A Unified Energy-Efficient Topology for Unicast and Broadcast

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1 A Unified Energy-Efficient Topology for Unicast and Broadcast Xiang-Yang Li Dept. of Compter Science Illinois Institte of Technology, Chicago, IL, USA Wen-Zhan Song School of Eng. & Comp. Sci. Washington State Uniersity Vancoer, WA, USA Weizhao Wang Dept. of Compter Science Illinois Institte of Technology, Chicago, IL, USA ABSTRACT We propose a noel commnication efficient topology control algorithm for each wireless node to select commnication neighbors and adjst its transmission power, sch that all nodes together self-form a topology that is energy efficient simltaneosly for both nicast and broadcast commnications. We proe that the proposed topology is planar, which garantees packet deliery if a certain localized roting method is sed; it is power efficient for nicast the energy needed to connect any pair of nodes is within a small constant factor of the minimm nder a common power attenation model; it is efficient for broadcast: the energy consmption for broadcasting data on top of it is asymptotically the best compared with strctres constrcted locally; it has a constant bonded logical degree, which will potentially redce interference and signal contention. We frther proe that the aerage physical degree of all nodes is bonded by a small constant. To the best of or knowledge, this is the first commnication-efficient distribted algorithm to achiee all these properties. Preiosly, only a centralized algorithm was reported in []. Moreoer, by assming that the ID and position of eery node can be represented in O(log n) bits for a wireless network of n nodes, or method ses at most 1n messages, where each message is of O(log n) bits. We also show that this strctre can be efficiently pdated for dynamical network enironment. Or theoretical reslts are corroborated in the simlations. Categories and Sbject Descriptors C..1 [Network Architectre and Design]: Wireless commnication, Network topology; G.. [Graph Theory]: Network problems, Graph algorithms General Terms Algorithms, Design, Theory The work of the athor is partially spported by NSF CCR Permission to make digital or hard copies of all or part of this work for personal or classroom se is granted withot fee proided that copies are not made or distribted for profit or commercial adantage and that copies bear this notice and the fll citation on the first page. To copy otherwise, to repblish, to post on serers or to redistribte to lists, reqires prior specific permission and/or a fee. MobiCom 05, Ag. 8-Sept., 005, Cologne, German Copyright 005 ACM /04/ $5.00. Keywords Graph theory, wireless ad hoc networks, topology control, power efficient, low weight, interference, nicast, broadcast. 1. INTRODUCTION A wireless ad hoc network consists of a distribtion of radios in a certain geographical area. Unlike celllar wireless networks, there is no centralized control in the network, and wireless deices (called nodes hereafter) can commnicate ia mlti-hop wireless channels: a node can reach all nodes inside its transmission range while two far-away nodes commnicate throgh the relaying by intermediate nodes. An important reqirement of these networks is that they shold be self-organizing, i.e., transmission ranges and data paths are dynamically restrctred with changing topology. Energy conseration and network performance are probably the most critical isses in wireless ad hoc (and sensor) networks, becase wireless deices are sally powered by batteries only and hae limited compting capability and memory. A wireless ad hoc or sensor network is modelled by a set V of n wireless nodes distribted in a two-dimensional plane. Each node has the same maximm transmission range R meters, e.g., a typical 80.11g wireless LAN adapter has a transmission range arond 100m 500m. By a proper scaling, we assme that all nodes hae the maximm transmission range eqal to one nit. These wireless nodes define a nit disk graph UDG(V ) in which there is an edge between two nodes iff the Eclidean distance between them is at most one nit. In other words, we assme that two nodes can always receie the signal from each other directly if the Eclidean distance between them is no more than one nit. Notice that, in practice, the transmission region of a node is not necessarily a perfect disk. As done by most reslts in the literatre, for simplicity, we model it by disk in order to first explore the nderlying natre of ad hoc networks. Hereafter, UDG(V ) is always assmed to be connected. We also assme that all wireless nodes hae distinctie identities(ids). Additionally we assme that each node knows the relatie position of its one-hop neighbors. The relatie position of neighbors can be estimated by the direction of signal arrial and the strength of signal. The geometry location of a wireless node can also be obtained by a localization method, sch as [5, 8, 15]. We here assme that the localization is low cost or it is already reqired by some other protocols. Notice that the higher these localization costs, the less desirable the adocated approach to design any protocol based on nodes geometry location.

2 We adopt the most common power-attenation model from literatre: the power needed to spport a link is assmed to be β, where is the Eclidean distance between and, β is a real constant between and 5 depending on the wireless transmission enironment. Throghot this paper we only focs on the transmission power of all nodes. This energy model only acconts for the emission power. This can be a good approximation of what happens in case of long range techniqes althogh the actal energy consmption is gien by a fixed part (receiing power and the power needed to keep the electric circits on) pls the emission power component. In other words, we assme that the transmission range is large enogh sch that the emission power is the major component and the receiing power is negligible. Notice that, as pointed ot by an anonymos reiewer, een if the energy cost of receiing a packet is high, there are a nmber of ways of redcing this cost by redcing the nmber of packets receied by bt not intended for a node. It incldes, bt is not limited to, the following approaches: (1) signals are sent with special small preambles that identify the intended recipient ; () the radios are freqency-agile and can choose different freqency channels to commnicate with different neighbors () the radios hae directional antennas which limit the olme oer which their signals are receied ; (4) faoring rotes that traerse sparser portions of the network. The (localized) topology control techniqe lets each wireless deice (locally) adjst its transmission range and select certain neighbors for commnication, while maintaining a decent global strctre to spport energy efficient roting and to improe the oerall network performance. A distribted method is localized if it rns in constant nmber of ronds [4]. By enabling each wireless node to shrink its transmission power (which is sally mch smaller than its maximm transmission power) sfficiently enogh to coer its farthest selected neighbor, topology control schemes can not only sae energy and prolong network life, bt also improe the network throghpt throgh mitigating the MACleel medim contention. Unlike traditional wired and celllar networks, the moement of wireless deices dring the commnication cold change the network topology to some extent. Hence, it is more challenging to design a topology control algorithm for ad hoc wireless networks: the topology shold be locally and self-adaptiely maintained sing low commnication cost, withot affecting the whole network s performance. The main contribtions of this paper are as follows. We present the first commnication efficient algorithm to constrct a nified energy-efficient topology for nicast and broadcast in wireless ad hoc/sensor networks. In one single strctre, we garantee the following network properties: 1. power efficient nicast: gien any two nodes, there is a path connecting them in the strctre with total power cost no more than ρ+1 times of the power cost of any path connecting them in the original network. Here ρ > 1 is some constant that will be specified later.. power efficient broadcast: the power consmption for broadcast is within a constant factor of optimm among all locally constrcted strctres. Notice that or strctre cannot garantee that the energy consmption is within a constant factor of the global optimm. Essentially, we proe that the strctre is lowweighted: its total edge length is within a constant factor of that of Eclidean Minimm Spanning Tree.. bonded logical node degree: each node has to commnicate with at most k 1 logical neighbors, where k 9 is an adjstable parameter; 4. bonded aerage physical node degree: the aerage physical node degree is bonded from aboe by a small constant. Here the physical degree of a node in a strctre H is defined as the nmber of nodes inside the disk centered at with radis max H. 5. planar: there are no edges crossing each other. This enables seeral localized roting algorithms, sch as [, 18, 4, 5], to be performed on top of this strctre to garantee the packet deliery withot roting table. 6. neighbors Θ-separated: the directions between any two logical neighbors of any node are separated by at least an angle θ, which as we will see redces the signal interference. It also can be sed to redce the receiing power cost when directional antenna is sed. In graph theoretical terminologies, gien a nit disk graph modelling the wireless ad hoc networks, we propose an commnication efficient distribted method to bild a low-weighted planar power-spanner with a bonded logical node degree. Here a geometric strctre is called low-weighted if its total edge length is no more than a small constant factor of that of the Eclidean minimm spanning tree. To the best of or knowledge, it is the first known commnication efficient algorithm to constrct sch a single strctre with all these desired properties. Preiosly, only a centralized algorithm was reported in []. Moreoer, by assming that the ID and position of eery node can be represented in O(log n) bits for a wireless network of n nodes, we show that the strctre can be initially constrcted sing at most 5n to 1n messages. In addition, we proe that the expected aerage node interference (which is defined as the nmber of nodes within its adjsted transmission range) in the strctre is bonded by a small constant. This is significant in its own de to the following reasons: it has been taken for granted that a network topology with a small logical node degree will garantee a small interference and recently Brkhart et al. [5] showed that this is not tre generally. Or reslts show that, althogh generally a small logical node degree cannot garantee a small interference, the expected aerage interference is indeed small if the logical commnication neighbors are chosen careflly. All or theoretical reslts are corroborated in simlations. We also show that or strctre can be easily pdated in a dynamic enironment when a node moes or dies after the battery power is drained. When a node moes, the topology can be dynamically self-maintained withot affecting the whole network, since each node adjsts its transmission range and selects neighbors only according to its neighbor information. To facilitate the efficient constrction of sch a nified energy-efficient topology, in the paper, we will first gie an improed method to constrct degree-bonded planar spanner by sing relatie positions only. The new strctre has the same power spanning ratio ρ = β 1 ( sin π k )β as the strctre proposed in [4]. Here k 9 is a cstomizable parameter. In addition, the directions between any two neighbors of each node are separated by at least a certain angle θ depending on the parameter k. Simlations show that the

3 node interference in or new strctre is indeed smaller than the strctre proposed in [4]. The rest of the paper is organized as follows. In Section, we reiew some prior arts in topology control, and smmarize some preferred properties of network topology for nicast and broadcast. Section presents an improed algorithm to bild a degree-bonded planar spanner with Θ-separation property. We then propose, in Section 4, the first commnication efficient algorithm to constrct planar spanner with bonded-degree and low weight. We stdy the expected interference of arios strctres in Section 5. In Section 6, we briefly stdy how to dynamically pdate the strctre if nodes are mobile. In Section 7, we condct extensie simlations to alidate or theoretical reslts. Finally, we conclde or paper in Section 8.. CURRENT STATE OF KNOWLEDGE.1 Energy-Efficient Unicast Topology Seeral strctres hae been proposed for topology control in wireless ad hoc networks. The relatie neighborhood graph, denoted by RNG(V ) [4], consists of all edges sch that the intersection of two circles centered at and and with radis does not contain any ertex w from the set V. The Gabriel graph [1] GG(V ) contains an edge if and only if disk(, ) contains no other points of V, where disk(, ) is the disk with edge as a diameter. For conenience, we also denote GG and RNG as the intersection of GG(V ) and RNG(V ) with UDG(V ) respectiely. Both GG and RNG are planar, are connected, and contain the Eclidean minimm spanning tree(emst ) of V if UDG is connected. RNG is not power efficient for nicast, since the power stretch factor of RNG is n 1 in the worst case [48] and not bonded by a constant een for n nodes randomly distribted (or proof is similar to the proof in [4] and omitted here de to space limit). Both RNG and GG are not degree-bonded. The Yao graph [51] with an integer parameter k > 6, denoted by Y Gk, is defined as follows. At each node, any k eqally-separated rays originated at define k cones. In each cone, choose the shortest edge UDG(V ) among all edges emanated from, if there is any, and add a directed link. Ties are broken arbitrarily or by ID. The reslting directed graph is called the Yao graph. It is well-known that the Yao strctre is power efficient for nicast. Seeral ariations [1] of the Yao strctre cold hae bonded logical node degree also. Howeer, all Yao related strctres are not planar. Li et al. [9] proposed the Cone Based Topology Control (CBTC) algorithm to first focs on seeral desirable properties, in particlar being an energy spanner with bonded degree. It is basically similar to the Yao strctre for topology control. Each node finds a power p,α sch that in eery cone of degree α srronding, there is some node that can reach with power p,α. Here, neertheless, we assme that there is a node reachable from by the maximm power in that cone. Notice that the nmber of cones to be considered in the traditional Yao strctre is a constant k. Howeer, nlike the Yao strctre, for each node, the nmber of cones needed to be considered in the method proposed in [9] is, where each neighboring node cold contribte two cones on both side of segment. Then the graph G α contains all edges sch that can commnicate with sing power p,α. They proed that, if α 5π 6 and the UDG is connected, then graph G α is a connected graph. On the other hand, if α > 5π, they showed that the 6 connectiity of G α is not garanteed by giing some conterexample [9]. Unlike the Yao strctre, the final topology G α is not necessarily a bonded degree graph. Bose et al. [] proposed a centralized method with rnning time O(n log n) to bild a degree-bonded planar spanner for a two-dimensional point set. It constrcts a planar t- spanner with low-weight for a gien nodes set V, for t = (1 + π) C del 10.0, sch that the node degree is bonded from aboe by 7. Hereafter, we se C del <.6 to denote the spanning ratio of the Delanay trianglation [11, 0, 19]. Howeer, a straightforward distribted implementation of this centralized method takes O(n ) commnications in the worst case for a set V of n nodes. Wang and Li [46] proposed the first efficient distribted algorithm to bild a degree-bonded planar spanner BP S for wireless ad hoc networks. Thogh their method can achiee three desirable featres: planar, degree-bonded, and power efficient, the theoretical bond on the node degree of their strctre is a large constant. For example, when α = π/6, the theoretical bond on node degree is 5. In addition, the commnication cost of their method can be ery high, althogh it is O(n) theoretically, which is achieed by applying the method in [6] to collect -hop neighbors information. The hidden constant is large: it is seeral hndreds. Recently, Song et al. [4] proposed two methods to constrct degree-bonded power spanner, by applying the ordered Yao strctres on Gabriel graph. They achieed better performance with mch lower commnication cost, compared with the method in [46]. One method in [4] only costs n messages for the constrction, and garantees that there are at most one neighbor node in each of the k = 9 eqal-sized cones. Notice that the strctres constrcted by the methods proposed in [46, 4] are not garanteed to be low-weighted. Both strctres are planar and degree-bonded. The strctre constrcted in [4] is only a power-spanner, while the strctre constrcted in [46] is also a length-spanner. Notice that it is known that a length-spanner is always a power spanner [1]. The main contribtion of this paper is that we propose the first method to constrct a single topology that is planar, length-spanner, bonded-degree, and lowweighted. In smmary, for energy efficient nicast roting, the topology is preferred to hae following featres: 1. Power Spanner: Formally speaking, a sbgraph H is called a power spanner of a graph G if there is a positie real constant ρ sch that for any two nodes, the power consmption of the shortest path in H is at most ρ times of the power consmption of the shortest path in G. Here ρ is called the power stretch factor or spanning ratio.. Degree Bonded: It is also desirable that the logical node degree in the constrcted topology is bonded from aboe by a small constant. A small node degree cold redce the MAC-leel contention and interference, also may help to mitigate the well known hidden and exposed terminal problems. In addition, a strctre with a small degree will improe the oerall network throghot []. Bonded degree strctres also find applications in Bletooth wireless networks since a master node can hae only 7 actie slaes.

4 . Planar: A network topology is also preferred to be planar (no two edges crossing each other in the graph) to enable some localized roting algorithms work correctly and efficiently, sch as Greedy Face Roting (GFG) [], Greedy Perimeter Stateless Roting (GPSR) [18], Adaptie Face Roting(AFR) [4], and Greedy Other Adaptie Face Roting (GOAFR) [5]. Notice that with planar network topology as the nderlying roting strctre, these localized roting protocols garantee the message deliery withot sing a roting table: each intermediate node can decide which logical neighboring node to forward the packet sing only local information and the position of the sorce and the destination.. Energy-Efficient Broadcast Topology Broadcast is also a ery important operation in wireless ad hoc/sensor networks, as it proides an efficient way of commnication that does not reqire global information and fnctions well with topology changes. For example, many nicast roting protocols [17, 6, 9, 8, 41] for wireless mlti-hop networks se broadcast in the stage of rote discoery. Similarly, seeral information dissemination protocols in wireless sensor networks se some forms of broadcast/mlticast for solicitation or collection of sensor information [14, 16, 5]. Since sensor networks mainly [1] se broadcast for commnication, how to delier messages to all the wireless deices in a scalable and power-efficient manner has drawn more and more attention. Not ntil recently hae research efforts been made to deise power-efficient broadcast strctres for wireless ad hoc networks. Notice that, a broadcast roting protocol can be interpreted as flood-based broadcasting on a certain sbgraph of the original commnication networks, since any broadcast roting can be iewed as an arborescence (a directed tree) T, rooted at the sorce node of the broadcasting, that spans all nodes. The tree T contains a directed edge if node receied the first copy of the broadcast data from node. Once a broadcast strctre H is constrcted, the broadcast is a simple flooding on top of H: once a node got the broadcast message from any of its logical neighbors, say, for the first time, it will simply forward it to all its logical neighbors (maybe except the node ) either throgh one-to-one or oneto-all commnications. Let f H (p) denote the transmission power of the node p reqired by broadcasting message on top of a broadcast strctre H. We assme that the tree H is a directed graph rooted at the sorce of the broadcasting session: link pq H denotes that node p forwarded message to node q. For any leaf node p of H, clearly we hae f H (p) = 0 since it does not hae to forward the data to any other node. For any internal node p of H, f H (p) = max pq H pq β nder or energy model P if an one-to-all commnication model is sed; and f H (p) = pq H pq β nder or energy model if an one-to-one commnication P model is sed. The total energy reqired by H is p V fh (p). In the literatre, the one-to-all commnication model is typically assmed. Minimm-energy broadcast roting in a simple ad hoc networking enironment has been addressed in [9, 1, 49]. It is known [9] that the minimm-energy broadcast roting problem is NP-hard, i.e., it cannot be soled in polynomial time nless P=NP. Three greedy heristics were proposed in [49] for the minimm-energy broadcast roting problem: EMST (minimm spanning tree), SPT (shortest-path tree), and BIP (broadcasting incremental power). Wan et al. [44, 45] showed that the approximation ratios of EMST and BIP are between 6 and 1 and between 1 and 1 respectiely; on the other hand, the approximation ratio of SPT is at least n, where n is the nmber of nodes. The approximation ratio of EMST is improed to 8 recently by Flammini et al. [1]. Unfortnately, none of the aboe strctres can be formed and pdated sing only linear nmber of messages, nor locally. RNG, which can be constrcted locally, has been sed for broadcasting in wireless ad hoc networks [40]. Howeer, an example was gien in [0] to show that the total energy sed by broadcasting on RNG cold be abot O(n β ) times of the minimm-energy sed by an optimm method. Seeral practical broadcasting protocols [50, 7, 47] are proposed recently, howeer, all of them did not proide their theoretical performance bond on the energy consmption. In fact, Li [0] showed that, no deterministic localized algorithm can find a strctre that approximates the total energy consmption of broadcasting within a constant factor of the optimm. Frthermore, in the worst case, for any broadcast based on a locally constrcted and connected strctre, there is a network configration of n nodes sch that its energy consmption is Θ(n β 1 ) times the optimm. On the other hand, gien any low-weighted strctre H, i.e., ω(h) O(1) ω(emst ), they proed the following lemma Lemma 1. [0] ω β (H) O(n β 1 ) ω β (EMST ), where H is any low-weighted strctre. Here P ω(g) is the total length of the links in G, i.e., ω(g) = G, and ω β(g) P is the total power consmption of links in G, i.e., ω β (G) = G β. Conseqently, lowweighted strctre is asymptotically optimal for broadcasting among any connected strctres bilt in a localized manner. Notice that, the aboe analysis is based on the assmption that eery link is sed dring the broadcast (one-to-one commnication), sch as sing the TDMA scheme. Een considering that the broadcast signal sent by a node can be receied by all nodes in its transmission region simltaneosly (one-to-all commnication), the aboe claim is also correct. The reason is basically as follows. Let B s (H) be the total energy consmed by broadcasting on a strctre H with sender s sing the one-to-all commnication model. Clearly, any flood-based broadcast based on a strctre H P consmes energy at most e i H eβ i if the message receied by an intermediate node is not forwarded to its parent, i.e., the node that jst forwarded P this message to ; and the total energy is at most e i H eβ i if an intermediate node may also forward the message to its parent. On the other hand, P the total energy B s (H) sed by any strctre H is at least e i EMST eβ i /1 [45]. Ths, B s (EMST ) X e i EMST e β i /1 = ω β(emst )/1. Then, if H is a low-weighted X strctre, we hae B s(h) e β i = O(nβ 1 ) ω β (EMST ) e i H 1 O(n β 1 ) B s (EMST ) Recall that B s(emst ) is no more than a constant ( 8) times of the optimm in an one-to-all commnication model [45, 1]. Conseqently, we hae the following lemma.

5 Lemma. The broadcast based on any low-weighted strctre H consmes energy at most O(n β 1 ) times of the optimm in both one-to-one and one-to-all commnication models. And the bond O(n β 1 ) is tight. In smmary, to enable energy efficient broadcasting, the constrcted topology is also preferred to be low-weighted, in addition to the three properties for nicast: 4. Low Weighted: the total link length of final topology is within a constant factor of that of EMST. Recently, seeral localized algorithms [0, ] hae been proposed to constrct low-weighted strctres, which indeed approximate the energy efficiency of EMST as the network density increasing. Howeer, none of them is power efficient for nicast roting. In this paper we will present the first efficient distribted method to constrct a planar, bonded degree spanner that is also low-weighted.. POWER-EFFICIENT UNICAST: SPANNER, PLANAR AND BOUNDED-DEGREE The ltimate goal of this paper is to constrct a nified topology that is power-efficient for both nicast and broadcast, in addition to be planar and hae a constant bonded logical node degree. To achiee this ltimate goal, in this section, we first present a new method that can constrct a power-efficient topology for nicast. We will proe that the constrcted strctre is a power-spanner, planar and has bonded node degree. Frthermore, it has an extra property: any two neighbors of each node are separated by at least a certain angle θ. Hereafter, we call it the Θ-separation property. As we will see later that this property frther redces the interference, especially when adopting directional antennas for transmission. This property also makes the proof mch easier that the strctre constrcted in the next section is also power-efficient for broadcast. One possible way to constrct a degree-bonded planar power spanner is to apply the Yao strctre on Gabriel graph, since GG is already planar and has a power stretch factor exactly 1. In [1], Li et al. showed that the final strctre by directly applying the Yao strctre on GG is a planar power spanner, called Y aogg, bt its in-degree can be as large as O(n), as in the example shown in Figre 1(b). In [4], Song et. al proposed two new methods to bond node degree by applying the ordered Yao strctres on Gabriel graph. The strctre SY aogg in [4] garantees that there is at most one neighbor node in each of the k eqal-sized cones. In this section, we will propose an improed algorithm to frther redce the medim contention by selecting less commnication neighbors and separating neighbors wider. Before we gie the algorithm, we first define a concept called θ-dominating Region. Definition 1. θ-dominating Region: For each neighbor node of a node, the θ-dominating region of is the θ-cone emanated from, with the edge as its axis. Figre illstrates the θ-dominating region of a node in the transmission disk of node. Using the concept of θ-dominating region instead of absolte cone partition in SYaoGG [4], we are able to proe that any two neighbors of each node are garanteed to be separated by at least θ θ Figre : Node s θ-dominating Region with respect to node. an angle θ. The final topology will be called SΘGG. Intitiely, the commnication interference in SΘGG will be smaller than the interference in SY aogg, which is also erified later by simlations as shown in Figre 10(c) and (d). Algorithm 1 SΘGG: Power-Efficient Unicast Topology 1: First, each node self-constrcts the Gabriel graph GG locally. Initially, all nodes mark themseles White, i.e., nprocessed. : Once a White node has the smallest ID among all its White neighbors in N(), it ses the following strategy to select neighbors: 1. Node first sorts all its Black neighbors (if aailable) in N() in the distance-increasing order, then sorts all its White neighbors (if aailable) in N() similarly. The sorted reslts are then restored to N(), by first writing the sorted list of Black neighbors then appending the sorted list of White neighbors.. Node scans the sorted list N() from left to right. In each step, it keeps the crrent pointed neighbor w in the list, while deletes eery conflicted node in the remainder of the list. Here a node is conflicted with w means that node is in the θ- dominating region of node w. Here θ = π/k (k 9) is an adjstable parameter. Node then marks itself Black, i.e. processed, and notifies each deleted neighboring node in N() by a broadcasting message UpdateN. : Once a node receies the message UpdateN from a neighbor in N(), it checks whether itself is in the nodes set for deleting: if so, it deletes the sending node from list N(), otherwise, marks as BLACK in N(). 4: When all nodes are processed, all selected links { N(), GG} form the final network topology, denoted by SΘGG. Each node can shrink its transmission range as long as it sfficiently reaches its farthest neighbor in the final topology. The basic idea of or method is as follows. Since the Gabriel graph is planar and power-spanner, we will remoe some links of GG to bond the nodal degree while not destroying the power-spanner property. The basic approach of bonding the nodal degree is to only keep some shortest link in the θ-dominating region for eery node. We process the nodes in a certain order. A node is marked White if it is nprocessed and is marked Black if it is processed. Originally all nodes are marked White. Initially, a node elects itself to start processing its neighbors if its ID 1 is smaller than all its 1 It is not necessary to se ID here. We can also se some other mechanism to elect a certain node to perform the remaining pro-

6 (a) UDG (b) RNG, GG (c) BP S (d) OrdY aogg (e) SY aogg (f) SΘGG Figre 1: Seeral planar power spanners for UDG shown in (a). Here k = 9 for constrcting SY aogg, SΘGG. nprocessed logical neighbors in the Gabriel graph. Assme that a node is to be processed. We frther assme that there are already some processed logical neighboring nodes, say 1,, t, among its neighbors in GG. It keeps the link to the closest processed neighbor, say 1, in GG, and remoes all links to all neighbors in the θ-dominating region of 1. In other words, the neighbor 1 dominates all other neighbors in its θ-dominating region. It then repeats the aboe procedre ntil no processed logical neighbors in GG are left. Assme that node also has some nprocessed logical neighbors, i.e., marked White. The node then keeps the link to the closest nprocessed neighbor, say w, in GG if there is any, and then remoes the links to all neighbors in the the θ-dominating region of w. It then repeats the aboe procedre ntil no nprocessed neighbors in GG are left. Node then marks itself Black and then informs its logical neighbors in GG abot its change of stats. The algorithm terminates when all nodes are marked processed. The remaining links form the final strctre, called SΘGG. In or new algorithm, a data strctre will be sed: N() is the set of neighbors of each node in the final topology, which is initialized as the set of neighbor nodes in GG. We are now ready to present or Algorithm 1, which constrcts a bonded degree planar power spanner. Notice that the final topology based on Yao graph, sch as SY aogg [4], may ary as the choice of the direction of cones aries. Here, SΘGG does not rely on the absolte cone partition by adopting the new concept of θ-dominating region. Hence, gien the point set V, SΘGG is niqe. In addition, the aerage node degree, interference and transmission range of SΘGG is expected to be smaller than SY aogg too. Frthermore, it is interesting to notice that the theoretical bond on the spanning ratio for SΘGG, that we can proe, is same as SY aogg, as proed later in Theorem 4. Lemma. Graph SΘGG is connected if the nderlying graph GG is connected. Frthermore, gien any two nodes and, there exists a path {, t 1,..., t r, } connecting them sch that all edges hae length less than. Proof. We proe the connectiity by contradiction. Sppose a link is the shortest link in UDG whose connectiity is broken by Algorithm 1. W.l.o.g, assme the link is remoed while processing node, becase of the existence of another node w. As shown in Figre, there are only two cases (ties are broken by ID) that the link can be remoed by node : cedres first. For example, we can se the RTS/CTS mechanism proided in the MAC layer to achiee this: the node that first sccessflly sent a RTS signal within its one-hop neighborhood will be elected. In this paper, we se ID jst for the sake of an easy presentation. w (a) w < w (b) w > Figre : Two cases when is remoed while processing. Case a: w <. Notice that w θ < π/4, hence w <. In other words, both link w and w are smaller than link. Since there are no paths according to the assmption, either the path w or w is broken. That is to say, either the connectiity of w or w is broken. Ths, is not the shortest link whose connectiity is broken, it is a contradiction. Case b: w >. It happens only when node w is processed and node is nprocessed. Similarly, w θ < π/4 < w (otherwise w > π/ iolates the Gabriel graph property), hence w <. Since node w is a processed node and node decides to keep link w, the link w will be kept in SΘGG. According to assmption that and are not connected in SΘGG, w and are not connected either. That is to say, is not the shortest link whose connectiity is broken. It is a contradiction. This finishes the proof of connectiity. Notice that the aboe proof implies that the shortest link in UDG is kept in the final topology. Clearly, the shortest link is in GG. Link cannot be remoed in or algorithm de to the case illstrated by Figre (a). Assme, for the sake of contradiction, that is remoed de to the case (b) where w > and w is processed when processing. Then w < is a contradiction to that is the shortest link in UDG. We then show by indction that, gien any link in UDG, there is a path connecting them sing edges with length at most. Assme is remoed when processing and de to the existence of link w. We bild a path connecting and by concatenating w and w, as shown in Figre. It is not difficlt to see that the longest segment of the path is less than, which occrs in case (b). In this case, the link w mst be kept becase both endpoints are processed, and w <. This finishes the proof. The property that for any link, there is a path connecting them sch that the links on the path hae length at most

7 is crcial for or later proof that or Algorithm bilds a low-weighted bonded degree planar spanner. Theorem 4. The strctre SΘGG has node degree at most k 1 and is planar power spanner with neighbors Θ- separated. Its power stretch factor is at most ρ = where k 9 is an adjstable parameter. β 1 ( sin π k )β, Proof. The proof wold be similar with the proof of SY aogg in [4]. The only difference is that, we sed the concept of dominating cones instead of Yao graph. While the power stretch factor remains the same theoretically, the degree bond is redced from k to k 1. Obiosly, the links in SΘGG are Θ-separated, in other words, the direction of any two neighbors of a node is Θ-separated. Figre 1 (e) and (f) show the difference of SY aogg and SΘGG. Compared with SY aogg, SΘGG is more eenly distribted and has a lower node degree. 4. UNIFIED POWER-EFFICIENT TOPOLOGY: DEGREE-BOUNDED PLANAR SPANNER WITH LOW WEIGHT To the best of or knowledge, so far, no commnication efficient topology control algorithm has achieed all the desirable properties smmarized in Section : degree-bonded, planar, power spanner, low-weighted. Those properties are not only interesting in terms of comptational geometry, bt also hae important applications in wireless ad hoc networks, as shown in section : enable energy efficient nicast and broadcast rotings in same strctre. Recall that, spanner property ensres that an energy efficient path is always kept for any pair of nodes, hence it is a necessary condition to spport energy efficient nicast. While low-weighted strctre is optimal for broadcast among any connected strctres bilt locally. Unfortnately, all the known spanners, inclding Yao [51], GG [1] and the recent deeloped degreebonded planar spanners BP S [46], SY aogg, OrdY aogg [4] and SΘGG, are not low-weighted. As illstrated in Figre 1, all of them will keep at least n 1 links between the two circles, while EMST (in Figre 5(b)) will keep only one link between them. Hence the weight of any of them is at least O(n) w(emst ). It is worth to clarify that, in this section, we are interested in finding a sbgraph to enable efficient broadcast rotings, een based on the simple-flooding method. We do not aim to sbstitte the known broadcasting protocols. In fact, the methods sed in those broadcasting protocols [50, 7] can be applied on the low-weighted strctres to consere more energy. The main contribtion of low-weighted strctre is that it bonds the worst case performance for broadcasting. Seeral known localized algorithms are gien in [0, ] to generate low-weighted graphs. In their algorithms, gien a certain strctre G, for any two links and xy of a graph G, they remoe xy if xy is the longest link among qadrilateral xy. They proed that the final strctres are low-weighted if G is RNG [0] or LMST []. Obiosly, they are not spanners. In fact, their techniqes can not be applied on spanner graph to bond the weight withot losing its spanner property. Figre 4 illstrates an example by applying their algorithms to SΘGG. The node ID of i is i, 1 4 < θ and 1 > 4 > max( 1, 4 ). While constrcting SΘGG, first node θ ε 1 Figre 4: The graph cold be disconnected if applying the preios method to bild low-weighted strctre on SΘGG. 1 selects 1 and 1 as its incident logical links and node selects 1 and 4, then node selects 1 and deletes 4. Hence 4 / SΘGG. If applying the rle described in [0, ], the link 1 will also be deleted becase 1 > max( 1, 4, 4 ). Then the graph will be disconnected. Then we can conclde that simple extension of methods in [] on top of SΘGG does not een garantee the connectiity, nor to say power-spanner property. Indeed, the spanner property and low-weight property are not easy to be achieed at same time. Intitiely, the spanner property reqires to keep more links, while the lowweight property reqires to keep less links from original graph. In the following, we will describe a noel algorithm to bild a low-weighted strctre from SΘGG, while keeping enogh links to garantee the power efficiency. Figre 5 illstrates the difference of LSΘGG from SΘGG and LMST. 4 (a) SΘGG (b) LMST (c) LSΘGG Figre 5: Three different strctres. Algorithm presents or new method that constrcts a bonded degree planar power-spanner that is also lowweighted. Althogh or algorithm prodces only powerspanner here, it can be extended to prodce also the lengthspanner if it is needed. To get a length-spanner, we constrct the strctre LDel (defined in [7]) instead of the Gabriel graph sed in or algorithm. It was proed in [7] that LDel is a planar, length-spanner, and can be constrcted sing only O(n) messages. The basic idea of or new method is as follows. Since the graph SΘGG is already planar, power-spanner, and has bonded-degree, we will remoe some of its edges to garantee that the reslting topology is low-weighted while not destroying the power-spanner property. Notice that remoing edges will not break the planar property and the bonded-degree property. In all preios methods presented in the literatre, a node x decides to remoe or keep links that are incident on x, i.e., it only cares abot the incident edges. While, in the method presented here, a node x will decide whether to keep or remoe links for not only incident links, bt also the links that are incident on one of its neighbors. To garantee a lowweight property the methods presented in [0, ] remoe

8 1 n 1 n 1 n 1 n 1 n 1 n (a) original graph SΘGG (b) graph reslted sing method in [0] (c) graph based on or method Figre 6: A seqence of links are recrsiely remoed. Here solid and dashed links represent the links from the original graph and the dashed links represent the links that are remoed by a topology control algorithm, while solid links represent the final strctre constrcted by a certain method. Here we assme that i i = R and the ID of link i i is less than the ID of link i+1 i+1. some links from a certain strctre sch that the remaining links satisfy the isolation property: for each remaining link xy, the disk centered at the midpoint of xy sing a radis proportional to xy does not intersect with any other remaining links. They achieed this property by remoing a link xy if there is another link sch that xy is the longest link in the qadrilateral yx. Howeer, this simple heristic cannot garantee the spanner property. Consider a link xy in some shortest path from s to t. See Figre 7 for an illstration. Link xy will be remoed de to the existence of link. Link cold also later be remoed de to the existence of another link 1 1, which cold also be remoed de to the existence of another link, and so on. See Figre 6 (b) for an illstration of the sitation where a seqence of links will be remoed: all links i i, for i will be remoed. Conseqently, the shortest path connecting nodes n and n cold be arbitrarily long in the reslting graph. Ths, instead of blindly remoing all sch links xy wheneer it is the longest link in a qadrilateral yx, we will keep sch a link when some links in its certain neighborhood hae been remoed. To do so, among all links from a graph, sch as SΘGG, that is planar, bonded-degree, power-spanner, we implicitly define an independent set of links. A link is in this independent set, which will be kept at last, if it has the smallest ID among nselected links from its neighborhood. Specifically, we implicitly define a irtal graph G oer all links in SΘGG: the ertex set of G is the set of all links in SΘGG and two links xy and of SΘGG are connected in G if one end-point of is in the transmission range of one end-point of link xy (they will interfere with each other if transmit simltaneosly). For example, the links 1 1 and are not independent in network topology illstrated by Figre 6 (a); while the links 1 1 and n n are independent. Notice that links 1 1 and 1 are independent since they do not form a for ertices conex hll. Notice that in or method presented later, we did not explicitly define sch graph G, nor compte the maximal independent set of sch graph G explicitly. We will proe that the selected independent set of links in SΘGG indeed is low-weighted and still preseres the power-spanner property, althogh with a larger power spanning ratio. Or method will keep link 1 1 since it has the smallest ID among all links that are not independent. When link 1 1 is kept, all links that are not independent (here are and ) will be remoed. Then link 4 4 will be kept. The aboe procedre will be repeated ntil all links are processed. The final strctre reslted from or method is illstrated by Figre 6 (c). Obiosly, the constrction is consistent for two endpoints of each edge: if an edge is kept by node, then it is also kept by node. The ID of a link is defined as following: ID() = {, min(id(), ID()), max(id(), ID())}. As we will see later that the nmber in criterion of Algorithm is careflly selected. xy > max(, x, y ) Algorithm Constrct LSΘGG: Planar Spanner with Bonded Degree and Low Weight 1: All nodes together constrct the graph SΘGG in a localized manner, as described in Algorithm 1. Then, each node marks its incident edges in SΘGG nprocessed. : Each node locally broadcasts its incident edges in SΘGG to its one-hop neighbors and listens to its neighbors. Then, each node x can learn the existence of the set of -hop links E (x), which is defined as follows: E (x) = { SΘGG or N UDG(x)}. In other words, E (x) represents the set of edges in SΘGG with at least one endpoint in the transmission range of node x. : Once a node x learns that its nprocessed incident edge xy has the smallest ID among all nprocessed links in E (x), it will delete edge xy if there exists an edge E (x) (here both and are different from x and y), sch that xy > max(, x, y ); otherwise it simply marks edge xy processed. Here assme that yx is the conex hll of,, x and y. Then the link stats is broadcasted to all neighbors throgh a message UpdateStats(xy). 4: Once a node receies a message UpdateStats(xy), it records the stats of link xy at E (). 5: Each node repeats the aboe two steps ntil all edges hae been processed. Let LSΘGG be the final strctre formed by all remaining edges in SΘGG. Theorem 5. The strctre LSΘGG is a degree-bonded planar spanner. It has a power spanning ratio ρ + 1, where ρ is the power spanning ratio of SΘGG. The node degree is bonded by k 1 where k 9 is a cstomizable parameter in SΘGG. Proof. The degree-bonded and planar properties are obiosly deried from the SΘGG graph, since we do not add any links in Algorithm. To proe the spanner property,

9 we only need to show that the two endpoints of any deleted link xy SΘGG are still connected in LSΘGG with a constant spanning ratio path. We will proe it by indction on the length of deleted links from SΘGG. x Figre 7: The path between x and y is at most (ρ + 1) xy in LSΘGG if xy SΘGG. Assme xy is the shortest link of SΘGG which is deleted by Algorithm becase of the existence of link with smaller length. Obiosly, path x y can be constrcted throgh the concatenation of path x, link and path y, as shown in Figre 7. Since xy > max( x, y ) and link xy is the shortest among deleted links in Algorithm, we hae p(x ) < ρ x β and p( y) < ρ y β. Hence, p(x y) < β + ρ x β + ρ y β < (ρ + 1) xy β. Sppose all the i-th (i t 1) deleted shortest links of SΘGG hae a path connecting their endpoints with spanning ratio ρ + 1. For the t-th deleted shortest link xy SΘGG, according to Algorithm, it mst hae been deleted becase of the existence of a link : sch that xy > max(, x, y ) in a conex hll yx. Now, we hae p(x ) < (ρ + 1) x β and p( y) < (ρ + 1) y β. Ths, p(x y) = β + p( x) + p( y) < β + (ρ + 1) x β + (ρ + 1) y β < xy β + (ρ + 1)( xy /) β + (ρ + 1)( xy /) β (ρ + 1) xy β Ths, LSΘGG has a power spanning ratio ρ + 1. We then show that graph LSΘGG is low-weighted. To stdy the total weight of this strctre, inspired by the method proposed in [0], we will show that the edges in LSΘGG satisfy the isolation property [10]. Theorem 6. The strctre LSΘGG is low-weighted. See the appendix for the proof. We contine to analyze the commnication cost of Algorithm 1 and. First, clearly, bilding GG in Algorithm 1 can be done sing only n messages: each message contains the ID and geometry position of a node. Second, to bild SΘGG, initially, the nmber of edges, say p, in Gabriel Graph is p [n, n 6] since it is a planar graph. Remember that we will remoe some edges from GG to bond the logical node degree. Clearly, there are at most n sch remoed edges since we keep at least n 1 edges from the connectiity of the final strctre. Ths the total nmber of messages, say q, sed to inform the deleted edges from GG is at most q [0, n]. Notice that p q is the edges left in the final strctre, which is at least n 1 and at most n 6. Thirdly, in the marking process described in Algorithm, the commnication cost of broadcasting its incident edges (or its neighbors) and pdating link stats are both (p q). Therefore the total y commnication cost is n + 4p q [5n, 1n]. Then the following theorem directly follows. Theorem 7. Assming that both the ID and the geometry position can be represented by log n bits each, the total nmber of messages dring constrcting the strctre LSΘGG is in the range of [5n, 1n], where each message has at most O(log n) bits. Compared with preios known low-weighted strctres [0, ], LSΘGG not only achiees more desirable properties, bt also costs mch less messages dring constrction. To constrct LSΘGG, we only need to collect the information E (x) which costs at most 6n messages. Or algorithm can be generally applied to any known degree-bonded planar spanner to make it low-weighted while keeping all its preios properties, except increasing the spanning ratio from ρ to ρ + 1 theoretically. 5. EXPECTED INTERFERENCE IN RANDOM NETWORKS This section is deoted to stdy the aerage physical node degree of or strctre when the wireless nodes are distribted according to a certain distribtion. For aerage performance analysis, we consider a set of wireless nodes distribted in a two-dimensional nit sqare region. The nodes are distribted according to either the niform random point process or homogeneos Poisson process. A point set process is said to be a niform random point process, denoted by X n, in a region Ω if it consists of n independent points each of which is niformly and randomly distribted oer Ω. The standard probabilistic model of homogeneos Poisson process is characterized by the property that the nmber of nodes in a region is a random ariable depending only on the area (or olme in higher dimensions) of the region. In other words, The probability that there are exactly k nodes appearing in any region Ψ of area A is (λa)k k! e λa. For any region Ψ, the conditional distribtion of nodes in Ψ gien that exactly k nodes in the region is joint niform. Definition. Gien a strctre H, the adjsted transmission range r H() is defined as max H, i.e., the longest edge of H incident on. The physical node degree of in H is defined as the nmber of nodes inside the disk disk(, r H ()). The node interference, denoted by I H (), of a node in a strctre H is simply the physical node degree of. The maximm node interference of a strctre H is defined as max I H (). The aerage node interference of a strctre H is defined as P I H()/n. Theorem 8. For a set of nodes prodced by a Poisson point process with density n, the expected maximm node interference of any connected strctre, e.g., EMST, GG, RNG, Yao and LSΘGG, is at least Θ(log n). Proof. Let d n (H) be the longest edge of a strctre H of n points placed independently in -dimensions according to standard poisson distribtion with density n. Obiosly, d n (EMST ) is the smallest among all connected strctres H. For simplicity, let d n = d n (EMST ). In [7], they showed that lim n Pr(nπd n log n α) = e e α.