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208 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 22, NO. 2, FEBRUARY 2011 Maximizing the Number of Broadcast Operations in Random Geometric Ad Hoc Wireless Networks Tiziana Calamoneri, Andrea E.F. Clementi, Emanuele G. Fusco, and Riccardo Silvestri Abstract—We consider static ad hoc wireless networks whose nodes, equipped with the same initial battery charge, may dynamically change their transmission range. When a node v transmits with range rðvÞ, its battery charge is decreased by rðvÞ2 , where > 0 is a fixed constant. The goal is to provide a range assignment schedule that maximizes the number of broadcast operations from a given source (this number is denoted by the length of the schedule). This maximization problem, denoted by MAX LIFETIME, is known to be NP-hard and the best algorithm yields worst-case approximation ratio Âðlog nÞ, where n is the number of nodes of the network. We consider random geometric instances formed by selecting n points independently and uniformly at random from a square of side length pﬃﬃﬃ n in the euclidean plane. We present an efficient algorithm that constructs a range assignment schedule having length not smaller than 1=12 of the optimum with high probability. Then we design an efficient distributed version of the above algorithm, where nodes initially know n and their own position only. The resulting schedule guarantees the same approximation ratio achieved by the centralized version, thus, obtaining the first distributed algorithm having provably good performance for this problem. Index Terms—Energy-aware systems, wireless communication, graph algorithms, network problems. Ç 1 INTRODUCTION I N static ad hoc wireless networks, nodes have the ability to vary their transmission ranges (and thus, their energy consumption) in order to provide good network connectiv- range values. For this reason, we will assume that nodes have the ability to choose their transmission range from a finite set À ¼ f0; r1 ; r2 . . . ; rk g (with 0 < r1 < r2 < Á Á Á < rk ) ity and low energy consumption at the same time. More that depends on the particular adopted technology (see [7], precisely, the transmission ranges determine a (directed) [8], [24]). Clearly, the maximal range value rk in À must communication graph over the set V of nodes. Indeed, a be sufficiently large to guarantee that at least one feasible node v, with range r, can transmit to another node w if and solution exists. Further technical constraints on À will be only if w belongs to the disk of radius r centered in v. The given and discussed in Section 2. transmission range of a node depends, in turn, on the energy power supplied to the node. In particular, the power 1.1 Range Assignments in Ad Hoc Wireless Pv required by a node v to correctly transmit data to another Networks station w must satisfy the inequality (see [24]): A fundamental class of problems, underlying any phase of a dynamic resource allocation algorithm in ad hoc wireless Pv ! ; networks, is the one known as range assignment problems. distðv; wÞ2 Given a specific graph connectivity property Å, the where distðv; wÞ is the euclidean distance between v and w, objective of these problems is to find a transmission range while is a constant that, without loss of generality, can be assignment r : V ! À such that: 1) r induces a communica- fixed to 1. tion graph satisfying Å and 2) its overall cost In several previous theoretical works [1], [10], [17], [22], it X is assumed that nodes can arbitrarily vary their transmis- costðrÞ ¼ rðvÞ2 ; sion range over the set fdistðv; wÞ j v; w 2 V g. However, in v2V some network models (like sensor networks), the adopted required to deploy the assignment [17], [22], is minimized. technology allows to have only few possible transmission Several research works [1], [10], [17] have been devoted to the case where Å requires the communication graph to . T. Calamoneri, E.G. Fusco, and R. Silvestri are with the Dipartimento di contain a directed spanning tree rooted at a given source ` Informatica, Sapienza Universita di Roma, Via Salaria 113, 00198 Roma, s 2 V (a broadcast tree from s). The relevance of this Italy. E-mail {calamo, fusco, silvestri}@di.uniroma1.it. problem, denoted by MIN ENERGY BROADCAST, is due to ` . A.E.F. Clementi is with the Dipartimento di Matematica, Universita degli studi di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, the fact that any communication graph satisfying the above Italy. E-mail clementi@mat.uniroma2.it. property allows the source to perform a broadcast operation. Manuscript received 7 Jan. 2009; revised 21 Apr. 2009; accepted 25 Sept. Broadcast is a task initiated by the source aiming to send a 2009; published online 2 Apr. 2010. message to all nodes. This task constitutes a fundamental Recommended for acceptance by S. Olariu. operation in real-life multihop wireless networks [2], [3], For information on obtaining reprints of this article, please send e-mail to: tpds@computer.org, and reference IEEECS Log Number TPDS-2009-01-0005. [17]. MIN ENERGY BROADCAST is known to be NP-hard Digital Object Identifier no. 10.1109/TPDS.2010.77. even when jÀj ¼ 3 and r1 is a small positive constant [10]. 1045-9219/11/$26.00 ß 2011 IEEE Published by the IEEE Computer Society CALAMONERI ET AL.: MAXIMIZING THE NUMBER OF BROADCAST OPERATIONS IN RANDOM GEOMETRIC AD HOC WIRELESS NETWORKS 209 A series of constant-factor approximation algorithms is 1.3 Our Results available in the literature (see, e.g., [1], [4], [10], [19]). The To the best of our knowledge, previous analytical results on best known approximation factor is close to 4 and it is given MAX LIFETIME concern worst-case instances only. Some in [6]. A more general version of MIN ENERGY BROADCAST experimental studies on MIN ENERGY BROADCAST have is studied in [5], where a nonuniform node efficiency function been done on random geometric instances [11], [19]. Such e : V ! Rþ is considered. Hence, the energy cost required to input distributions turn out to be very important in the transmit from node v to w is given by distðv; wÞ2 =eðvÞ. This study of range assignment problems. On one hand, they nonsymmetric version of MIN ENERGY BROADCAST seems represent the most natural random instance family, where to be harder: the best known algorithm is given in [5] and greedy heuristics (such as the MST-based one, see [17]) have yields approximation ratio Âðlog nÞ. a bad behavior [19]. On the other hand, random geometric distributions provide a good model for well-spread networks 1.2 The MAX LIFETIME Problem located on two-dimensional regions [7], [8], [17], [21]. The MIN ENERGY BROADCAST problem does not consider We study MAX LIFETIME in random geometric instances some important ad hoc wireless network scenarios, where of arbitrary size: the set V is formed by n nodes selected nodes are equipped with batteries of limited charge: the uniformly and independently at random from the two- goal here is to maximize the number of broadcast pﬃﬃﬃ dimensional square of side length b nc. Such instances will operations. Indeed, in a network where each device has its own battery, minimizing only the overall power be simply denoted by random sets. Note that the maximal consumption can cause early power drains in few key euclidean distance between two nodes in random sets is pﬃﬃﬃﬃﬃﬃ nodes, hence, disconnecting the network. This important 2n, soﬃﬃﬃﬃﬃﬃ maximal range value rk can be assumed to be at p the range assignment problem has been first analytically most 2n. studied in [5] and it is the subject of our paper. A natural and important open question is to establish Time is divided into (time) periods. Period t is devoted to whether efficiently constructible range assignment sche- broadcast the tth message from the source s. All nodes are dules exist for MAX LIFETIME having provably good length initially equipped with the same battery charge B > 0. on random sets. Moreover, the design of efficient distributed A range assignment schedule S is a sequence of range implementations of such schedules is of particular rele- assignment frt : V ! À; t ¼ 1; . . . ; mg. vance in ad hoc wireless networks. The length m of a range assignment schedule is the To this aim, as a first step, we provide an upper bound number of periods. At every period t, the battery charge of on the length of an optimal range assignment schedule S for each node v is reduced by amount rt ðvÞ2 , where rt ðvÞ any finite set V in the two-dimensional plane. Note that this denotes the range assigned to node v during t and > 0 is a upper bound holds for any instance, not only for random fixed constant depending on the adopted technology. So, a sets. When V is a random set, we present an efficient range assignment schedule is said to be feasible if, at any centralized algorithm that, with high probability, returns a period t, rt yields a broadcast tree from s, and for any v 2 V , feasible schedule of length which is not smaller than 1=12 of it holds that the optimum. Here and in the sequel, the term with high probability means that the event holds with probability at X m least 1 À 1=nc for some constant c > 0. rt ðvÞ2 B: t¼1 We then exploit our algorithm in order to design a fully distributed protocol for MAX LIFETIME. The protocol acts in In this paper, we assume that ¼ 1; however, all our results discrete time slots1 and assumes that every node initially can be easily extended to any > 0. The MAX LIFETIME knows n, its unique label in ½1; . . . ; n (our results remain problem requires to find a feasible range assignment valid even if labels are chosen in ½1; . . . ; N, where schedule of maximal length. N 2 OðnÞ), and its euclidean position. In [5], MAX LIFETIME is shown to be NP-hard. In the This assumption is reasonable in static ad hoc wireless same paper, by means of a rather involved reduction to networks since the node position can be either stored in the MIN ENERGY BROADCAST with nonuniform node effi- node during the deployment phase or it can be locally ciency, a polynomial-time algorithm is provided, yielding computed using a GPS system in a setup phase. This approximation ratio Âðlog nÞ. This positive result also holds operation is not too expensive in terms of energy consump- when the initial node battery charges are not uniform. tion since it is performed only once during the setup phase. A static version of MAX LIFETIME has been studied in We then show that the resulting schedule is equivalent to [21]: the broadcast tree is fixed during the entire schedule the one yielded by the centralized version, and hence, when and the quality of solutions returned by the MST-based applied to random sets, it achieves, with high probability, algorithm is investigated. Such results and techniques are constant approximation ratio as well. We thus get the first not useful for solving MAX LIFETIME problem, as the distributed algorithm for MAX LIFETIME having provably broadcast tree may change at each period. good performance. Several other problems concerning network lifetime have We remark that in the analysis of our protocol, we been studied in the literature [7], [8], [21]. Their definitions consider both the costs due to the construction of the range vary depending on the particular node technology (i.e., assignment schedule and that due to its use for the broadcast fixed or adjustable node power) and on the required operations. Furthermore, our protocol is designed to take connectivity or covering property. However, both results and techniques (most of them being experimental) are not 1. Thus, periods are, in turn, divided into discrete time slots: a full related to ours. description of the distributed model is given in Section 5. 210 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 22, NO. 2, FEBRUARY 2011 care about message collisions yielded by the interference the square grows. In this paper, we adopt the former problems, thus, no cost is hidden in the analysis. approach to simplify calculations; nevertheless, our results The MAX LIFETIME problem does not consider the goal can be easily translated to suit the latter approach—by of minimizing the completion time of the broadcast opera- means of a unit conversion—and thus, handle networks of tions, i.e., the number of time slots required to get all the growing sizes where nodes have fixed ranges. nodes informed (a node is said to be informed if it has Assumptions on range set À. We recall that pﬃﬃﬃﬃﬃﬃ À¼ received the source message). However, the completion f0; r1 ; r2 . . . ; rk g is such that 0 < r1 < r2 < Á Á Á < rk 2n. time is clearly an important measure to evaluate the In addition, we assume that 1 r1 < CTðnÞ. This condition efficiency of a solution. We thus provide an analysis of is motivated by our choice of studying random sets. Indeed, the amortized completion time of each broadcast operation define Cs as the connected component containing s in the yielded by our protocol. If the length of the range assign- disk graph GðV ; r1 Þ. If r1 ! CTðnÞ, then MAX LIFETIME on ment schedule generated by the protocol is T , then the random sets admits a trivial schedule which is, with high amortized completion time is given by the overall number probability, optimal: Since the source must transmit in of elapsed time slots divided by T . It turns out that our every period with range at least r1 and Cs , with high protocol has amortized completion time probability, contains all nodes, then an optimal schedule is pﬃﬃﬃ pﬃﬃﬃ r2 n n n obtained by assigning range r1 to all other nodes at every O þ r2 þ : T 2 r2 period. This motivates our assumption on r1 . The other values in À can be arbitrarily chosen provided Assume that r2 2 À is close to the connectivity threshold of that all of them are not smaller than CTðnÞ and at least one of randompﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ geometric graphs [16], [20], [25], [26], i.e., pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ them is larger than 2 2c log n, where c > is a small r2 ¼ Âð log nÞ. Then, the worst scenario for our protocol is constant that will be defined later in Lemma 4.1. Informally when the initial battery charge B is very small so that T is speaking, we require that at least one value in À is a bit small as well. Indeed, if T 2 Oð1Þ, from thepﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ formula, previous larger than the connectivity threshold. This is reasonable amortized completion time Oðn n log nÞ, which is we get anpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ and relevant in energy problems related to random a factor n log n larger than the best known distributed broadcasting time, i.e., OðnÞ [16]. However, this protocol geometric wireless networks since this value is the minimal does not take into account node energy costs, and thus, the one achieving global connectivity with high probability. lifetime of the network. Our protocol, instead, trades Further discussion on such assumptions can be found in completion time of each broadcast operation with global Section 6. network lifetime. This fact clearly arises whenever B is large pﬃﬃﬃ nÞ number of broadcast operations: enough to allow T 2 ð pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 THE UPPER BOUND in this case, we get Oðn log nÞ amortized completion time, In this section, we provide an upper bound on the length of which is very close to the completion time of the best known distributed broadcast algorithm. any feasible range assignment schedule for a set V . Lemma 3.1. Given a set V and a source s 2 V , it holds that optðV ; sÞ B=r2 . Furthermore, if the size k1 of Cs is less 2 PRELIMINARIES 1 than n, then A random set V is formed by n nodes selected uniformly and independently at random from the square Q of side length optðV ; sÞ pﬃﬃﬃ & ' n. The source node s can be any node in V . The length of a B BÀ 2 2 2 Á maximum feasible range assignment schedule (in short, min 2 4 k1 r2 þ r1 À k1 r1 : r1 r2 schedule) for an input ðV ; sÞ is denoted by optðV ; sÞ. Given a set V of n nodes in the two-dimensional euclidean Proof. Since the source must transmit with range at least r1 plane and a positive real r, the disk graph GðV ; rÞ is the at any period, the first upper bound follows easily. symmetric graph where edge ðu; vÞ exists iff distðv; wÞ r. If k1 < n, then consider any feasible range assignment When V is a random set, the resulting disk graph distribution schedule S. Let l1 and l2 be the number of periods, where is known as geometric random graphs, an important and deeply the source transmits with range r1 and with range at least studied model in wireless networks [16], [20], [25], [26]. It is r2 , respectively. It must hold that known that for sufficiently large n, a random geometric graph GðV ; rÞ is connected with high probability if and only if l1 r2 þ l2 r2 1 2 B: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ for r ! log n, where ¼ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ any constant > 0 [20], [25], Since k1 < n, then, in each of the l1 periods of S, there is [26]. The value CTðnÞ ¼ log n is known as the connectivity at least one node in Cs À fsg having radius at least r2 . threshold of random geometric graphs. This yields Note that the connectivity threshold grows when the area of the square grows as a consequence of the l1 r2 ðk1 À 1ÞB: 2 assumption of having a fixed node density for squares of any size. A more realistic assumption would be to consider The maximum value of l1 þ l2 is achieved when l1 ¼ node ranges as fixed. As a consequence, higher node ðk1 À 1ÞB=r2 and l2 ¼ B=r2 À ðk1 À 1ÞBr2 =r4 . As l1 þ l2 is 2 2 1 2 densities would be required in order to maintain the the length of the schedule, we obtain the following upper network connected with high probability while the area of bound on the number of periods of S: CALAMONERI ET AL.: MAXIMIZING THE NUMBER OF BROADCAST OPERATIONS IN RANDOM GEOMETRIC AD HOC WIRELESS NETWORKS 211 Fig. 1. Execution example for algorithm BS. The source message is sent through nodes in Cs to the first pivot, using range r1 . When a pivot transmits with range r2 , all nodes in its cell and in the neighboring cells receive the message. l1 þ l2 let Cs be the connected component in GðV ; r1 Þ that & ' B BÀ 2 2 2 Á contains s; min 2 ; 4 k1 r2 þ r1 À k1 r1 : if jCs j r2 then r1 r2 2 u t Ws Cs ; else Note that if V is a random set, then, since r1 < CTðnÞ, it Ws any connected subgraph of Cs s.t. jWs j ¼ r2 and 2 holds with high probability k1 < n. s 2 Ws ; construct an arbitrary ordering of Ws ; 4 THE ALGORITHM for any period t ¼ 1; . . . do if node with index t mod jWs j in Ws has remaining In this section, we present a simple and efficient algorithm for battery charge at least r2 then 2 MAX LIFETIME and then analyze its performance. For the it is selected as pivot and range r2 is assigned to it; of sake pﬃﬃﬃ simplicity, we restrict ourselves to the case pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ else r2 ! 2 2c log n. Nevertheless, it is easy to extend all our the algorithm stops; results to the more general assumption described in Section 2. for any cell Qj do The following algorithm partitions Q into square cells if node with index t mod jVj j in Qj has remaining and selects, for every period, a set of pivots, i.e., nodes battery charge at least r2 then 2 having assigned range r2 . Each set of pivots is responsible it is selected as pivot and range r2 is assigned to it; for spreading the message of its period. This message is else delivered to one of the pivots from the source by means of the algorithm stops; transmissions with range r1 , thus, exploiting the subgraph all nodes in Ws not selected yet have radius r1 ; Cs . Observe that we here assume every node knows the all nodes in V n Ws not selected yet have range 0. positions of all the other nodes and the cell partition: the algorithm is thus centralized: In order to analyze the performance of Algorithm BS, we will use the following lemma whose proof is a simple See Fig. 1 for an example of execution of Algorithm BS. application of Chernoff’s bound (an alternative proof can Algorithm 1. BS (Broadcast Schedule) also be obtained from [18, Lemma 1]): Input: Set V Q of n nodes; a source s 2 V ; a battery Lemma 4.1. There exist two positive constants, c and , such that charge B > 0; the range set À ¼ f0; r1 ; r2 . . . ; rk g. the following holds. Given a random set V Q of n nodes, and Output: A range-assignment schedule S. pﬃﬃﬃ partition of pﬃﬃﬃinto square cells of side length ‘, where apﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Q Partition Q into square cells of side length r2 =ð2 2Þ; c log n ‘ n, every cell contains at least ‘2 nodes with for any cell Qj , let Vj be the set of nodes in Qj ; high probability. The constants can be set as c ¼ 12 and construct an arbitrary ordering in Vj ; ¼ 5=6. 212 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 22, NO. 2, FEBRUARY 2011 Proof. Given a fixed cell, let Xi be the random variable that During the schedule, every node v in Ws will have range gets the value 1 if node i falls into the cell and 0 either r1 or r2 . Let jWs j ¼ k, then the energy spent by v is otherwise. Observe that: 1) the event Xi ¼ 1 has at most probability ‘2 =n and 2) the random variables Xi s, 1 i n, are independent. T 8T P þ r2 þ T r2 : ð2Þ Let X ¼ n Xi . The expected value of X, i.e., the i¼1 k r2 22 1 average number of nodes in a cell of side ‘, is E½X ¼ ‘2 . Indeed, in (2), we have considered that a node in Ws can By applying Chernoff’s bound [23], we get have range r2 because it has been selected as pivot either Pr½X < ‘2 < eÀ‘ ð1À Þ =2 : 2 2 of its cell or of Ws . pﬃﬃﬃ Now, two cases may arise: There are at most b n=‘c2 n=‘2 cells fully contained in Q, and thus, the probability of having one such cell with . If k ! ðr2 Þ2 , since r1 ! 1, from (2), the amount of r1 less than ‘2 nodes is at most ðn=‘2 ÞeÀ‘ ð1À Þ =2 by 2 2 spent energy is at most 2T r2 þ 8= T r2 ð2 þ 1 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8= Þ. We require B to exceed the latter value, so T applying the Union Bound. As ‘ ! c log n, this prob- ability is at most can be any value such that 2 log nð1À Þ2 =2þlnðn=ðc2 log nÞÞ B eÀc < T : ð3Þ r2 ð2 1 þ 8= Þ 2 2 log nð1À Þ =2þlogðn=ðc log nÞÞ 2 < eÀc : Observe that every value T that satisfies (3) also pﬃﬃﬃﬃﬃﬃﬃﬃ B By setting ¼ 5=6 and c ! 144 ¼ 12, we obtain satisfies (1). So T can assume value r2 ð2þ8= Þ , and 1 from Lemma 3.1, we have that Pr½X < ‘2 < optðV ; sÞ < eÀ144=72 log nþlogðn=ð144 log nÞÞ T! : 2 þ 8= eÀ2 log nþlogðn=ð144 log nÞÞ < 1=n: u t . If k < ðr2 Þ2 , according to the definition of Ws , we r1 have k ¼ k1 . From (2) and some simple calcula- tions, the energy spent by v 2 Ws is at most Theorem 4.2. Let V Q be a random set of n nodes and s 2 V be any source node. Then, with high probability, the range r4 þ k1 r2 r2 þ ð8= Þk1 r2 2 1 2 2 T ; assignment schedule returned by Algorithm BS is feasible and r2 k1 þ r2 À k1 r2 2 1 1 it has length at least optðV ; sÞ, where ¼ 1=12. where we used the fact that r2 À k1 r2 0. Observe 1 1 Proof. Let us consider any period of the algorithm’s also that, since k1 < ðr2 Þ2 and r1 ! 1, we have r1 schedule. The component Ws is not empty since it contains at least s. Hence, it contains a pivot which is k1 r2 r2 þ ð8= Þk1 r2 1 2 2 connected to s by a path using ranges of size r1 only. From Lemma 4.1, all cells are nonempty with high 4 8 8 r2 1 þ 2 r4 1 þ 2 : probability. So, a pivot is selected in every cell with high r1 probability. This implies that, with high probability, the set of pivots forms a strongly connected subgraph It thus follows that the energy spent by v is at whose r2 -disks cover all nodes in V . Moreover, Algo- most rithm BS assigns, to every node, an energy power which r4 ð2 þ 8= Þ is never larger than the current battery charge of the T 2 : 2 k þ r2 À k r2 r2 1 node. So the range assignment schedule is feasible, with 1 1 1 high probability. Hence, T can be any value such that We now evaluate the length T of the scheduling produced by Algorithm BS. Observe that T equals the r2 k1 þ r2 À k1 r2 2 1 1 index t of the last period performed by Algorithm BS on T B: ð4Þ input ðV ; sÞ. r4 ð2 þ 8= Þ 2 Let w be any node in V n Ws ; then, from Lemma 4.1, in its cell, there are at least r2 =8 nodes with high Similar to the previous case, every value T that 2 probability. So, w spends at most energy satisfies (4) also satisfies (1). Finally, by combin- ing (4) and Lemma 3.1, we get again 8T r2 : optðV ; sÞ r2 2 2 T ! : 2 þ 8= Hence, T can be any value such that So, the theorem is proved for ¼ 1=ð2 þ 8= Þ > B T : ð1Þ 1=12. t u 8 CALAMONERI ET AL.: MAXIMIZING THE NUMBER OF BROADCAST OPERATIONS IN RANDOM GEOMETRIC AD HOC WIRELESS NETWORKS 213 We conclude this section by observing that when every Observe that at the end of the Preprocessing phase, s has node has full knowledge of the node positions, the time full knowledge of Ws . pﬃﬃﬃ complexity of Algorithm BS is Oðr2 þ T Á ð n=r2 ÞÞ. Indeed, 2 Broadcast operations: the organization of the nodes according to the cell partition for t ¼ 0; 1; . . . =Ã periods Ã = do and the construction of an arbitrary order of the nodes in Execute Procedure BROADCAST(mt ) every cell require linear time. The construction of Ws can be performed by a standard graph search, so it can be done in Procedure BROADCAST(mt ) Oðnr2 Þ time. Finally, for each period t, constant time is 2 required to activate the right pivot in every cell. The activation Nodes in Ws only: of those cells having the same r2 -hop distance from the source can be done in parallel. Since the maximum hop distance from s selects the ðt mod minfjWs j; r2 gÞ-th node in Ws as pﬃﬃﬃ 2 the source to any cell is Oð n=r2 Þ, we get the bound. pivot (range r2 will be assigned to it); s transmits, with range r1 , hmt ; P i where P is the 5 THE DISTRIBUTED VERSION path in Tree from s to the pivot. When a node in Ws receives hmt ; P i, it checks In this section, we present the distributed version of whether its label is the first in P . If this is the case, Algorithm BS. it transmits, with range r1 , hmt ; P 0 i where P 0 is the According to the standard radio communication model residual path to the pivot. [2], [13], [15], we assume that nodes act in discrete uniform time slots and are nonspontaneous (but the source, the other When the selected pivot p of Ws receives nodes are activated when they get the source message). hmt ; P ¼ ðpÞi, it transmits, with range r2 , hmt ; ii However, we assume a weaker, local synchronous model: if, where i is the index of its cell. at a given time slot t, the range of a message transmission covers a cell, then, at time slot t þ 1, the nodes of that cell All nodes: (included the ones in Ws ) are activated, and so, they will agree on the same time. We assume that every node v knows the number n of points, the If ðt ð =8Þr2 Þ then 2 range À, its own unique label, and its relative coordinates in - When a node v receives for the first time hmt ; ii the square grid Q. in the period t from the pivot of a neighbor cell Our protocol is based on the same partition in cells of i, it becomes active. Algorithm BS. Cells are numbered with consecutive integers. - An active node, at every time slot, increments The aim of our protocol is to replicate the behavior of counter by one and checks whether its label is Algorithm BS in a distributed fashion. In order to do so, equal to the value of its counter. If this is the nodes need to acquire some knowledge of the network case, it becomes the pivot of its cell and within minimal cost in terms of energy, and as we shall see, transmits, with range r2 , hmt ; ii where i is the time costs. index of its cell. Let hðWs Þ be the eccentricity of the source s in Ws , i.e., - When an active node in cell i receives hmt ; ii, it the maximum distance between s and a node in Ws . The tth (so the pivot as well) records in P ½t the current message sent by s is denoted by mt . We assume that mt value of counter c, i.e., the label of the pivot, contains the value of period t. The protocol is described in and becomes inactive. Algorithm 2. else (i.e., ðt > ð =8Þr2 Þ) 2 - When a node v receives for the first time hmt ; ii Algorithm 2. DBS (Distributed Broadcast Schedule) in period t from the pivot of a neighbor cell i, it Preprocessing: construction of Ws Cs such that checks if its label is equal to P ½t mod ð =8Þr2 . If hðWs Þ r2 2 2 this is the case, it becomes the pivot of its cell One-to-All and transmits, with range r2 , hmt ; ji where j is Starting from s, use round robin and range transmission r1 the index of its cell. to inform all nodes in Cs that are at most within r2 hops 2 from s: such nodes will form Ws . Our protocol has the following properties that are a key- The one-to-all operation induces a spanning tree Tree of ingredient in the performance analysis: Ws rooted at s. Fact 5.1. Even though they initially do not know each other, all All-to-One nodes in the same cell are activated (and disactivated) at the By a simple bottom-up process on Tree and using round same time slot, so their local counters share the same value at robin on each level, s collects all node labels and the every time slot. Furthermore, after the first ð =8Þr2 broadcast 2 structure of Tree. operations, all nodes in the same cell know the set P of pivots Initialization: of their cell and the relative order of its elements. Every node sets a local counter counter ¼ À1. Furthermore, each node has a local array P of length Proof. A node in a cell is activated when it receives a ð =8Þr2 where it will store the ordered list of the first 2 message from the pivot of a neighboring cell and is set ð =8Þr2 labels belonging to its own cell. This array is 2 inactive when it receives the message sent by the pivot of initially empty. its cell. Each point in a cell is within distance r2 from all 214 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 22, NO. 2, FEBRUARY 2011 points in the same cell and in the neighboring cells; further messages that are sent are the ones required to moreover, the pivots transmit with range r2 . So activa- reach the elected pivot in Ws : the overall number of them tions and deactivations of nodes in a cell happen in the is bounded by r2 . 2 t u same time slots. This proves the first part of the claim. Theorem 5.5. The overall number of time slots required by The set of pivots is learned by all nodes in a cell as a Algorithm DBS to perform T broadcast operations is consequence of the transmissions made by the pivots. À pﬃﬃﬃ À pﬃﬃﬃ ÁÁ Since this set has size bounded by ð =8Þr2 , the proof of 2 O r2 n n þ T Á r2 þ n=r2 : 2 the claim is completed. t u Proof. For a single broadcast operation performed by More practically, the above claim implies that if l0 < l1 < Algorithm DBS, we define the delay of a cell as the l2 < Á Á Á < lt are the labels of the nodes in a cell, then, during number of time slots from its activation time to the the first ð =8Þr2 broadcast operations (i.e., periods), the 2 selection of its pivot. Observe that the sum of delays pivot of the cell at period t will be the node having label lt . introduced by a cell during the first ð =8Þr2 broadcasts is 2 In order to evaluate the length of the broadcast schedule at most n À ð =8Þr2 . Indeed, when a (new) pivot 2 yielded by Algorithm DBS, observe that the distributed transmits, a new entry in the pivot-array is determined; version performs, in parallel, two tasks: 1) it constructs a while in every time slot there is silence, nodes in the cell broadcast communication subgraph starting from the learn that the node with the corresponding label is not in source and 2) transmits the source message along this the cell. As each cell contains at least ð =8Þr2 nodes with 2 subgraph to all nodes. In our next analysis, all node costs high probability, the delay introduced by each cell before due to both the above tasks are taken into account: completing the set is with high probability at most whenever a node transmits any message with range r, its n À ð =8Þr2 . Once the set of pivots is completed (i.e., after 2 battery charge is decreased by r2 . the first ð =8Þr2 periods), the delay of every cell becomes 2 The following lemma states the equivalence between the 0 for all the remaining broadcasts. Moreover, a broadcast performance of Algorithm BS and that of Algorithm DBS: pﬃﬃﬃ can cross at most Oð n=r2 Þ cells (as the side length of pﬃﬃﬃ Lemma 5.2. Given a random set V Q and any source s 2 V , if each cell is Âðr2 Þ, while the diameter of Q is Âð nÞ). By the length of the broadcast schedule yielded by Algorithm BS is assuming the worst scenario, i.e., a maximal length cell pﬃﬃﬃ T , then the length of the broadcast schedule yielded by path (this length being Âð n=r2 Þ) together with maximal Algorithm DBS is at least T À 2. cell delay can be found in each of the first Proof. Note that the only difference in terms of power minfð =8Þr2 ; T g broadcasts, we can bound the maximal 2 pﬃﬃﬃ consumption between Algorithms BS and DBS lies in the overall delay with Oðr2 n nÞ time slots. Preprocessing phase required by the latter one. In that In the Preprocessing phase, Algorithm DBS uses phase, at most two messages with range r1 are sent by a round-robin to avoid collisions. During the All-to-One node to discover Ws . Hence, in the worst case, the phase, each node needs to collect all messages from its distributed version performs two broadcasts less than children before sending a message to its parent in Tree. the centralized algorithm. Note that, due to Fact 5.1, the if Hence, the whole phase is completed in Oðnr2 Þ time 2 branch of the Broadcast procedure spends time instead of slots, as the height of Tree is bounded by r2 .2 power in order to discover the set of Pivots of each cell. t u Finally, the number of time slots required by every broadcast without delays and Preprocessing time is pﬃﬃﬃ By combining the above lemma and Theorem 4.2, we Oðr2 þ n=r2 Þ, since r2 is the upper bound on the height 2 2 easily get the following: of Tree and the length of any path on the broadcast tree pﬃﬃﬃ outside Ws is Oð n=r2 Þ. Corollary 5.3. Let V Q be a random set of n nodes and s 2 V By combining the three contributions, we get the be any source node. Then, with high probability, the range theorem bound without considering collisions among assignment schedule yielded by Algorithm DBS is feasible and pivots of adjacent cells. In order to avoid such collisions, it has a length at least optðV ; sÞ À 2, where ¼ 1=12. we further organize Algorithm DBS into iterative phases: in every phase, only cells with not colliding pivot We now evaluate message and time complexity of transmissions are active. Since the number of cells that Protocol DBS. can interfere with a given cell is constant, this further Lemma 5.4. The overall number of node transmissions (i.e., the scheduling will increase the overall time of DBS by a message complexity) of Algorithm DBS is OðjWs j þ T Á constant factor only. This iterative process can be ððn=r2 Þ þ r2 ÞÞ, where T is the length of the schedule. 2 2 efficiently performed in a distributed way since every Proof. Observe that in the Preprocessing phase, only nodes node knows n and its position, so it knows its cell. u t in Cs exchange messages. In particular, all nodes (the source as well) in Cs within r2 hops from s send only one From Theorem 5.5, the amortized completion time of a 2 message; all other nodes, at hop distance from 1 to r2 À 1 single broadcast operation performed by Algorithm DBS is 2 from s, send two messages. It follows that the message pﬃﬃﬃ pﬃﬃﬃ r2 n n 2 n complexity of the Preprocessing phase is ÂðjWs jÞ. During O þ r2 þ : T r2 each broadcast, exactly one message per cell is sent (see Fact 5.1). As there are Oðn=r2 Þ cells, in each period, an 2 Since our protocol returns an almost maximal number T Oðn=r2 Þ number of messages are exchanged. The only 2 of broadcast operations with high probability, unless the CALAMONERI ET AL.: MAXIMIZING THE NUMBER OF BROADCAST OPERATIONS IN RANDOM GEOMETRIC AD HOC WIRELESS NETWORKS 215 available battery charge of nodes is small, the analysis we [10] A. Clementi, P. Crescenzi, P. Penna, G. Rossi, and P. Vocca, “On the Complexity of Computing Minimum Energy Consumption made at the end of Section 1.3 on the amortized completion Broadcast Subgraphs,” Proc. 18th Ann. Symp. Theoretical Aspects of time of Algorithm DBS is likely to fall in a scenario in which Computer Science (STACS), pp. 121-131, www.dia.unisa.it/ T is large enough to significantly shrink the gap between ~penna, Feb. 2001. [11] A. Clementi, G. Huiban, P. Penna, G. Rossi, and Y.C. Verhoeven, our distributed algorithms and the best known distributed “On the Approximation Ratio of the MST-Based Heuristic for the broadcasting time [16]. Energy-Efficient Broadcast Problem in Static Ad-Hoc Radio Networks,” Proc. Int’l Parallel and Distributed Processing Symp. (IPDPS ’03), vol. 222, 2003. 6 OPEN PROBLEMS [12] A.E.F. Clementi, A. Monti, and R. Silvestri, “Distributed Broadcast in Radio Networks of Unknown Topology,” Theoretical Computer In this paper, we provided efficient solutions for the MAX Science, vol. 302, nos. 1-3, pp. 337-364, 2003. LIFETIME problem on random sets. Further interesting [13] M. Chrobak, L. Gasieniec, and W. Rytter, “Fast Broadcasting and future studies should address other basic operations such as Gossiping in Radio Networks,” J. Algorithms, vol. 43, no. 2, pp. 177-189, 2002. the gossiping operation which is known to be NP-hard as [14] W. Chu, C.J. Colbourn, and V.R. Syrotiuk, “The Effects of well [5]. A more technical problem, left open by our work, is Synchronization on Topology Transparent Scheduling,” Wireless the study of MAX LIFETIME when À contains more than one Networks, vol. 12, pp. 681-690, 2006. [15] A. Czumaj and W. Rytter, “Broadcasting Algorithms in Radio positive value smaller than the connectivity threshold CTðnÞ Networks with Unknown Topology,” J. Algorithms, vol. 60, no. 2, of random geometric graphs. This case seems to be very pp. 115-143, 2006. hard since it concerns the size and the structure of the [16] A. Dessmark and A. Pelc, “Broadcasting in Geometric Radio Networks,” J. Discrete Algorithms, vol. 5, no. 1, pp. 187-201, 2007. connected components of such random graphs under the [17] A. Ephremides, G.D. Nguyen, and J.E. Wieselthier, “On the connectivity threshold [20], [25]. Construction of Energy-Efficient Broadcast and Multicast Trees in Finally, we emphasize that after the presentation of the Wireless Networks,” Proc. IEEE INFOCOM, pp. 585-594, 2000. [18] A.D. Flaxman, A.M. Frieze, and J.C. Vera, “On the Average Case conference version of this work, a new protocol for MIN Performance of Some Greedy Approximation Algorithms for the ENERGY BROADCAST has been given in [9]. This new Uncapacitated Facility Location Problem,” Proc. ACM Symp. protocol is inspired by ours and achieves provably good Theory of Computing (STOC ’05), pp. 441-449, 2005. performances on random-grid instances yielded by nonuni- [19] M. Flammini, A. Navarra, and S. Perennes, “The Real Approx- imation Factor of the MST Heuristic for the Minimum Energy form node distributions. Broadcast,” Proc. Int’l Workshop Experimental and Efficient Algo- rithms (WEA ’05), pp. 22-31, 2005. [20] P. Gupta and P.R. Kumar, “Critical Power for Asymptotic ACKNOWLEDGMENTS Connectivity in Wireless Networks,” Stochastic Analysis, Control, Optimization and Applications, pp. 547-566, Birkhauser, 1999. An extended abstract of this work has been presented at [21] I. Kang and R. Poovendran, “Maximizing Network Lifetime of OPODIS 2007. This research is partially supported by the Wireless Broadcast Ad Hoc Networks,” J. ACM Mobile Networks European Union under the Project IP-FP6-015964 AEOLUS. and Applications, vol. 10, no. 6, pp. 879-896, 2005. [22] L.M. Kirousis, E. Kranakis, D. Krizanc, and A. Pelc, “Power Consumption in Packet Radio Networks,” Theoretical Computer Science, vol. 243, pp. 289-305, 2000. REFERENCES [23] M. Mitzenmacher and E. Upfal, Probability and Computing. [1] C. Ambuehl, “An Optimal Bound for the MST Algorithm to Cambridge Univ. Press, 2005. Compute Energy Efficient Broadcast Trees in Wireless Networks,” [24] K. Pahlavan and A. Levesque, Wireless Information Networks. Proc. Int’l Colloquium Automata, Languages and Programming Wiley-Interscience, 1995. (ICALP ’05), pp. 1139-1150, 2005. [25] M. Penrose, Random Geometric Graphs. Oxford Univ. Press, 2003. [2] R. Bar-Yehuda, O. Goldreich, and A. Itai, “On the Time- [26] P. Santi and D.M. Blough, “The Critical Transmitting Range for Complexity of Broadcast in Multi-Hop Radio Networks: An Connectivity in Sparse Wireless Ad Hoc Networks,” IEEE Trans. Exponential Gap between Determinism and Randomization,” Mobile Computing, vol. 2, no. 1, pp. 25-39, Jan.-Mar. 2003. J. Computer and System Sciences (JCSS), vol. 45, pp. 104-126, 1992. [3] R. Bar-Yehuda, A. Israeli, and A. Itai, “Multiple Communication Tiziana Calamoneri received the graduate in Multi-Hop Radio Networks,” SIAM J. Computing (SICOMP), degree in mathematics in 1992 and the PhD vol. 22, no. 4, pp. 875-887, 1993. degree in computer science in 1997 from the [4] G. Calinescu, X.Y. Li, O. Frieder, and P.J. Wan, “Minimum-Energy University of Rome “La Sapienza,” Italy. Since Broadcast Routing in Static Ad Hoc Wireless Networks,” Proc. 2006, she is an associate professor in the IEEE INFOCOM, pp. 1162-1171, Apr. 2001. Department of Computer Science, University of [5] G. Calinescu, S. Kapoor, A. Olshevsky, and A. Zelikovsky, Rome “La Sapienza,” where she was also an “Network Lifetime and Power Assignment in Ad Hoc Wireless assistant professor from 2000 to 2006. Her Networks,” Proc. European Symp. Algorithms (ESA ’03), pp. 114-126, research interests include sensor networks, 2003. parallel and sequential graph algorithms, layout [6] I. Caragiannis, M. Flammini, and L. Moscardelli, “An Exponential of networks topologies and optimal routing schemes, and channel Improvement on the MST Heuristic for the Minimum Energy assignment in wireless networks. Broadcast Problem,” Proc. Int’l Colloquium Automata, Languages and Programming (ICALP ’07), pp. 447-458, 2007. [7] M. Cardei and D.Z. Du, “Improving Wireless Sensor Network Lifetime through Power Organization,” Wireless Networks, vol. 11, pp. 333-340, 2005. [8] M. Cardei, J. Wu, and M. Lu, “Improving Network Lifetime Using Sensors with Adjustable Sensing Ranges,” Int’l J. Sensor Networks, vol. 1, nos. 1/2, pp. 41-49, 2006. [9] T. Calamoneri, A. Clementi, A. Monti, G. Rossi, and R. Silvestri, “Minimum-Energy Broadcast in Random-Grid Ad-Hoc Networks: Approximation and Distributed Algorithms,” Proc. 11th ACM Int’l Symp. Modeling, Analysis and Simulation of Wireless and Mobile Systems (MSWiM), 2008. 216 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 22, NO. 2, FEBRUARY 2011 Andrea E.F. Clementi received the laurea Riccardo Silvestri received the degree in degree in mathematics in 1990 and the PhD mathematics (magna cum laude) from the degree in computer science in 1994 from the University of Rome in 1987 and the PhD degree University “La Sapienza” of Rome. From 1996 to in computer science in 1992. He is currently an 1998, he was an assistant professor at “La associate professor in the Department of Com- Sapienza.” From 1998 to 2002, he was an puter Science. In 1994, he was first appointed at associate professor at the University “Tor Ver- the University of Rome “La Sapienza.” Before gata” of Rome. Since 2002, he is a full professor joining “La Sapienza” at Rome, he worked from at the same university. His main research activity 1987 to 1988 in two Italian software companies. focuses on algorithms and complexity theory In 1991, he was a guest at the University of with special interest in randomized and/or distributed models. He Rochester, New York. His main research interests include algorithms published more than 50 papers in the most important international and computational complexity, especially randomized algorithms and journals and conferences of the area. He has visited several universities, distributed models and algorithms. He is an author of more than such as UCSD of San Diego, University of Geneva (where he was an 60 scientific papers in the main international journals and conferences. assistant professor for two years), and the French Research Institute INRIA (Sophia Antipolis). He has been the local coordinator of several national and international research projects. He has been a member of . For more information on this or any other computing topic, the Program and/or Organizing Committees of several international please visit our Digital Library at www.computer.org/publications/dlib. conferences such as RANDOM/APPROX, ARACNE, ACM DIALM, WMAN. ALGOSENSORS, the IEEE/ACM Distributed Computing in Sensor Systems, ACM-IEEE MSWIM, and SIROCCO. Emanuele G. Fusco received the graduate degree in computer science in 2005 and the PhD degree in computer science in 2009 from Sapienza, University of Rome. He is currently a research fellow in the Computer Science De- partment at Sapienza, University of Rome. His research interests include distributed algorithms and graph theory.

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