Distributed Multicast Tree Construction in Wireless Sensor Networks

Transcription

1 Distributed Multicast Tree Construction in Wireless Sensor Networks Hongyu Gong, Luoyi Fu, Xinzhe Fu, Lutian Zhao 3, Kainan Wang, and Xinbing Wang Dept. of Electronic Engineering, Shanghai Jiao Tong University, China. Eail:{ann, Dept. of Coputer Science, Shanghai Jiao Tong University, China. Eail:{yiluofu, fxz4, 3 Dept. of Matheatics, Shanghai Jiao Tong University, China. Eail: Abstract Multicast tree is a key structure for data disseination fro one source to ultiple receivers in wireless networks. Miniu length ultica odeled as the Steiner Tree Proble, and is proven to be NP-hard. In this paper, we explore how to efficiently generate iniu length ult wireless sensor networks (WSNs), where only liited knowledge of network topology is available at each node. We design and analyze a siple algorith, which we call Toward Source Tree (TST), to build ulticast trees in WSNs. We show three etrics of TST algorith, i.e., running and energy efficiency. We prove that its running tie is O( n ), the best aong all existing solutions to our best knowledge. We prove that TST tree length is in the sae order as Steiner tree, give a theoretical upper bound and use siulations to show the ratio be only.4 when nodes are uniforly distributed. We evaluate energy efficiency in ters of essage coplexity and the nuber of forwardin prove that they are both order-optial. We give an efficient way to construct ulticast tree in support of transission of voluinous data. I. INTODUCTION Wireless Sensor Network (WSN) is a network of wireless sensor nodes into which sensing, coputation and counication functions are integrated. Sensors are self-organizing and deployed over a geographical region []. Multicasting, i.e., one-to-any essage transission, is one of the ost coon data transission patterns in WSNs. Tree is the topology for non-redundant data transission. To enable efficient ulticast, ulticast tree has been proposed and widely used. It has not only been used for ulticast capacity analysis in wireless networks [3] [5], but in practice, ulticast supports a wide range of applications like distance education, ilitary coand and intelligent syste [6]. Many researchers have been working on constructing efficient ulticast trees [7] [9], []. They have proposed a nuber of algoriths so as to iniize the routing coplexity as well as achieve the tie and energy efficiency (for details, please refer to the next section), but ost of the did not focus on an iportant perforance easure: the tree length. This is a critical etric since larger tree length clearly results in longer delay. To enable the essages to be forwarded farther, sensors have to increase their transission power, A previous version of this work appears in MobiHoc 5 []. Copyright (c) 4 IEEE. Personal use of this aterial is peritted. However, perission to use this aterial for any other purposes ust be obtained fro the IEEE by sending a request to pubs-perissions@ieee.org. causing ore energy consuption as well as ore serious interference to neighboring nodes. Besides, as the transission distance increases, the essages suffer fro higher probability of transission failure. ecently, GEographic Multicast (GEM), inspired by Euclidean Steiner Tree, was proposed for routing in dense wireless networks []. Forally, given a network G = (V, E), the weight of each edges, and a set of terinals S V, the Steiner Tree Proble is to find a tree in G that spans S with the iniu total weight [3]. This proble has been proven to be NP-hard [4], and has not been visited for a long tie. Forer fors of its approxiate ipleentation were not appropriate for constructing ulticast trees in WSNs for various reasons (details will be discussed in the following section.) In GEM, the authors took the first step to utilize the Steiner tree for constructing ulticast trees in WSNs, achieving routing scalability and efficiency. This approach can potentially reduce the tree length, but this very siple for of utilization only considers the hop count in an unweighted graph, but not the total length of the ulticast tree in a weighted graph. Further, as for the perforance analysis, the statistical properties were all under the assuption that all nodes are uniforly distributed, aking it difficult to tell its efficiency under a realistic network environent. In this paper, inspired by taking the advantage of the Steiner tree property, we design a novel distributed algorith to construct an approxiate iniu-length ulticast tree for wireless sensor networks, aiing at achieving energy efficiency, ease of ipleentation and low coputational coplexity, at an affordable cost on the sub-optiality of tree length. In what follows, we call our design Toward Source Tree Algorith, or TST for short. We quantitatively evaluate TST algorith perforance under general node distribution, and show that TST has the following satisfactory etrics: Its running tie is O( n ), the best aong all existing solutions for large ulticast groups. Its tree length is in the sae order as Steiner tree, and siulation shows the constant ratio between the is only.4 with uniforly distributed nodes. Its essage coplexity (which we will forally define later) and the nuber of nodes that participate in forwarding are both order-optial, yielding high energy efficiency for sensor networks.

2 Its theoretical properties and distributive nature render it suitable for sensor network architecture and protocol design for perforance iproveent. The rest of paper is organized as follows. Section II states related work. In Section III, we introduce our network odel. In Section IV, we present our Toward Source Tree Algorith to construct a ulticast tree. In Section V, VI and VII, we evaluate the perforance of our algorith ainly fro three aspects: ulticast tree length, running tie and energy efficiency separately. In Section VIII, we use extensive siulations to further evaluate the perforance and also illustrate how to apply our TST algorith to practical applications. We conclude this paper and present discussions of future works in Section IX. II. ELATED WOK We review related works in three categories: the application of iniu ulticast tree, ulticast routing and the approxiate Steiner tree. Miniu-length Multicast Tree. The ost significant advantage of iniu ulticast tree in wireless sensor networks is energy efficiency. As sensor nodes are often powered by batteries that drain rather fast and are difficult to replace, energy conserving is extreely crucial in sensor networks. Furtherore, nodes in sensor network consue ost of its energy in counication [8]. Hence, iniu ulticast tree-based routing is desirable in any cases. Specifically, it is extensively used in the following two applications in sensor networks - user query and data aggregation. As wireless sensor networks are ostly data centric [9], users have to query for inforation and disseinate it in the network. To spread the query in a network as energy-efficient as possible, we need to build a iniu ulticast tree and route the data following the trees topology []. To achieve this, soe existing works apply a Steiner tree-based approach []. Data aggregation is to integrate the data fro different sources and route for eliinating redundancy. It saves energy by reducing the nuber of transissions []. Miniu-length tree topology is a widely used technique to solve the iplosion probles in data centric routing []. In data aggregation, the routing pattern of a sensor network is siilar to a reverse ulticast tree []. Achieving the optial data aggregation, i.e., constructing the iniu ulticast tree, is also treated as a Steiner tree proble []. Besides iprove energy efficiency and extending network lifetie, routing on a iniu ulticast tree also has underlying erits, such as indirectly reducing network delay [3]. Since the total length of tree is iniized, it is obvious that the path won t be too long between the source and any destination. Multicase tree construction. Many studies focus on ulticast routing in wireless networks, and useful techniques for routing have been proposed in WSN. Sanchez et al. proposed Geographic Multicast outing (GM), a heuristic neighborhood selection algorith based on local geographic inforation [7]. Later Park et al. [4] cobined distributed geographic ulticasting with beaconless routing. In Localized Energy-Efficient Multicast Algorith (LEMA), forwarding eleents apply the MST algorith locally for routing [8]. Dijkstra-based Localized Energy-Efficient Multicast Algorith (DLEMA) finds energy shortest paths leading through nodes with axial geographical advance towards desired destinations [9]. Hierarchical geographic ulticast routing (HGM) tries to cobine the advantages of geographic ulticast routing (GM) and hierarchical rendezvous point ulticast (HPM) []. It achieves transission ties close to GM, encoding overhead close to HPM, and good packet delivery ratio in siulations. Our concern for delay and the running tie of ulticast tree construciton is orthogonal to the evaluation etrics in HGM. In suary, few works have considered iniizing the distance of ulticast routing or providing coprehensive quantitative analysis theoretically on the perforance of routing policies. Approxiate Steiner Tree. Shortest Path Heuristic (SPH) and Kruskal Shortest Path Heuristic (KSPH) add new nodes to existing subtrees through the shortest path [5]. Average Distance Heuristic (ADH) joins subtrees that contain receivers by a path passing non-receivers with inial average distance to existing subtrees [6]. Santos et al. pushed forward distributed dual ascent (DA) algorith, achieving good perforance in practice [7]. The coparison of these algoriths with our TST algorith is shown in Table.. These algoriths were proposed for point-to-point networks. In this paper, we consider the Steiner Tree Proble in wireless sensor networks that are broadcast in nature. In addition, each node has liited coputation and storage capability. Devices are usually battery-powered, therefore energy-efficiency is of great iportance. Due to these specific features and requireents, existing algoriths for PP are not suitable for WSN. To su up, there have been extensive existing works focusing on ulticast tree construction or the approxiate Steiner tree probles, but we have not found a perfect adoption of Steiner tree into constructing ulticast trees. III. NETWOK MODEL Let us first use atheatical odel to capture a wireless sensor network. We assue the network consists of n nodes in total (or we call the network size is n), distributed independently and identically in a unit square. Each node is assigned with a unique identifier to be distinguished fro others. Each tie when a source needs to transit essages, it chooses receivers randoly. In other words, is the nuber of nodes that participate in a ulticast transission, or we call it the ulticast group size. For our statistical analysis, we focus on the dense network and large ulticast group where and n are both very large, and n. This is particularly suitable to describe a wireless sensor network. The geographical distribution of nodes is described by a density function f(x) where x is the position vector. Here we allow x to be of any diension; in the rest of this paper we let it be a two-diensional vector for ease of presentation, but it does not hurt any generality. We assue f(x) is independent of n and. We also assue that < ɛ f(x) ɛ where ɛ and ɛ are both constants, i.e., a node has a positive probability to be located in any region of this area.

3 3 TABLE.: Coparison of Distributed Algoriths for Approxiate Steiner Construction Algorith Expected tree length Expected tie Expected essages Assuptions SP H [5] -approxiation O(» n ) O(n) Known shortest paths; KSP G [5] Steiner tree, O( O(» n Point-to-point network. ) ADH [6] O(» n ) O(n + n) DA [7] O( ) O(n ) O(n ) Unknown shortest paths; Applied in point-to-point network. T ST O( ) O( n ) O(n) Unknown shortest paths; Applied in wireless network. To ensure the connectivity of (» the whole ) network, we set the transission range r = Θ n [5]. For all nodes, r is the sae and fixed. We assue that two nodes u and v can counicate with each other directly if and only if the Euclidean distance between the, d uv, is no larger than r. Every node can obtain its own geographical location, e.g., via the Global Position Syste (GPS). However, nodes do not know the exact location of other nodes until they receive essages containing that piece of inforation. n r r c L V L M TABLE 3.: Notations and Definitions the total nuber of nodes in the network the nuber of receivers transission range coverage range of receiver searching the length of teporary tree the length of ulticast tree IV. ALGOITHM In this section, we describe our Toward Source Tree algorith in detail. This algorith consists of three phases. In the first phase, the source broadcasts a essage and wakes up all receivers it chooses. In the second phase, every receiver chooses the closest neighboring receiver that has shorter Euclidean distances to the source node than the receiver node itself, and then a teporary tree can be established aong all receivers. However teporary tree is a virtual topology since ulticast group ebers ay not be connected directly to each other given liited transission range. We select appropriate relays to keep these ebers connected while controlling the tree length. However, till the end of this stage cycles ight exist. Hence we eliinate these cycles to further reduce tree length and avoid redundant transissions in the third phase. In what follows we describe the process in detail, and we will use an exaple to illustrate how to generate such a tree at the end of this section. A. Phase : Identifying eceivers Each node has a label indicating its role in the ulticast tree: S stands for the source and for receivers. In this phase, a essage containing all receivers identifiers is sent fro the source so that all nodes in the network can be aware whether they are receivers. Upon receiving this essage, receivers then wake up, label theselves with and be ready to participate Algorith Neighbor equest fro Multicast Mebers : for all receiver in a ulticast group do : the nuber of request session: k 3: coverage range: r c r 4: tie out interval: T Θ ( k ) 5: set the node sequence as {} 6: total hop: H 7: path length: p 8: forward the request essage to its neighborhood 9: while no response is received when tie is out for the k th request session do : k k + : r c k r : T k+ T k 3: set the node sequence as {} 4: H 5: p 6: forward the request essage to its neighborhood 7: end while 8: end for in the ulticast routing. The source will also specify its own location in this essage. This step is necessary for ulticast routing since no one except the source knows which nodes the essages are destinated for. In this phase, the broadcast inforation will notify the nodes who are selected into the ulticast group, and all receivers will be awakened. B. Phase : Connecting All eceivers In this phase, we first build a teporary tree consisting of only the ulticast group ebers, and then find the iniuhop shortest path between each pair of ebers that are directly connected in the teporary tree. All ulticast group ebers will be connected with the newly added relays. Step : Searching eceivers in the Neighorhood In this step, each ulticast eber chooses an appropriate neighbor to connect to. The neighboring eber selection criteria is: each eber chooses the closest one fro the set of ebers that have shorter Euclidean distances to the source node than this node itself. If no such neighboring eber can be found, then this ulticast eber directly connects to the source. When a eber tries to contact its neighbor ebers, it is regarded as the sender that sends request essage. Its

4 4 for is: <sender id, sender location, location of previous hop, coverage range r c, node sequence, total hop H, path length p>. Sender id is used to identify the ulticast ebers sending the request essage, and path length can be updated with the location of previous hop and current hop. The coverage range r c sets the range within which the ulticast eber searches for its neighboring ebers. The Euclidean distance between the sender and current node can be calculated with sender location, and essages will be discarded if the distance is larger than r c. Node sequence records in order the nodes through which this essage has passed, which acts as a guide for response fro neighboring receivers so that the response can be routed via the available path. The hop count H is the nuber of hops the essage has passed through, and p is the path length the essage have been through when it reaches the current node. In each search session, the eber broadcasts the request essage within search coverage range. The sender sets an appropriate tieout interval. Once the sender receives replies fro neighboring nodes, the search session terinates. Then it enters step. However, if tie runs out and no reply is obtained, it eans that no appropriate neighboring ebers are found. The sender then doubles its search range and initiates another search. In Algorith, we show how a ulticast group eber connects to their neighboring ebers or the source. A node ay receive ore than one request essage fro the sae sender. If it is within coverage range, it will choose the one with the fewest hops aong all the essages. If the nubers of hops are the sae, it picks out the essage with the shortest path length. Then it odifies this essage. It adds itself to the node sequence, increases the hop count by and calculates new path length given the location of the previous hop. With these inforation updated, it forwards the essage. Algorith describes how nodes deal with request essages in detail. When a ulticast eber finds it closer to the source than the sender of the request essage, it ight be chosen as the neighbor by the sender. Therefore, this eber will choose a path to the sender and respond with the respond essage. The for of the respond essage is: <sender id, respondent id, node sequence, total hop H, path length p>. The respond essage can be routed with the path inforation provided by the node sequence. Step : Connecting to the Nearest Neighbor With respond essages, every eber selects the closest neighbor. Once a neighbor is chosen, the connect essage is forwarded via the iniu-hop shortest path. The connect essage is used to establish a connection between nodes in the ulticast group. At the sae tie, all relay nodes on the iniu-hop shortest path record this pair of ebers, previous hop and the next hop on the path. When all receivers send the connect essage, a teporary tree aong all utlicast group ebers including the source is constructed. C. Phase 3: Eliinating Cycles In Phase, we construct a teporary tree ade up of ulticast group ebers. However, when other nodes are Algorith equest Forwarding : for all node u receiving request essage do : dist = location of u - sender location 3: if dist < r c then 4: add u to node sequence 5: H H + 6: newdist = location of u - location of previous hop 7: p p + newdist 8: forward the request essage to its neighborhood 9: if u is in the ulticast group then : nodesourcedist = location of u - source location : sendersourcedist = sender location - source location : if nodesourcedist < sendersourcedist then 3: send respond essage back to the sender 4: end if 5: end if 6: end if 7: end for added to it as relays, cycles ight be fored. In particular, when paths connecting different pairs of ulticast ebers share the sae relay nodes, such node ay receive redundant inforation, which indicates that cycles coe into being. Therefore, we check the existence of cycles in this phase and eliinate the if any. Suppose a node u acts as a relay for k (k > ) pairs of nodes in the ulticast group, which are directly connected in the teporary tree, denoted as (, ), (, ),..., ( k, k ). Let us assue that in each pair, i is closer to source than i ( i k). A relay stores its previous and the next hop of the path fro i to i, and they are denoted as P H i and NH i respectively. Then it chooses one pair randoly, say, ( j, j ) and keeps the inforation: ( j, j, P H j, NH j ). For other pairs ( i, i ) where i j, the relay odifies their inforation as ( j, i, P H j, NH i ). Define a set Q, where Q = {q q =< i, i, P H i >, i j }. Last, it sends Eliinate essage Q and its previous hops delete unnecessary edges accordingly. In Algorith 3, we show how to wipe out the cycles. D. Proof of Tree Topology The previous subsections describe how we can connect utlicast group ebers using our TST algorith. Now let us prove that the topology constructed by TST algorith is exactly a tree. We first show that teporary tree fored in the second phase has a tree topology in Lea. But relays are added into the teporary tree to connect receivers, which ight result in the existence of cycles. In Lea we show that cycle eliination can in fact guarantee the tree topology. Lea : The teporary tree connecting receivers has a tree topology. Proof: Assue that each wireless node is identified with a unique label n i. We order these nodes based on their distance fro the source node, and we have an ordered set:

5 5 {n, n,..., n } such that n j is closer to the source than n i for j < i. It is easy to show that n i can only connect to node in the set {n,..., n i } according to our construction algorith. Such ordering guarantees that the generated topology is acyclic. Besides, every ulticast eber tries to connect to another eber that is closer to source, so every eber can find a path to the source. Naturally the generated topology is also connected. A connected and acyclic graph is a tree. S Algorith 3 Cycle Eliination : for all node u engaged in paths between k pairs of ebers do : choose an integer j such that j k 3: for all i such that i k and i j do 4: forward Eliinate essage Q i =< i, i, P H i > 5: end for 6: end for 7: for all node w receiving Eliinate essage Q i do 8: if w is exactly P H i then 9: if w is not in the ulticast group then : P H i previous hop of w on path ( i, i ) : forward the odified Q i to the previous hop : eliinate inforation: ( i, i, P H i, NH i ) 3: end if 4: end if 5: end for Based on Lea, we have the following lea: Lea : The topology connecting nodes generated by TST algorith is a tree that spans all ulticast group ebers. Proof: The existence of cycles eans that soe nodes in the ulticast tree ay receive redundant essages, i.e., soe nodes have ore than one previous hop. For these nodes, they send Eliinate essages and ensure that they have only one previous hop. When all nodes in the ulticast tree have only one previous hop, no cycle exists. Multiple previous hops also indicate that ultiple paths ay exist between two nodes. Once soe previous hops are unnecessary, the paths involving these hops can also be eliinated. Thus Algorith 3 can eliinate these unnecessary paths, and this copletes our proof. E. Illustration We use an exaple to illustrate our TST algorith in Figure 4.. Nodes are distributed in the unit square as shown in Figure (a). Solid nodes represent source nodes labeled by S, or ulticast ebers labeled by. The hollow nodes can be chosen as relays. The first step is to build a teporary tree spanning all ulticast ebers. The dashed lines denote virtual connections between two ebers. Then nodes on the inial-hop shortest path are engaged as relays between two neighboring ebers. They for the topology as shown in Figure (b). Note that there exists a cycle arked with dotted rectangular box. The last step is to eliinate unnecessary edges as is done in Figure (c). Finally we obtain the ulticast tree as is shown in Figure (d). (a) Building a teporary tree spanning ulticast group ebers (b) Adding relay nodes (c) Eliinating redundant edges and (d) Constructing the ulticast tree aintaining the topology of tree with relays added Fig. 4.: Steps of the TST algorith V. LENGTH ANALYSIS The previous section described our Toward Source Tree algorith. In the next three sections, we will discuss its perforance in ters of tree length, tie coplexity, and energy efficiency. In this section, we discuss the length of TST. We first obtain the length of teporary tree, first assuing unifor distribution nodes and then extending to a general setting. Next we explore the length of inial-hop path that connects two receivers. Cobining the length of teporary tree and the path, we can derive the upper bound for the ulticast tree length. A. Teporary Tree in Unifor Distribution We start by discussing the tree length of the teporary tree. Lea 3: Assue nodes are uniforly distributed in a unit square. The expected length of the teporary tree spanning receivers is upper bounded by c, where c = 5.6. Proof: See Appendix A. B. Teporary Tree in General Distribution Based on the conclusions of tree length in unifor distribution, we further study the case that nodes are non-uniforly distributed. We partition the unit square into k sall squares, where = k +γ and < γ <. We construct trees aong nodes in each square, and then connect nodes in different cells so that all nodes in the network are connected. For each square, the source is outside the square and we still apply the TST algorith for the tree construction. Lea 4 can estiate the intra-square edge length, and we study the inter-square edge

6 6 length in Lea 5. With both inter- and intra- square edge estiation, we derive upper bound for teporary tree length in general distribution. S S y Fig. 5.: Approxiate neighbor region when the source is located outside the square Lea 4: (Intra-square edges) Let nodes be independently distributed in a unit square with density function f(x). The source S is located outside the square. Let each node connect to the closest neighbor that has shorter Euclidean distance to S than the node it self. If no such receiver exists, it does not connect to other nodes. A tree can be constructed aong nodes, and the expected length of such a tree is upper bounded by c, where c = 5.6. Proof: For those nodes that not closest to S, they can always find another node to connect to. For the node that is closest to S, it will be connected to by other nodes. We can prove that the topology fored by nodes is exactly a tree with Lea. We denote this tree as T. There are two differences of this lea fro Lea 3. One is that the source is located outside the square, and the other is that a node won t connect to others when it can t find another one that has shorter Euclidean distance to the source. Now we find a point S that is closest to S in the boundary of square region, as is shown in Figure 5.. With S as the source, a teporary tree as entioned in TST algorith can be established spanning all nodes in the network. We denote the teporary tree as T. In the following we deonstrate that the tree length of T can be used to estiate the upper bound of length of T. For a node, we use N to denote the regions where nodes ight be selected by as a its neighbor. We use a rectangular region as approxiate neighbor region N, and N N. The approxiate neighbor region is the region arked with parallel lines in Figure 5.. We use the ethod adopted in the proof of Lea 3 to estiate the length of T. It can be observed that the approxiate neighbor regions are the sae in both cases that we take S as the source and that we take S as the source. There are soe details that need to be clarified. Firstly, when we only consider the nodes in approxiate neighbor region, the estiated tree length is larger than actual length, because we ignore the nodes that are closer to node. Secondly, if neighbors exist in the x approxiate region, the estiations of edge length are the sae for both T and T. Thirdly, if no neighbor is found in approxiate region for a node, we assue that it does not connect to others in T but it connects to the source in T in our calculation. Fro the analysis above, we can conclude that length of T is upper bounded by the length of T. Also recall that in our proof of Lea 3, and 5.6 is the upper bound for teporary tree length wherever S is. In suary, we can directly use estiated tree length in Lea 3 as the upper bound of the tree length of T. This copletes our proof. Lea 5: (Inter-square edges) Let nodes be independently distributed in a unit square with density function f(x). The unit square [, ] [, ] can be partitioned into k square cells with edge length of k, where = k +γ and < γ <. The length of inter-square edges connecting k cells in the unit square is o( ). Proof: We know that the expected nuber of nodes in each square cell is greater than k ɛ = k γ ɛ. To copute the inial distance between two nodes in adjacent squares, we partition the cell with edge length of k into saller grids with edge length of k, where α > α. We clai that if α γ <, the inial length between two adjacent cells is in an order of o Ä ä k. This coes fro the observation that we can connect adjacent cells by connecting nodes in adjacent grids whose edge length is k, as is shown α in Figure 5.. In this figure, the yellow and the black squares are two adjacent cells with edge length of k. The blue grids contained in the are the saller squares with edge length of k. Green lines are used to show that nodes in the adjacent α grids are connected. As we can see fro Figure 5., for two adjacent cells with edge length of k, k α / pairs of nodes in adjacent grids ight exist. Denote P as the probability that a node exists in a grid with edge length of /k α. Since the area of each square is very sall, we can regard nodes in the sae square uniforly distributed. We have P = ( ɛ ) k kα. Denote P as the probability that nodes exist in both of the adjacent grids. P = P. There are k α / pairs of nodes in adjacent grids, and we denote P as the probability that at least one pair exist. We have P = P kα /. Hence we have P = ( ( ( ɛ ) k kα ) ) kα / (5-) In order to let k squares connected by inter-square edges, it should hold that P k. Therefore, we need the following condition k k α / ( ( ɛ ) k kα ). (5-)

7 7 The expression that r (k) r (k) eans that r (k)/r (k) as k. Condition (5-) is equivalent to condition (5-3). log k, (5-3) k /+γ+α ɛ kα Condition (5-3) can be satisfied when α < γ +. With < α < γ +, we can evaluate P. By (5-) it can be verified that P exp( ɛ k / α+γ ɛ log k/ α ). (5-4) It is easy to show that P k with the expression (5-4), which eans such pairs of nodes exist for all adjacent cells with high probability. Since ( P ) is exponentially decaying to zero, the expectation of total path length needed to connect k cells is k α P k + k / k( P ) k α = o(k / ) (5-5) Due to the fact that k = o(), the expected path length for inter-square connection is in the order of o( ). Fig. 5.: Inter-square edges between nodes in adjacent square cells Lea 6: Let nodes be independently distributed with density function f(x). The expectation for the total length of teporary tree E[L V ] is saller than c. We have E[L V ] c x [,] f(x)dx, where c 5.6. Proof: See Appendix B. C. Path With Minial Hops eceivers are connected by the inial-hop path. In this part, we study the relationship between the path length and Euclidean distance between two nodes. Lea 7: Let n nodes be independently and identically distributed over [, ] [, ] with distribution function f(x). Suppose that the Euclidean distance between two nodes u and v is x. The following properties hold: (a) The expectation of fewest relays that are needed to connect u and v converges to x r as n approaches ; (b) The length expectation of the path connecting uv and involving the fewest relays converges to Euclidean distance x. Proof: See Appendix C. D. Multicast Tree We divide the [, ] [, ] network region into k squares. In each square, we construct a tree and connect nodes with intrasquare edges. Adjacent squares are connected by the intersquare edges. All nodes are connected by intra- and intersquare edges, and they can be used to estiate the tree length. Theore : Let n nodes be independently and identically distributed in a unit square and their distribution satisfies the density function f(x). We construct a ulticast tree spanning receivers as well as the source with TST algorith. When and n are both very large, the expected length of the tree is upper bounded by c x [,] f(x)dx, where c = 5.6. Proof: Denote e i,j as the edge connecting eceivers i and j in the teporary tree T V, l i,j as the length of the inial-hop path between the two receivers. Since redundant edges will be eliinated, E(L M ) E(l i,j ). And the path length e i,j T V converges to Euclidean distance as network size goes to according to Lea 7. So we have: E(L M ) c x [,]» f(x)dx. (5-6) eark: We derive an upper bound for the Toward Source Tree, but it is not a tight bound. In Section VIII, we will show that TST algorith has even better epirical perforance than our theoretical bound. Lea 8: Suppose X i, i <, are independent rando variables with distribution µ having copact support in d, d. If the onotone function ψ satisfies ψ(x) x α as x for soe < α < d, then with probability li n n (d α)/d M(X X,..., X n ) = c(α, d) f(x) (d α)/d dx d (5-7) Here f denotes the density of the absolutely continuous part of µ and c(α, d) denotes a strictly positive constant which depends only on the power α and the diension d [6]. Given a graph with soe nodes and edges, building a inial length tree spanning a subset of nodes with relays appropriately added is forulated as Steiner Tree Proble. If no relay nodes are allowed, then the tree with inial length is called inial spanning tree. However, Steiner tree can only optiize tree length by a constant ratio copared with the inial spanning tree. Lea 9: Let P be a set of n points on the Euclidean plane. Let l s (P ) and l (P ) denote the lengths of the Steiner iniu tree and the iniu spanning tree on P respectively. The inequality holds: [8] 3 l s (P ) l (P ) (5-8) Cobining the two leas above, we can conclude that the length of Steiner tree spanning receivers is: 3 L ST c [,]» f(x)dx (5-9)

8 8 Here c is the constant equal to c(, ) entioned in Lea 8. oberts estiated that c =.656 [7]. Fro (5-6) and (5-9), we prove that the length of Toward Source Tree is in the sae order as that of Steiner tree, and the difference between the is only a constant ratio no larger than. eark : Many works have shown the lengths of any well-designed spanning tree consisting of nodes can reach the order of O( ), but the order-optiality of our Toward Source Tree in tree length is not a trivial result. TST is not a siple spanning tree of ulticast group ebers, since other relay nodes ust be selected carefully and be involved in ulticasting due to the liited transission range of wireless sensors. Hence iniu ulticast tree is forulated as a Steiner tree proble instead of iniu spanning tree proble. Constructing a iniu spanning tree takes polynoial tie, while constructing a Steiner tree is proved to be NPhard. Asyptotic tree length of TST is a result based on quanlitative analysis of this hard proble in graph theory. As far as we know, few works have given asyptotic length of their ulticast trees. One ay also doubt that when the network is dense, we can choose infinitely any relay nodes such that the path length between two ulticast ebers approaches their Euclidean distance. In this way, the tree can also reach the order-optiality in length. However, we should notice that it would bring terribly long delay and large energy consuption since too any extra nodes are involved in ulticasting. The technique to balance the tree length and delay is that we choose iniu-hop shortest path between ulticast ebers, as is described in Section IV. The iniu length ulticast tree has potential benefits of energy efficiency and delay reduction. The asyptotic length of TST not only indicates its order-optiality, but also shows that TST is quite a good approxiation of Steiner tree since the approxiation ratio is no larger than. VI. UNNING TIME ANALYSIS Tie efficiency is another iportant aspect to evaluate the quality of ulticast routing algoriths. In practice, it is expected that the ulticast tree can be constructed with sall tie costs. Now let us derive the tie coplexity of TST. Theore : Let n nodes be independently and identically distributed in unit square. The running tie of TST algorith is O( n ). Proof: There are three serial phases in TST algorith, so we discuss the tie cost of each phase one by one. In Phase, the essages containing location inforation of the source are broadcast in the network. The furthest distance between the source and another node is O(), so at ost O ( ) r relays are needed for a essage to reach one node. In expectation, there are πr ɛ n = O ( nr ) nodes within transission range of a node and hence a node has to wait for O ( nr ) tie slots to transit a essage. The tie needed for Phase is: O Ç å E(t ) O (nr). (6-) r In Phase, the doinant tie cost is searching for neighboring receivers. In the k th search session, the coverage range is k r. We need O ( k) relays to forward request essages fro one receiver to any other nodes within its search coverage range. Since the coverage range does not exceed, the nuber of search sessions cannot be ore than log r. Ö log è r E(t ) O i nr O(nr). (6-) i= In Phase 3, the worst case is that relays on the path whose length is O() for cycles. Tie for cycle eliination is Å ã E(t 3 ) = O r nr = O(nr). (6-3) The total running tie is E(t) = i=3 Å ã n O which copletes our proof. Approxiation ratio of tree length Infeasible region { i= E(t i ), so we have E(t) O( n ). (6-4) n O(n/log n ) O(n log n) O(c ) Coplexity Fig. 6.: elationship between tree length and tie coplexity eark: For any algorith to construct a ulticast tree aong a group of nodes, broadcast in Phase is necessary. Since no node has a knowledge of the ulticast group except the source, such inforation has to be forwarded to every node in the network so that they can know whether they should participate in ulticasting. The lower ) bound of tie for ulticast tree construction is O (» n. Since TST achieves the tie coplexity upper bounded by O( n ), the inial tie cost to construct a ulticast tree is also upper bounded by O( n ). Hence the tie coplexity of TST algorith shares the sae upper and lower bounds as the inial tie cost, and the ratio between these two bounds is only O(). The length of ulticast trees have a great influence on counication quality in ters of transission delay and wireless interference. Construction of iniu-length trees is an NP-hard proble, and takes exponential tie. Approxiate algoriths achieve larger tree length with lower tie coplexity. Now we explore the relationship between tree length and tie coplexity in Figure 6.. Since the lower bound of

9 9 (» ) tie needed for ulticast tree construction is O n, (» ) the region with tie coplexity saller than O n is infeasible. Accurate solution to Steiner tree proble achieves the approxiation ratio of at the cost of exponential tie, and our algorith achieves the ratio of. The approxiation ratio of other algoriths like those in Table. approaches but they have larger tie costs. VII. ENEGY EFFICIENCY Energy is a priary consideration in wireless sensor networks since sensors are battery-powered and their energy is liited. We consider the following factors: ) the energy consued to construct such ulticast trees; and ) the energy needed to send essages along the tree constructed by this algorith. The forer one is usually easured by the aount of exchanged essages to run distributed routing algoriths; and the latter directly depends on the nuber of nodes participating in the transission. We focus on both aspects. A. Message Coplexity The following theore quantifies the essage coplexity in TST. Theore 3: Let n nodes be independently and identically distributed in the unit square. The essage coplexity of TST algorith is O(n). Proof: See Appendix D. eark: Since each node needs a essage telling the whether they are chosen as receivers, the lower bound of essage coplexity is O(n). Hence TST algorith is an order-optial solution in ters of essage coplexity. B. Nuber of Forwarding Nodes Since the transission range is fixed, the nuber of transitters in the tree deterines the energy consuption for inforation propagation. We evaluate the nuber of forwarding nodes in this subsection. Theore 4: Let n nodes be independently and identically distributed in the unit square. The nuber of forwarding nodes in the ulticast tree is N T ST = (» Θ n Θ(), ), = O Ä n = ω Ä n ä ; ä (7-). When = O (n/ ), the nuber of forwarding nodes is order-optial. Proof: Let T V be the virtual tree, e i,j be an edge in the virtual tree connecting two receivers i and j, and d i,j be the Euclidean distance between the. When is sall, relay nodes for the doinant part of the forwarding nodes in our ulticast tree. The total nuber of transitting Ç nodes, ån T ST d in the Toward Source Tree is: N T ST = Θ ij r e i,j T V = Θ Ä ä r. As grows larger, receivers are close to each other and thus fewer relay nodes are added. Therefore, receivers are doinant in the ulticast tree, N T ST = Θ(). We should discuss the nuber of forwarding nodes in two cases, and there exists a critical value for that deterines in which case it should be discussed. The critical value satisfies: Θ Ä ä c r = Θ(c ), so c = Θ ( ) r. Denote N in as the inial nuber of relay nodes that are engaged in propagating the essages fro one source to receivers. [3] gives the lower bound of N in under the assuption that all nodes are uniforly distributed. Now we use its ethod and explore N in in the case of general distribution. When is sall, the distance between two receivers is large copared with the transission range. N in = Ω Ä ä r. This lower bound is achievable with our algorith, so N in = Θ Ä ä r. When is very large, there exist any receivers within the transission range of one node, so that one transission can deliver essages to a large nuber of receivers. In this case, we only need to choose a connected doinating set fro receivers, and N in is exactly the size of iniu connected doinating set. We will give the definitions of both connected doinating set and iniu connected doinating set. Definition (Connected doinating set): D is the connected doinating set of a graph G if and only it satisfies two properties: (a) Any node in D can reach any other node in D by a path that stays entirely within D. (b) Every vertex in G either belongs to D or it is adjacent to a vertex in D. Definition (Miniu connected doinating set): MD is the iniu connected doinating set of graph G if MD is the connected doinating set containing the sallest nuber of nodes. We still need to discuss N in exists a critical value d, and Θ( d ) = Θ Ä d r Θ Ä n ä. N in = (» Θ n Ω Ä ä n, in two cases. ä There also, so d = ), = O Ä n = ω Ä n ä ; ä (7-). Fro (7-) and (7-), we can find when = O Ä n ä, the nuber of forwarding nodes in the ulticast tree is optial in order sense. eark:when = ω Ä n ä, the nuber of forwarding nodes in TST tree ay not be order-optial. However, in graph theory, finding the iniu connected doinating set of a given graph is proved to be NP-coplete [9]. And it also requires global inforation of network topology. So we consider it an acceptable sacrifice of energy to achieve the feasibility and tie-efficiency in practice. VIII. SIMULATIONS AND APPLICATIONS We first perfor extensive siulations to evaluate the epirical perforance of Toward Source Tree algorith, in ters of the length of the ulticast tree, essage coplexity and the nuber of forwarding nodes engaged in the tree, and then present concrete exaples of how Toward Source Tree algorith can be applied to realistic scenarios. In the siulations, we ainly consider two coon distribution patterns: unifor distribution and noral distribution.

10 Steiner Tree Toward Source Tree Steiner Tree Toward Source Tree 8 Tree Length 5 Tree Length (ulticast group size) (ulticast group size) (a) Unifor distribution (b) Noral distribution Fig. 8.: Tree length coparison 8 n= n=6 n= 8 n= n=6 n= Messages(x) 6 4 Messages(x) (ulticast group size) (a) Unifor distribution Fig. 8.: Message coplexity 5 5 (ulticast group size) (b) Noral distribution 8 8 Node nuber 6 4 Node nuber (ulticast group size) (a) Unifor distribution (ulticast group size) (b) Noral distribution Fig. 8.3: Nuber of forwarding nodes A. Perforance Evaluation ) Unifor Distribution: We first consider nodes are uniforly distributed» in a unit square and transission range is set to be r = n. We explore the effect of ulticast group size on the tree length. Assuing that the network size is fixed as, we obtain the lengths of the Stenier tree and TST tree when the value of varies. The length of the Steiner tree can be obtained via NewBossa in [3]. Two curves in Figure 8.(a) describe the relationship between and the length of the Toward Source Tree as well as the Steiner Tree. It is shown that the length of TST tree is larger than that of the Steiner Tree but quite close to it. According to siulation statistics, the ratio of the tree length achieved by the two algoriths is.4 on average. When nodes are uniforly distributed, Toward Source Tree is a good approxiation of the Steiner Tree. Then we evaluate the essage coplexity in the construction of TST tree, and explore the relationship aong the network size n, the ulticast group size and essage coplexity. We set the network size n =, 6, respectively, and record the quantity of exchanged essages when ulticast group size varies. Different curves correspond to different network sizes in Figure 8.(a). As can be seen in the figure, the quantity of exchanged essages increases with the ulticast group size as well as the network size. It is quite intuitive that the larger network size can result in ore exchanged essages. Since the transission power necessary to aintain the connectivity is less in dense networks than in sparse networks, ore relays are engaged in

11 ulticasting as the network size increases. Hence essages used to contact nodes and inquire routing inforation becoe ore. We find that ore essages are exchanged when we fix the network size and add ore ulticast group ebers. This result is not so intuitive. On the one hand, ulticast group ebers becoe closer to each other when the ulticast group size increases, so they need to search for the appropriate neighbors in a saller coverage range. Fewer nodes are inquired within the coverage range, and fewer request essages are sent. On the other hand, when a ulticast group eber looks for neighbors, ore other ebers ight find they are closer to the source. Hence ore response essages ight be sent back. Total essages increase as ore nodes join the ulticast group. Finally, we consider the nuber of forwarding nodes engaged in ulticasting. To derive the statistical properties of TST, we set the network size as,. In Figure 8.3(a), when the ulticast group is sall, the first part of the curve indicates that the nuber of forwarding nodes is O( ). As there are ore ulticast group ebers, the nuber of forwarding nodes grows linearly with the group size. ) Non-unifor Distribution: We randoly choose the location of source, x s, within the unit square first. For the case of non-unifor distribution, we consider that nodes satisfy the noral distribution: f(x) = π e x xs, where x x s is the Euclidean distance between the node and the source. It is possible that the nodes are scattered outside unit square during siulations. If this happens, we relocate these nodes until they are within this unit square. We evaluate the TST algorith still in ters of tree length, essage coplexity and the nuber of forwarding nodes. We find the results are quite siilar to those in the unifor distribution. In Figure 8.(b), the length of TST tree is only a little larger than the optial length in our siulations. The statistics show that the ratio between the is. on average. As for the essage coplexity, Figure 8.(b) shows that ore essages are exchanged aong ore ulticast group ebers or in denser networks, and the quantity of essages is still O(n). As is shown in Figure 8.3(b), the nuber of transitting nodes in the ulticast tree is linear with, and becoes linear with when ore nodes participate in the ulticast group assuing that the network size keeps unchanged. B. Practical Applications TST algorith can be ipleented in practical systes as well as be integrated into sensor network architecture and protocol design to iprove the network perforance. Many sensor network systes need ulticast transission as they involve different kinds of sensors and interactions between various odules such as the LED lighting syste in [3], the localization syste in [3] and the building onitoring syste in [33]. These ake ulticast groups naturally for within the systes and ulticasting through a iniu ulticast tree is a desirable way for these energy-constrained sensor applications. Therefore, we can apply TST algorith for ulticast routing within the aforeentioned systes. In addition, during the tree construction process, TST algorith only requires single-hop transission, which confors to the counication protocols (e.g. zigbee) used in these systes. Besides, currently, a notable trend in sensor network study is to adopt the IPv6-based architecture [34], in which ulticasting is frequently used for scope addressing, discovery and configuration [34]. In those IPv6-based architectures, TST algorith exhibits an apparent advantage for ulticast routing over other algoriths in the sense that the sender does not need to have any prior knowledge of geographical locations of intended destinations before the tree construction. Theses locations can be acquired in the first phase of TST algorith. Such property caters to the ost coon situation of ulticast in IPv6 [35]. IX. CONCLUSION AND FUTUE WOKS In this paper, we propose a novel algorith, which we call Toward Source Tree, to generate approxiate Steiner Trees in wireless sensor networks. The TST algorith is a siple and distributed schee for constructing low-cost and energy-efficient ulticast trees in the wireless sensor network setting. We prove its perforance easures in ters of tree length, tie coplexity, and energy efficiency. We show that the tree length is in the sae order as, and is in practice very close to, the Steiner tree. We prove its running tie is the shortest aong all existing solutions. We prove that its essage coplexity and the nuber of nodes that participate in forwarding are both order-optial, yielding high energy efficiency for applications. For future research, we present several directions to extend the current study of iniu ulticast tree construction as follows. Our present work ainly focuses on the optiization of path length, energy and coputation costs in ulticast of wireless sensor networks. It is also interesting to develop an algorith which optiizes other etrics jointly such as the throughput [39], load balancing [37] and congestion control [37]. The idea that utilizes ultiple ulticast trees to provide backup routing paths for load balancing [37] is particularly enlightening. Our algorith is currently designed for static networks. It is of great interest to consider ulticast tree construction in tie-varying networks [38], [39] or obile networks, where the ajor technique of tree construction ay differ greatly fro that in static networks. The proposed TST algorith in our work is a distributed algorith, and the analysis is perfored in the setting of stochastic networks. It suggests an inspiring direction to cobine cobinatorial optiization ethods with stochastic optiization ethods [38], [39] to solve ulticast routing probles in sensor networks. ACKNOWLEDGMENT This work was supported by NSF China (No. 653, 635, 679, 656, 6485, 6633 and 94385) and China Postdoctoral Science Foundation.

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Hui and D. Culler. IP is dead, long live IP for wireless sensor networks. In Proc. ACM SenSys, 8. [35] X. Wang. Multicast for 6LoWPAN wireless sensor networks. In IEEE Sensors Journal, Vol. 5, No. 5, pp , 5. [36] E. Baccelli, M. Philipp and M. Goyal. The pp-rpl routing protocol for ipv6 sensor networks: Testbed experients. In Proc. of IEEE SoftCo,. [37] S. Sarkar and L. Tassiulas. A fraework for routing and congestion control for ulticast inforation flows. In IEEE Trans. on Infor. Theory, Vol. 48, No., pp.69-78,. [38] M. J. Neely. Energy optial control for tie-varying wireless networks. In IEEE trans. on Infor. Theory, Vol. 5, No.7, pp , 6. [39] A. Sinha, G. Paschos, C-P. Li and E. Modiano. Throughputoptial broadcast on directed acyclic graphs. In Proc. of IEEE INFOCOM, 5. APPENDIX A TEE LENGTH IN THE UNIFOM CASE When we analyze the teporary tree ade up of the source and receivers, transission range can be ignored since it does nothing with the teporary tree construction. We establish a two-diensional coordinate syste shown in Figure.. Let the source be the origin, and X-axis as well as Y-axis parallel to the square edge. The whole network is divided into square cells, and the edge length of each cell is. In the coordinate syste, the edge length is noralized to be, so the intersections in the network have integer coordinates. We use the coordinate of the vertex that is farthest

13 3 fro the origin in the square cell as the the coordinate of this cell, and hence cells also have integer coordinates. In Figure., the red node is the source node S and the blue node is the receiver node whose coordinate is (x, y). Without loss of generality, suppose that x and y are both positive. Then we set the radius to be the Euclidean distance between S and, and draw two circles with the center in S and in respectively. The intersected region of the unit square and two circles overlapping area is the shaded region in Figure.. According to criteria for neighboring receiver selection, the shaded region in figure is where the possible neighbor receivers of are located. If no receiver lies in this shaded area, can only connect to S directly. y Source eceiver Fig..: Two-diensional coordinate syste, actual and approxiate neighbor regions Now we illustrate how the neighbor is selected by the receiver in a ore detailed way. Suppose that the coordinate of the possible neighboring receiver is (a, b) (a and b don t need to be integers). And there are three conditions that should be satisfied: a + b < x + y, (x a) + (y b) < x + y (a, b) is within the unit square x (-) Let N denote all the nodes whose coordinates satisfy the conditions above, i.e., N = {(a, b) node (a,b) ight be connected to by the receiver }. For siplicity of analysis, we reduce the size of the set N and get a new set N, an approxiate neighbor region which is the rectangular region arked with parallel lines, which is shown in Figure.. N = {(a, b) a x, b y } (-) Now nodes in N are the ones that are possibly chosen as neighbors by in our analysis later. Note that in our algorith, it is possible that soe nodes in other cells with coordinates not in N ight also be connected to by the receiver. With the constraints above, the estiated length of teporary tree aong receivers is larger than that of the tree built with our algorith since we ignore soe nodes that are close to receiver. Upper bound of the teporary tree length built with our algorith can thus be derived. Besides shrinking the neighbor region, we also rearrange the nodes in each cell. Suppose that the coordinate of a node is (a, b), then we ove it to the point with coordinate ( a, b ). All nodes are now farther fro receiver after rearrangeent. Let M be the set of all receivers and p i,j be the probability that the node in the cell whose coordinate is (i, j) is chosen by the receiver as a neighbor to connect to. Let p S be the probability that the receiver directly connects to the source. We assue that the nodes in cell (i, j) are chosen if all cells (i, j ) are epty, where i i x and j j y (but i = i and j = j are not satisfied at the sae tie). This assuption can also ake the edge length estiation larger than the actual edge length. Since soe nodes that are closer to the receiver ight also be chosen as a neighbor receiver but they are not taken into consideration under this assuption, the estiated probability that one node is chosen is larger than actual probability. p i,j p(all (i, j )s are epty) p(all (i, j )s and (i, j) are epty) (-3) Å ã ( x + i)( y + j) Å ã ( x + i)( y + j) (-4) We define new variables a and b as: a = x + i and b = y + j. Note that a and b are both integers. We can obtain the following expression with (-4). c c ( < c < c < ), ab = Θ(); p i,j = e ab (e ), ab = o(). (-5) Since (c c ), we can oit the case that ab = Θ() when we calculate the length expectation. x y p S ( ). (-6) The expected length of teporary tree is: E(L V ) x y E ( p i,j»(x i + ) + (y j + ) M i= j= + p S x + y ) (-7) Cobining (-5), (-6) and (-7), we have E(L V ) x y E ( e ab (e ) M a= b=»(a + ) + (b + ) + e x y x + y ) (-8)

14 4 For the first part in (-7), we use the integration to evaluate its pattern. x y a= b= e ab» (a + ) + (b + ) (a,b) [,x] [,y]\[,] [,] y e + b= + e b» 4 + (b + ) + e ab (a + b)dadb (-9) x a= e a» 4 + (a + ) + e 4 3 (-) η η x η y ηx x ηy y + ηxy x + η + 3 η 4 + ηxy y + w (x, y) e ab (a + b)dadb. (-) Here η = e +/ in (-). Inequality (-) holds because when a and b (a, b does not equal to at the sae tie), (a + ) + (b + ) a + b. Then we take the coponents for suation at a = and b =. We ust add the value at a = b = since (-) does not hold at this point. The substraction in (-) coes fro the fact that we have already add this value at a = b =, so we ust subtract the corresponding ter in this part. For clarity, we replace soe expressions with functions. J(x, y) = η η x η y ηx x ηy y + ηxy x + ηxy y, y w (x, y) = η b» + (b + ) + η a» + (a + ), b= w (x, y) = η x y x + y, x a= c 3 = η + 3 η 4 (e + )(e ) /e 4. The expected length of teporary tree can be expressed as: E(L V ) e + E(J(x, y) + w (x, y) + c 3 ) M E(w (x, y)) (-) M The coordinate (x,y) of a receiver is dependent on the location of source. The farther the receiver is fro the source, the larger the value of J(x, y)+w (x, y) is. We can obtain the axiu of E(J(x, y)+w (x, y)), when the source is located at one of the square s four vertices. Thus the receiver coordinate ranges fro (, ) to (, ). For E(w (x, y)), there exist constraints for both x and y. That is, x and y. So the integral of w (x, y) is written as: E(w (x, y)) M w (x, y)dxdy. (-3) We use discrete suation to evaluate the expression on the right side of the inequality (-3). w (x, y)dxdy 4 η a» ( + a) a= x η a (a + 4/3) + a=5 y η b (b + 4/3) b=5 (-4) It is not hard to check the suation on the right side of (-4). We find that ( i= w (x, y) 4 e i» ( + i) 9e e 4 (e ) ). (-5) Wherever the source is, the part, can be ignored. This is because M E(w (x, y)) M E(w (x, y)), e x y (x + y)dxdy e In the end, we consider the part E(J(x, y)) in E(L V ). E(J(x, y)) M (-6) J(x, y)dxdy ( ) η e (-7) With (-5), (-6) and (-7), we can derive the upper bound of E(L V ) by (-). This copletes our proof. E(L V ) 5.6. (-8) APPENDIX B TEE LENGTH IN THE GENEAL DISTIBUTION We partition the unit square into k square cells with edge length of k. = k +γ and < γ <. In each cell, the expected nuber of points is approaching infinity (actually is greater than ɛ /k). First, we prove an assertion for a scaled unifor distribution. With Lea 3, we know that length of teporary tree spanning receivers is upper bounded by c, when nodes are uniforly distributed in a unit square. In a scaled square region with edge length of k, the expected length of teporary tree is saller than c k. We first construct a tree consisting of intra-square and intersquare edges. Intra-square edge eans that a receiver chooses to connect to another receiver located in the sae square, while inter-square edge eans that a receiver connects to another located in a different square. Let us first consider the intra-square edges. Source is located inside one of k cells, and outside (k ) cells. For the cell

15 5 containing the source, we build a teporary tree aong all nodes in it. For other cells, we let each node in it connect to the closest neighbor that has shorter Euclidean distance to source and it does not connect to any nodes if no such neighbor exists. As is proved in Leas and 5, trees can be established within these k cells. Then we consider the inter-square edges. For any cell, there is always an adjacent cell that is closer to source. We let a node fro each cell connect to another node in adjacent cells and the inter-square edge between the has the inial distance aong all node pairs. So all squares are connected by inter-square edges. With intra-square edges and inter-square edges, all of receivers for a tree topology. We denote this tree as T a. We consider the length of intra-square edges within the unit square. Suppose that Sq i is one of k cells in the set {Sq,, Sq k } and its edge length of k. According to Leas 3 and 5, we can conclude that total length» of intra-square edges within Sq i is is upper bounded by c k Sq i f(x)dx with scaling ethod. The expected length of intra-square edges within the whole network can be obtained by the suation: k» c k Sq i f(x)dx. Since the integration over the sall i= square is saller than k ax x Sq i f(x), we take the square root of this expression and by the definition of ieann- Stiejies integration, we know that the su is k i= c k f(x)dx c x [,]» f(x)dx. Sq i (-) The total length of inter-square edges within the unit square is o( ) as we have proved in Lea 5. So the expected length of T a is upper bounded by c x [,]» f(x)dx. Based on neighbor selection criteria in TST algorith, each node always connects to the closest node that has shorter Euclidean distance to the source. However, for intra-square edges in T a, the node is also required to connect to the neighbor in the sae square. More liitations are iposed on neighbor selection, so the edge length between two nodes ight be larger than that in teporary tree. Therefore, the length of T a can be regarded as the upper bound of that of teporary tree. The expected length of teporary tree is: E(L V ) c x [,]» f(x)dx. (-) This copletes our proof. APPENDIX C MINIMUM HOP PATH Fro work [5], ) we know that transission range is set as r = Θ, in order to ensure that network is connected. (» n There are about Θ() nodes within the transission range of one node. Finding the path with inial hops between two nodes can be regarded as a connectivity proble, which is a Poisson process. Since the expression for the area where two circles intersect is hard to estiate, we estiate it in the infinite nor without affecting the results. (a) -nor (b) infinite-nor Fig. 3.: We estiate the probability of points on the red line instead of green arc The transission range is set as r =» n. There are two receivers u and v, and the left circle in Figure 3.(a) shows the transission range of u. The circuscribed square is ebedded properly in this left circle. The right circle with center in v has a radius of x. The intersected part of square and the right circle is the green arc. For siplicity, we use the red line rather than the green arc, and this will overestiate the path length. We can use -nor in Figure 3.(b) for estiation instead of the nor. ɛ» Let A be the event that next point exists with distance s, and B be the event that there is a point within the transission region. It is easy to see that P (A B) = e nɛ(r s)r ( e nɛsr ) e nɛr = e nɛr e nɛr (e nɛsr ) (3-) Denote E r (x) as the expected nuber of relay nodes on the path with inial hops to a receiver with distance x. Here we consider that r in the nor are contained by r in the nor, so we have» the following functional equation when plugging in r = ɛ n : r E r (x) = ( + E r (x s)) nɛ re nɛr e nɛsr ds e nɛr = nɛ r r n ( + E r (x s))e nsr ds (3-) = + nɛ r n E r (x αr)e nɛrα dα = + n E r (x αr)n α dα (3-3) Clearly the lower bound of hop count is x/r. As for the upper bound of hop count, E r (x), we can also prove that E r (x) d(x/r)+ for a fixed d > and a large enough n by induction. We know that E r (x) = when x (, r), so it is satisfied that E r (x) d(x/r)+. Suppose that E r (x αr)

16 6 d x αr r +, we can show E r (x) = + n + n E r (x αr)n α dα d x αr n α dα (3-4) r + d x r d n n + d d(x/r) + (3-5) Inequality (3-5) holds for d (n ) n ( ) +. (3-6) We can conclude that E r (x) converges to x/r uniforly when n is very large. The first property has been proved. And because the transission range is r, the total tree length converges to x. So the second property also holds. APPENDIX D MESSAGE COMPLEXITY We only consider the approxiate neighbor region shown by the shaded area in Figure 4.(a) as in length analysis. We rearrange the cells in the feasible region along with the nodes in these cells, and put the in a colun. We first copare x-coordinates of the cells, and cells with larger x-coordinates will be put at the top of the colun. If they have the sae x-coordinates, cells with larger y-coordinates will be put at the top. Figure 4.(b) illustrates this arrangeent. We have x y cells rearranged in a colun. These cells are nubered fro to x y. After rearrangeent, it is easy to see that the actual distance between nodes in the i th cell and the receiver ust be saller than i+3. y Source eceiver (a) Approxiate neighbor region x (b) earrangeent of cells Fig. 4.: Shrinking neighbor region and rearranging cells There are five types of essages are exchanged in TST algorith, and their quantities are denoted as Msg i respectively ( i 5). In phase, all nodes participate in essage propagation, so E(Msg ) = n. In Phase, three types of essages are sent during this process: request essage, respond essage and connect essage. We rearrange the cells as in Figure 4.(b). The i th cell is represented by g i, and the probability that one node is located in the i th cell is assued as p(g i ). The probability that a node in the i th cell is chosen as a neighbor by the receiver (x, y) is denoted as p i. We have i i p i = ( p(g i )) ( p(g i )) = e j= i ( ) p(g j) j= e ( ) j= i j= p(g j) (4-) e (i )ɛ e iɛ (4-) When no appropriate receivers are found in this region, receiver connects to the source directly. Its probability is denoted as p S. p S ( x y ɛ ) (4-3) If the receiver in the i th cell is selected by receiver, k search sessions are needed in total for receiver, where k = i+3 log r. E(Msg ) i+3 x y log r p i i= k= x log +y r + k= π( k r) ɛ n p S π( k r) ɛ n (4-4) Cobining (4-), (4-3) and (4-4), we have Ñ é x y E(Msg n ) O (e (i )ɛ e iɛ )(i + ) i= ( n ) + O e x y ɛ (x + y ) (4-5) It is easy to prove that x y i= (e (i )ɛ e iɛ )(i + ) = O(). (4-6) e x y ɛ (x + y ) = O(). (4-7) Cobining (4-5), (4-6) and (4-7), we can obtain ( n E(Msg ) = O, (4-8) ) so the expected nuber of request essages is: E(Msg ) = E(Msg ) = O (n). Next coes respond essage. If M the neighbor search ends up in the k th session, the expected nuber of responding receivers is at ost 3 4 π(k r) ɛ 4 π(k r) ɛ π( k r) ɛ. If the response is fro the receiver in the i th cell, as we have proved, there are O( i+ r ) relays on the inialhop path connecting two receivers. E(Msg 3 ) O Ñ x y i= ( p i π( k r) ɛ ) k= k i+3 log r é

17 7 Ñ +O ps é π(k r) k k= log x +y r Lutian Zhao received the B.Sc. degree in Mathe- (4-9) é ä (i + 3)3 X Ä O e (i ) e i r i= Ç å 3 (x + y ) + O e bxcbyc (4-) r Å ã O. (4-) r atics fro Shanghai Jiao Tong University. Since Septeber 5, he has been a Ph.D candidate Ñ bxcbyc with the Departent of Matheatics, University of Illinois at Urbana-Chapaign. His current research interests are in the area of algebraic geoetry and string theory, especially in oduli space of stable objects in derived category, the atheatical description of several curve counting invariants arising fro string theory and their relation with each other. The total nuber of respond essages is: Å ã X. E(M sg3 ) = E(M sg ) = O r Kainan Wang received the B.S. degree in Coputer M Science fro Shanghai Jiao Tong University. During The last type of essage in the second phase is connect essage, Ä it äcan not be larger than respond essages. E(M sg4 ) O r. In cycle eliination, all relays participate in forwarding eliinate essage for the worst case. E(M sg5 ) O (n). (4-) Xinbing Wang and Prof. Xiaotie Deng in different research areas. Since Septeber 5, he has been a graduate student in Wireless Networking Group (WiNG) directed by Prof. Songwu Lu in Departent of Coputer Science at University of California, Los Angeles. His research areas include Coputer Theory, Algoriths and Networking. The total essage coplexity is E(M sg) = O(n). his undergraduate studies, he was advised by Prof. (4-3) Xinbing Wang received the B.S. degree (with hons.) This copletes our proof. in Autoation fro Shanghai Jiao Tong University, Shanghai, China, in 998, the M.S. degree in co- Hongyu Gong received the B.E. degree in Electrical puter science and technology fro Tsinghua Univer- Engineering fro Shanghai Jiao Tong University in sity, Beijing, China, in, and the Ph.D. degree 5. Since August 5, she has been a graduate with a ajor in electrical and coputer engineering student in the Departent of Electrical and Co- and inor in atheatics fro North Carolina State puter Engineering at the University of Illinois at University, aleigh, in 6. Currently, he is a Professor in the Departent of Urbana Chapaign. She has worked on wireless Electronic Engineering, and Departent of Coputer Science, Shanghai Jiao counications, with research involving topology Tong University, Shanghai, China. Dr. Wang has been an Associate Editor for design and algorith analysis in wireless sensor networks. Her current re- IEEE/ACM TANSACTIONS ON NETWOKING,IEEE TANSACTIONS search interests also include Natural Language Processing, ainly in seantic ON MOBILE COMPUTING, and ACM Transactions on Sensor Networks. analysis with achine learning ethods. He has also been the Technical Progra Coittees of several conferences including ACM MobiCo,4, ACM MobiHoc -7, IEEE Luoyi Fu received her B. E. degree in Electronic Engineering fro Shanghai Jiao Tong University, China, in 9 and Ph.D. degree in Coputer Science and Engineering in the sae university in 5. She is currently working with Prof. Xinbing Wang as a postdoc in Departent of Electronic and Engineering in Shanghai Jiao Tong University. Her research of interests are in the area of scaling laws analysis in wireless networks, connectivity analysis, sensor networks and social networks. Xinzhe Fu is an undergraduate student in Departent of Coputer Science at Shanghai Jiao Tong University, China. He is currently working as an research intern supervised by Prof. Xinbing Wang. His research interests include cobinatorial optiization, asyptotic analysis and privacy protection in social networks. INFOCOM 9-7.