An Energy-Efficient Distributed Algorithm for Minimum-Latency Aggregation Scheduling in Wireless Sensor Networks

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1 An Energy-Efficient Distributed Algorithm for Minimum-Latency Aggregation Scheduling in Wireless Sensor Networks Yingshu Li, Longjiang Guo, and Sushil K. Prasad Department of Computer Science Georgia State University {yli, lguo, Abstract Data aggregation is an essential yet timeconsuming task in wireless sensor networks (WSNs). This paper studies the well-known Minimum-Latency Aggregation Schedule (MLAS) problem and proposes an energy-efficient distributed scheduling algorithm named Clu-D based on a novel cluster-based aggregation tree. Our approach differs from all the previous schemes where Connected Dominating Sets or Maximal Independent Sets are employed. We prove that Clu-D has a latency bound of 4R + 2 2, where is the maximum degree and R is the inferior network radius which is smaller than the network radius R. Clu-D has comparable latency as the previously best centralized algorithm E-PAS, while Clu-D consumes 78% less energy as shown by the simulation results. Clu-D outperforms the previously best distributed algorithm whose latency bound is 16R + 14 on both latency and energy consumption. On average, Clu-D transmits 67% fewer total messages than does. We also propose an adaptive strategy for updating the schedule to accommodate dynamic network topology. Keywords: Cluster-based Aggregation Tree, MLAS, Inferior Network Radius, Maximum Degree. I. INTRODUCTION Wireless sensor networks (WSNs) have proven their success in various applications such as battlefield surveillance, traffic monitoring and forest fire monitoring. In some realtime applications, e.g. forest fire monitoring and aquiculture surveillance, users want to extract aggregate data information from WSNs with low latency to the sink. When two or more sensors send data to a common neighbor at the same time, data collision occurs at the common neighbor. Communication collision is a primary reason for long latency in data aggregation. Previous researchers of innetwork data aggregation do not consider the interference problem but leave it to the MAC layer which incurs a large amount of energy consumption and time latency during data aggregation. Recently, some researchers have begun to study the Minimum-Latency Aggregation Schedule (MLAS) problem [1] - [5] and have tried to find a minimum-latency schedule to overcome collisions during aggregation. MLAS, an NPhard problem [5], is defined as follows. Given a WSN that consists of a number of sensors and a sink, with each sensor having a piece of data to be aggregated and transmitted to the sink, the MLAS problem is to decide a collision free transmission schedule of data aggregation for all sensors such that the total time latency for aggregated data to reach the sink is minimized. Extensive research has been conducted on the MLAS problem. In general, these works can be classified into two categories: centralized algorithms and distributed algorithms. Centralized algorithms consist of three state-of-theart scheduling algorithms [1], [4], [5]. Chen et al. [5] proved that the MLAS is NP-hard. They designed an algorithm named SDA (Shortest Data Aggregation) based on Shortest Path Tree with a latency bound of ( 1)R, where is the maximum degree and R is the network radius. Huang et al. [4] proposed an algorithm based on Maximal Independent Set (MIS) which has a latency bound of 23R Wan et al. [1] proposed three algorithms: SAS, PAS and E- PAS. These algorithms have latency bounds of 15R + 4, 2R + O(logR) + and (1 + O(logR/ 3 R))R +, respectively. Although these centralized algorithms have an important contribution in theory, they are not practically applicable to WSNs, since they require the sink to compute the schedule and disseminate it to the sensors. Once all the sensors receive the schedule, they work according to the schedule. Since topology changes often occur in WSNs due to reasons such as node failures and nodes active/sleep state switching, the sink has to frequently gather new global network topology information, recompute a schedule and disseminate it. This consumes lots of energy making these algorithms less attractive. To overcome the above problems, researchers have proposed several distributed algorithms [2], [3]. Yu et al. [3] proposed a distributed algorithm named that has a latency bound of 24D , where D is the network diameter. Xu et al. [2] presented a distributed algorithm that has a latency bound of 16R + 14, where R is the inferior network radius which satisfies R R D 2R (see Sections IV-A and V-A). The two state-of-the-art distributed algorithms adopt Connected Dominating Sets (CDSs) [6] to construct aggregation trees. The substaintial difference between [2] and [3] is the analysis of time latency. They are not compared with the best centralized algorithm [1] on the aspects of energy consumption and time latency to prove

2 the fact that distributed algorithms can conserve more energy and extend network lifetime. In this paper, we present an energy efficient distributed algorithm named Clu-D. Clu-D constructs a novel Cluster-based Data Aggregation Tree (Clu-DAT) which is different from the commonly used CDS-based or MIS-based aggregation trees. The aggregation latency of Clu-D is 4R which is smaller than those presented in [2] and [3] in typical scenarios. But if is sufficiently large compared to R, the latency of [2] is smaller. In this case, a sensor s transmission range is relatively large compared to the area in which the sensors are deployed. However, this case rarely occur. The aggregation latency of Clu-D is better than all the previous algorithms except the best centralized algorithm E-PAS in [1] in most cases. Moreover Clu-D can save much more energy than E-PAS and all the other previous algorithms. To explore its practicability, extensive simulations to investigate aggregation latency and energy consumption were conducted, whereas in [1], no simulation results are presented. Simulation results show that Clu-D has a much better performance in practice than all the previous algorithms. To the best of our knowledge, Clu-D is so far the most practical algorithm for WSNs. II. PROBLEM DESCRIPTION We consider a WSN consisting of stationary sensor nodes with one sink in an Euclidean plane. All the sensors are homogeneous. We assume that the transmission coverage of any sensor node is a circle with unit radius centered at the node. For simplicity, a WSN with sink node s can be represented as a graph G = (V, E), where V denotes all the sensor nodes in the network and s V. An edge (u, v) E indicates that u and v can communicate, i.e., u lies in v s transmission range and v lies in u s transmission range. We also assume that G is connected. Definition 1: Neighbor Set. For a node u, if there exists another node v such that v lies in u s transmission area, then v is called u s neighbor. All of u s neighbors form a set, which is called u s Neighbor Set, denoted by Nbr(u). Nbr(u) does not include u itself. Data sent by a node u simultaneously reaches all the nodes in N br(u). Definition 2: Transmission Schedule. u v is called a Transmission Schedule, where u is the sender, and v is the receiver. u v denotes that u transmits data to v. Interference Model: In each time-slot, any node cannot send and receive data simultaneously, i.e., any node either sends data or receives data. In the protocol interference model [7], each node has a transmission range r and an interference range r I r. For any two transmission schedules u v and x y, we say u v and x y are conflicting if and only if u y r I or v x r I, where u y denotes the distance between u and y. As assumed by [2] [5], r I = r = 1. This assumption is for theoretically analyzing the time latency. Under this interference model, Figure 1. Three types of collisions. if two or more nodes are sending in the same time-slot and there exists a node v in their overlapped transmission area, then v cannot successfully receive any data since all transmissions are interfering with each other. This situation is called a collision. Examples of collisions are shown in Fig.1. Definition 3: Conflicting Transmission Schedules. u v and x y are called Conflicting Transmission Schedules if and only if v Nbr(x) or y Nbr(u). For example, in Fig.1(a), (b), and (c), u v and x y are conflicting transmission schedules. In Fig.1(d), u v and x y are not conflicting transmission schedules, despite a collision occurs at node w. Definition 4: Transmission Schedule Set. SH is called a Transmission Schedule Set if any pair of transmission schedules u v and x y in SH are not conflicting transmission schedules, i.e., v / Nbr(x) and y / Nbr(u). Definition 5: Sender Set of Transmission Schedule Set. SH = {u 1 v 1, u 2 v 2,..., u n v n } is a transmission schedule set. Sender(SH) = {u 1, u 2,..., u n } is called a Sender Set of SH. A data aggregation schedule is a sequence of transmission schedule sets {SH 1, SH 2,..., SH l }, where SH i (1 i l) is a transmission schedule set satisfying the following conditions: 1) i j, Sender(SH i ) Sender(SH j ) = ; 2) l i=1 Sender(SH i) = V {r}. All sensor nodes in V are organized as a data aggregation tree, where r is the root of the data aggregation tree. l is called the data aggregation latency. 3) Data are aggregated from Sender(SH k ) to V k i=1 Sender(SH i) in time-slot k, for all k = 1,..., l and all the data are aggregated to r in l time-slots. The MLAS problem is defined as follows. Given a graph G = (V, E) representing a WSN and the sink node s V, find a data aggregation schedule with the minimum latency. III. RELATED WORK Extensive research has been conducted on data aggregation. A distributed cross-layer scheduling protocol for data aggregation was proposed in [8], in which each node negotiates with its parent to decide its time-slots for transmission and constructs a schedule for its query processing. Chipara et al. [9] developed a dynamic scheduling scheme supporting

3 different kinds of aggregation queries. Yu et al. [10] studied the energy-latency tradeoff of scheduling for data aggregation. Huang et al. [11] studied packet loss and focused on reliability issues in data aggregation. Zhang et al. [12] addressed the bursty convergecast in real-time applications. Lee et al. [13] proposed a collision-free scheduling method for data collection. These works aimed at minimizing the overall energy consumption subject to the latency constraint and they do not consider the MLAS problem. The most relevant works for the MLAS problem are on the theoretical side. Kesselman et al. [14] proposed a randomized and distributed algorithm for aggregation in an n-node sensor network with an expected latency of O(logn). In their model, there are two assumptions. One is that each sensor has the capability of detecting whether a collision occurs after transmitting data. Another one is that sensors can adjust their transmission ranges at will. These assumptions pose some challenging issues for hardware design and the latter assumption is almost impossible when the scale of a WSN is very large. As research on the MLAS, the two kinds of algorithms, centralized algorithms [1], [4] and [5] and distributed algorithms [2] and [3], have already been discussed in detail in Section I. IV. CLU-D DISTRIBUTED ALGORITHM The key behind-the-scene idea of Clu-D is to construct a Cluster-based Data Aggregation Tree (Clu-DAT) to avoid conflicting transmissions among neighboring clusters. Transmissions among different clusters are concurrent and conflicting free. This can reduce the aggregation latency. Firstly, Clu-D constructs a Clu-DAT which is a clusterbased tree. In this process, some clusters are formed. The cluster heads are then connected by some intermediate nodes. For those nodes, they are neither cluster heads nor cluster members, but leaves of the Clu-DAT. Then, Clu- D decides an aggregation schedule based on the Clu- DAT in a distributed way. The process of constructing a Clu- DAT is presented in IV-A. Schedule generation is presented in IV-B. A. Construct a Clu-DAT In this phase, we first construct a Clu-DAT in a distributed way. For a given graph G = (V, E) and the sink node s V, we choose node v c V, the network center of G as the root of a Clu-DAT. A node v c is called the network center [2] in G, if v c = arg min v {max u {d G (u, v)}}, where d G (u, v) denotes the distance between u and v in G. It is actually the minimum number of hops from u to v. For example, in Fig.2, node 0 is the network center. As assumed by the previous works such as [2], the network center node is known and static. The computation of the network center is out of the scope of this paper. After the network center v c gathers the aggregated data from all the nodes, the aggregation result is forwarded to the sink s V using the shortest path from v c to s. Definition 6: Inferior Network Radius. The Inferior Network Radius of G, denoted by R = max{d G (u, v c ) u V }, which is the maximum distance between v c and any other node u in G. For example, in Fig.2, R = 4. We assume that v c is known in advance and each node u maintains the following data structures. 1) u s ID, u s level initialized to +. v c.level is set to 0. 2) u s parent in the Clu-DAT initialized to NULL. 3) u s color in the Clu-DAT initialized to NONE. 4) u s cluster head ID, clu-head-id, initialized to N U LL. It indicates u belongs to cluster clu-head-id. It can be set to NONE indicating u does not belong to any cluster. 5) u s neighbor list nbr-list. This list is a set of u s neighbors data structures identified by neighbors IDs. 6) u s neighbor s data structures include neighbor s ID, neighbor s level, and neighbor s 1-hop neighbor ID set. 7) u s children list, children-list, initialized to. The property of this list is the same as nbr-list. 8) u s forwarding flag, flag. If flag = 0, it indicates that u never forwards message BLD-CLU(u, v), where BLD-CLU(u, v) is the building cluster message that contains u s and v s data structures, and v is a cluster head to which u belongs; if flag = 1, it indicates that u has already forwarded BLD-CLU. This flag guarantees that u forwards BLD-CLU no more than once. Initially, each node initializes its data structures and broadcasts a message containing its ID to its 1-hop neighbors to exchange neighbors information. Constructing a Clu- DAT involves three main tasks: (1) Find cluster heads (represented by BLACK nodes) and corresponding cluster members (represented by WHITE nodes); (2) Connect cluster heads; (3) Link nodes which are neither cluster heads nor cluster members to a node which is already in the Clu-DAT. The first task follows the following rules: a) If a node u is a cluster head, then any v Nbr(u) is u s member. b) A node v cannot concurrently belong to two or more clusters. The Clu-DAT construction starts at v c, which is the root of the Clu-DAT at level 0. After the Clu-DAT construction is completed, any node in G is included in the Clu-DAT and has a color (BLACK/WHITE/BLUE/YELLOW/GREEN). v c first designates itself as a cluster head and colors itself BLACK. Then v c broadcasts BLD-CLU(v c, v c ) to its 1-hop neighbors in G. To accomplish the first task, all the nodes runs the following: For any node v (v v c ), upon receiving BLD-

4 CLU(u, u) from u, if v does not belong to any cluster, it calls DST(v, u, u, WHITE, u.level+1, u) (shown in Algo.1) to become cluster u s member and broadcasts BLD- CLU(v, u) to v s 1-hop neighbors to announce that v joins cluster u and regards u as its parent in the Clu-DAT. v also sets its flag to 1. if v belongs to a certain cluster x or v is a cluster head, according to the above mentioned rules, v simply discards the received BLD-CLU(u, u). For any node v (v v c ), upon receiving BLD- CLU(u, w), (w NULL and u w) from u, If v does not belong to any cluster, it calls DST(v, NONE, NULL, NONE, u.level + 1, u). If v.flag = 0, i.e. v never forwarded a BLD-CLU message, it broadcasts BLD-CLU(v, N U LL) to its 1-hop neighbors to announce that v does not join any cluster and v sets its flag to 1. Otherwise, v discards the received BLD-CLU(u, w). If v belongs to a certain cluster x, it calls DST(v, x, NULL, NONE, u.level + 1, u) and simply discards BLD-CLU(u, w). If v is a cluster head, it adds u to its children-list and nbr-list, and v discards BLD-CLU(u, w). For any node v (v v c ), upon receiving BLD- CLU(u, NULL) from u, If v belongs to a certain cluster x, it calls DST(v, x, NULL, NONE, u.level + 1, u) and discards BLD-CLU(u, NULL). If v does not belong to any cluster, it calls DST(v, NONE, NULL, NONE, u.level + 1, u). v checks whether all of its neighbors do not belong to any cluster, and whether v has the largest number of neighbors and the smallest ID. If so, v sets itself as a cluster head via DST(v, v, N U LL, BLACK, u.level + 1, u) and broadcasts BLD- CLU(v, v) to its 1-hop neighbors. Otherwise, if v.flag = 0, it broadcasts BLD-CLU(v, NULL) to its 1-hop neighbors and sets its flag to 1. If v.flag = 1, it simply discards the received BLD- CLU(u, NULL). The following lemma shows the property of BLACK nodes. Lemma 1: There are at least 3 hops between any two BLACK nodes in a Clu-DAT. Proof: Without loss of generality, suppose u is a BLACK node, and u has the most neighbors among its neighbors and all u s neighbors do not belong to any cluster. Then, u broadcasts BLD-CLU(u, u) to u s 1-hop neighbors. For any u s 1-hop neighbor w, when w receives BLD-CLU(u, u), w joins u s cluster, so w cannot be BLACK since w is u s cluster member. After joining in u s cluster, w forwards BLD-CLU(w, u) to w s 1-hop neighbors. For any w s 1-hop Algorithm 1 : DST(v, c-id, parent, color, level, nbr) Require: v: node ID; c-id: the cluster head ID to which v belongs; parent: v s parent in the Clu-DAT; color: v s color in the Clu-DAT; level: v s level in the Clu-DAT; nbr: v s neighbor. Ensure: Data structure maintained by node v. 1: v.clu-head-id c-id; 2: v.parent parent; 3: v.color color; 4: if level < v.level then 5: v.level level; 6: end if 7: v.nbr-list v.nbr-list {nbr}; neighbor x, x cannot be a BLACK node since otherwise, x s 1-hop neighbor w belongs to cluster x, which conflicts the fact that w has already joined cluster u. Therefore, u s 2- hop-away neighbors cannot be BLACK. The second task is to look for nodes that can connect BLACK nodes. These connecting nodes are colored BLUE or YELLOW. Once a node is colored BLACK, it tries to find some connecting nodes to get connected to another BLACK node whose level is less than u s level. u first finds a neighbor v from u.nbr-list which has the smallest level and the smallest ID, then sends CHILD(u, u) to v and regards v as it s parent, where CHILD(u, u) is the message to build a parent-children relationship in a Clu-DAT. The message means that u is in the cluster headed by u. The message contains u s data structures. The process below is executed at every non-black node which is a candidate to connect two BLACK nodes. All the BLACK nodes can be connected in the end. For each node v, upon receiving CHILD(u, u) from u, 1) v colors itself BLUE and adds u into its children list. 2) v finds a neighbor y from v.nbr-list which has the smallest level and the smallest ID and v sends CHILD(v, v.clu-head-id) to y and regards y as it s parent. The message CHILD(v, v.clu-head-id) means that v is in the cluster headed by v.cluhead-id. For each node v, upon receiving CHILD(u, u.clu-headid) from u, 1) v adds u into its children list. 2) If (v.clu-head-id).level < v.level < u.level then v colors itself YELLOW and discards CHILD. Otherwise, v colors itself BLUE and adds u into its children list. Then v finds a neighbor z from v.nbr-list which has the smallest level and the smallest ID and regards z as its parent. Finally, v sends CHILD(v, v.clu-head-id) to z.

5 Figure 2. An example Clu-DAT. If a node v finds that all its neighbors are colored WHITE, BLUE or YELLOW by checking v.nbr-list, and it still does not join any cluster, then v colors itself GREEN and randomly chooses a neighbor u from v.nbr-list which has the smallest level. v regards u as its parent by sending CHILD(v, N ON E) to u. If a BLACK/BLUE/YELLOW node w receives message CHILD(v, NONE) from v, w adds v to its children-list. These actions are for the third task. Table I shows nodes relationships in a Clu-DAT. Table I NODES RELATIONSHIPS IN A CLU-DAT u s color p(u) s color u s children s color GREEN WHITE, BLUE or YELLOW No children WHITE BLACK GREEN BLACK BLUE WHITE or YELLOW BLUE BLUE or YELLOW BLUE, GREEN or BLACK YELLOW BLACK BLUE or GREEN Fig.2 shows an example Clu-DAT for network G, where the number in each circle is the node ID, and the number beside each circle is the final level of the node. Node 15 is the sink and node 0 is the network center. The solid and dashed lines represent network links. Solid lines are the edges in the final Clu-DAT. B. Distributed Aggregation Scheduling In this section, we generate an aggregation schedule based on a Clu-DAT in a distributed way. The scheduling algorithm takes an input of a network topology G = (V, E) and a corresponding Clu-DAT. After scheduling, each node u is allocated a time slot to transmit its data after u has collected data from all of its children in the Clu-DAT. Definition 7: Competitor and Competitor Set. For a node u, a node v is called a Competitor of u if v cannot send data while u is sending data due to collision. The set of all the competitors of u in a Clu-DAT T is called u s Competitor Set [3] with respect to T, denoted by CS(u) = Nbr(p(u)) ( v Nbr(u)\Ch(u) Ch(v)) \ {p(u), u}, where p(v), Ch(u), and Nbr(v) are v s parent in T, u s children set in T, and v s 1-hop neighbor set, respectively. The competitor set of each node can be simply computed by nbr-list and children-list maintained by each node. Given a Clu-DAT T, each node u maintains the following information: 1) The number of u s children that have not been scheduled, denoted by N child(u), which is initialized to the number of u s children. 2) u s competitor set in T denoted by CS(u). 3) u s assigned time slot denoted by ts(u), which is initialized to 1. 4) u s schedule state DON E initialized to false. If DONE = true, it denotes that u has already been assigned a time slot ts(u). 5) u s ready competitor set RCS(u) = {v v CS(u) and Nchild(v) = 0}. Each node v RCS(u) is ready to make its schedule. RCS(u) is initialized by null set. The distributed scheduling algorithm determines ts(u) for node u based on u s color. Node color decides scheduling priority. The descending order of the priorities is: GREEN, WHITE, BLACK, BLUE and YELLOW. For any two nodes u and v, we say u s priority is less than v s priority, i.e. (u.color, u.id) < (v.color, v.id) if (1) u.color < v.color or (2) u.color = v.color and u.id < v.id. If a leaf node u has a color and u finds that all of its neighbors have their colors, then u starts the distributed scheduling algorithm. For each node u, if u s schedule state DONE is true, then u does nothing. For each node u, if N child(u) = 0, it does the following: 1) u sends a READY message containing u s ID to all the nodes in CS(u) and receives READY or REPLY from CS(u). 2) If node u gets the READY or REPLY messages from all the nodes in CS(u), then a) If RCS(u) =, then u sends COMPLETE(u, ts(u)) to all the nodes in CS(u) and sets DONE to true. b) If RCS(u), then it checks if u s priority is the largest in RCS(u). If (u.color, u.id) > (w.color, w.id) for each w RCS(u), if so, u sends COMPLETE(u, ts(u)) to all the nodes in CS(u) {p(u)} and sets DONE to true. For each node u, upon receiving READY from v, if v CS(u), u adds v to its RCS(u). Then, if Nchild(u) 0, u sends REPLY to v. For each node u, upon receiving REPLY from v, if v CS(u), then u records that it has already received

6 Figure 3. An example schedule. REPLY from v. For each node u, upon receiving COMPLETE(v, ts(v)) from v, 1) u deletes v from RCS(u). 2) If v is a child of u, then u reduces its Nchild(u) by 1. 3) u updates its ts(u) to max{ts(u), ts(v)+1} since u s assigned time slot must be posterior to ts(v). When Nchild(v c ) becomes 0, i.e. all the nodes have been scheduled, the algorithm ends. For data aggregation, all the nodes send data to its parent in the Clu-DAT in their assigned time slot. Fig.3 shows an example schedule based on the Clu-DAT in Fig.2, where the number in brackets is the assigned time slot. Node 0 is the last one to receive the final result. In the end, node 0 sends the final result to the sink 15 via the shortest path. A. Time Latency Analysis V. PERFORMANCE ANALYSIS Our distributed aggregation scheduling starts from the leaf nodes of a Clu-DAT. Once a node has sent COMPLETE and got its time slot, its schedule state DONE is set to true. Logically, such a node is removed from the Clu-DAT, resulting in some new leaves. This process is repeated until v c is the only node left. For simplicity, the rest part of Clu-DAT after removing GREEN leaves and WHITE nodes is called Rest-Clu-DAT. Fig.4 shows an example Rest-Clu- DAT based on the Clu-DAT in Fig.3. Leaves are scheduled according to their priorities. GREEN leaves are scheduled first. Next, WHITE leaves are scheduled. Third, the nodes in Rest-Clu-DAT are scheduled. We now estimate the time latency for these scheduling phases separately: (1) GREEN nodes scheduling; (2) WHITE nodes scheduling; (3) Nodes in Rest-Clu-DAT scheduling. (1) GREEN nodes scheduling. Lemma 2: For any leaf node u, suppose v i and v j are u s any two neighbor nodes, and v ik, v jl are any two neighbor nodes of v i, v j, respectively, such that v ik {Nbr(v i ) {p(v i ), u}}, v jl {Nbr(v j ) {p(v j ), u}}, v ik, v jl / (Nbr(v i ) {p(v i ), u}) (Nbr(vj) {p(v j ), u}). If v ik v i, v jl v j are scheduled before u p(u), then v ik v i, v jl v j are two non-conflicting transmission schedules. Proof: We prove this by contradiction. Suppose that v ik v i, v jl v j are two conflicting transmission schedules. According to Definition 3, v ik Nbr(v j ) or v jl Nbr(v i ). In addition, we know that v ik p(v j ) and v jl p(v i ), since otherwise, v ik, v jl cannot be scheduled before u since v ik and v jl are not leaves, whereas u is a leaf node. So, v ik {Nbr(v j ) {p(v j ), u}} or v jl {Nbr(v i ) {p(v i ), u}}, which conflicts with the fact v ik, v jl / (Nbr(v i ) {p(v i ), u}) (Nbr(v j ) {p(v j ), u}). Lemma 3: Let u be the last leaf scheduled in a Clu-DAT and p(u) be its parent. It takes at most max{ Nbr(v) 1 v Nbr(u)} time slots when u finishes sending data to p(u). Proof: We prove this by induction on the number of u s neighbors, N br(u). Base case: N br(u) = 1. u has only one neighbor p(u). u is scheduled if and only if u s priority is the largest in RCS(u), i.e. before ts(u), there are at most N br(p(u)) {p(p(u)), u} RCS(u) nodes that have already been scheduled. Thus, it takes at most 1 + Nbr(p(u)) {p(p(u)), u} RCS(u) time slots for u to send data to p(u), which is less than Nbr(p(u)) 1 = max{ Nbr(v) 1 v Nbr(u)}. Inductive hypothesis: For Nbr(u) = k > 1, Lemma 3 holds. Inductive step: Assume the inductive hypothesis is true for Nbr(u) = k > 1, we need to show Lemma 3 is true for Nbr(u) = k + 1. Suppose Nbr(u) = {v 1, v 2,..., v k, v k+1 }, Nbr(v i ) = {v i1, v i2,..., v i Nbr(vi) }, for simplicity, let N i = Nbr(v i ) {p(v i ), u}. For u s previous k neighbors v 1,..., v k, according to the inductive hypothesis, if u finishes sending data to p(u), then it takes at most max{ Nbr(v) 1 v (Nbr(u) {v k+1 })} time slots. N k+1 is divided into two parts: N k+1 = (N k+1 k i=1 (N k+1 N i )) k i=1 (N k+1 N i ). For all the nodes in k i=1 (N k+1 N i ), since k i=1 (N k+1 N i ) k i=1 N i, these nodes will be scheduled while the nodes in k i=1 N i are scheduled. Thus, they will not affect u s time slot ts(u). For any node v k+1j N k+1 k i=1 (N k+1 N i ) and v ml k i=1 N i (1 m k), since v k+1j / k i=1 (N k+1 N i ), v ml / N k+1, hence v k+1j / N k+1 N m, v ml / N k+1 N m. According to Lemma 2, v k+1j v k+1 and v ml v m are non-conflicting schedules. v k+1j and v ml can be scheduled simultaneously. Hence, u is the last scheduled leaf.

7 The number of time slots is at most: max{max{ N br(v) 1 v (Nbr(u) {v k+1 })}, N k+1 k i=1 (N k+1 N i ) }, which is no more than max{ Nbr(v) 1 v Nbr(u)}. Let u be the last GREEN leaf scheduled in a Clu-DAT, according to Lemma 3, it takes at most max{ Nbr(v) 1 v Nbr(u)} 1 time slots when u finishes sending data to p(u). (2) WHITE nodes scheduling. Let u be the last WHITE leaf scheduled in a Clu-DAT after removing GREEN leaves. According to Lemma 3, it takes at most max{ Nbr(v) 1 v Nbr(u)} 1 time slots when u finishes sending data to p(u). (3) Nodes in Rest-Clu-DAT scheduling. Suppose v 0 is the farthest BLACK node with respect to the network center v c. Now we estimate the time latency of aggregation from v 0 to v c. Consider the path {v 0...v 1...v 2...v k } (v k = v c ) from v 0 to v k in a Rest-Clu- DAT, where v i (i = 0, 1,..., k) are BLACK and there are some BLUE and YELLOW nodes between two adjacent BLACK nodes. We first estimate the number of BLUE nodes neighbors in a Rest-Clu-DAT. The following Wegner Theorem [15] is used. Theorem 4: Wegner Theorem [15]. The area of the convex hull of any n 2 non-overlapping unit-radius circular disks is at least: 2 3(n 1)+(2 3) 12n 3 3 +π. Lemma 5: Let u be a BLUE node in a Rest-Clu-DAT, then u has at most 4 BLUE neighbors. Proof. Its complete proof is available in [16]. Lemma 6: Let u be a YELLOW node in a Rest-Clu-DAT, then u has at most 4 BLUE neighbors. Proof. The proof is similar to Lemma 5. See [16]. Lemma 7: Let u be a BLACK node in a Rest-Clu-DAT, then u has at most 6 YELLOW neighbors. Proof. Its complete proof is available in [16]. Lemma 8: It takes at most 3(k + 2) time slots when v i finishes sending data to v i+1 in a Rest-Clu-DAT through subpath {v i, b 1,..., b }{{ k, y, v } i+1 }, where y is a YELLOW node, k b 1,..., b k (k 2) are BLUE nodes, and b j+1 is b j s father. Proof. Its complete proof is available in [16]. Lemma 9: Let v 0 be the farthest BLACK node with respect to the network center v c in a Rest-Clu-DAT. The time latency from v 0 to v c is at most 3R. Proof. Its complete proof is available in [16]. The following theorem estimates the total time latency. Theorem 10: The time latency of Clu-D is at most 4R Proof. Its complete proof is available in [16]. In the remainder of this section, we clarify the relationship of R, R, D and 2R which is R R D 2R. The aim is to unify the metric of aggregation time latency. Table II shows the upper bound of the latencies of all the previous works, where A = logr/ R, B = log(2r )/ 2R. Figure 4. An example Rest-Clu-DAT. Table II SUMMARY OF ALL WORKS Ref. Algo. Bound of latency Estimated upper bound of latency [5] SDAT (R 1) (2R 1) [4] Scott 23R R + 18 [1] SAS 15R R + 4 [1] PAS 2R + O(logR) + 4R + O(log(2R )) + [1] E-PAS (1 + O(A))R + (1 + O(B))2R + [3] 24D R [2] 16R R + 14 This Clu-D 4R R paper Definition 8: Network Radius. The Network Radius of G, denoted by R, is the maximum distance between the sink s and any other node in G. Definition 9: Network Diameter. The Network Diameter of G, denoted by D = max{max{d G (u, v) u V } v V }, is the maximum distance between any two nodes in G. For example, in Fig.2, R = 4, R = 7, and D = 8. Proposition 11: R R D 2R. Proof. Based on Definitions 6, 8 and 9, we have R = min{max{d G (u, v) u V } v V }; R = max{d G (u, s) u V }; D = max{max{d G (u, v) u V } v V }. Hence, R R D. Without loss of generality, suppose there are two nodes u 1 and v 1, such that D = d G (u 1, v 1 ) = max{max{d G (u, v) u V } v V }, where d G (u 1, v 1 ) denotes the minimum number of hops from u 1 to v 1, thus D = d G (u 1, v 1 ) d G (u 1, v c ) + d G (v c, v 1 ) 2R. B. Message Complexity Analysis In this section, we analyze the message complexity. The Clu-DAT construction needs O(n + (n/ )R ) messages, where n is the number of nodes in a WSN. This is because that each node requires O(n ) message transmissions to get neighbors information. There are totally O(n ) BLD- CLU messages forwarded by each node. Connecting cluster heads requires O((n/ )R ) messages. This is because

8 finding nodes to connect cluster heads starts from BLACK nodes. Suppose there are x BLACK nodes and y leaves in a Clu-DAT. Each BLACK node has cluster members, so we have x + y n. Thus x n/. A CHILD message is forwarded from each BLACK node from the lowest level to upper levels through R hops, so it needs O((n/ )R ) messages. Thus, the Clu-DAT construction requires O(n + (n/ )R ) messages. Next, the distributed scheduling algorithm needs O(n ) messages. The reason is that each node sends and receives messages from its competitors and the number of messages transmitted by each node can be bounded by the number of a node s neighbors. From Lemmas 3 through 7, we know that the number of neighbors of GREEN, WHITE, BLACK, BLUE and YELLOW nodes are 1, 1, 6, 4, and 4 respectively. For each BLACK node, there exist a corresponding BLUE node and a corresponding YELLOW node, since BLUE node is a BLACK node s parent and YELLOW node is BLUE node s parent. Thus, the number of BLUE/YELLOW nodes is no more than the number of BLACK nodes which is n/. Therefore, the total number of messages is at most (3(n/ )) 6 + z( 1), where z is the sum of the number of GREEN and WHITE nodes in a Clu-DAT. Since z n, 18(n/ ) + z( 1) O(n ). For the previous distributed algorithms in [2], [3], the one in [3] requires O(n ) total messages. The one in [2] focuses on reducing the time latency and does not give the analysis of message complexities in details. In fact, the message complexity of the works in [2], [3] are the same, this is because both of them employ the same CDS-based approach [6] to construct an aggregation tree and their scheduling algorithms are the same. The essential difference between these two works is the analysis of time latency. According to the above analysis, our work has the same upper bound of message complexity as [2], [3], but our work has less message transmissions in practice since 18(n/ ) + z( 1) O(n ). For a centralized algorithm, the sink needs to gather topology information from a WSN. This requires O(nR) messages. The sink computes the schedule and disseminates it to all the sensors. This also requires O(nR) messages. A centralized algorithm is not practically applicable to WSNs, especially for large scale WSNs where R >> is common. VI. AN ADAPTIVE SCHEDULING OF CLU-D In this section we consider the maintenance of a Clu-DAT and an adaptive scheduling of Clu-D considering dynamic topology. We focus on stationary WSNs. To conserve energy, sensor nodes need to switch between the sleep/wakeup states periodically, thus the topology changes often. We assume that the network is always connected. When a node u wakes up, it must join in a Clu-DAT, u sends a JOIN message. All the nodes in u s transmission area will receive the message. For any node v receiving JOIN, it sends back an ACK including its color. After u collects all ACK from its neighbors, it checks if there are messages from BLACK nodes. If so, u picks any one, say w, as its cluster head and sends a CHILD message to w to join w s cluster. u becomes a cluster member of w and colors itself WHITE. Otherwise, if u has BLUE, YELLOW, or WHITE neighbors, u picks any one, say p, as its parent and sends a CHILD message to p to be p s leaf and u colors itself GREEN. If u has only GREEN neighbors, u makes itself a BLACK node and u picks any GREEN neighbor, say q, as its BLUE parent and sends a CHILD message to q to be q s child. When a node q receives a CHILD message from u, it colors itself BLUE. If q s parent is WHITE, then q notifies its parent to become YELLOW. When a node u is going to sleep or its energy is going to be exhausted, u broadcasts a SLEEP message containing u s ID and u s color. All u s 1-hop neighbors receive the SLEEP message. If a u s neighbor v receives the SLEEP message, v knows that u should be removed from the topology and the following corresponding actions will be carried out. If u is GREEN or WHITE, and v is u s parent, then v removes u from its child list. If u is a WHITE non-leaf node, and v is u s GREEN child, then v can be regarded as new joining node. If u is BLACK, and v is u s WHITE child, then v can be regarded as a new joining node. If v is u s BLUE parent, then v removes u from its child list and if v does not have any other children, then v turns its color to GREEN. If u is YELLOW, and v is u s GREEN child, then v can be regarded as a new joining node. If v is u s BLUE child, v will find a new parent p from v.nbr-list which has the smallest level and the smallest ID and v sends CHILD to p and regards p as it s new parent. If v is u s BLACK parent, v removes u from its child list. If u is BLUE, and v is u s parent, then v removes u from its child list. If v is u s GREEN child, then v can be regarded as a new joining node. If v is u s BLUE child, then v will find a new parent p from v.nbr-list that has the smallest level and the smallest ID. v sends CHILD to p and regards p as it s new parent. If v is u s BLACK child, then v will find a new parent p from v.nbr-list which has the smallest level and the smallest ID and v sends CHILD to p and regards p as it s new BLUE parent. Any node who loses its parent can always find a new parent since the network is connected. After the Clu-DAT has been updated, those nodes that have changed their parents recompute their new competitor set and send their updated info to their new competitors. After that the nodes who

9 have changed parents mark themselves renewed and send a RENEW message to their parents. Every node receiving RENEW marks itself renewed and forwards RENEW to its parent. If a node is an unrenewed node, then the node sends its scheduled time slot to its renewed competitors. Each renewed node u needs to run Clu-D again to get a new ts(u). In this way, a Clu-DAT is maintained when topology changes without incurring much traffic. VII. SIMULATION RESULTS In our simulations, we randomly and uniformly deployed N sensors into a square region of size 200m 200m. The topology simulator takes in an input of R, and a transmission range of sensor nodes. All the sensors have the same transmission range r. The sink is always the node with ID 0. Its position is random. We compared our Clu- D with SAS, PAS and E-PAS (the best centralized algorithm) proposed in [1], and (the best distributed algorithm) proposed in [2]. We evaluate the performance from two aspects: average aggregation time latency and total messages. For average aggregation time latency, we conducted our simulations from two points of view: (a) the effects of R and on the average aggregation latency. (b) the effects of N and r on the average aggregation latency. First, is set to 26 and r is set to 30m. R varies from 10 to 150 with an increment of 10. For each R, we generated 30 networks. The size of each network is proportional to R. We computed the average aggregation latency for these networks. As can be seen from Fig.5(a), the average aggregation latency is proportional to R. On average, Clu-D has 30% less average time latency than the previously best distributed algorithm, and 5% more average time latency than the previously best centralized algorithm E-PAS. Second, R is fixed to 25, and r is set to 30m. varies from 22 to 78 with an increment of 4. For each fixed, we generated 30 random networks and computed the average aggregation latency for these networks. As shown in Fig.5(b), the average aggregation latency is also proportional to. On average, Clu-D has 8% less average time latency than, and 4% more average time latency than E-PAS. Third, r is set to 30m. N varies from 200 to 1150 with an increment of 50. For each N, we generated 30 networks and computed the average aggregation latency for these networks. As shown in Fig.5(c), on average, Clu-D has 8% less average time latency than, and 3% more average time latency than E-PAS. Fourth, N is fixed to 200. r varies from 25 to 67 with an increment of 3. For each r, we generated 30 networks and computed the average aggregation latency for these networks. In Fig.5(d), it shows that on average, Clu- D has 8% less average time latency than, and 2% less average time latency than E-PAS. In this group of simulations, for Clu-D, most nodes become cluster members, and GREEN leaves are very few, which makes Clu-D outperforms E-PAS on average time latency. Although increscent is unfavorable for Clu-D, Clu- D still has less average time latency than and E- PAS. For total messages, the network topology is dynamic. 500 nodes were randomly deployed in a fixed region of size 200m 200m. We use node switching rate α to indicate the topology changing frequency, which is defined as the ratio of the number of the nodes switching from active/sleep state to sleep/active state over the total number of the nodes. Since SAS, PAS and E-PAS are all centralized algorithms, and they work similarly. Their energy consumptions are almost the same, so we only compare Clu-D with E-PAS for this group of simulations. First, r is set to 30m. α varies from 30% to 72% with an increment of 3%. In Fig.5(e), it shows on average, Clu- D has 48% fewer total messages than, and 70% fewer total messages than E-PAS. Second, r is set to 30m. We name the duration for a complete data aggregation for all the sensors a round. Topology varies in each round. α is set to 40%. In this group of simulations, we measured the total messages of each algorithm with round varies from 10 to 100 with an increment of 5. As shown in Fig.5(f), on average, Clu- D has 67% fewer total messages than, and 78% fewer total messages than E-PAS. Obviously, Clu-D can save much more transmission energy, which helps a lot with extending network lifetime. This result conforms to our analysis in Section V-B, i.e. the main reason of saving energy of Clu-D have been discussed in details in Section V-B. The simulation results show that Clu-D has comparable time latency as the previously best centralized algorithm E-PAS, while Clu-D consumes much less energy for transmission. For WSNs, especially for large scale WSNs, distributed algorithms are greatly preferred. Compared with the previously best distributed algorithm, Clu-D demonstrates better performances on both time latency and energy conservation. VIII. CONCLUSION In this paper, we investigate the problem of MLAS. We proposed the techniques of constructing Clu-DAT and designed a distributed scheduling algorithm based on a Clu- DAT with a latency bound of 4R We theoretically proved that Clu-D has a latency bound of 4R +2 2, where is the maximum degree and R is the inferior network radius which is smaller than the network radius R. The simulation results indicate that Clu-D has comparable latency as the previously best centralized algorithm E-PAS, while Clu-D consumes much less energy. Clu-D outperforms the previously best distributed algorithm

10 Aggregation time latency SAS PAS E-PAS CLU-D = R (a) Aggregation time latency SAS PAS E-PAS CLU-D R= ( b) Aggregation time latency SAS PAS E-PAS CLU-D r = N (c) Aggregation time latency 65 SAS PAS 60 E-PAS 55 CLU-D N= r (d) Total messages E-PAS CLU-D α (e) Total messages E-PAS CLU-D α = 40% Rounds (f) Figure 5. Simulation results. whose latency bound is 16R + 14 on both latency and energy consumption. We also proposed an adaptive strategy for updating the schedule to accommodate dynamic network topology. ACKNOWLEDGMENT This work is supported by the NSF under grant No. CCF and CCF It is partly supported by the National Natural Science Foundation of China for Young Scholar under grant No , the China Postdoctoral Science Foundation under grant No , the Science and Technology Innovation Research Project of Harbin for Young Scholar under grant No.2008RFQXG107, the Heilongjiang Postdoctoral Science Foundation under grant No.LRB08-021, the Science and Technology Key Research of Heilongjiang Educational Committee under grant No.1154Z1001. REFERENCES [1] P.-J. Wan, S. C.-H. Huang, L. Wang, Z. Wan, and X. Jia, Minimum-latency aggregation scheduling in multihop wireless networks, ACM MobiHoc [2] X. H. Xu, S. G. Wang, X. F. Mao, S. J. Tang, and X. Y. Li, An improved approximation algorithm for data aggregation in multi-hop wireless sensor networks, FOWANC [3] B. Yu, J. Li, and Y. Li, Distributed data aggregation scheduling in wireless sensor networks, IEEE INFOCOM [4] S. C.-H. Huang, P.-J. Wan, C. T. Vu, Y. Li, and F. Yao, Nearly constant approximation for data aggregation scheduling in wireless sensor networks, IEEE INFOCOM [5] X. Chen, X. Hu, and J. Zhu, Minimum data aggregation time problem in wireless sensor networks, MSN [6] P.-J. Wan, K. M. Alzoubi, and O. Frieder, Distributed construction of connected dominating set in wireless ad hoc networks, Mobile Networks and Applications, vol.9, pp , [7] P. Gupta and P.Kumar, The capacity of wireless networks, IEEE Transaction on Information theory, vol.46, pp , [8] H. Wu, Q. Luo, and W. Xue, Distributed cross-layer scheduling for in-network sensor query processing, IEEE PERCOM [9] O. Chipara, C. Lu, and J. Stankovic, Dynamic conflict-free query scheduling for wireless sensor networks, IEEE ICNP [10] Y. Yu, B. Krishnamachari, and V. K. Prasanna, Energy-latency tradeoffs for data gathering in wireless sensor networks, IEEE INFOCOM [11] Q. Huang and Y. Zhang, Radial coordination for convergecast in wireless sensor networks, IEEE LCN [12] H. Zhang, A.Arora, Y.-R. Choi, and M. G. Gouda, Reliable bursty convergecast in wireless sensor networks, ACM Mobi- Hoc [13] H. Lee and A. Keshavarzian, Towards energy-optimal and reliable data collection via collision-free scheduling in wireless sensor networks, IEEE INFOCOM [14] A. Kesselman and D. Kowalski, Fast distributed algorithm for convergecast in ad hoc geometric radio networks, WONS [15] G. Wegner, Uber ber endliche kreispackungen in der ebene, Studia Scientiarium Mathematicarium Hungarica, vol.21, pp.1-28, [16]