeslot assigned to a node u l Number of nodes scheduled at a timeslot t π t Assignment set at a timeslot t Π Schedule L Length of schedule Π z Number of rows (columns) of a partitioned network C i,j Cell at row i and column j (1 i, j z) h Head node of a cell A. Directional Antenna Model We adopt the switch beam directional antenna system [17], where each node is equipped with switch beam directional sending antenna and omnidirectional receiving antenna. In the system, each node u has K fixed broadcasting sectors, denoted by sec k (u), 1 k K, whose central angle is θ (0, 2π), and it switches on one of the sectors for transmission (i.e., each node is scheduled to emit its radio signal to the sec k (u) direction). Let us denote the set of K sectors of u by S(u) = {sec k (u) 1 k K}. Commercially available sectored antennas are typically designed for beam-widths of π, 2π/3, π/2, π/3, and π/4 [18].

3 B. Network Model A WSN consists of a set V of sensor nodes deployed in a plane. Each node u V is equipped with a switch beam directional sending antennas with a beamwidth θ (0, 2π) and an omnidirectional receiving antenna, a timeslot t(u), a transmission power level p(u) (0, p max ], where p max is assumed to be big enough to cover whole network, and an antenna orientation ω(u), which switches on (activates) one of the antenna sectors using p(u). Then, the transmission range r(u) of u can be defined as the radius of the broadcasting sector sec k (u) covered by the signal sent by u using p(u). A directed edge (u, v) is said to exist from node u to node v, if v sec k (u). collision occurs at node w such that w sec k (u) and w sec k (u ), if there exist concurrently sending nodes u and u, where t(u) = t(u ). C. Problem Definition The MLAS problem is defined as follows. Given a set V of nodes in a plane, we assign each node u a timeslot t(u), a power level p(u), and an antenna orientation ω(u). The goal is to compute a schedule with a minimum number of timeslots such that data from all non-sink nodes are aggregated to a sink node s V with no collisions. A schedule is defined as a sequence of such timeslots at each t i of which, several senders s ti, 1 i l, can be scheduled, and every s ti utilizes its ω(s ti ) to send its data to its neighbors in the activated sec k (s ti ) using its p(s ti ) with no collisions. Formally, at each timeslot t, we have an assignment set π t = {(s ti, ω(s ti ), p(s ti )) 1 i l}, where l denotes the number of nodes scheduled at timeslot t. An aggregation schedule is a sequence of assignment sets Π = (π 1, π 2,..., π L ), where L is the length of the schedule, also called latency. An aggregation schedule Π is successful if data from every node u V \ {s} is aggregated to a sink node s V. Input: A set V of nodes Output: A successful minimum latency schedule Π III. ALGORITHM In this section, we start by describing our algorithm, named Hierarchical Agglomerative Aggregation Scheduling (HAAS), for the MLAS problem under the non-uniform power model with power control in directional WSNs. We assume that every node u V is equipped with a switch beam directional antenna with a fixed beam-width θ = π/2, and an omnidirectional receiving antenna. Accordingly, broadcasting sector of each node u is partitioned into four sectors (i.e., K = 4), each of which is identified as sec k (u), k {1, 2, 3, 4} as shown in Fig. 1. A. Algorithm As described in Algorithm 1, HAAS starts with the set V = V of sender nodes (Step 1) and sets the timeslot t = 1 (Step 2). Then, the following steps (Steps 4 10) are repeated. Fig. 1: Broadcasting sectors of u with the beam-width θ = π/2 Algorithm 1 HAAS Input: A set V of nodes Output: Schedule Π 1: V V /* V is the set of senders. */ 2: t 1 3: repeat 4: Mark all nodes in V as non-head nodes. 5: Partition the network into z 2 square cells such that 0 C i,j 4, where C i,j denotes the set of nodes which reside in the cell at row i and column j (1 i, j z). 6: GRAY {C i,j i % 2 0 and j % 2 0} {C i,j i % 2 = 0 and j % 2 = 0} 7: W HIT E {C i,j i % 2 0 and j % 2 = 0} {C i,j i % 2 = 0 and j % 2 0} 8: t Scheduling(t, GRAY ) 9: t Scheduling(t, W HIT E) 10: V {v v V, v head or v s} 11: until z = 1 12: return Π (π 1, π 2, π L ) 1) (Step 4) Mark all nodes as non-head nodes. 2) (Step 5) The network is divided into z 2 square cells such that 0 C i,j 4, where C i,j denotes the set of nodes which reside in the cell at row i and column j (1 i, j z). Note that V = {u u C i,j, 1 i, j z}. 3) (Steps 6 7) The cells are divided into two groups, GRAY and W HIT E, where GRAY = {C i,j i % 2 0 and j % 2 0} {C i,j i % 2 = 0 and j % 2 = 0} and W HIT E = {C i,j i % 2 0 and j % 2 = 0} {C i,j i % 2 = 0 and j % 2 0} See an example of grouping of cells in Fig. 2. 4) (Step 8) Scheduling (Algorithm 2) is called to schedule the nodes in the cells of GRAY with the starting timeslot t start. Specifically, for each cell C i,j GRAY, the following steps are applied. a) (Step 2) It sets the first timeslot t for the cell with t start. b) (Steps 3 4) Scheduling uses a divider which is parallel to y-axis (or x-axis) to divide C i,j vertically (or horizontally) into two subcells, denoted by C A and C B, in each of which at most two nodes reside. (See Fig. 3.) The nodes in C A (C B ) are labeled as a and

4 Fig. 2: Partitioning a network into cells, and grouping the cells into GRAY and W HIT E a (b and b ), if available. Without loss of generality, let us assume that there are two nodes in each subcell, from now on. Here, node a (node b ) is the node that is closer to the divider than the other node a (node b). If sink is a (b), then swap sink s label with a (b ). c) (Steps 5 8) Node a is assigned the starting timeslot t start, and the power level d(a, a ) to send its data to a, where d(u, v) denotes the Euclidian distance between u and v. Accordingly, its transmission range r(a) = d(a, a ). The antenna orientation ω(a) is set to activate sec k (a) such that sec k (a) S(a), a sec k (a), and sec k (a) does not cover any other GRAY cells. Similarly, node b is assigned the next timelot t start +1, and the power level d(b, b ). The antenna orientation ω(b) is set to activate sec k (b) such that sec k (b) S(b), b sec k (b), and sec k (b) does not cover any other GRAY cells. d) (Steps 9 12) Scheduling selects one node h {a, b }, which is closer to s than the other node, as the head node of the cell C i,j. Without loss of generality, let us assume that h = a. Then the non-head node b is assigned the next timeslot t start + 2, and the power level d(b, a ) to send its data to the head a. The antenna orientation ω(b ) is set to activate sec k (b ) such that sec k (b ) S(b ), b sec k (b ), and sec k (b ) does not cover any other GRAY cells. At the end of the Scheduling call, every head node h {a, b } in each C i,j GRAY receives aggregated data from all the other nodes in C i,j, and Scheduling returns the next timeslot t next that will be the starting timeslot for group W HIT E. Note that the maximum number of timeslots used for Scheduling with GRAY is 3, i.e., t next t start ) (Step 9) Again, HASS calls Scheduling to schedule the nodes in the cells of W HIT E with the starting timeslot t start. Then, a) d) Scheduling assigns timeslots, and power levels, and antenna orientations to nodes in every cell C i,j W HIT E as for GRAY. Note that every antenna orientation for a node selects a sector that does not cover any other W HIT E cells. At the end of the Scheduling call for W HIT E, every head node h {a, b } in each C i,j W HIT E Algorithm 2 Scheduling Input: Starting timeslot t start, and a group GROUP Output: Next timeslot t next for the other group 1: for each cell C i,j GROUP do 2: t t start 3: Partition C i,j into two subcells C A and C B, in each of which at most two nodes reside. 4: Label nodes in C A as a and a, and nodes in C B as b and b as shown in the Fig. 3. 5: t(a) t, p(a) d(a, a ), ω(a) {sec k (a) sec k (a) S(a), a sec k (a), and sec k (a) does not cover any other GROUP cells.} 6: π t π t (a, ω(a), p(a)), t t + 1 7: t(b) t, p(b) d(b, b ), ω(b) {sec k (b) sec k (b) S(b), b sec k (b), and sec k (b) does not cover any other GROUP cells.} 8: π t π t (b, ω(b), p(b)), t t + 1 9: Pick node h {a, b } which is closed to sink s is the shortest as the head node for the cell. 10: h {u u {a, b }, u h} 11: t(h ) t, p(h ) d(h, h), ω(h ) {sec k (h ) sec k (h ) S(h ), h sec k (h ), and sec k (h ) does not cover any other GROUP cells.} 12: π t π t (h, ω(h ), d(h, h)) 13: end for 14: t next max{t(u) u C i,j, C i,j GROUP } : return t next receives aggregated data from all the other nodes in C i,j W HIT E, and Scheduling returns the next timeslot t next that will be the starting timeslot for GRAY. Note that the maximum number of timeslots used for Scheduling with W HIT E is also 3, i.e., t next t start ) (Step 10) HASS updates the sender set V by removing the scheduled nodes. Note that the updated V = {u u C i,j, 1 i, j z, u head or u s}. The HASS repeats the steps from 1) to 6) until every node except s is scheduled. B. Correctness of Algorithm In this section, we prove that the proposed algorithm, HASS, produces a successful schedules. Lemma 1: Nodes in the same group are scheduled with no collisions by Scheduling. Proof: Let us prove that there is no collision among nodes in the same group. Without loss of generality, consider one group, say GRAY. For each C i,j GRAY, HASS uses C i,j different timeslots to schedule every non-head node in C i,j at a distinct timeslot. Therefore, there is no collision among nodes in each C i,j. For every C i,j GRAY, every ω(u), u C i,j, activates sec k (u) that does not cover any other GRAY cells.

5 (a) (b) (c) Fig. 3: Dividing a cell C i,j into two subcells C A and C B (a) vertically or (b) (c) horizontally using a divider. (a) (c) Labeling nodes as a, a, b and b, and selecting one as a header. Therefore, a non-head node in one GRAY cell does not interfere with other non-head nodes in other GRAY cells. Thus, any two nodes in GRAY (or W HIT E) do not interfere each other. Lemma 2: Nodes in the different groups are scheduled with no collisions by Scheduling. Proof: Let us prove that there is no collision among nodes in different groups. Assume that the first and last timeslots used for GRAY (W HIT E) are t start and t last, respectively, where t start t last. After scheduling every non-head node in GRAY (W HIT E), Scheduling uses t last + 1 as the starting timeslot for scheduling W HIT E (GRAY ). Thus, all non-head nodes in GRAY have been already scheduled at different timeslots before the first non-head node in W HIT E is scheduled. By Lemmas 1 and 2, we can have the following corollary. Corollary 3: Nodes scheduled at the same timeslot by Scheduling do not interfere among others. Finally, we conclude with the following theorem. Theorem 4: The algorithm HASS produces a successful schedules. Proof: Given a network partitioned into cells (Steps 4 7 in Algorithm 1) with the sender set V, let us consider the iteration (steps 4 10) in which Scheduling (steps 8 9 in Algorithm 1) schedules every non-head node so that their data is aggregated to their head nodes without any collisions (Corollary 3). Next iteration starts with the reduced network that consists of only the head nodes at previous iteration. Similarly, the nodes are partitioned, and aggregated so that they are scheduled without any collision (Corollary 3). The aforementioned steps (Steps 3 11 in Algorithm 1) are repeated until there remains only one cell, in which the sink becomes the head node in the remaining cells. As a result, the sink receives the aggregated data from all the other nodes successfully. IV. SIMULATION In this section, we evaluate the performance of our proposed algorithm, HASS (Algorithm 1), in terms of latency and power level. Also, we compare it with the algorithm by Liu et al. [17] which studied the MLAS problem in the directional WSNs but without power control. As the algorithm by Liu et al. requires an initial power assignment of nodes with a uniform power level, we computed the minimally required power level r to get a connected graph as done in [19], and set the uniform power level p(u) for every u V to be r. In the Euclidean plane of dimension , we randomly generated a set G = {G n n = 100, 200,, 500}, where G n = {G i n 1 i 100} consists of 100 different networks,, each of which has n nodes. The same simulated networks were used for both the two algorithms. For each G n G, we averaged the latencies over 100 networks. As shown in Table III and Fig. 4, HASS performs better than Liu et al. s algorithm in terms of both average latencies and average power consumption. G 1 n, G 2 n,, G 100 n TABLE III: Average latencies and power levels Latency Average power level n HASS Liu et al. HASS Liu et al V. CONCLUSION In this paper, we studied the MLAS problem under the nonuniform power model with power control in the directional WSNs. Unlike most existing works that utilize trees, the proposed algorithm, named HASS, does not require to create trees. Specifically, HASS employs hierarchical agglomerative steps, where a whole network is repeatedly partitioned into smaller networks and the smaller networks are systematically agglomerated to achieve the two desirable properties. Our simulation results showed that HASS performs better than an existing algorithm by Liu et al. [17]. To the best of our knowledge, ours is the first study addressing the MLAS problem in directional WSNs under non-uniform power model with power control. As to the future work, we plan to analyze the complexity of the proposed algorithm and also study other related problems

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