Capacity of Data Collection in Arbitrary Wireless Sensor Networks

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1 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. 1 Capacity of Data Coection in Arbitrary Wireess Sensor Networks Siyuan Chen Minsu Huang Shaojie Tang Yu Wang Abstract Data coection is a fundamenta function provided by wireess sensor networks. How to efficienty coect sensing data from a sensor nodes is critica to the performance of sensor networks. In this paper, we aim to understand the theoretica imits of data coection in a TDMA-based sensor network in terms of possibe and achievabe maximum capacity. Previousy, the study of data coection capacity [1] [6] has concentrated on arge-scae random networks. However, in most of the practica sensor appications, the sensor network is not uniformy depoyed and the number of sensors may not be as huge as in theory. Therefore, it is necessary to study the capacity of data coection in an arbitrary network. In this paper, we first derive the upper and ower bounds for data coection capacity in arbitrary networks under protoco interference mode and disk graph mode. We show that a simpe BFS tree based method can ead to order-optima performance for any arbitrary sensor networks. We then study the capacity bounds of data coection under a genera graph mode, where two nearby nodes may be unabe to communicate due to barriers or path fading, and discuss performance impications. Finay, we provide discussions on the design of data coection under physica interference mode or Gaussian channe mode. Index Terms capacity, data coection, arbitrary networks, wireess sensor networks. I. INTRODUCTION Due to their wide-range potentia appications in various scenarios such as battefied, emergency reief and environment monitoring, wireess sensor networks have recenty emerged as a premier research topic. The utimate goa of a sensor network is often to deiver the sensing data from a sensors to a sink node and then conduct further anaysis at the sink node. Thus, data coection is one of the most common services used in sensor network appications. In this paper, we study some fundamenta capacity probems arising from data coection in wireess sensor networks. We consider a wireess sensor network where n sensors are arbitrariy depoyed in a finite geographica region. Each sensor measures independent fied vaues at reguar time intervas and sends these vaues to a sink node. The union of a sensing vaues from n sensors at a particuar time is caed a snapshot. The task of data coection is to deiver these snapshots to a singe sink. Due to spatia separation, severa sensors can successfuy transmit at the same time if these transmissions do not cause any destructive wireess interference. As in the iterature, we first adopt the protoco interference mode in our anaysis and assume that a successfu S. Chen, M. Huang and Y. Wang are with Department of Computer Science, University of North Caroina at Charotte, Charotte, NC, USA. S. Tang is with Department of Computer Science, Iinois Institute of Technoogy, Chicago, Iinois, USA. transmission over a ink has a fixed data-rate W bit/second. Later, we reax these assumptions to more reaistic modes: physica interference mode and Gaussian channe mode. The performance of data coection in sensor networks can be characterized by the rate at which sensing data can be coected and transmitted to the sink node. In particuar, the theoretica measure that captures the imits of coection processing in sensor networks is the capacity of many-toone data coection, i.e., the maximum data rate at the sink to continuousy receive the snapshot of data from sensors. Data coection capacity refects how fast the sink can coect sensing data from a sensors with interference constrain. It is critica to understand the imit of many-to-one information fows and devise efficient data coection agorithms to improve the performance of wireess sensor networks. Capacity imits of data coection in random wireess sensor networks have been studied in the iterature [1] [6]. In [1], [2], Duarte-Meo et a. first introduced the many-to-one transport capacity in dense and random sensor networks under protoco interference mode. Both E Gama [3] and Barton and Zheng [4] investigated the capacity of data coection with compex physica ayer techniques, such as antenna sharing, channe coding and cooperative beam-forming. Liu et a. [5] recenty studied the capacity of a genera someto-some communication paradigm under protoco interference mode in random networks with mutipe randomy seected sources and destinations. Chen et a. [6] studied the capacity of data coection under protoco interference mode with mutipe sinks. However, a the research above shares the standard assumption that a arge number of sensor nodes are either ocated on a grid structure or randomy and uniformy distributed in a pane. Such an assumption is usefu to simpify the anaysis and derive nice theoretica imits, but may be invaid in many practica sensor appications. In this paper, we focus on deriving capacity bounds of data coection for arbitrary networks, where sensor nodes can be depoyed in any distribution and can form any network topoogy. We summarize our contributions as foows: For arbitrary sensor networks under protoco interference mode and disk graph mode (if two sensors are within the transmission ranges of each other then they can communicate), we propose a simpe data coection method which performs data coection on branches of the Breadth First Search (BFS) tree. We prove that this method can achieve coection capacity of Θ(W ) which matches the theoretica upper bound. Since the disk graph mode is ideaistic, we aso consider a more practica network mode: genera graph mode. Digita Object Indentifier /TPDS /11/$ IEEE

2 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. 2 In the genera graph mode, two nearby nodes may be unabe to communicate due to various reasons such as barriers and path fading. We first show that Θ(W ) may not be achievabe for a genera graph. Then we prove that a greedy scheduing agorithm on BFS tree can achieve capacity of Θ( λ W ) whie the capacity is bounded by Θ( W Δ ) from above. Here, Δ, λ, and λ are three new interference reated parameters defined in Section V. Finay, we discuss the data coection capacity under more genera communication modes, physica interference mode and Gaussian channe mode. For physica interference mode, we prove that the capacity of data coection is in the same order as the one under protoco interference mode. For Gaussian channe mode, we derive an upper bound of data coection capacity. The resuts above not ony hep us to understand the theoretica imits of data coection in sensor networks, but aso provide practica and efficient data coection methods (incuding how to construct data coection structure and how to schedue data coection) to achieve near-optima capacity. Even though we are focusing on arbitrary networks, a of our soutions can be appied to random networks since any random network is just a specia case of arbitrary networks. The rest of this paper is organized as foows. We first review reated work in Section II, and then describe our network mode in Section III. We study the data coection capacity under disk graph mode and protoco interference mode in Section IV. In Section V, we reax the disk graph mode in our anaysis and derive the bounds of data coection capacity in a genera graph mode. We discuss the coection capacity under physica interference mode and Gaussian channe mode in Section VI, and concude the paper in Section VII. A preiminary conference version of this paper appeared in [7]. Due to space imit, some detaied proofs and simuation resuts are ignored here, and provided as Suppementa Materia. II. RELATED WORK Gupta and Kumar initiated the research on capacity of random wireess networks by studying the unicast capacity in the semina paper [9]. A number of foowing papers studied capacity under different communication scenarios in random networks: unicast [10] [12], muticast [13] [15], broadcast [16], [17]. In this paper, we focus on the capacity of data coection in a many-to-one communication scenario. Capacity of data coection in random wireess sensor networks has been investigated in [1] [6]. Duarte-Meo et a. [1], [2] first studied the many-to-one transport capacity in random sensor networks under protoco interference mode. They showed that the overa capacity of data coection is Θ(W ). E Gama [3] studied data coection capacity subject to a tota average transmitting power constraint. They reaxed the assumption that every node can ony receive from one source node at a time. It was shown that the capacity of random networks scaes as Θ(og nw ) when n goes to infinity and the tota average power remains fixed. Their method uses antenna sharing and channe coding. Barton and Zheng [4] aso investigated data coection capacity under more compex physica ayer modes (non-cooperative SINR mode and cooperative time reversa communication (CTR) mode). They first demonstrated that Θ(og nw ) is optima and achievabe using CTR for a reguar grid network in [18], then showed that the capacities of Θ(og nw ) and Θ(W ) are optima and achievabe by CTR when operating in fading environments with power path-oss exponents that satisfy 2 <β<4 and β 4 for random networks [4]. Recenty, Chen et a. [6] have studied data coection capacity with mutipe sinks. They showed that with k sinks the capacity increases to Θ(kW) when k = O( n nw n og n ) or Θ( og n ) when k = Ω( og n ). Liu et a. [5] atey introduced the capacity of a more genera some-to-some communication paradigm in random networks where there are s(n) randomy seected sources and d(n) randomy seected destinations. They derived the upper and ower bounds for such a probem. However, a research above shares the standard assumption that a arge number of sensor nodes are either ocated on a grid structure or randomy and uniformy distributed in a pane. Such an assumption is usefu to simpify the anaysis and derive nice theoretica imits, but may be invaid in many practica sensor appications. To our best knowedge, our paper is the first to study data coection capacity for arbitrary networks. III. NETWORK MODELS AND COLLECTION CAPACITY A. Basic Network Modes In this paper, we focus on the capacity bound of data coection in arbitrary wireess sensor networks. For simpicity, we start with a set of simpe and yet genera enough modes. Later, we wi reax them to more reaistic modes. We consider an arbitrary wireess network with n sensor nodes v 1,v 2,,v n and a singe sink v 0. These n sensors are arbitrariy distributed in a fied. At reguar time intervas, each sensor measures the fied vaue at its position and transmits the vaue to the sink. We first adopt a fixed data-rate channe mode where each wireess node can transmit at W bits/second over a common wireess channe. We aso assume that a packets have unit size b bits. The time is divided into time sots with t = b/w seconds. Thus, ony one packet can be transmitted in a time sot between two neighboring nodes. TDMA scheduing is used at MAC ayer. Under the fixed data-rate channe mode, we assume that every node has a fixed transmission power P. Thus, a fixed transmission range r can be defined such that a node v j can successfuy receive the signa sent by node v i ony if v i v j r. Here, v i v j is the Eucidean distance between v i and v j. We ca this mode disk graph mode. We further define a communication graph G =(V,E) where V is the set of a nodes (incuding the sink) and E is the set of a possibe communication inks. We assume graph G is connected. Due to spatia separation, severa sensors can successfuy transmit at the same time if these transmissions do not cause any destructive wireess interferences. As in the iterature, we first mode the interference using protoco interference mode. A nodes have a uniform interference range R. When node v i transmits to node v j, node v j can receive the signa successfuy if no node within a distance R from v j is

3 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. 3 transmitting simutaneousy. Here, for simpicity, we assume that R r is a constant α which is arger than 1. Let δ(v i) be the number of nodes in v i s interference range (incuding v i itsef) and Δ be the maximum vaue of δ(v i ) for a nodes v i, i =0,,n. We summarize a notations used in this paper in a tabe given in Section VI of Suppementa Materia. vj R v x v y v z B. Capacity of Data Coection We now formay define deay and capacity of data coection in wireess sensor networks. Reca that each sensor at reguar time intervas generates a fied vaue with b bits and wants to transport it to sinks. We ca the union of a vaues from a n sensors at particuar samping time a snapshot of the sensing data. The goa of data coection is to coect these snapshots from a sensors to the sinks. It is cear that the sink prefers to get each snapshot as quicky as possibe. In this paper, we assume that there is no correation among a sensing vaues and no network coding or aggregation technique is used during the data coection. Definition 1: The deay of data coection D is the time used by the sink to successfuy receive a snapshot, i.e., the time needed between competey receiving one snapshot and competey receiving the next snapshot at the sink. Definition 2: The capacity of data coection C is the ratio between the size of data in one snapshot and the time to receive such a snapshot (i.e., nb D ) at the sink. Thus, the capacity C is the maximum data rate at the sink to continuousy receive the snapshot data from sensors. Here, we require the sink to receive the compete snapshot from a sensors (i.e., data from a sensors need to be deivered). Notice that data transport can be pipeined in the sense that further snapshots may begin to transport before the sinks receiving prior snapshots. In this paper, we focus on capacity anaysis of data coection in an arbitrary sensor network. IV. COLLECTION CAPACITY FOR DISK GRAPH MODEL Upper Bound of Coection Capacity: It has been proved that the upper bound of capacity of data coection for random networks is W [1], [2]. It is obviousy that this upper bound aso hods for any arbitrary network. The sink v 0 cannot receive at rate faster than W since W is the fixed transmission rate of individua ink. Therefore, we are interested in design of data coection agorithm to achieve capacity in the same order of the upper bound, i.e. Θ(W ). In this section, we propose a simpe BFS-based data coection method and demonstrate that it can achieve the capacity of Θ(W ) under our network mode: disk graph mode. Our data coection method incudes two steps: data coection tree formation and data coection scheduing. A. Data Coection Tree - BFS Tree The data coection tree used by our method is a cassica Breadth First Search (BFS) tree rooted at the sink v 0. The time compexity to construct such a BFS tree is O( V + E ). Let T be the BFS tree and v1,,vc be a eaves in T. For each eaf vi, there is a path P i from itsef to the root r/2 Pi Fig. 1. Proof of Lemma 1: on a path P i in BFS T, the interference nodes for a node v j is bounded by a constant. v 0. Let δ P i (v j ) be the number of nodes on path P i which are inside the interference range of v j (incuding v j itsef). Assume the maximum interference number Δ i on each path P i is max{δ Pi (v j )} for a v j P i. Hereafter, we ca Δ i path interference of path P i. Then we can prove that T has a nice property that the path interference of each branch is bounded by a constant. Lemma 1: Given a BFS tree T under the protoco interference mode, the maximum interference number Δ i on each path P i is bounded by a constant 8α 2, i.e., Δ i 8α 2. Proof: We prove by contradiction with a simpe area argument. Assume that there is a v j on P i whose δ P i (v j ) > 8α 2. In other words, more than 8α 2 nodes on P i are ocated in the interference region of v j. Since the area of interference region is πr 2, we consider the number of interference nodes inside a sma disk with radius r 2. See Figure 1 for iustration. πr The number of such sma disks is at most 2 π( = 4α 2 r 2 )2 inside πr 2. By the Pigeonhoe principe, there must be more than 8α2 4α =2nodes inside a singe sma disk with radius r 2 2. In other words, three nodes v x, v y and v z on the path P i are connected to each other as shown in Figure 1. This is a contradiction with the construction of BFS tree. As shown in Figure 1, if v x and v z are connected in G, then v z shoud be visited by v x not v y during the construction of BFS tree. This finishes our proof. Fig. 2. sot 1 sot 2 sot 3 data Path P i with Δ= i 3 Sot 1 Sot 2 Sot 3 V 0 Scheduing on a path: after Δ i sots the sink gets one data. B. Branch Scheduing Agorithm We now iustrate how to coect one snapshot from a sensors. Given the coection tree T, our scheduing agorithm basicay coects data from each path P i in T one by one. First, we expain how to schedue coection on a singe path. For a given path P i, we can use Δ i sots to coect one data in the snapshot at the sink. See Figure 2 for iustration. V 0 V 0 (a) (b) (c) (d)

4 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. 4 V a B 2 B 4 P P P P B 1 V b P 1 P 1 P 2 V4 V B 4 3 V V 3 V V 2 V 3 V 1 V 2 V 3 V 1 V 2 V 3 V 1 V 2 V 3 1 V 2 1 (a) BFS Tree T (b) Branches in T (c) Step 1 (d) After Step 1 (e) Step 2 P 4 P 4 P 2 P 3 P V 3 V 3 V 3 V 3 V 3 V V V V V (f) After Step 2 (g) Step 3 (h) After Step 3 (i) Step 4 (j) After Step 4 Fig. 3. Iustrations of our scheduing on the data coection tree T. In this figure, we assume that R = r, i.e., ony adjacent nodes interfere with each other. Thus Δ i =3. Then we coor the path using three coors as in Figure 2(a). Notice that each node on the path has unit data to transfer. Links with the same coor are active in the same sot. After three sots (Figure 2(d)), the eaf node has no data in this snapshot and the sink got one data from its chid. Therefore, to receive a data on the path, at most Δ i P i time sots are needed. We ca this scheduing method Path Scheduing. Now we describe our scheduing agorithm on the coection tree T. Remember T has c eaves which define c paths from P 1 to P c. Our agorithm coects data from path P 1 to P c in order. We define that i-th branch B i is the part of P i from v i to the intersection node with P i+1 for i =[1,c 1] and c-th branch B c = P c. For exampe, in Figure 3(b), there are four branches in T : B 1 is from v 1 to v a, B 2 is from v 2 to v 0, B 3 is from v 3 to v b, and B 4 is from v 4 to v 0. Notice that the union of a branches is the whoe tree T. Agorithm 1 shows the detaied branch scheduing agorithm. Figure 3(c)-(j) give an exampe of scheduing on T. In the first step (Figure 3(c)), a nodes on P 1 participate in the coection using the scheduing method for a singe path (every Δ 1 sots, sink v 0 receives one data). Such coection stops unti there is no data in this snapshot on branch B 1, as shown in Figure 3(d). Then Step 2 coects data on path P 2. This procedure repeats unti a data in this snapshot reaches v 0 (Figure 3(j)). Agorithm 1 Branch Scheduing on BFS Tree Input: BFS tree T. 1: for each snapshot do 2: for t =1to c do 3: Coect data on path P i. A nodes on P i transmit data towards the sink v 0 using Path Scheduing. 4: The coection terminates when nodes on branch B i do not have data for this snapshot. The tota sots used are at most Δ i B i, where B i is the hop ength of B i. 5: end for 6: end for C. Capacity Anaysis We now anayze the achievabe capacity of our data coection method by counting how many time sots the sink needs to receive a data of one snapshot. Theorem 2: The data coection method based on pathscheduing in BFS tree can achieve data coection capacity of Θ(W ) at the sink. Proof: In Agorithm 1, the sink coects data from a c paths in T. In each step (Lines 3-4), data are transferred on path P i and it takes at most Δ i B i time sots. Reca that Path Scheduing needs at most Δ i k time sots to coect k packets from path P i. Therefore, the tota number of time sots needed for Agorithm 1, denoted by τ, is at most c i=1 Δ i B i. Since the union of a branches is the whoe tree T, i.e., c i=1 B i = n. Thus, τ c i=1 Δ i B i c Δ B i=1 i Δn. Here Δ = max{δ 1,, Δ c }. Then, the deay of data coection D = τt Δnt. The capacity C = nb D Δnt nb = W Δ. From Lemma 1, we know that Δ is bounded by a constant. Therefore, the data coection capacity is Θ(W ). Remember that the upper bound of data coection capacity is W, thus our data coection agorithm is order-optima. Consequenty, we have the foowing theorem. Theorem 3: Under protoco interference mode and disk graph mode, data coection capacity for arbitrary wireess sensor networks is Θ(W ). V. COLLECTION CAPACITY FOR GENERAL GRAPH MODEL So far, we assume that the communication graph is a disk graph where two nodes can communicate if and ony if their distance is ess than or equa to transmission range r. However, a disk graph mode is ideaistic since in practice two nearby nodes may be unabe to communicate due to various reasons such as barriers and path fading. Therefore, in this section, we consider a more genera graph mode G =(V,E) where V is the set of sensors and E is the set of possibe communication inks. Every sensor sti has a fixed transmission range r such that the necessary condition for v j to receive correcty the signa from v i is v i v j r. However, v i v j r is not the sufficient condition for an edge v i v j E. Some inks do not beong to G because of physica barriers or the

5 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. 5 seection of routing protocos. Thus, G is a subgraph of a disk graph. Under this mode, the network topoogy G can be any genera graph (for exampe, setting r = and putting a barrier between any two nodes v i and v j if v i v j / G). Notice that even though we sti consider the protoco interference mode, our anaysis sti hods for arbitrary interference graph. In genera graph mode, the capacity of data coection coud be W n in the worst-case. We consider a simpe straight-ine network topoogy with n sensors as shown in Figure 4(a). Assume that the sink v 0 is ocated at the end of the network and the interference range is arge enough to cover every node in the network. Since the transmission on one ink wi interfere with a the other nodes, the ony possibe scheduing is transferring data aong the straight-ine via a inks. The tota time sots needed are n(n +1)/2, thus the capacity is nb n(n+1)t/2 =Θ(W n at most ). Notice that in this exampe, the maximum interference number Δ of graph G is n. It seems the upper bound of data coection capacity coud be W Δ.Wenow show an exampe whose capacity can be much arger than W Δ. Again we assume a n nodes with the sink interfering with each other. The network topoogy is a star with the sink v 0 in center, as shown in Figure 4(b). Ceary, a scheduing that ets every node transfer data in order can ead to a capacity W which is much arger than W Δ = W n. Fig. 4. V 2 V n V 2 V 0 (a) Straight-ine Topoogy V n (b) Star Topoogy The optimum of BFS-based method under two extreme cases. A. Upper Bound of Coection Capacity We first present a tighter upper bound of data coection capacity for genera graph mode than the natura one W. Consider a packets from one snapshot, we use p i to represent the packet generated by sensor v i. For any v i, et (v i ) be its eve in the BFS tree rooted at the sink v 0 ( which is the minimum number of hops required for packet p i or a packet at v i to reach v 0 ). We use D(v 0,) to represent a virtua disk centered at the sink node v 0 with radius of hop distance. The critica eve (or caed the critica radius) is the greatest eve such that no two nodes within eve from v 0 can receive a message in the same time sot, i.e., = max{ v i,v j D(v 0,) cannot receive packets at the same time}. The region defined by D(v 0,) is caed critica region. See Figure 5 for iustration. For any packet p i originated at node v i,we define { λ (vi ) if v i = i D(v 0, ) +1 otherwise. Here, λ i gives the minimum number of hops needed to reach the sink v 0 after packet p i reaches the critica region around v 0. Let λ = max i {λ i }. Then we can prove the foowing emma on the ower bound of deay for data coection. * V j * V q V k V s V i (V i ) (a) critica region around sink v 0 V j V q V k V s (b) a tree view of critica region V i =0 =1 =2 =(V ) i Fig. 5. Iustration of the definition of critica region, i.e.. The grey area is the critica region, where no any two nodes can receive a message in the same time sot due to interference around v 0. Lemma 4: For a packets from one snapshot, the deay to coect them at sink v 0 D t λ i. Proof: It is cear the critica region around the sink v 0 is a botteneck for the deay. Any packet inside the critica region can ony move one step at each time sot. First, the tota deay must be arger than the deay which is needed for the case where a packets originated outside critica region are just one hop away from the critica region. In other words, assume that we can move a packets originated outside critica region to the surrounding area without spending any time. Then each packet p i needs λ i time sots to reach the sink. By the definition of the critica region, no simutaneous transmissions around the critica region (1-hop from it) can be schedued in the same sot. Therefore, the deay is at east the summation of λ i. i λ i i Let Δ = n, we have a new upper bound of data coection capacity, C W Δ W. Notice that Δ 1 and it represents the imit of scheduing due to interference around the sink (and its critica region). B. Lower Bound of Coection Capacity The data coection agorithm based on branch-scheduing in BFS tree can sti achieve the capacity of W Δ. However, in genera graph mode we can not bound Δ by a constant any more, and it coud be O(1) or O(n). Though this simpe method can match the tight upper bounds Θ( W n ) and W of exampes shown in Figure 4, it is sti not a tight bound. We show such an exampe and discuss a tighter ower bound based on this method in Section I of Suppementa Materia. Now we introduce a new greedy-based scheduing agorithm which is inspired by [19]. The scheduing agorithm sti uses the BFS tree as the coection tree. A messages wi be sent aong the branch towards the sink v 0.Forn messages from one snapshot, it works as foows. In every time sot, it sends

6 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. 6 each message aong the BFS tree from the current node to its parent, without creating interference with any higher-priority 1 message. The priority ρ i of each packet p i is defined as (v i ). It is cear that packets originated from the chidren of the sink have the highest priority ρ i =1whie packets originated from other nodes have ower priority ρ i < 1. For two packets with the same priority (on the same eve in the BFS tree), ties can be broken arbitrariy. Given a schedue, et vj τ be the node of packet p j in the end of time sot τ. The detaied greedy agorithm is given in Agorithm 2. Agorithm 2 Greedy Scheduing on BFS Tree Input: BFS tree T. 1: Compute the priority ρ i =1/(v i ) of each message p i. 2: for each snapshot do 3: whie p j such that vj τ v 0 do 4: for a such p i in decreasing order of priority ρ i do 5: if sending p i from node vi τ wi not create interference with any higher-priority messages that are aready schedued for this time sot then 6: node vi τ sends p i to its parent par(vi τ ) in T. 7: end if 8: end for 9: τ = τ : end whie 11: end for Now we anayze the capacity achieved by this greedy data coection method. Before presenting the anaysis, we first introduce some new notations. For two nodes v i and v j, h(v i,v j ) denotes the shortest hop number from v i to v j in graph G. The deay of packet p j is defined as the time unti it reach the sink v 0, i.e., D j = t min{τ : vj τ = v 0}. Vj Fig. 6. Iustration of the definitions of λ i. λ i V i Let λ i be the minima hops that a packet needs to be forwarded from node v i before a new packet at v i can be safey forwarded aong the BFS tree. So λ i = max{ v j,h(v i,v j )= and transmission from v i to par(v i ) interferes with transmission from v j to par(v j )} +1. Here par(v i ) is the parent of v i in T. See Figure 6 for iustration. Here λ i =4for v i.we define that λ = max i {λ i }. Both λ and λ i are integers (hop counts). In addition, we can prove that λ λ. A detaied proof is provided in Section II of Suppementa Materia. Packet p j is said to be bocked in time sot τ if, in time sot τ, p j is not sent out. We define the foowing bocking reation on our greedy agorithm schedue: p k p j if in the ast time sot in which p j is bocked by the transmission of higher priority packets in that time sot, p k is the one cosest to p j in term of hops among these packets (ties broken arbitrariy). The bocking reation induces a directed bocking tree T D where nodes are a message p i and edge (p k,p j ) representing p k p j. The root p r of the tree T D is a message with highest priority (originated in a chid of v 0 ) which is never bocked. Let P (j) the path in T D from p r to p j and h(j) be the hop count of P (j). We then derive an upper bound on the deay D j of packet p j in the greedy agorithm. Lemma 5: For each packet p j in the snapshot, its deay D j t p i P (j) min{(v i),λ}. Proof: We prove this emma by induction on h(j). For any packet p j,ifh(j) =0, which means p j is the root p r of T D, it wi not be bocked. So D j = t (v j ). Then consider the right side of the inequation t p i P (j) min{(v i),λ} = t min{(v j ),λ}. Since p j is packet with highest priority, (v j )= 1 and (v j ) λ. Thus, t p i P (j) min{(v i),λ} = t (v j ) and the caim in this emma hods for the case where h(j) =0. If h(j) > 0, i.e., p j p r, et τ be the ast time sot in which p j is bocked by packet p k, i.e., p k p j. Notice that t h(vk τ,v 0) D k t τ, otherwise p k woud not reach v 0 by time D k. Aso h(vj t,vt k ) λ 1 since after p k moves one hop p j is safe to move. From time sot τ +1, p j may be forwarded towards v 0 over one hop in each time sot, and reach v 0 at the eariest time sot, D j t (τ +1+h(vj,v t 0 )) t (τ +1+h(vk,v t 0 )+h(vj,v t k)) t t (τ +1)+D k t τ + t λ 1 = D k + t λ. On the other hand, D j D k + t (v j ) because after p k reaches the sink v 0, p j needs at most (v j ) to reach the sink. Consequenty, D j D k + t min{(v j ),λ}. This competes our proof. Lemma 6: The data coection capacity of our greedy agorithm is at east λ W. Proof: Let p j be the packet having maximum D j.by Lemma 5 and λ λ, D j t min{(v i ),λ} λ λ t min{(v i ),λ } p i T D p i P (j) λ λ t( v i D(v 0, ) (v i )+ v i / D(v 0, ) ( +1)) = λ λ t λ i = λ λ ntδ. i Thus, the capacity achieved by our greedy agorithm is at east nb D j = λ λ W Δ. Remark: In summary, we show that under protoco interference mode and genera graph mode data coection capacity for arbitrary sensor networks has the foowing bounds: Theorem 7: Under protoco interference mode and genera graph mode, data coection capacity for arbitrary sensor networks is at east λ W W and at most Δ. Here λ describes the interference around the sink v 0, whie λ describes the interference around a node v i. Since λ λ, λ λ 1. For disk graph mode, λ λ is a constant. However,

7 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. 7 for genera graph mode it may not, thus, there is sti a gap between the ower and upper bounds (such an exampe is given in Section I of Suppementa Materia). We eave finding tighter bounds to cose the gap as one of our future works. For two exampes in Figure 4, the greedy method matches the optima soutions in order. For the straight-ine topoogy in Figure 4(a), λ = λ = n and Δ = Θ(n). Thus, the capacity λ W =Θ( W n ) matches the upper bound. For the star topoogy in Figure 4(b), λ = λ = 1 and Δ λ = 1. In this case, W = Θ(W ) aso matches the upper bound. Compared with the branch scheduing method, greedy method can achieve much better capacity in practice, since greedy agorithm aows packet transmissions among mutipe branches of the BFS tree in the same time sot. This is confirmed by our simuation resuts on random networks (Section V of Suppementa Materia). VI. DISCUSSIONS ON OTHER MODELS A. Physica Interference Mode So far, we ony consider the protoco interference mode, which is an idea and simpe mode. We can extend our anaysis to the physica interference mode by appying a technique introduced by Li et a. [8] when they studied the broadcast capacity of wireess networks. In physica interference mode, node v j can correcty receive signa from a sender v i if and ony if, given a constant η > 0, the SINR (Signa to Interference pus Noise Ratio) P v i v j β B N 0 + k I P v η, k v j β where B is the channe bandwidth, N 0 is the background Gaussian noise, I is the set of activey transmitting nodes when node v i is transmitting, β>2is the pass oss exponent, and P is the fixed transmission power. We can prove the foowing theorem which indicates that data coection capacity under physica interference mode is sti Θ(W ). Theorem 8: Under physica interference mode and disk graph mode, data coection capacity for arbitrary wireess sensor networks is Θ(W ). Due to space imit, the detaied proof of this theorem is given in Section III of Suppementa Materia. B. Gaussian Channe Mode For both protoco interference mode and physica interference mode, as ong as the vaue of a given conditiona expression (such as transmission distance or SINR vaue) beyond some threshod, the transmitter can send data successfuy to a receiver at a specific constant rate W due to the fixed rate channe mode. Whie widey studied, fixed rate channe mode may not capture we the feature of wireess communication. We now discuss the capacity bounds under a more reaistic channe mode: Gaussian channe mode. In such mode, it determines the rate under which the sender can send its data to the receiver reiaby, based on a continuous function of the receiver s SINR. Again, we assume every node transmits at a constant power P. Any two nodes v i and v j can estabish a direct communication ink v i v j, over a channe of bandwidth W, of rate P v i v j W ij = W og 2 (1+ β ) N 0 + k I P v k v j β. This mode assigns a more reaistic transmission rate at arge distance than the fixed rate channe mode with protoco or physica interference mode. In order to derive an upper bound for the capacity of data coection under Gaussian channe mode, we consider the congestion at the sink node. In particuar, we prove that whatever scheduing scheme is impemented, the tota transmission rate of a the incoming inks at the sink node is upper bounded by some vaue. As a botteneck, the capacity of the whoe network is aways bounded by that vaue. Our proof basicay foows the same idea proposed in [12] [13], which is firsty used to study the capacity bound for muticast session under Gaussian channe mode. Due to space imit, the detaied proof is given in Section IV of Suppementa Materia. Theorem 9: An upper bound for data coection capacity under Gaussian channe mode is at most max(w i0 )+W og 2 (n). i The first part of this upper bound depends on the rate of the shortest incoming ink at sink, whie the second part depends on the tota number of nodes. Notice that max i (W i0 ) W og 2 (1 + P N 0 ). Thus, which part in the bound paying an important roe depends on the reationship between n and 1+ P N 0. When the network is a reguar grid or a random homogeneous topoogy, it is satisfied that i0 n γ for some constant γ < 0. Then we have max i (W i0 )=O(Wog n). Therefore, the tota rate of a incoming inks at sink node v 0 is at most O(og n W ). A ower bound of data coection capacity in this mode is sti open. VII. CONCLUSION In this paper, we study the theoretica imits of data coection in terms of capacity for arbitrary wireess sensor networks. We first propose a simpe data coection method based on BFS tree to achieve capacity of Θ(W ), which is order-optima under protoco interference mode and disk graph mode. However, when the underying network is a genera graph, we show that Θ(W ) may not be achievabe. We prove that a new BFS-based method using greedy scheduing can sti achieve capacity of Θ( λ W ) and aso give a tighter upper bound Θ( W Δ ). At ast, we discuss the coection capacity under more genera modes, physica interference mode or Gaussian channe mode. Tabe I summarizes our resuts. A of our methods can achieve these resuts for random networks too. We aso provide some simuation resuts on random networks in Section V of Suppementa Materia. There are sti severa open probems eft as our future work. First, we woud ike to cose the gap of upper and ower bounds of data coection capacity for genera graph; Second, the ower bound of data coection capacity under Gaussian channe mode is sti open. We pan to design new data coection schemes to approximate the upper bound better. Third, even though the capacity of data aggregation for arbitrary networks

8 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. 8 has been studied in [20], the author ony considered the worst case capacity. It is interesting to study aggregation capacity for any arbitrary network. Fourth, different coection methods may cost different amount of energy. It is desired to study the trade-off between the achievabe capacity and the energy consumption for data coection in sensor networks. Recent study [21] provides a nice start on this direction. Last, we aso pan to study the coection capacity under more practica modes (considering data correation, fading effects, and time varying channes). TABLE I SUMMARY OF DATA COLLECTION CAPACITY Network Mode Interference Mode Capacity C Disk Graph Protoco Interference C =Θ(W ) Disk Graph Physica Interference C =Θ(W ) Genera Graph Protoco Interference Θ( λ W ) C Θ( W Δ ) Genera Graph Gaussian Channe C max i (W i0 )+W og 2 (n) VIII. ACKNOWLEDGMENTS The work of S. Chen, M. Huang and Y. Wang is supported in part by the US Nationa Science Foundation (NSF) under Grant No. CNS , CNS , and CNS , and by Tsinghua Nationa Laboratory for Information Science and Technoogy (TNList). REFERENCES [1] E.J. Duarte-Meo and M. Liu, Data-gathering wireess sensor networks: Organization and capacity, Computer Networks, 43, , [2] D. Marco, E.J. Duarte-Meo, M. Liu, and D.L. Neuhoff, On the manyto-one transport capacity of a dense wireess sensor network and the compressibiity of its data, in Proc. Int Workshop on Information Processing in Sensor Networks, [3] H.E. Gama, On the scaing aws of dense wireess sensor networks: the data gathering channe, IEEE Trans. on Information Theory, vo. 51, no. 3, pp , [4] R. Zheng and R.J. Barton, Toward optima data aggregation in random wireess sensor networks, in Proc. of IEEE Infocom, [5] B. Liu, D. Towsey, and A. Swami, Data gathering capacity of arge scae mutihop wireess networks, in Proc. of IEEE MASS, [6] S. Chen, Y. Wang, X.-Y. Li, and X. Shi, Capacity of data coection in randomy-depoyed wireess sensor networks, ACM Springer Wireess Networks (WINET), to appear, Short version in Proc. of IEEE SECON, [7] S. Chen, S. Tang, M. Huang, and Y. Wang, Capacity of data coection in arbitrary wireess sensor networks, in Proc. of IEEE Infocom, [8] X.-Y. Li, J. Zhao, Y.W. Wu, S.J. Tang, X.H. Xu, and X.F. Mao, Broadcast capacity for wireess ad hoc networks, in Proc. of IEEE MASS, [9] P. Gupta and P.R. Kumar, The capacity of wireess networks, IEEE Trans. on Information Theory, 46(2), , [10] M. Grossgauser and D. Tse, Mobiity increases the capacity of ad-hoc wireess networks, in Proc. of IEEE Infocom, [11] B. Liu, P. Thiran, and D. Towsey, Capacity of a wireess ad hoc network with infrastructure, in Proc. of ACM MobiHoc, [12] Franceschetti, M. and Dousse, O. and Tse, D.N.C. and Thiran, P. Cosing the gap in the capacity of wireess networks via percoation theory, IEEE Trans. on Information Theory, 53(3), , [13] Keshavarz-Haddad, A. and Riedi, R.H. Bounds for the capacity of wireess mutihop networks imposed by topoogy and demand, in Proc. of ACM MobiHoc, [14] X.-Y. Li, S.-J. Tang, and O. Frieder, Muticast capacity for arge scae wireess ad hoc networks, in Proc. of ACM MobiCom, [15] S. Shakkottai, X. Liu, and R. Srikant, The muticast capacity of arge mutihop wireess networks, in Proc. of ACM MobiHoc, 2007 [16] A. Keshavarz-Haddad, V. Ribeiro, and R. Riedi, Broadcast capacity in mutihop wireess networks, in Proc. of MobiCom, [17] B Tavi, Broadcast capacity of wireess networks, IEEE Communications Letters, 10, 68-69, [18] R.J. Barton and R. Zheng, Order-optima data aggregation in wireess sensor networks using cooperative time-reversa communication, in Proc. of Annua Conf. on Information Sciences and Systems, [19] V. Bonifaci, P. Korteweg, A. Marchetti-Spaccamea, and L. Stougie, An approximation agorithm for the wireess gathering probem, Operations Research Letters, 36, , [20] T. Moscibroda, The worst-case capacity of wireess sensor networks, in Proc. of ACM IPSN, [21] X.-Y. Li, Y. Wang, and Y. Wang, Compexity of data coection, aggregation, and seection for wireess sensor networks, IEEE Trans. on Computers, to appear, Siyuan Chen received his B.S. degree from Peking University, China in He is currenty a PhD student in the University of North Caroina at Charotte, majoring in computer science. His current research focuses on wireess networks, ad hoc and sensor networks, and agorithm design. Minsu Huang received his BS degree in computer science from Centra South University in 2003 and his MS degree in computer science from Tsinghua University in He is currenty a PhD student in the University of North Caroina at Charotte, majoring in computer science. His current research focuses on wireess networks, ad hoc and sensor networks, and agorithm design. Shaojie Tang has been a PhD student of Computer Science Department at the Iinois Institute of Technoogy since He received BS degree in Radio Engineering from Southeast University, China, in His current research interests incude agorithm design and anaysis for wireess ad hoc network and onine socia network. Yu Wang is an Associate Professor of Computer Science at the University of North Caroina at Charotte. He received his PhD degree (2004) in computer science from Iinois Institute of Technoogy, his BEng degree (1998) and MEng degree (2000) in computer science from Tsinghua University, China. His current research interests incude wireess networks, ad hoc and sensor networks, and agorithm design. He is a recipient of Raph E. Powe Junior Facuty Enhancement Awards from ORAU. He is a member of ACM and senior member of IEEE.