Ad Hoc Networks. Latency-minimizing data aggregation in wireless sensor networks under physical interference model

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1 Ad Hoc Networks (4) 5 68 Contents lists available at SciVerse ScienceDirect Ad Hoc Networks journal homeage: Latency-minimizing data aggregation in wireless sensor networks under hysical interference model Hongxing Li a,, Chuan Wu a, Qiang-Sheng Hua b, Francis C.M. Lau a a Deartment of Comuter Science, The University of Hong Kong, Pokfulam Road, Hong Kong b Institute for Interdiscilinary Information Sciences, Tsinghua University, Beijing, China article info abstract Article history: Received May Received in revised form Setember Acceted 6 December Available online December Keywords: Data aggregation Wireless sensor networks Physical interference model Minimum-latency Minimizing latency is of rimary imortance for data aggregation which is an essential alication in wireless sensor networks. Many fast data aggregation algorithms under the rotocol interference model have been roosed, but the model falls short of being an accurate abstraction of wireless interferences in reality. In contrast, the hysical interference model has been shown to be more realistic and has the otential to increase the network caacity when adoted in a design. It is a challenge to derive a distributed solution to latency-minimizing data aggregation under the hysical interference model because of the simle fact that global-scale information to comute the cumulative interference is needed at any node. In this aer, we roose a distributed algorithm that aims to minimize aggregation latency under the hysical interference model in wireless sensor networks of arbitrary toologies. The algorithm uses O(K) time slots to comlete the aggregation task, where K is the logarithm of the ratio between the lengths of the longest and shortest links in the network. The key idea of our distributed algorithm is to artition the network into cells according to the value K, thus obviating the need for global information. We also give a centralized algorithm which can serve as a benchmark for comarison uroses. It constructs the aggregation tree following the nearest-neighbor criterion. The centralized algorithm takes O( logn) and O(log n) time slots when couled with two existing link scheduling strategies, resectively (where n is the total number of nodes), which reresents the current best algorithm for the roblem in the literature. We rove the correctness and efficiency of our algorithms, and conduct emirical studies under realistic settings to validate our analytical results. Ó Elsevier B.V. All rights reserved.. Introduction Data aggregation is a habitual oeration of many wireless sensor networks, which transfers data (e.g., temerature) collected by individual sensor nodes to a sink node. The aggregation tyically follows a tree toology rooted at the sink. Each leaf node would deliver its collected data to its arent node. Intermediate sensor nodes of the tree may otionally erform certain oerations (e.g., sum, maximum, Corresonding author. Tel.: ; fax: addresses: hxli@cs.hku.hk (H. Li), cwu@cs.hku.hk (C. Wu), qshua@mail.tsinghua.edu.cn (Q.-S. Hua), fcmlau@cs.hku.hk (F.C.M. Lau). minimum, mean, etc.) on the received data and forward the result. Because the wireless medium is shared, transmissions to forward the data need to be coordinated in order to reduce interference and avoid collision. The fundamental challenge can be stated as: How can the aggregation transmissions be scheduled in a wireless sensor network such that no collision may occur and the total number of time slots used (referred to as aggregation latency) is minimized? This is known as the Minimum-Latency Aggregation Scheduling (MLAS) roblem in the literature [ 5]. The MLAS roblem is tyically aroached in two stes: (i) data aggregation tree construction and (ii) link transmission scheduling. For (ii), we assume the simlest mode /$ - see front matter Ó Elsevier B.V. All rights reserved. doi:.6/j.adhoc...4

2 H. Li et al. / Ad Hoc Networks (4) in which every non-leaf node in the tree will make only one transmission, after all the data from its child nodes have been received. A correct solution to the MLAS roblem requires that no concurrent transmissions interfering with each other should take lace. If stes (i) and (ii) are carried out simultaneously in a solution, we have a joint design. To model wireless interference, existing literature mostly assume the rotocol interference model, in which a transmission is successful if and only if its receiver is within the transmission range of its transmitter and outside the interference range of any other concurrent transmitters. The best results known for the MLAS roblem or similar roblems ([ 5]) under the rotocol interference model bound the aggregation latency in O(D + R) time slots, where R is the radius of the sensor network in hos and D is the maximal node degree (i.e., the maximum number of nodes in any node s transmission range). The rotocol interference model however has been found to be too simlistic and cannot serve as an accurate abstraction of wireless interferences. Instead, the hysical interference model [6], which catures the reality more accurately, is becoming more oular. Little research however has so far been done to address the MLAS roblem under the hysical interference model. The rotocol interference model considers only interferences within a limited region, whereas the hysical interference model tries to cature the cumulative interference due to all other concurrently transmitting nodes in the entire network. More recisely, in the hysical interference model, the transmission of link e ij can be successful if the following condition regarding the Signal-to-Interference-Noise-Ratio (SINR) is satisfied: P ij =d a ij N þ P e gh K ij fe ij g P gh=d a gj P b: ðþ Here K ij denotes the set of links that transmit simultaneously with e ij. P ij and P gh denote the transmission ower at the transmitter of link e ij and that of link e gh, resectively. d ij (d gj ) is the distance between the transmitter of link e ij (e gh ) and the receiver of link e ij. a is the ath loss ratio, whose value is normally between and 6. N is the ambient noise. b is the SINR threshold for a successful transmission, which is at least. We give an examle, in Fig., to demonstrate the advantage of the hysical interference model over the traditional rotocol interference model, with which the network caacity is underestimated (data aggregation time is longer). In the figure, six nodes are located on a line, where sink a aggregates data from the other five nodes, b f. The number on a link is the distance between the two nodes joined by the link. Under the rotocol interference model, any two concurrent transmissions conflict with each other, and therefore five time slots are needed to aggregate all the data to the sink a, such as by the sequence f? e? d? c? b? a. On the other hand, with Fig.. A data aggregation examle. the hysical interference model, three time slots are enough: at time slot, the transmissions b? a, d? c, and f? e can be scheduled concurrently, using transmission ower N b 6 a. At time slots and, e? c and c? a can be scheduled consecutively with transmission ower N b6 a and N b4 a, resectively. It can be easily verified that the above link scheduling and ower assignment satisfy the SINR condition () at each receiver under tyical network settings, e.g., a = 4 and b =. In this aer, we investigate the MLAS roblem under the hysical interference model. A solution to the MLAS roblem can be a centralized one, a distributed one, or mixed. For a large sensor network, a distributed solution is certainly the desired choice. Distributed scheduling algorithm design is significantly more challenging with the hysical interference model, as global information in rincile is needed by each node to comute the cumulative interference at the node. We are only aware of one study [7] which resents a distributed solution to the MLAS roblem under the hysical interference model; they derived a latency bound of O(D + R) in a network, where sensors are uniformly randomly deloyed. One of the drawbacks of this work is that the efficiency guarantee is not rovided for arbitrary toologies. In this aer, we tackle the minimum-latency aggregation scheduling roblem under the hysical interference model by designing both a centralized and a distributed scheduling algorithm. Our algorithms are alicable to arbitrary toologies. The distributed algorithm we roose, Cell-AS, circumvents the need to collect global interference information by artitioning the network into cells according to a arameter called the link length diversity (K), which is the logarithm of the ratio between the lengths of the longest and the shortest links. Our centralized algorithm, NN-AS, combines our aggregation tree construction algorithm with either one of the link scheduling strategies roosed in [8,9] to achieve the best aggregation erformance in the current literature. Our main focus in this aer is on the distributed algorithm; the centralized algorithm is included for comleteness and to serve as a benchmark in the erformance comarison. For situations in ractice, where centralization is not a roblem, the centralized algorithm may be a useful choice. We conduct theoretical analysis to rove the correctness and efficiency of our algorithms. We show that the distributed algorithm Cell-AS achieves a worst-case aggregation latency bound of O(K) (where K is the link length diversity), and the centralized algorithm NN-AS achieves worst-case bounds of O(logn) and O(log n) when couled with the link scheduling strategies in [8,9], resectively (where n is the total number of sensor nodes). In addition, we derive a theoretically otimal lower bound for the MLAS roblem under any interference model log (n). Given this otimal bound, the aroximation ratios are O(K/logn) with Cell-AS, O() with NN-AS and the link scheduling in [8], and O(log n) with NN-AS and the link scheduling in [9]. We also comare our distributed algorithm with Li et al. s algorithm in [7] both analytically and exerimentally. We show that both algorithms have an O(n) latency uer bound in their resective worst cases, while Cell-AS

3 54 H. Li et al. / Ad Hoc Networks (4) 5 68 can be more effective, with latency O(logn), when alied to Li et al. s worst case examles. Our exeriments under realistic settings demonstrate that Cell-AS can achieve u to a 5% latency reduction as comared to Li et al. s. Besides, we have found that in uniform toologies, the aggregation latencies for NN-AS (with the link scheduling in [9]) and Li et al. s algorithm can be reduced to O(log n) and O(log 7 n), resectively, while Cell-AS s latency is between O(log 5 n) and O(log 6 n). The contribution of this aer can be summarized as follows:. We investigate the Minimum-Latency Aggregation Scheduling (MLAS) roblem under the hysical interference model for arbitrary toologies, and roose a distributed algorithm, Cell-AS, to avoid the need of global information about interference with a latency bound of O(K), where K is the link length diversity (the logarithm of the ratio between the lengths of the longest and the shortest links).. We also roose a centralized algorithm, NN-AS, for comleteness and to serve as a benchmark in the erformance comarison. The worst-case latency bounds of the centralized algorithm can be O(logn) and O(log n) when couled with the link scheduling strategies in [8,9], resectively (where n is the total number of sensor nodes).. A theoretically otimal lower bound for the MLAS roblem under any interference model is derived log (n). Given this otimal bound, the aroximation ratios are O(K/logn) with Cell-AS, O() with NN-AS and the link scheduling strategy in [8], and O(log n) with NN-AS and the link scheduling strategy in [9]. Thus, our centralized algorithm, NN-AS, with link the scheduling strategy in [8] achieves an asymtotically otimal latency erformance, which is the current best result in the literature.. Both analytical and exerimental comarisons are conducted between our distributed algorithm and Li et al. s algorithm in [7] to demonstrate the efficiency of our roosed algorithm. The remainder of this aer is organized as follows. We discuss related work in Section and formally resent the roblem model in Section. The Cell-AS and NN-AS algorithms are resented in Sections 4 and 5, resectively. An extensive theoretical analysis is given in Section 6. We reort our emirical studies of the algorithms in Section 7. Finally, we conclude the aer in Section 8.. Related work.. Data aggregation Data aggregation is an imortant roblem in wireless sensor network research. There exist a lot of exciting work investigating the roblem [ 5,7,,], among which minimizing aggregation time via transmission scheduling is a common toic. To the best of our knowledge, all excet one aer [7] assume the rotocol interference model. Chen et al. [] roose a data aggregation algorithm with a latency bound of (D )R, where R is the network radius in ho count and D is the maximal node degree. The NP-hardness roof of the MLAS roblem is also resented. The current best contributions [ 5,] bound the aggregation latency by O(D + R). [] is the first work that converts D from a multilicative factor to an additive one. The algorithm is built on the basis of maximal indeendent set, which is also used in [5]. The latter work rovides a distributed solution to the roblem. In [], the MLAS roblem is dealt with in the context of multi-ho wireless networks and with the assumtion that each node has a unit communication range and an interference range of q P. Xu et al. [4] roose a distributed aggregation schedule and rove a lower bound of max{logn, R} on the latency of data aggregation under any grah-based interference model, where n is the network size. Different from the above work, where connected dominating sets or maximal indeendent sets are emloyed, a novel aroach of distributed aggregation with latency bound O(D + R ) is introduced in []. Here, R is the inferior network radius satisfying R 6 R 6 D 6 R, where D is the network diameter in ho-count. The MLAS roblem is extended to the case with multile sinks in [] with a latency bound of O(D + kr), where k is the number of sinks. The only solution to the MLAS roblem under the hysical interference model is by Li et al. [7]. They roose a distributed aggregation scheduling algorithm with constant ower assignment, which can achieve a latency bound of O(D + R) when the transmission range is set as dr. <d < is a configuration arameter and r is the maximum achievable transmission range under the hysical interference model with ower assignment P and P=ra N ¼ b. No deterministic latency bound can be derived when the transmission range is changed to r, for which robabilistic analysis has been conducted. The efficiency of Li et al. s algorithm may not be guaranteed when alied to arbitrary toologies, which is a consequence of constant ower assignment. A detailed comarison of data aggregation algorithms is given in Table... Link scheduling under the hysical interference model The hysical interference model has received increasing attention in recent years, as a more realistic abstraction of wireless interferences [6]. It has also been shown that it can significantly imrove the network caacity [9, 5], as comared to the rotocol interference model. An imortant track of existing studies focuses on the Minimum Length link Scheduling (MLS) roblem [9,4 8], which is to find the minimum amount of time to schedule the transmissions in a given link set without collision. The MLS roblem is closely related to the link scheduling ste of the MLAS roblem. Moscibroda et al. are the first to formally define and investigate the link scheduling comlexity over a connected structure in wireless networks [4]. They further study toology control for the MLS roblem under the

4 H. Li et al. / Ad Hoc Networks (4) Table Comarison of data aggregation algorithms. Algorithm Latency Centralized vs. distributed Interference model [] (D )R Centralized Protocol [] R + D 8 Centralized Protocol [] 5R + D 4 Centralized Protocol [5] 4D +6D + 6 Distributed Protocol [4] 6R + D 4 Distributed Protocol [] 4R +D Distributed Protocol [7] O(D + R) Distributed Physical This aer O(K) Distributed Physical hysical interference model and obtain a theoretical uer bound on the scheduling comlexity in arbitrary wireless network toologies [5]. In [9], Moscibroda rooses a link scheduling algorithm for connected structures, with a scheduling comlexity of O(log n). The scheduling comlexity of the connected structure is further reduced to O(logn) in[8]. Hua et al. [9] extend the MLS roblem for connected structures to ultra-wideband networks and derive a scheduling algorithm with comlexity O(log (n/m) log n), where m is the rocessing gain. They further [] solve the MLS roblem at the cost of moderately exonential time. Halldórsson et al. [] give a distributed solution to the MLS roblem with O(logn) aroximation. They then resent a constant-factor aroximation for the MLS roblem with any given link set and length-monotone, sub-linear ower assignment in []. A unified algorithmic framework is built to develo aroximation algorithms for link scheduling with or without ower control under the hysical interference model in []. Wan et al. [4] show a constant-aroximation in the simlex mode. Kesselheim et al. [5] roose another constant aroximation in fading metrics and an O(logn) aroximation in the general metric sace. In [6], a new measurement called disturbance is roosed to address the difficulty of finding a short schedule. Goussevskaia et al. [7] make the milestone contribution of roving the NP-comleteness of a secial case of the MLS roblem. In [8], Fu et al. extend the MLS roblem by introducing consecutive transmission constraints. An NP-hardness roof is rovided for this extended roblem.. The roblem model We consider a wireless sensor network of n arbitrarily distributed sensor nodes, v, v,..., v n, and a sink node, v n. Let directed grah G =(V,E) denote the tree constructed for data aggregation from all the sensor nodes to the sink, where V ={v, v,..., v n } is the set of all nodes, and E ={e ij } is the set of transmission links in the tree with e ij reresenting the link from sensor node v i to its arent v j. Our roblem at hand is to ick the directed links in E to construct the tree and to come u with an aggregation schedule S ={S, S,..., S T }, where T is the total time san for the schedule and S t denotes the subset of links in E scheduled to transmit in time slot t, "t =,..., T. A correct aggregation schedule must satisfy the following conditions. First, any link should be scheduled exactly once, i.e., S T t¼ S t ¼ E and S i \ S j = ;, where i j. Second, a node cannot act as a transmitter and a receiver in the same time slot, in order to avoid rimary interference. Let T(S t ) and R(S t ) denote the transmitter set and receiver set for the links in S t, resectively. We need to guarantee T(S t ) \ R(S t )=;, "t =,..., T. Third, a non-leaf node v i transmits to its arent only after all the links in the subtree rooted at v i have been scheduled, i.e., T(S i ) \ R(S j )=;, where i < j. Finally, each scheduled transmission in time slot t, i.e., link e ij S t, should be correctly received by the corresonding receiver under the hysical interference model, considering the aggregate interference from concurrent transmissions of all links e gh S t {e ij }, i.e., the condition P ij =d P a ij N þ P e gh S t fe ij g gh =da gj P b should be satisfied. The minimum-latency aggregation scheduling roblem can be formally defined as follows: Definition ( Minimum-latency aggregation scheduling). Given a set of nodes {v, v,..., v n } and a sink v n, construct an aggregation tree G =(V, E) and a link schedule S ={S, S,..., S T } satisfying S T t¼ S t ¼ E; S i \ S j ¼;, where i j, and T(S i ) \ R(S j )=;, where i 6 j, such that the total number of time slots T is minimized and all transmissions can be correctly received under the hysical interference model. Without loss of generality, we assume that the minimum Euclidean distance between each air of nodes is. As our algorithm design targets at arbitrary distribution of sensor nodes, we assume that the uer bound on the transmission ower at each node is large enough to cover the maximum node distance in the network, such that no node would be isolated. Each node in the network knows its location. This is not hard to achieve during the bootstraing stage in a network, where the sensors are stationary. Imortant notations are summarized in Table for ease of reference. 4. Distributed aggregation scheduling Our main contribution is an efficient distributed scheduling algorithm called Cell Aggregation Scheduling (Cell-AS) for solving the MLAS roblem with arbitrary distribution of sensor nodes. Our distributed algorithm features joint tree construction, link scheduling, and ower control, and executes in a hase-by-hase fashion to achieve the minimum

5 56 H. Li et al. / Ad Hoc Networks (4) 5 68 Table Notations. Symbol V E v n v i e ij S S t T(S t ) R(S t ) K R D n N a b P ij d ij K ij aggregation latency. In contrast, the tree construction and link scheduling are disjoint stes in [7]. We first resent the key idea behind our algorithm and then discuss imortant techniques to imlement the algorithm in a fully distributed fashion. 4.. Design idea Definition Node set including the sink Link set The sink node Node i Link from node v i to v j Aggregation schedule Set of links scheduled at time slot t Transmitter set for link set S t Receiver set for link set S t Link length diversity Network radius in terms of ho count Maximum node degree Number of sensor nodes in the network Background noise Path loss ratio SINR threshold Transmission ower at the transmitter of link e ij Distance between node v i and v j Set of links scheduled simultaneously with e ij Initially, the entire area can be seen as being divided into many small areas. Our distributed algorithm first aggregates data from sensor nodes in each small area, where the transmission links are short, and then aggregates data in a larger area by collecting from those small ones with longer transmission links; this rocess reeats until the entire network is covered by one large area. We divide the lengths of all ossible transmission links in the network into K + categories: [, ], (, ],...,( K, K ], where K is bounded by the maximum node distance D in the network with K < D 6 K. A link from node v i to node v j falls into category k if the Euclidean distance between these two nodes lies within ( k, k ] with k =,..., K, or [, ] with k =. We refer to K as the link length diversity which is roortional to the logarithm of the ratio between the lengths of the longest and the shortest links in the network. In our design, aggregation links in category k are treated and their transmissions are scheduled (to aggregate data in the smaller areas) before links in category k + are rocessed (to aggregate data in the larger areas). The algorithm is carried out in an iterative fashion: In round k (k =,..., K), the network is divided into hexagonal cells of side length k. In each cell, a node with the shortest distance to the sink is selected as the head, resonsible for data aggregation; the other nodes in the cell directly transmit to the head, one after another, with links no longer than k. In the next round k +, only the head nodes in the revious round remain in the icture. The network is covered by hexagonal cells of side length k+ and a new head is selected for data aggregation in each cell. After K + rounds of the algorithm, only one node remains, which will have collected all the data in network, and will transmit the aggregated data to the sink node in one ho. Fig. gives an examle of the algorithm in a sensor network with three link length categories, in which selected head nodes are in black. In each round k of the algorithm, links of length category k are scheduled as follows to avoid interference and to minimize the aggregation latency. We assign colors to the cells and only cells with the same color can schedule their link transmissions concurrently in one time slot. To bound the interference among concurrent transmissions, cells of the same color need to be sufficiently far aart. We use 6 X þ X þ 7 colors in total, such that cells of the same color are searated by a distance of at least a =a, (X + ) k with X ¼ 6b þ ffiffi þ a as illustrated in Fig.. (The solid cells are of the same color. A F are six cones to be referred to in the analysis in Section 6.) We will show in Section 6 that by using these many colors, we are able to bound the interferences and thus rove the correctness and efficiency of our algorithm. Inside each cell, the transmission links from all other nodes to the head are scheduled sequentially. Note that each round of the algorithm may take multile time slots. The Cell-AS algorithm is summarized in Algorithm, where the scheduling of links in cells of the same color is carried out according to Algorithm. Algorithm. Distributed Aggregation Scheduling (Cell-AS) Inut: Node set V with sink v n. Outut: Tree link set E and link schedule S. : k:¼; t:¼; V:¼V {v n }; E:¼ ;; S:¼;; a =a; : X :¼ 6b þ ffiffi a þ : while jvj do 4: Cover the network with cells of side length k and color them with 6 X þ X þ 7 colors; 5: for i:¼ to 6 X þ X þ 7 do 6: E i :¼;, where E i is link set in cells of color i; 7: for each cell j with color i do 8: Select node v m in cell j closest to sink v n as head; 9: Construct links from all other nodes in cell j to v m ; : Add the links to E i and E; : Remove all the nodes in cell j excet v m from V; : end for : (PS i, t) :¼ Same-Color-Cell-Scheduler (E i, t); 4: S:¼S [ PS i ; 5: end for 6: k:¼k +; 7: end while 8: v m :¼ the only node in V; Construct link e mn from v m to v n ; 9: E:¼E [ {e mn }; S:¼S [ {{e mn }}; : return E and S.

6 H. Li et al. / Ad Hoc Networks (4) (a) Round. (b) Round. (c) Round. Fig.. The iterations of Cell-AS: an examle with three link length categories and one sink in the center. y (X+) k (,) x 4... Location and synchronization In the bootstraing hase, a middle osition of the sensor network is assigned to be the origin (,). Each node is then assigned its location coordinates (x, y) relative to the origin with such techniques as GPS. In fact, only a small number of nodes need to be assigned their coordinates initially, as the others can obtain their coordinates through relative ositioning (e.g., [6]). Each node in the sensor network carries out the distributed algorithm in a synchronized fashion, i.e., it knows the start of each round k and each time slot t. Such synchronization can be achieved using one of the ractical synchronization algorithms in the literature (e.g., [7]). Fig.. Link scheduling in one time slot of Cell-AS: cells with the same color are searated by a distance of at least (X + ) k, where a =a. X ¼ 6b þ ffiffi þ a Algorithm. Same-color-cell-scheduler Inut: Link set E i and time slot index t. Outut: Partial link schedule PS i for links in E i, and t. a =a; : X :¼ 6b þ ffiffi a þ : Define constant c:¼n bx a ; : PS i :¼;; 4: while E i ; do 5: S t :¼;; 6: for each cell j with color i do 7: Choose one non-scheduled link e gh in cell j; 8: Assign transmission ower P gh :¼ c d a gh ; 9: S t :¼S t [ {e gh }; E i :¼E i {e gh }; : end for : PS i :¼PS i [ {S t }; t:¼t +; : end while : return PS i and t. 4.. Distributed imlementation The algorithm can be imlemented in a fully distributed fashion Neighbor discovery In each round k, the network is divided into cells of side length k in the manner as illustrated in Fig.. Each node can determine the cell it resides in the current round based on the node s location. It can then discover its neighbors in the cell via local broadcasting [8]. The broadcasting range is k+, such that all nodes in the same cell can be reached Head selection The head of a cell in round k is the node in the cell closest to the sink. All the nodes are informed of the sink s location in the bootstraing stage of the algorithm, or even before they are laced in the field. Since each node knows the location information of all its neighbors in the same cell, it can easily identify the head Distributed link scheduling In each round k, coloring of the cells is done as illustrated in Fig.. As each node knows which cell it resides in, it can comute color i of its cell in this round. Cells of the same color are scheduled according to the sequence of their color indices, i.e., cells with color i schedule their transmissions before those with color i +. The head node in a cell is resonsible to decide when the other nodes in its cell can start to transmit, and to announce the comletion of transmissions in its cell to all head nodes within distance (X + ) k. A head node in a cell with color i + waits until it has received comletion notifications from all head nodes in

7 58 H. Li et al. / Ad Hoc Networks (4) (a) Round (b) Round (c) Round Fig. 4. The stes of NN-AS: an examle with six sensor nodes. cells of color i within distance (X + ) k. It then schedules the transmission of all the other nodes in its cell one by one, by sending ulling messages. For a non-head node in the cell, it waits for the ulling message from the head node and then transmits its data to the head. When the algorithm is executed round after round, only the nodes that have not transmitted (the heads in revious rounds) remain in the execution, until their transmission rounds arrive. 5. Centralized aggregation scheduling Assuming global information is available at each sensor, then a centralized scheduling algorithm can be constructed, which can achieve the best aggregation latency for the MLAS roblem. We resent in the following such a centralized algorithm, Nearest-Neighbor Aggregation Scheduling (NN-AS). Our centralized algorithm rogresses in a hase-byhase fashion, with joint tree construction and link scheduling. In each round, the algorithm finds a nearest neighbor matching among all the sensor nodes that have not transmitted their data, and schedule all the links in the matching. The algorithm is started with all the sensor nodes in V {v n }. It finds for each node v i the nearest neighbor node v j, where neither v i nor v j has already been included in the matching, and a directed link from v i to v j is established. For examle, in Fig. 4 showing a sensor network of six nodes, the matching identified in round contains two links,? and 4? 6. The links in matching M (of round ) are then scheduled, using either the link scheduling algorithm roosed in [8] or the one in [9], both of which schedule a set of links with guaranteed scheduling correctness under the hysical interference model. After all transmissions in round are scheduled, all the nodes that have transmitted are removed, and the algorithm reeats with the remaining nodes. In Fig. 4b, nodes,, 5, and 6 remain, and two links are generated based on the nearest neighbor criterion and then scheduled for transmission. The rocess reeats until only one sensor node remains, which will transmit its aggregate data to the sink node in one ho. The centralized algorithm is summarized as Algorithm, where Phase-Scheduler- and Phase-Scheduler- call uon Algorithm 4 rovided in [8] and Algorithm 5 rovided in [9], resectively, to generate the schedule for links in matching M k in round k. InAlgorithm 4, f() is the Riemann zeta function [9]. InAlgorithm 5, the re-rocessing (M k ) rocedure assigns two values, i.e., s ij and c ij related to link length d ij, for each link e ij M k, while the check (e ij,s t ) rocedure checks whether link e ij can transmit concurrently with links in S t and returns a Boolean value. Algorithm. Centralized Aggregation Scheduling (NN-AS) Inut: Node set V with sink v n. Outut: Tree link set E and link schedule S. : k:¼; t:¼; E:¼;; S:¼;; V = V {v n }; : while jvj do : M k :¼;; 4: for each v i V do 4: if v i R T(M k ) [ R(M k ) then 5: Find v i s nearest-neighbor v j V; 5: if v j R T(M k ) [ R(M k ) then 6: Construct link e ij from v i to v j ; M k :¼M k [ {e ij }; 7: end for 8: E:¼E [ M k ;(PS k,t):¼ Phase-Scheduler-(M k, t) or Phase-Scheduler-(M k, t); S:¼S [ PS k ; 9: V:¼V T(M k ); k:¼k +; : end while : v i :¼ the only node in V; Construct link e in from v i to v n ; : E:¼E [ {e in }; S:¼S [ {{e in }}; : return E and S. Algorithm 4. Phase-Scheduler- [8] Inut: Link set M k and time slot index t. Outut: Partial link schedule PS k for links in M k, and t. : Define constant integer b:¼d(6 a+ f(a/) b) /(a ) e; PS k :¼;; : Let R max :¼ max eij M k fd ij g; R min :¼ min eij M k fd ij g; : for each integer v with 6 v 6 b do 4: Sv:¼;; 5: end for 6: for each link e ij M k do 7: P ij :¼Nb (R max ) (a )/ (d ij ) (a+)/ ; 8: u :¼ blog ðd ij =R min Þc; q ¼ u mod b; l :¼ b modb ffiffi þb y cmodb; u R min 9: S lq :¼S lq S {eij }; : end for : for each integer v with 6 v 6 b do : if Sv ; then : PS k :¼PS k S {S v}; t:¼t +; 4: end if 5: end for 6: return PS k and t. ffiffi x c u R min

8 H. Li et al. / Ad Hoc Networks (4) Algorithm 5. Phase-Scheduler- [9] Inut: Link set M k and time slot index t. Outut: Partial link schedule PS k for links in M k, and t. Phase-Scheduler-(M k,t) : re-rocessing (M k ); : Define a large enough constant c ; PS k :¼;; n:¼n (a )/(a ); : for m =to nd log (nb)e do 4: Let E m :¼{e ij M k jc ij = m}; 5: while not all links in E m have been scheduled do 6: S t :¼;; 7: for each e ij E m in decreasing order of d ij do 7: if check(e ij,s t ) then S 8: S t :¼S t feij g; E m :¼E m fe ij g; P ij :¼d a ij ðnbþs ij ; 9: end for S : PS k :¼PS k {St }; t:¼t +; : end while : end for : return PS k and t. re-rocessing (M k ) : Please refer to [9] for details. check (e ij,s t ) : Please refer to [9] for details. 6. Analysis In this section, we rove the correctness of our distributed and centralized algorithms, and analyze their efficiency with resect to the bound of aggregation latency. 6.. Correctness We first rove that 6 X þ X þ 7 colors are enough to searate the cells of the same color by a distance of at least (X +)d, where d = k is the side length of cells in category k. Lemma. At most 6 X þ X þ 7 hexagons with size length of d can cover a disk with radius (X + )d. Proof. As shown in Fig., we divide the disk into six equal-sized non-overlaing cones. It is clear that the maximum number of hexagons to cover the disk is at most six times of that to cover each cone. Take cone A for instance. We have at most 6 hexagons in range of d; 6 þ hexagons in range of d; 6 þ þ hexagons in range of 7 d, etc. So it is not hard to rove by induction that we have at most =6 þ P j þj i hexagons in range of i¼ d in one cone. So in a range of (X +)d, for which j 6 4ðXþÞ, we have at most =6 þ 4ðXþÞ 4ðXþÞ ð þ Þ hexagons in one cone, which means at most 6 X þ X þ 7 in the disk. h Proof. Algorithm guarantees that each sensor node transmits exactly once and will not serve as a receiver again after the transmission. Hence the resulting transmission links constitute a tree. The link scheduling guarantees that a node would not transmit and receive at the same time and a non-leaf node transmits only after all the nodes in its subtree have transmitted. We next rove that each transmission is successful under the hysical interference model. In [], a safe CSMA rotocol under the hysical interference model is resented. The core idea is to searate each air of concurrent transmitters by a redefined distance, such that the cumulative interference in the network can be bounded. However, the background noise is not considered in []. We revise the conclusion of [] to adat their result to the hysical interference model in this aer. We know that any two concurrent transmitters of links in the same category k are searated by at least (X + ) k, a =a. where X ¼ 6b þ ffiffi a þ For any scheduled link with length r, the ower assigned for transmission is P = N b X a r a. According to the conclusion of [], the cumulative interference I at any receiver of a link in category k satisfies I 6 6 a þ a N bx a ð k Þ a ffiffiffi X a ð k Þ a ¼ 6 þ a ffiffiffi N a b ¼ N ðx a Þ: So the SINR value for any scheduled link with length r should satisfy P=r a N þ I P N bx a N þ N ðx a Þ ¼ b: We can conclude that each link transmission is successful under the hysical interference model. h Theorem (Correctness of NN-AS). The centralized algorithm NN-AS (Algorithm ) can construct a data aggregation tree and correctly schedule the transmission under the hysical interference model. Proof. The algorithm in Algorithm guarantees that each node will be removed from the node set V after selected for transmission, and hence it will be a transmitter exactly once. At the end of each round, receivers and other nonscheduled nodes remain in V, and all aggregated data reside in the remaining nodes. Therefore, the generated transmission links correctly construct a data aggregation tree. For the link scheduling, Algorithm alies either one of the algorithms in [8,9], whose correctness under the hysical interference model are roven. h Theorem (Correctness of Cell-AS). The distributed algorithm Cell-AS (Algorithm ) can construct a data aggregation tree and correctly schedule the transmissions under the hysical interference model. 6.. We now analyze the efficiency of the algorithms. We also derive a theoretically otimal lower bound of the

9 6 H. Li et al. / Ad Hoc Networks (4) 5 68 aggregation latency for the MLAS roblem under any interference model and show the aroximation ratios of our algorithms with resect to this bound Distributed Cell-AS We now analyze the efficiency of the distributed Cell-AS algorithm. Theorem (Aggregation Latency of Cell-AS). The aggregation latency for the distributed algorithm Cell-AS (Algorithm ) is uer bounded by 6 X þ X þ 7 K X 7X 9 ¼ OðKÞ, where K is the link length diversity and a =a X ¼ 6b þ ffiffi a þ is a constant. Proof. We first show that if the minimum distance between any node air is, there can be at most seven nodes in a hexagon with side length. We rove by utilizing an existing result from []: Suose U is a set of oints with mutual distances at least in a disk of radius r; then juj 6 ffiffiffi r þ r þ : A hexagon of side length can be included in a disk of radius r = at the center. Then we derive juj 6 ffiffiffi þ þ ¼ 7:769 < 8: ðþ Hence there can be at most seven nodes with mutual distance of in the unit disk, and therefore in the hexagon. An examle is given in Fig. 5, with seven nodes in one hexagon of side length d =. From the above result, we know that there can be at most six links transmitting to the head node in each cell of side length. Each cell of side length k with k > covers at most cells of side length k (an illustration is given in Fig. b and c). Therefore, at most six time slots are needed for scheduling the transmissions in a cell of side length, and at most for the cells of side length k (k > ), to avoid the rimary interference. As we cover cells of the same size with 6 X þ X þ 7 colors, at most 6 X þ X þ 7 rounds are needed to schedule all the cells in the same link length category. Thus at most 6 6 X þ X þ 7 time slots are needed for scheduling all the cells with side length, and 6 X þ X þ 7 time slots for cells of side length k (k > ). Since K P D (the maximum node distance in the network), cells of side length K can cover the entire network. There can be only one cell of this size, and so at most time slots are needed for scheduling its links. In summary, at most 6 6 X þ X þ 7 þ 6 X þ Xþ 7ÞðK Þþ ¼ 6 X þ X þ 7 K X 7X time slots are needed to schedule all the transmissions in the data aggregation tree. One additional time slot is required to transmit the aggregated data to the sink. Therefore the overall aggregation latency is at most 6 X þ X þ 7 K a =a X 7X 9. Since X ¼ 6b þ ffiffi a þ is a constant, the overall aggregation latency is O(K). h 6... Centralized NN-AS We first rove a few lemmas before analyzing the efficiency of the centralized NN-AS algorithm. Lemma. The data aggregation tree can be constructed with l m at most log7n 6 rounds in NN-AS. Proof. We give the roof by first showing that each node can be the nearest neighbor of at most six other nodes on a euclidean lane. We rove this claim by contradiction. Fig. 5 gives an examle that one node (node ) can be the nearest neighbor of six other nodes. Suose that a node can be the nearest neighbor of seven other nodes, e.g., node in Fig. 6. Let d ij reresent the distance between node i and j in the figure. We have d 6 d and d 6 d, and thus \ P \ and \ P \. Since \ + \ + \ =, we have \ P. Similarly, we can derive \ P ; \4 P ; \45 P ; \56 P ; \67 P, and \7 P. Therefore \þ \ þ \4 þ \45 þ\56 þ\67 þ\7 P 7 >, which is a contradiction. Therefore a node can be the nearest neighbor of at most six nodes. In each round of NN-AS, each node v i V is the nearest neighbor of at most six nodes. Then at least one link will be established from or to one of these seven nodes, and at Fig. 5. Seven nodes in a hexagon cell. Fig. 6. Node as nearest neighbor of seven other nodes: a contradiction.

10 H. Li et al. / Ad Hoc Networks (4) least one node out of these seven nodes will be removed from V at the end of this round. Therefore at least 7 jvj nodes are removed from V in total. From the above discussion, at most 6 7 jvj nodes are left in V after each round of the algorithm. The algorithm terminates when only one node remains in V. Let k be the maximum number of rounds which the algorithm l m l m executes. We have 6 7k n ¼, and thus k ¼ log7 6 n. h n n n Lemma. The link scheduling latency in each round of NN- AS is O() with Phase-Scheduler- in Algorithm 4 and O(log n) with Phase-Scheduler- in Algorithm 5. Proof. In each round of NN-AS, the number of links to be scheduled is equal to exactly the number of nodes removed from V, i.e., at least jvj. Meanwhile, as each node can either 7 be the transmitter or the receiver but not both in one round, the number of links to be scheduled is uer bounded by jvj. Since jvj 6 n, we have O(n) links to schedule in each round. As the link set generated in each round is based on the nearest-neighbor mechanism, we can aly the link scheduling strategy roosed in [8] to schedule them with constantly bounded time slots. On the other hand, the link scheduling algorithm achieves a latency of O(log n) with n links [9]. Therefore, the link scheduling latency in each round of NN-AS is O() with Phase-Scheduler- in Algorithm 4 and O(log n) with Phase-Scheduler- in Algorithm 5. h Theorem 4 ( of centralized NN-AS). The aggregation latency of the centralized algorithm NN-AS (Algorithm ) is uer bounded by O(logn) with Phase-Scheduler- in Algorithm 4 and O(log n) with Phase-Scheduler- in Algorithm 5. Proof. From Lemmas and, we know that NN-AS is executed l m for at most log7n 6 rounds and the link scheduling latency in each round is O() with Phase-Scheduler- in Algorithm 4 and O(log n) with Phase-Scheduler- in Algorithm 5. In total, NN-AS schedules the data aggregation l m in O log7n time slots, which is equivalent to O(logn), 6 l m with Phase-Scheduler- in Algorithm 4 and O log7n 6 log nþ time slots, which is equivalent to O(log n), with Phase-Scheduler- in Algorithm 5. h 6... Otimal lower bound We next derive the otimal lower bound of the aggregation latency, and the aroximation ratios of our algorithms with resect to this bound. n Theorem 5 (Otimal lower bound of aggregation latency). The aggregation latency for the MLAS roblem under any interference model is lower bounded by logn. Proof. Under any interference model, as a node cannot transmit and receive at the same time, at most jvj links can be scheduled for transmission in one time slot. Since each node only transmits exactly once, at most jvj nodes comlete their transmissions in one time slot. Suose l mwe need k time slots to aggregate all the data. We have n ¼, and thus k = dlogne, i.e., the aggregation k latency under any interference model is at least logn. h Comaring to the otimal lower bound, our distributed Cell-AS achieves an aroximation ratio of O(K/logn), and the centralized NN-AS achieves an aroximation ratio of O() with the link scheduling strategy in [8] and O(log n)/logn, which is equivalent to O(log n), with the link scheduling strategy in [9]. We show in Aendix A that O(K) is between O(logn) and O(n). 6.. Comarison with Li et al. s Algorithm [7] n Fig. 8. Worst case II for Li et al. s algorithm. n n Fig. 9. An worst case for both Cell-AS and Li et al. s algorithm. We next analytically comare our distributed Cell-AS with the distributed algorithm roosed by Li et al. [7], which is the only existing work addressing the MLAS roblem under the hysical interference model. Toology Center r n n n r r n r n Fig. 7. Worst case I for Li et al. s algorithm.

11 6 H. Li et al. / Ad Hoc Networks (4) 5 68 beta= beta= beta=5 beta= beta= beta= beta=5 beta= beta= beta= beta=5 beta= (a) α =, Uniform (b) α = 4, Uniform (c) α = 5, Uniform beta= beta= beta=5 beta= beta= beta= beta=5 beta= beta= beta= beta=5 beta= (d) α =, Poisson (e) α = 4, Poisson (f) α = 5, Poisson beta= beta= beta=5 beta= beta= beta= beta=5 beta= beta= beta= beta=5 beta= (g) α =, Cluster (h) α = 4, Cluster (i) α = 5, Cluster Fig.. with Cell-AS in different toologies. Li et al. s algorithm has four consecutive stes: Toology Center Selection: the node with the shortest network radius in terms of ho count is chosen as the toology center. Breadth First Sanning (BFS) Tree Construction: using the toology center as the root, breadth-first searching is executed over the network to build a BFS tree. Connected Dominating Set (CDS) Construction: a CDS is constructed as the backbone of the aggregation tree with an existing aroach [], based on the BFS tree. Link Scheduling: the network is divided into grids with side length l ¼ dr= ffiffiffi, where < d < is a configuration arameter assigned before execution, and r is the maximum achievable transmission range under the hysical interference model with constant ower assignment P and P=ra N ¼ b. The grids are colored with a ffiffi 4bsPl a þ þ ffiffiffi e colors and links are scheduled ð Þ a Pl a bn with resect to grid colors. Here, s ¼ aðþ a Þ a þ a ða Þ Aggregation Latency. Li et al. s algorithm solves the MLAS roblem in O(D + R) time slots, where R is the network radius in ho count and D is the maximum node degree. In the worst case, either R or D can be O(n), e.g., in the examles in Figs. 7 and 8 to be discussed shortly, and R = O(logn) in the best case. Our Cell-AS achieves an aggregation latency of O(K), which is also equal to O(n) in the worst case, e.g., in the examle in Fig. 9, and O(logn) in the best case (see Aendix A). Therefore the two algorithms have the same orders of worst-case and best-case aggregation latencies. Comutational and Message Comlexity. Both the comutational comlexity and the message comlexity of our Cell-AS algorithm are uer bounded by O(min{Kn, K }). Since K = n in the worst case, both are at most O(n ). Li et al. s algorithm has a comutational comlexity of O(njEj) and message comlexity of O(n + jej). As jej = n in the worst case, the comutational comlexity and

12 H. Li et al. / Ad Hoc Networks (4) beta= beta= beta=5 beta= beta= beta= beta=5 beta= beta= beta= beta=5 beta= (a) α =, Uniform (b) α = 4, Uniform (c) α = 5, Uniform beta= beta= beta=5 beta= beta= beta= beta=5 beta= beta= beta= beta=5 beta= (d) α =, Poisson (e) α = 4, Poisson (f) α = 5, Poisson beta= beta= beta=5 beta= beta= beta= beta=5 beta= beta= beta= beta=5 beta= (g) α =, Cluster (h) α = 4, Cluster (i) α = 5, Cluster Fig.. with NN-AS in different toologies. message comlexity of Li et al. s algorithm are O(n ) and O(n ), resectively. We can see that Cell-AS enjoys a better comutational comlexity while having the same order of message comlexity with Li et al. s algorithm. More details on the analysis of the comlexities of our algorithm and Li et al. s algorithm can be found in Aendix B. Case Study. We next show that Cell-AS can outerform Li et al. s algorithm in its worst cases. The minimum link length is set to one unit in the following examles, without loss of generality. Fig. 7 is a worst case of Li et al. s algorithm. Nodes are located along the line with distance r = between neighboring nodes. The toology center should be the center of the line, which leads to R ¼ n. According to the latency bound O(D + R), Li et al. s algorithm takes O(n) time slots to comlete data aggregation. On the other hand, the maximum node distance in Fig. 7 is n. Therefore, the link length diversity K with our n algorithm should be log. According to the latency bound O(K), the scheduling latency should be O(logn) with Cell-AS, which is better than O(n). Fig. 8 is another worst case for Li et al. s algorithm, in which all nodes reside on the circle with unit distance between neighboring nodes, excet for node in the center. The radius of the circle is r >. Therefore, node has the maximum node degree D of n. With resect to latency bound O(D + R), O(n) time slots are required to comlete aggregation with Li et al. s algorithm. Meanwhile, the maximum node distance in Fig. 8 is r. Since the distance between any neighboring nodes on the circle is, we have r n with large values of n, which is an aroximation of the circle s erimeter. Then the link diversity K should be about log. n Therefore, the aggregation latency is O(logn) with Cell-AS, which is better than O(n) with Li et al. s algorithm.

13 64 H. Li et al. / Ad Hoc Networks (4) 5 68 beta= beta= beta=5 beta= beta= beta= beta=5 beta= beta= beta= beta=5 beta= beta= beta= beta=5 beta= beta= beta= beta=5 beta= beta= beta= beta=5 beta= (a) α =, Uniform (b) α = 4, Uniform (c) α = 5, Uniform beta= beta= beta=5 beta= beta= beta= beta=5 beta= (d) α =, Poisson (e) α = 4, Poisson (f) α = 5, Poisson beta= beta= beta=5 beta= (g) α =, Cluster (h) α = 4, Cluster (i) α = 5, Cluster Fig.. with Li et al. s algorithm in different toologies. Fig. 9 is a worst case examle for both Cell-AS and Li et al. s algorithm. In this examle, the maximum node distance is n between nodes and n while the minimum node distance is between nodes and. Thus, K ¼ log n with Cell-AS. As for Li et al. s algorithm, 4 D = n since the transmission range should be at least n to maintain connectivity. Both Cell-AS and Li et al. s algorithm will take n time slots to comlete the data aggregation. On the other hand, our centralized NN-AS algorithm can erform better than this and achieve an aggregation latency of O(logn) oro(log n) according to Theorem Emirical study We have imlemented our roosed distributed algorithm Cell-AS, centralized algorithm NN-AS, as well as Li et al. s algorithm, and carried out extensive simulation exeriments to verify and comare their efficiency. It should be noted that the link scheduling algorithm in [8] achieves a worst-case latency bound of b (8logn +)=O(logn), where n is the number of nodes and b is a constant integer related to the ath-loss-ratio a and the SINR threshold b. b is the number of colors to color the grids that cover the whole network. Since the value of b is too large with any (a,b) airs, the number of required colors inhibits the alication of the link scheduling algorithm roosed in [8] in tyical networks of limited sizes. As a result, in the emirical study, we only imlement the Phase-Scheduler- algorithm based on [9] in NN-AS. In our exeriments, three tyes of sensor network toologies, namely Uniform, Poisson and Cluster, are generated with the number of nodes n = to, which are distributed in a square area of 4, square meters ( meters times meters). The nodes are uniformly randomly distributed in the Uniform toologies, and are distributed with the Poisson distribution in the Poisson toologies. In the Cluster toologies [], the centers of n C clusters are uniformly randomly located in the square and, for each cluster, n n C

14 H. Li et al. / Ad Hoc Networks (4) NN AS Cell AS Li et al. NN AS Cell AS Li et al (a) α = 4, β =, Uniform (b) α = 5, β =, Uniform NN AS Cell AS Li et al. NN AS Cell AS Li et al (c) α = 5, β = 4, Uniform (d) α = 5, β = 6, Uniform Fig.. comarison of the three algorithms in selected network settings. nodes are uniformly randomly distributed within a disk of radius r C at the center. We use the same settings as in [] in our exeriments, where n C = and r C =. We set N to the same constant value. as in [7] (which nevertheless would not affect the aggregation latency). The transmission ower in our imlementation of Li et al. s algorithm is assigned the value that would result in a transmission range of 4 to maintain the connectivity of the resective network with high robability, while d is set to.6 in comliance with the simulation settings in [7]. Since < a < 6 and b P, we exeriment with a set to, 4 and 5, and b to values between and, resectively. We imlement the three algorithms in C++ and run the rograms on a Solaris server with an 8- core CPU (.6GHZ) and 8G RAM. All our results resented are the average of trials. We first comare the aggregation latency of the three algorithms with different combinations of a and b values in the three tyes of toologies. The results are resented in Figs., resectively. Fig. shows that the aggregation latency with Cell-AS is larger with smaller a, which reresents less ath loss of ower and thus larger interference from neighbor nodes, and with larger b, corresonding to higher SINR requirement. We however observe in Fig. that, with NN-AS, the latency curves tend to overla under the same node distribution even when values of a and b vary, but they show marked differences with different node distributions. This shows that network toology is the dominant influential factor in aggregation latency for NN-AS, which can be exlained by the algorithm s nearest-neighbor mechanism in tree construction and non-linear ower assignment [9] for link scheduling. For Li et al. s algorithm, Fig. shows that most of the curves roduced at different b values are straight or nearly straight lines that overla, excet in the following cases with Uniform toologies: b = when a =4; b =, b =4 and b = 6 when a = 5. The reason behind the linearity of the lines is that each grid is scheduled one by one without any concurrency with Li et al. s algorithm in the cases of the Poisson and Cluster toologies, as well as the Uniform toologies with smaller a and larger b values. The no-concurrency henomenon is due to the fact that since the 4bsPl number of colors is ffiffi a a ffiffiffi þ þ with ð Þ a Pl a bn l ¼ dr= ffiffiffi a þ ; s a ¼ þ a P=ra and a ða Þ N ¼ b (see Section 6. for detailed discussion of Li et al. s algorithm), smaller a and larger b values lead to a larger number of colors needed. On the other hand, in the Poisson and Cluster toologies, the nodes are not evenly distributed, thus a larger r is requested to maintain the network connectivity, which leads to a smaller number of grids since the side length of each grid is dr= ffiffiffi. In these cases, the number of required colors in the algorithm, as decided by a and b, is larger than the total number of grids in the network (which is roortional to /r). Therefore, each grid is actually scheduled one by one. In comarison, the number of cells in our Cell-AS algorithm is only related to the link length

15 66 H. Li et al. / Ad Hoc Networks (4) NN AS Cell AS Li et al NN AS Cell AS Li et al (a) Divided by log n (b) Divided by log 5 n.4 x NN AS Cell AS Li et al. NN AS Cell AS Li et al (c) Divided by log 6 n (d) Divided by log 7 n Fig. 4. Asymtotic aggregation latency of the three algorithms (a = 4,b = ). diversity, but not r. Therefore, our algorithm can execute with much more concurrency in link scheduling across different cells, leading to the sublinear curves in Fig.. Figs. show that concurrent link scheduling (across different cells/grids) occurs with all three algorithms only in the following four cases in the Uniform toologies: () a =4,b = ; () a =5,b = ; () a =5,b =4; and (4) a =5, b = 6. We next comare the aggregation latencies achieved by the three algorithms in these four cases. Fig. shows that our centralized NN-AS achieves a much lower aggregation latency as comared to the other two algorithms, such that the changes in its curves are almost unobservable. The erformance of our distributed Cell-AS is similar to that of Li et al. s algorithm when n 6, but is u to 5% better than the latter when the network becomes larger. To obtain a better understanding of the asymtotic erformance of each algorithm, we further divide the aggregation latency in Fig. by log n, log 5 n, log 6 n, and log 7 n, resectively, and lot the results in Fig. 4 (since the curves are similar in all four cases, we show the results obtained at a = 4 and b = as being reresentative). Our rationale is that, if the aggregation latency of an algorithm has a higher (lower) order than O(log i n), its curve in the resective lot should go u (down) with the increase of the network size, and a relatively flat curve would indicate that the aggregation latency is O(log i n). From Fig. 4a and d, we infer that the average aggregation latency of NN-AS and Li et al. s algorithm is O(log n) and O(log 7 n), resectively. The curves corresonding to the Cell-AS algorithm slightly go u in Fig. 4b and slightly go down in Fig. 4c, indicating that Cell-AS achieves an average aggregation latency between O(log 5 n) and O(log 6 n). Our analysis in Section 6 gives an aggregation latency uer bound of O(K) for Cell-AS and O(log n) for NN-AS with the link scheduling strategy in [9]. Our exeriments have shown that the average aggregation latency under ractical settings is better in the Uniform toologies with the algorithms. 8. Concluding remarks This aer tackles the minimum-latency aggregation scheduling roblem under the hysical interference model. Many results for the MLAS roblem under the rotocol interference model have been obtained in recent years, but they are not as relevant to real networks as any solution under the hysical interference model which is much closer to the hysical reality. The hysical interference model is favored also because of its otential in enhancing the network caacity when the model is adoted in a design [,,9,4,5]. Although the hysical interference model makes finding a distributed solution difficult, we roose a distributed algorithm to solve the roblem in networks of arbitrary toologies. By strategically dividing the network into cells according to the link length diversity (K), the algorithm obviates the need for global information and can be

16 H. Li et al. / Ad Hoc Networks (4) imlemented in a fully distributed fashion. We also resent a centralized algorithm which reresents the current most efficient algorithm for the roblem, as well as rove an otimal lower bound on the aggregation latency for the MLAS roblem under any interference model. Our analysis shows that the roosed distributed algorithm can aggregate all the data in O(K) time slots (with aroximation ratio O(K/ logn) with resect to the otimal lower bound), and the centralized algorithm in at most O(logn) time slots (with aroximation ratio O(), and using the link scheduling strategy in [8]) and O(log n) time slots (with aroximation ratio O(log n), and using the link scheduling strategy in [9]). Our emirical studies under realistic settings further demonstrate that both Cell-AS and NN-AS (using the link scheduling strategy in [9]) outerform Li et al. s algorithm in all three toologies tested. Furthermore, the Cell-AS and NN- AS algorithms (using the link scheduling strategy in [9]) can otentially achieve an average aggregation latency of between O(log 5 n) and O(log 6 n), and O(log n) in ractice, resectively. In our future work, we will investigate further reduction of the theoretical uer bound on the aggregation latency with distributed imlementations and study the latencyenergy tradeoff in data aggregation, e.g., the achievable asymtotic order of aggregation latency with constraint transmission ower in each time slot. Acknowledgements This roject is artially suorted by Hong Kong RGC GRF Grants 79E and 74, and National Natural Science Foundation of China Grant 686. Aendix A. Analysis of the range of K Fig. 9 is a worst case examle for Cell-AS. The minimum geometric node distance is and the maximum geometric node distance is P n i¼ i ¼ð n Þ=. So K ¼ log n, 4 which is O(n) in the worst case. Recall the existing result from []: suose the entire network is a disk of radius r = K, and the node set V is a set of oints with mutual distances at least ; then we have n 6 ffiffiffi r þ r þ ) n 6 ffiffiffi ð K Þ þ K þ ffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) K P log 4 þ 8!! ffiffiffi ðn Þ ¼Oðlog ffiffiffi n Þ: Since the aggregation latency low bound is O(logn) by Theorem 5, K is O(logn) in the best case instead of Oðlog ffiffiffi n Þ (otherwise, the aggregation latency with Cell AS is OðKÞ ¼Oðlog ffiffiffi n Þ, which contradicts with Theorem 5). Aendix B. More on the comutational and message comlexity of Cell-AS and Li et al. s algorithm B.. Comutational comlexity Cell-AS has three main function modules, i.e., neighbor discovery, head selection, and link scheduling. During neighbor discovery in each round, each node erforms exactly one local broadcast. There are n nodes in round and at most min{n, K k+ } nodes in round k >. So at most n þ P K k¼ minfn; K kþ g¼minfðk þ Þn; n þ ðk Þ g local broadcast oerations are involved in K + rounds. For head selection, the total numbers of location comarisons to decide the heads in round and in round k > are at most 7n and minfkn; P K k¼ K kþ g, resectively, as there are at most seven nodes in each cell in round, and er cell in round k >. Hence the overall comutational comlexity for head selection throughout the algorithm is at most 7n þ minfkn; 69ðK Þ g. Similarly, link scheduling also has a comutational comlexity of 7n þ minfkn; 69ðK Þ g. In summary, Cell-AS has an overall comutational comlexity of O(min{Kn, K }). Li et al. s algorithm is divided into four hases, i.e., toology center selection, breadth-first sanning (BFS) tree construction, connected dominating set (CDS) construction, and link scheduling. For toology center selection, the node with the shortest network radius in terms of ho count is chosen as the toology center. If the classical Bellman-Ford algorithm is alied to derive the routing matrix, the comlexity for this hase is O(jVjjEj). For BFS tree construction, the comlexity is O(jVj + jej). The CDS construction hase also has a comlexity of O(jVj + jej). Their link scheduling hase consists of an outer iteration on the nodes and an inner iteration on the colors. Let the number of colors be c; the comutational comlexity in this hase is O(cjVj). In summary, Li et al. s algorithm requires a comutational comlexity of O(jVjjEj). B.. Message comlexity Cell-AS: During both the neighbor discovery and the link scheduling hase, n nodes in round and at most min{n, K k+ } nodes in round k send messages to their neighbors. Thus, the message comlexity of either of these two functions is minfðk þ Þn; n þ ðk Þ g. 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Welzl, Caacity of arbitrary wireless networks, in: Proc. of INFOCOM 9, IEEE, 9. Hongxing Li received his B.Sc. and M.Engr. degrees in 5 and 8 from the Deartment of Comuter Science and Technology, Nanjing University, China. He is currently a Ph.D. candidate in the Deartment of Comuter Science, the University of Hong Kong, China. His research interests include wireless networks and network coding. Chuan Wu received her B.Engr. and M.Engr. degrees in and from Deartment of Comuter Science and Technology, Tsinghua University, China, and her Ph.D. degree in 8 from the Deartment of Electrical and Comuter Engineering, University of Toronto, Canada. She is currently an assistant rofessor in the Deartment of Comuter Science, the University of Hong Kong, China. Her research interests include measurement, modeling, and otimization of large-scale eer-to-eer systems and online/mobile social networks. She is a member of IEEE and ACM. Qiang-Sheng Hua received his B.Engr. and M.Engr. degrees in and 4 from the School of Information Science and Engineering, Central South University, China, and his Ph.D. degree in 9 from the Deartment of Comuter Science, The University of Hong Kong, China. He is currently an assistant rofessor in the Institute for Interdiscilinary Information Sciences, Tsinghua University, China. His research interests are on wireless networking and algorithms. Francis C.M. Lau received his PhD in comuter science from the University of Waterloo. He is currently a rofessor in comuter science in The University of Hong Kong. His research interests include comuter systems, networks, rogramming languages, and alication of comuting in arts.