Minimum-Energy Broadcast in All-Wireless Networks: NP-Completeness and Distribution Issues

Transcription

1 Minimum-Energy Broacast in All-Wireless Networks: NP-Completeness an Distribution Issues Mario Čagal LCA-EPFL CH-05 Lausanne Switzerlan Jean-Pierre Hubaux LCA-EPFL CH-05 Lausanne Switzerlan Christian Enz CSEM CH-2007 Neuchâtel Switzerlan ABSTRACT In all-wireless networks a crucial problem is to minimize energy consumption, as in most cases the noes are batteryoperate. We focus on the problem of power-optimal broacast, for which it is well known that the broacast nature of the raio transmission can be exploite to optimize energy consumption. Several authors have conecture that the problem of power-optimal broacast is NP-complete. We provie here a formal proof, both for the general case an for the geometric one; in the former case, the network topology is represente by a generic graph with arbitrary weights, whereas in the latter a Eucliean istance is consiere. We then escribe a new heuristic, Embee Wireless Multicast Avantage. We show that it compares well with other proposals an we explain how it can be istribute. Categories an Subect Descriptors C.2. [Computer-Communication Networks]: Network Architecture an Design; C.2.2 [Computer- Communication Networks]: Network Protocols; F.2.2 [Analysis of Algorithms an Problem Complexity]: Nonnumerical Algorithms an Problems General Terms Algorithms, Design, Performance Keywors Wireless a hoc networks, minimum-energy networks, energy efficiency, broacast algorithms, istribute algorithms The work presente in this paper was supporte (in part) by the National Competence Center in Research on Mobile Information an Communication Systems (NCCR-MICS), a center supporte by the Swiss National Science Founation uner grant number ( Permission to make igital or har copies of all or part of this work for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page. To copy otherwise, to republish, to post on servers or to reistribute to lists, requires prior specific permission an/or a fee. MOBICOM 02, September 23-26, 2002, Atlanta, Georgia, USA Copyright 2002 ACM X/02/ $ INTRODUCTION In recent years, all-wireless networks have attracte significant attention ue to their potential applications in civil an military omains [7, 8,, 4]. An all-wireless network consists of numerous evices that are equippe with processing, memory an wireless communication capabilities, an are linke via short-range a hoc raio connections. This kin of network has no pre-installe infrastructure, but all communication is supporte by multi-hop transmissions, where intermeiate noes relay packets between communicating parties. Each noe in such a network has a limite energy resource (battery), an each noe operates unattene. Consequently, energy efficiency is an important esign consieration for these networks [23, 26]. The broacast communication is an important mechanism to communicate information in all-wireless networks. This is because the network escribe above can be regare as a istribute system (istribute harware + istribute control + istribute ata), where broacast is an important communication primitive. In aition, many routing protocols for wireless a-hoc networks nee a broacast mechanism to upate their states an maintain the routes between noes [2]. In this paper, we focus on source-initiate broacasting of ata in static all-wireless networks. Data are istribute from a source noe to each noe in a network. Our main obective is to construct a minimum-energy broacast tree 2 roote at the source noe. Noes belonging to a broacast tree can be ivie into two categories: relay noes an leaf noes. The relay noes are those that relay ata by transmitting it to other noes (relaying or leaf), while leaf noes only receive ata. Each noe can transmit at ifferent power levels an thus reach a ifferent number of neighboring noes. Given the source noe r, we want to fin a set consisting of pairs of relaying noes an their respective transmission levels so that all noes in the network receive a message sent by r, an the total energy expeniture for this task is minimize. We call this broacasting problem the minimum-energy broacast problem. We base our work on the so calle noe-base multicast moel [25]. In this moel there is a trae-off between reach- Throughout the paper we refer to these evices as noes. 2 In this paper, the wors energy an power are use interchangeably.

2 ing more noes in a single hop using higher power an reaching fewer noes using lower power. This trae-off is possible ue to the broacast nature of the wireless channel. This paper is organize as follows. In Section 2, we overview relate work concerning minimum-energy broacast. In Section 3, we iscuss the system moel use. In Section 4, we prove that the minimum-energy broacast problem is NP-complete an show that it cannot be approximate better then O(log ) for a general graph, where is the maximum noe egree in a network; we also give a proof of the NP-completeness of the geometric version of the minimum-energy broacast problem. Then we escribe an approximation algorithm an its istribute implementation in Section 5. Performance evaluation results are presente in Section 6. Finally we conclue in Section RELATED WORK The problem of minimizing the energy consumption of allwireless networks has receive significant attention over the last few years [, 20, 9, 3, 27, 4, 22]. We are inspire by exciting results relate to the problem of minimum-energy broacasting in all-wireless networks [23, 24, 5, 2], an in particular by the work of Wieselthier et al. [25, 26]. In this work they have introuce the noe-base multicast moel for wireless networks upon which they have built several broacast an multicast heuristics. One of the most notable contributions of their work is the Broacast Incremental Power (BIP) algorithm. The main obective of BIP is to construct a minimum-energy broacast tree roote at the source noe. It constructs the tree by first etermining the noe that the source can reach with minimum expeniture of power. After the first noe has been ae to the tree, BIP continues by etermining which uncovere noe can be ae to the tree at minimum aitional cost. Thus at some iteration of BIP, the noes that have alreay inclue some noe in the tree can aitionally increase their transmission power to reach some other yet uncovere noe. The BIP algorithm can be regare as Prim s algorithm [3] for the formation of minimum spanning trees, but with the ifference that weights, with BIP, are ynamically upate at each step. Also notice that BIP is a centralize algorithm. In [24] Wan et al. have given the first analytical results for minimum energy broacast. By exploring geometric structures of an Eucliean minimum spanning tree (MST), they have prove that the approximation ratio of MST is between 6 an 2, an the approximation ratio of BIP is between 3 6 an 2. Wan et al. have also foun that for some instances BIP fails to use the broacast nature of the wireless channel. This happens because BIP as ust one noe at each iteration, the one that can be ae at minimum aitional cost. Thus BIP, although centralize, oesn t use all available information about the network. For this reason it may en up in a broacast tree that coincies with the shortest path tree of a network graph, where the broacast nature of the meia is completely ignore. A possible approach to cope with this is to allow an algorithm to a to a tree more than one noe at each iteration, an not necessarily at minimum aitional cost. However, in this case there must be another criterion for the selection of noes in a broacast tree. Another ifficulty with BIP is that it is not obvious how to istribute it, an accoring to the authors of BIP an the authors in [24] the evelopment of istribute algorithms is the maor challenge consiering the minimum energy broacast. However, Wan et al. [24] an Wieselthier et al. [25] o not really aress this challenge. In Sections 5. an 5.2 we will escribe a possible approach to the above problems. Li et al., in another closely relate work [2], also have recognize weaknesses of BIP an propose another centralize heuristic to attack the broacasting problem. However, they haven t consiere the issue of eveloping a istribute algorithm for a minimum energy broacast. Li et al. [2] have also given a sketch proof of the NP-harness of a general version of the minimum energy broacast. A proof of NP-harness of the minimum energy broacast problem in metric space has been given by Eğecioğlu et al. [5]. However, in their interpretation of the minimum energy broacast problem, they restrict a noe to select the transmission raius only from a set of integers, which captures very few instances of the problem in metric space. Very recently, it was brought to our attention that other researchers have also stuie this same problem of minimum-energy broacasting in all-wireless networks [2, 6, 5]. In the following section, we escribe a system moel for all-wireless networks that will be use throughout the paper. 3. SYSTEM MODEL We first give a wireless communication moel an then, base on it we evelop a graph moel, which will be use to assess the complexity of the minimum-energy broacast problem an to evelop an approximation algorithm. In our moel of a wireless network, noes are stationary. We assume the availability of a large number of banwith resources, i.e. communication channels. This is so because, in this paper, we are focuse only on minimum energy broacast communication an o not consier issues like contention for the channel, lack of banwith resources etc. We also assume that noes in a network are equippe with omniirectional antennas. Thus by a single transmission of a transmitting noe, ue to the broacast nature of wireless channels, all noes that fall in the transmission range of the transmitting noe can receive its transmission. This property of wireless meia is calle Wireless Multicast Avantage, which we refer to as WMA [25]. In this moel each noe can choose to transmit at ifferent power levels, which o not excee some maximum value p 0. Let P enote the set of power levels at which a noe can transmit 3. When a noe i transmits at some power level p P, we assign it a weight, which we call a noe power, that is equal to the power at which noe i has transmitte, that is p. The connectivity of the network epens on the transmission power. Noe i is sai to be connecte to noe if noe falls in the transmission range of noe i. This link is then assigne a link cost c i, which is equal to the minimum power that is necessary to sustain link (i, ). Next we give a graph moel for wireless networks that captures important properties of wireless meia (incluing the wireless multicast avantage). An all-wireless network can be moele by a irecte graph G = (V, E), where V represents the finite set of noes an E the set of communication links between the noes. Each ege (arc) (i, ) E 3 We assume the carinality of P to be finite; this oes not reuce the generality of our approach, as this carinality can be arbitrarily large.

3 has link cost c i R + assigne to it 4, while each noe i V is assigne a variable noe power p v i. The variable noe power takes a value from the set P efine above. Initially, the variable noe power assigne to a noe is equal to zero, an is set to value p P after the noe has transmitte at p. Let V i enote the set of neighbors of noe i. Noe is sai to be a neighbor of noe i if noe falls in the maximum transmission range of noe i, which is etermine by p 0. All noes V i that satisfy c i p v i are sai to be covere by noe i. Thus, if noe i transmits at power p 0, all the noes from V i will be covere. Now that we have the moel, we stuy in etail the intrinsic complexity of the minimum-energy broacast problem in the following section. 4. COMPLEXITY ISSUES The problem of fining a minimum energy broacast tree in wireless networks appears to be har to solve [25]. For example, a simple analysis can show that given an instance of the minimum-energy broacast problem, the number of possible broacast trees is exponential in the number of noes V (when each noe can reach all other noes). This is easy to see by assigning each noe a binary variable, which inicates whether the noe transmits or not, an then by calculating the number of possible combinations of transmitters. An even more ifficult problem is obtaine when noes are allowe to transmit at P ifferent power levels. Hence, acquiring insights into the complexity of the minimum-energy broacast problem is of great importance. In what follows we give an in-epth analysis of the complexity of the minimum-energy broacast problem. Let us first briefly remin a few concepts from complexity theory [8]. The problems polynomially solvable by eterministic algorithms belong to the P class. On the other han, all the problems solvable by noneterministic algorithms belong to the NP class. It can easily be shown that P NP. Also, there is wiesprea belief that P NPṪhe theory of complexity is esigne to be applie only to ecision problems, i.e., problems which have either yes or no as an answer. Notice that each optimization problem can be easily state as the corresponing ecision problem. Informally, a ecision problem Π is sai to be NP-complete if Π NP an for all other problems Π NP, there exists a polynomial transformation from Π to Π (we write Π Π) [8]. There are two important properties of the NPcomplete class. If any NP-complete problem coul be solve in polynomial time, then all problems in NP coul also be solve. If any problem in NP is intractable 5, then so are all NP-complete problems. Presently, there is a large collection of problems consiere to be intractable. In this section, we consier the problem of minimumenergy broacast in two ifferent graph moels, namely a general graph an a graph in Eucliean metric space. In general graphs, links are arbitrarily istribute, an have arbitrarily weights chosen from the set P. This graph moel is well suite for moeling wireless networks in inoor environments. On the other han, for graphs in Eucliean metric space, the existence an the weight of the link between two noes epens exclusively on the istance between the 4 We esignate with R + strictly positive reals. 5 We refer to a problem as intractable if no polynomial time algorithm can possibly solve it. noes an their transmission levels. This graph moel fits well for outoor scenarios. 4. General graph version In the following we show that a general graph version of the minimum-energy broacast problem is intractable, that is, it belongs to the NP-complete class. Because of its similarity to the well known Set Cover problem [0], which aims at fining the minimum cost cover for a given set of noes, we call it the Minimum Broacast Cover an refer to it as MBC. A ecision problem relate to the minimum broacast cover problem can be escribe as follows: Minimum Broacast Cover (MBC) Instance: A irecte graph G = (V, E), a set P consisting of all power levels at which a noe can transmit, ege costs c i : E(G) R +, a source noe r V, an assignment operation p v i : V (G) P an some constant B R +. Question: Is there a noe power assignment vector A = [p v p v 2... p v V ] such that it inuces the irecte graph G = (V, E ), where E = {(i, ) E : c i p v i }, in which there is a path from r to any noe of V (all noes are covere), an such that i V pv i B? Notice that the above question is equivalent to asking if there is a broacast tree roote at r with total cost B or less, an such that all noes in V are inclue in the tree (covere). We prove NP-completeness of MBC for a general graph by showing that a special case of it is NP-complete. In orer to obtain a special case of MBC, we specify the following restrictions to be place on the instances of MBC. Each noe is assigne ust one power level p P at which it can transmit. Consequently, the power level assigne to each noe is either 0 (the noe oesn t transmit) of p. We call this special case Single Power MBC. We prove NP-completeness of the Single Power MBC problem by reuction from the SET COVER (SC) problem, which is well known to be NPcomplete [8]. Set Cover (SC) Instance: A set I of m elements to be covere an a collection of sets S I, J = {,..., n}. Weights w for each J, an a constant B R +. Question: Is there a subcollection of sets C that form a cover, i.e., CS = I an such that C w B? First we escribe the construction of a graph G that represents any instance of the set cover problem. The graph G has a vertex set I {v, v 2,..., v n}, that is, G consists of elements of I an set vertices v representing sets S I, J = {,..., n}. There is an ege between an element e I an a set noe v i if the set S i contains the element. Each set noe v i is assigne the weight w i of the set S i the noe represents. All other noes an all eges are not weighte, that is, they have zero weight. Thus, G = (V, E) is a bipartite graph, as is illustrate in Figure a. The transformation from SC to Single Power MBC first consists in aing a source (root) noe r to G an making it aacent to all the set noes v. Then, a zero weight is assigne to the root noe r while all other weights are kept the same. The resulting graph, which we enote with G b = (V b, E b ), is illustrate in Figure b. It is easy to see that the transformation can be one in polynomial time. Notice that without any loss of generality we can use unirecte graphs for our purposes. This is because we can easily transform an unirecte graph to a irecte one by simply exchanging

4 each unirecte ege with two eges irecte in opposite irections. v v n e e 2 e m (a) v r v n e e 2 e m (b) Figure : The reuction of (a) SET COVER to (b) SINGLE POWER MINIMUM BROADCAST COVER Next we prove the following theorem. Theorem. SINGLE POWER MBC is NP-complete. Proof. The proof consists first in showing that Single Power MBC belongs to the NP class, an then in showing that the above polynomial transformation (Figure ) reuces SC to Single Power MBC. It is easy to see that Single Power MBC belongs to the NP class since a noneterministic algorithm nee only guess a set of transmitting noes (p v i > 0) an check in polynomial time whether there is a path from the source noe r to any noe in a final solution, an whether the cost of the final solution is B. We continue the proof by showing that given the minimum broacast cover C b of G b with cost cost(c b ), the set C b {r} always correspons to the minimum set cover C of G of the same cost (cost(c) = cost(c b )), an vice versa. Let C enote the minimum set cover of G. Let cost(c) = C w enote the cost of this cover. It is easy to see that all noes of G b can also be covere with total cost cost(c). This can be achieve by having the source noe r cover all the set noes v, J = {,..., n} at zero cost, an then by selecting among the covere noes those corresponing to the noes of G that satisfy v C as new transmitting noes, which we refer to as C b {r}. Therefore the minimum broacast cover of G b is C b with total cost cost(c b ) = cost(c). Conversely, suppose that we have the minimum broacast cover C b of G b with total cost cost(c b ). Then the minimum set cover C of G consists of noes corresponing to those noes of G b that satisfy v C b {r}. We prove this by contraiction. Let C enote the minimum set cover of G such that C C. In this case, by the same reasoning as before, G b can be covere by some C b that satisfies cost(c b) cost(c b ). However this contraicts the preceing assumption that C b is the minimum broacast cover of G b an conclues the proof. Since the Single Power MBC problem is a special case of the MBC problem, an MBC belongs to the NP class, which can be shown along the similar lines as for the Single Power MBC problem, we have the following corollary. Corollary. MINIMUM BROADCAST COVER (MBC) is NP-complete. We saw in the proof of Theorem that the minimum cover sets of SC an Single Power MBC iffer in only one item, namely, the source noe r. However, the weight assigne to r is zero an thus the costs of the minimum cover sets of SC an Single Power MBC are the same. Hence, the transformation from SC to Single Power MBC preserves approximation ratios that can be achieve either for SC or MBC (generalization of Single Power MBC). It is known that no polynomial-time approximation algorithm for SC achieves an approximation ratio smaller than O(log ) if P NP, where is the size of largest set S [0]. Thus, for a general graph an arbitrary weights, we cannot expect to obtain an approximation algorithm for MBC that achieves the approximation ratio better than O(log ), where represents the maximum noe egree in a graph. Fortunately, this is not necessarily true for all instances of the minimum-energy broacast problem. By exploring the geometric structure of the minimum-energy broacast problem, Wan et al. in [24] were able to show that the Eucliean minimum spanning tree approximates the minimum-energy broacast problem within a factor of 2. However, whether the geometric instances of the minimum-energy broacast problem can be solve in polynomial time was left as an open question. We provie answer in this section. 4.2 Geometric version In this section, we prove that the minimum-energy broacast problem in two-imensional Eucliean metric space is intractable. In metric space, the istance between points (noes) obey triangle inequality, that is, i ik + k, where i is the Eucliean istance between noes i an. We have seen that given the graph version of the minimumenergy broacast problem we coul have arbitrary costs of links between noes. This is because we haven t ha to worry about the istances between noes, an yet all links have been ictate by a given graph. However, in metric space links an their respective costs are ictate by the istances between noes an their transmission energies. The cost c i between two noes i an is given as c i = k α i where k R + is constant epening on the environment, i is the istance between the noe i an, an α is a propagation loss exponent that takes values between 2 an 5 [9]. We refer to this instance of the minimum-energy broacast problem as to the Geometric Minimum Broacast Cover problem an enote it with GMBC. A ecision problem relate to the GMBC problem can be formulate as follows: Geometric Minimum Broacast Cover (GMBC) Instance: A set of noes V in the plane, a set P consisting of all power levels at which a noe can transmit, a constant k R +, costs of eges c i = k α i where i is Eucliean istance between i an, a real constant α [2..5], a source noe r V, an assignment operation p v i : V (G) P an some constant B R +. Question: Is there a noe power assignment vector A = [p v p v 2... p v V ] such that it inuces the irecte graph G = (V, E), with an ege (arc) irecte form noe i to noe if an only if c i p v i, in which there is a path from r

5 to any noe of V (all noes are covere), an such that i V pv i B? We prove NP-completeness of GMBC by reuction from the planar 3-SAT problem, which is known to be NPcomplete [6]. Planar 3-SAT (P3SAT) Instance: A set of variables V = {v, v 2,..., v n} an a set of clauses C = {c, c 2,..., c m} (Boolean formulae) over V such that each c C has c 3. Furthermore, the bipartite graph G = (V C, E) is planar, where E = {(v i, c ) v i c or v i c } {(v i, v i+) i < n} 6. Question: Is there an assignment for the variables so that all clauses are satisfie? Theorem 2. GEOMETRIC MINIMUM BROADCAST COVER (GMBC) is NP-complete. Proof. The GMBC problem belongs to the NP-class for the same reason as the Single Power MBC (see the proof of Theorem ). We continue the proof by showing that P3SAT polynomially reuces to GMBC. Our proof of the NP-completeness of GMBC follows Lichtenstein s proof of the NP-completeness of the Geometric Connecte Dominating Set [6]. We encoe a Boolean formula C of P3SAT by a network representing an instance of GMBC such that given the source noe, all noes in the network can be covere at minimum cost if an only if C is satisfiable. We first escribe the Figure 3: The line that connects variables The variables are linke together by the connector shown in Figure 3. The connector passing through a variable is shown in Figure 4. The connector will follow the path taken by the arcs {(v i, v i+) i < n} from the P3SAT. The arrangement of the rhombus noes here ensures that the connector en noes transmit at k 0 an thus cover one roun an one square noe belonging to a variable triplet. The rhombus noe belonging to the variable triplet is move outsie the variable, since otherwise it woul be covere by the connector en noe, an woul not force at least one of the two nearby noes from the variable triplet to transmit Figure 2: The structure representing a variable of P3SAT structures we will be using in the rest of the proof. Let enote the istance that correspons to the maximum transmission range p 0 = max{p : p P }, that is is the farthest istance that can be reache by any noe. We encoe the variables of C by the structure shown in Figure 2. Let us call a group consisting of one roun noe, one square noe an one rhombus noe laying on the same line a variable triplet. Assume now that one variable triplet is covere. Then there are ust two ways to cover the structure representing the variable at minimum cost, specifically, either all the roun noes or all the square noes transmit. Notice that the minimum cost equals p0 times the total number 3 of noes in the variable. If all the roun or square noes transmit, this correspons to the variable being set to true or false, respectively. The rhombus noes force at least one of the two noes aacent to it to transmit. This structure can be arbitrary long. The istances (,,...) are selecte such that they ensure require properties of all the 0 40 structures we will be using in the proof. Notice that these istances are not unique in this regar. 6 We remove the ege (v n, v ) without any change in ifficulty with the problem. See [6]. Figure 4: The connecting line passing through a variable Clauses are represente by the kin of structure shown in Figure 5. Notice here that the rhombus noes only force the roun noes from the clause to transmit. For a clause c = (v v 2 v 3), one black noe is at istance of a roun noe in the structure representing the variable v, a secon one is at istance of a roun noe representing v 2, an a thir one is at istance of a square noe representing the variable v 3. It is important to emphasize that the black noes are place so that they are always in the transmission range of only a single roun (square) noe. Finally, as the source noe in an instance of the GMBC we choose one roun noe from the connector lines. Let us introuce the following notation: N var is the 3 number of noes in all the structures representing variables; N conn the number of noes that are force to transmit in all the connectors; N cls the number of noes that are force to transmit in all the clauses. Let e min = (N var + N conn + N cls + m)p 0. Recall that m is the total number of clauses an p 0 the maximum transmission power. Now that we have escribe the structure for reucing the P3SAT to GMBC, we prove that GMBC has a minimum broacast cover of the cost e min if an only if C is satisfiable. Let us assume that we have an assignment of the vari-

6 (8) (4) (5) 9 () 0 (2) 6 (5) 8 (4) (4) (5) Figure 5: The structure that encoes a clause ables from V that satisfies the Boolean formula C. Then the corresponing instance of the GMBC can be covere with e min. This can be achieve by selecting the following noes in the set of the transmitting noe. We select the roun (square) noes in variables accoring to whether the variable is true or false in the given assignment. Then we select one black noe in each clause that lies at the istance of a roun (square) noe alreay chosen. Finally, we select all the roun noes that are force to transmit by the corresponing rhombus noes in each clause an in each connector. Conversely, let us assume that we have an instance of the GMBC with a minimum cover of cost e min. We will show that in this case all the structure representing variables look right, that is, no variables switch from true to false or vice versa (i.e. some roun noes an some square noes of the same variable transmit in the same instance of the GMBC). As an immeiate consequence, all the clauses have to be satisfie, otherwise they coul not be covere. Let us assume that in the given instance of GMBC a variable switches from true to false. However, this woul incur a larger cost to cover the variable than in the case when either all the roun noes woul transmit or all the square noes (the total cost of the cover woul be e min). Consequently either all the roun noes of the variable transmit or all the square noes o, that is, the variable looks right. The builing blocks use in our construction are of polynomial size an it requires polynomial time to put all noes at consistent coorinates (i.e.,,,...). Hence, our 0 20 transformation can be one in polynomial time. This conclues the proof. We have seen that the problem of minimum energy broacast is intractable, even in two-imensional Eucliean metric space. For this reason, in the following section, we evise a heuristic algorithm that enables us to fin goo solutions to the problem at reasonable computation costs. 5. PROPOSED ALGORITHMS In this section, we will first provie the escription of a centralize heuristic algorithm. We will then show that it can easily be istribute. 5. A heuristic base approach Let us first provie an informal escription of the algorithm we propose. We begin with a feasible solution (an Figure 6: The network example an its MST (e MST = 23) initial feasible broacast tree) for a given network. Then we improve that solution by exchanging some existing branches in the initial tree for new branches so that the total energy necessary to maintain the broacast tree is lower. This is one so that the feasibility of the obtaine solution remains intact. We call the ifference in the total energies of the trees before an after the branch exchange a gain. In our heuristic, the notion of gain is use as the criterion for the selection of transmitting noes in a broacast tree. We use the link-base minimum spanning tree (MST) as the initial feasible solution. The main reason we take MST is that it performs quite well even as a final solution to our problem, which can be seen from the simulation results in Section 6. Notice that although we use link-base MST, which oesn t exploit WMA, the evaluation of its cost takes into consieration the WMA [25]. We will now escribe in etail our algorithm, which we call Embee Wireless Multicast Avantage an refer to as EWMA. An example is provie in Figure 6. Let us first introuce some notations. Let C enote the set of covere noes in a network, F the set of transmitting noes of the final broacast tree, an E the set of exclue noes. Noe i is sai to be an exclue noe if noe i is the transmitting noe in the initial solution but is not the transmitting noe in the final solution (i.e. i / F ). Notice that the contents of the above sets change throughout the execution of the EWMA, an that the sets o not hol any information about the MST. Initially, C = {r}, where r is the source noe (noe 0 in our example), an sets F an E are empty. In this example, we assume a propagation constant α = 2. After the MST has been built in the initialization phase, we know which noes in the MST are transmitting noes, an their respective transmission energies. In our example the transmitting noes are 0, 9, 6,, 8, an their transmission energies are 2, 8, 5, 4, an 4, respectively. The total energy of MST is e MST = 23. Notice here that we take into consieration the WMA in the evaluation of the cost of the MST. Notice also that C = {0}, an F = E = { }. In the secon phase, EWMA starts to buil a broacast tree from noes in the set C F E by etermining their respective gains. The gain of a noe v is efine as a ecrease

7 in the total energy of a broacast tree obtaine by excluing some of the noes from the set of transmitting noes in MST, in exchange for the increase in noe v s transmission energy. Notice that this increase of noe v s transmission energy has to be sufficient for it to reach all the noes that were previously covere by the noes that were exclue. Consequently, the feasibility of a solution is preserve. At this stage of the algorithm the set C F E contains ust the source noe 0. Thus for example, in orer to exclue noe 8, the source noe 0 has to increase its transmission energy by (see Figure 6): e 8 0 = max {e0,i} e0 = 3 2 = i {2,5} The gain (g 8 0) obtaine in this case is: 3 7 g 8 0 = e 6 + e 8 + e 9 e 8 0 = = 6 where e i, i = {6, 8, 9}, is the energy at which noe i transmits in MST. Notice that, in aition to noe 8, the noes 6 an 9 can also be exclue. Likewise, g 0 = e + e 6 + e 8 + e 9 e 0 = 5 g 6 0 = e 6 e 6 0 = 2 g 9 0 = e 6 + e 8 + e 9 e 9 0 = 6 Having the gains for all noes from C F E, our algorithm selects a noe with the highest positive gain in the set F. Our algorithm then as all the noes that this noe exclues to the set E. Thus the source noe 0 is selecte in the set F to transmit with energy that maximizes its gain, that is: e 0 = e 0 + arg max{g0}, i g e i 0 i 0 0 The source noe 0 transmits with energy e 0 = e 0 + e 8 0 = 2 + = 3 at which it can cover noes 6, 8, 9 an all their chil noes in MST. Noe is sai to be a chil noe of noe i if noe is inclue in a broacast tree by noe i. Hence, at this stage we have C = {, 2, 4, 5, 6, 7, 8, 9, 0}, E = {6, 8, 9} an F = {0}. If none of the noes from C F E has a positive gain, EWMA selects among them the noe that inclues its chil noes in MST at minimum cost (energy). The above proceure is repeate until all noes in the network are covere. In our example there is still one noe to be covere, namely noe 3. Again, EWMA scans the set C F E = {, 2, 4, 5, 7} an at last selects noe to be the next forwaring noe. When noe transmits with energy e = 4, all noes are covere (C = {, 2, 3, 4, 5, 6, 7, 8, 9, 0}) an the algorithm terminates. At the final stage we have E = {6, 8, 9} an F = {, 0}. The resulting tree, shown in Figure 7, has a cost e EW MA = 7, (e MST = 23). Notice that our algorithm always results in a broacast tree with the total energy e MST, which is, in the case of Eucliean MST, less then 2e opt [24]. The EWMA algorithm oes not perform an exhaustive search over all possible combinations of transmitting noes. Next we show that the running time of the EWMA algorithm is polynomial in the total number of noes n. Let enote the size of largest neighborhoo (i.e. the maximum noe egree), i the number of noes that are newly covere in iteration i, an m the total number of transmitting noes Figure 7: The broacast-tree obtaine by EWMA heuristic (e EW MA = 7) (i.e. F = m). Here we assume a straightforwar implementation of our algorithm. Thus, in orer for a noe to check if it can exclue some neighbor, the noe has to test all the neighbors of that neighbor, which takes O() time. Now, in orer to calculate the gain that can be attaine by this exclusion, the noe repeats the above proceure for all the remaining neighbors, which thus takes O( 2 ) time. Finally, the noe repeats all the above steps for all its neighbors an ecies to transmit with the energy that maximizes its gain, which takes O( 3 ) time. As this is repeate for, at most, all the covere noes up to an incluing some iteration i, the running time of the EWMA algorithm is boune by: m O( 3 ) i= i m O( 3 ) = = O( 3 ) O( 4 )m 2 i i= = m i Note that O( 4 )m 2 is not a tight boun on the running time of the EWMA algorithm. 5.2 Distribute implementation of EWMA One of the maor research challenges, with respect to the broacasting problem, is the evelopment of a istribute algorithm [25, 24]. In the following we escribe our solution. Let us first introuce the notation we will be using. Let noe i transmit at power level p P. We enote the set of noes that are covere by this transmission with V p i. Let noe be a neighbor of i, that is, V i. We enote with O p p i the set of noes belonging to Vi V an call it the overlapping set. We assume that each noe knows the cost of each ege aacent to itself, an the ientity of its neighbors. A noe maintains this information in a cost matrix. Once noe receives a message from noe i, it can learn which of the noes from its neighbor set V have also receive the message by calculating the overlapping set O p i. The neighbors of noe that have not yet receive the message are sai to be uncovere, an we enote this set with U where U = V O p i. If noe is a forwaring noe in the MST, then the set of yet uncovere chilren noes of noe in i=

8 the MST is enote with U mst where U mst = V mst O p i. Here, V mst is the set comprising all the chilren noes of in the MST. Finally, we enote with e mst the energy with which noe transmits in the MST. Now we escribe our istribute algorithm. The algorithm is ivie into two phases. In the first phase, all noes run a istribute algorithm propose by Gallager et al. [7] to construct a minimum-weight spanning tree. The total number of messages require for a graph of V noes an E eges is at most 5 V log 2 V + 2 E, an the time until completion is O( V log V ) [7]. Notice that Gallager et al. consiere the link-base moel, while we use the noebase multicast moel, which captures the wireless multicast avantage property [25]. As a consequence, the total number of messages require in our moel may be consierably lower. We require that at the en of the first phase, each noe has information about the cost of its two-hop neighbors relate to the built MST. This can be achieve by piggybacking information about the cost on regular messages. T max U mst V mst e e mst Label b: upon receive a msg from noe i O p i V p i if (U mst V mst U mst U mst = AND e > 0) then HALT else for all l V o calculate-gains g l g max max l {g l } if g max > 0 then e e + arg max e l {g l } O p i Tr g max else if e > 0 then Tr 2 e else HALT wait Ta T max + Tr Tr i if uring T act an before expiration of Ta same msg receive then goto b: else broacast the msg at power e Tprob Tcorr Tact i k l k l Figure 9: Upate session at noe - message reception from noe i i T r T a T r roun n roun n roun n 2 Figure 8: Synchronization of the secon phase In the secon phase, the final broacast tree is built up. The main ifficulty in this istribute setting is the unavailability of information about which noes have been covere up to a certain moment. In orer to cope with this problem we apply two techniques. First, we organize this secon phase in rouns. Secon, we require that the ientities of the noes on the transmission chain from the source to some noe an their respective transmission powers are propagate along that chain to the noe in question (source routing like technique). Each roun of the secon phase is T max long. Rouns are aitionally ivie into three time perios, namely, a probation perio (T prob ), a correction perio (T corr), an an active perio (T act), which are all known by network noes (Figure 8). Let noe i transmit at Tr i time from the beginning of the active perio of roun n. Noe receives this message an starts the upate proceure shown in Figure 9. Thus, noe calculates the overlapping set for the sener i an for other transmitters on this chain of transmitting noes for which noe has neighbors in common (recall that this information is propagate along the chain). If noe is a forwaring noe in the MST an it fins that the set of uncovere noes U mst is empty for the receive message, it will not re-broacast the message. Otherwise, (namely if U mst is non-empty or was a leaf noe in the MST), it calculates the gains it can achieve by covering yet uncovere noes (base on locally available information), an selects the maximum gain g max. In the case g max > 0, noe can contribute to the ecrease of the total cost of the broacast tree an its transmission energy increases as follows: e = e + arg max e l {g l }, otherwise (g max 0) its transmission energy remains unchange. Notice here that the leaf noes re-broacast a message only if they can achieve a positive gain. At this stage, noe waits for some time perio Ta before possibly re-broacasting the message. The waiting perio is given as follows: T a = T max + T r T i r where Tr = g max if g max > 0, an Tr = 2 e if g max 0 an e > 0. In the first case the waiting perio Ta is reciprocal to the gain, in orer to give avance to noes with higher positive gains over noes with lower positive gains. In the secon case, the waiting perio Ta is proportional to the transmission energy in orer to give avance to noes with lower transmission energies over noes with higher transmission energies. Aitionally, the noes with positive gains are preferable to the noes with low transmission energies (i.e. g max 2 e ). This fact is capture by setting appropriately the constants an 2. Since noe calculates the gains base on only locally available information, it can happen that in the calculation of the gains, it tries to exclue alreay exclue noes. In orer to prevent this, noe transmits a probe message uring the probation perio T prob of roun n +. Note that by knowing T act an Tr, i noe actually knows when roun n + starts. The probe message carries the aresses of all the noes by exclusion of which noe attains g max > 0, an it carries the starting time of the correction perio. If some of these noes have alreay been exclue, they will respon back to noe uring the correction perio. Noe

9 will accoringly upate its gain an the waiting perio Ta by taking into account the alreay elapse time of the waiting perio. The uration of the probation an correction perios shoul be such that any potential forwaring noe is given the chance to test its prospect of actually being the forwaring noe. It is important to stress that only those noes that obtain g max > 0 o the probation. Note that throughout the execution of the secon phase, the waiting perio Ta is counte own. Finally, noe enters into the active perio. Again, noe, base on the knowlege of T prob an T corr, knows when the active perio of roun n + starts. If uring that perio an before expiration of the waiting perio Ta noe receives a uplicate message, it repeats the upate proceure in Figure 9, otherwise, upon expiration of Ta, it rebroacasts the message with energy e, stores this value an marks itself as the forwaring noe. In our example shown in Figure 8, noe ecies to be the forwaring noe an broacasts a message at power e. By oing so, it initiates the upate proceure at noes k an l, which repeat the whole process. Recall that noe also sens information about all the transmitting noes on the chain from the source noe to noe. Next we show uner which conitions the waiting perio Ta expires solely uring the active perio of roun n +. From Figure 8 we can see that this happens if Ta conforms to the following conitions: T a T max T i r T a T max + T act T i r From the first inequality an the efinition of Ta we obtain that Tr 0, which is always satisfie. Along the same lines, from the secon inequality we obtain that Tr T act. Consequently, we efine the active perio as follows: T act = max F {T r } = max{ 2 e} F where the secon equality follows from the fact that g max 2 e. Now, since we alreay have ecie on an 2, we only have to fin the cost of the most expensive ege in the MST. Note that this information can be obtaine from the first phase of the algorithm. This in aition to the appropriate selection of the perios T prob an T corr, ensures a synchronous execution of the secon phase of the istribute algorithm. The uration of the secon phase is boune by F T max, where F is the set of the forwaring noes at the en of the secon phase. Thus, at the en of the secon phase, the broacast tree is built (i.e. we have a set of forwaring noes F an their respective transmission energies for a given source noe). Any subsequent broacast message can be isseminate along the tree in an asynchronous way (i.e. forwaring noes may re-broacast a message immeiately upon receiving it). 6. PERFORMANCE EVALUATION We performe a simulation stuy to evaluate our centralize algorithm (EWMA) an its istribute version. We compare the centralize version of our algorithm (EWMA) with BIP an MST algorithms. The simulations were performe using networks of four ifferent sizes: 0, 30, average normalize tree power average normalize tree power propagation loss exponent = 2 (confience interval 95%) MST BIP EWMA network size propagation loss exponent = 4 (confience interval 95%) MST BIP EWMA network size Figure 0: Normalize tree power for 00 network instances (confience interval 95%) an propagation loss exponent α = 2 (above) an α = 4 (below) 50 an 00 noes. The noes in the networks are istribute accoring to a spatial Poisson istribution over the same eployment region. Thus, the higher the number of noes, the higher the network ensity. The source noe for each simulation is chosen ranomly from the overall set of noes. The maximum transmission range is chosen such that each noe can reach all other noes in the network. The transmission power use by a noe in transmission ( α ) epens on the reache istance, where the propagation loss exponent α is varie. Similarly to Wieselthier et al. in [25], we ran 00 simulations for each simulation setup consisting of a network of a specifie size, a propagation loss exponent α, an an algorithm. The performance metric use is the total power of the broacast tree. Here we use the iea of the normalize tree power [25]. Let p i(m) enote the total power of the broacast tree for a network instance m, generate by algorithm i (i = {EW MA, BIP, MST }). Let p 0 be the power of the lowest-power broacast tree among the set of algorithms performe an all network instances (00 in our case). Then the normalize tree power associate with algorithm i an network instance m is efine as follows: p i(m) = p i(m) p 0. Let us first consier the performance of the algorithms

10 shown in Figure 0. In the figure we can see the average normalize tree power (shown on the vertical axis) achieve by the algorithms on networks of ifferent sizes (the horizontal axis). To estimate the average power, we have use an interval estimate with the confience interval of 95%. The figure shows that the solutions for the broacast tree obtaine by EWMA have, on the average, lower costs than the solutions of BIP an MST. (This is also true for α = 3, which is not shown in the figure). However, we can notice that for the propagation loss exponent of α = 4, the confience intervals of the algorithms overlap for certain cases, which means that the solutions provie by the algorithms are not significantly ifferent. Thus the figure also reveals that the ifference in performance ecreases as the propagation loss exponent increases. The main reason for such behaviour is that by increasing the propagation loss exponent, the cost of using longer links increases as well. Consequently, EWMA an BIP select their transmitting noes to transmit at lower powers, which is typical for the transmitting noes of MST. Hence, in a sense, EWMA an BIP s broacast trees converge to the MST tree when α increases. This inicates that in scenarios where α takes higher values, MST performs quite well. We also conucte a simulation stuy of the istribute algorithm presente in Section 5.2. The performance metric use here is the same as in the case of the centralize algorithm, an base again on the normalize tree power. However, here we o not consier the cost of builing a broacast tree, but only the cost of the final tree prouce by the istribute algorithm. The performance of the istribute algorithm is compare to that of the centralize algorithms, an is shown in Figure. We can see that broacast trees prouce by istribute EWMA have, on the average, lower costs than those obtaine by the centralize BIP an MST. Also, we can see that istribute EWMA performs slightly worse than its centralize counterpart. Note that the results for the centralize algorithms iffer between Figure 0 an Figure. This is because here we run another set of simulations for all the algorithms, an for each network the source noe is chosen at ranom. Base on our simulation results, we conclue that EWMA utilizes the wireless multicast avantage property at least as well as BIP. The main problem with BIP is that it is not easy to istribute. We showe in Section 5.2 that EWMA can be easily istribute by using the mechanism of the waiting perios. For these reasons, we think EWMA to be preferable to BIP. 7. CONCLUSION We have provie novel contributions on the two most relevant aspects of power-efficient broacast in all-wireless networks. First, we stuie the complexity of the problem. We iscusse two configurations, represente each by a specific graph: a general graph an a graph in Eucliean space (geometric case). For both, we provie a proof that the problem is NP-complete. Secon, we elaborate an algorithm, calle Embee Wireless Multicast Avantage (EWMA). We showe that this algorithm outperforms one of the most prominent proposals provie in the literature, BIP. Moreover, we escribe a fully istribute version of EWMA, a feature that other authors have reckone to be both necessary an challenging, an for which we coul not fin a solution in the average normalize tree power propagation loss exponent = 2 (confience interval 95%) MST BIP EWMA Distribute EWMA network size Figure : Distribute algorithm - normalize tree power for 00 network instances (confience interval 95%) an propagation loss exponent α = 2 literature. In terms of future work, we inten to explore how other mechanisms can be use to further reuce power consumption. Moreover, we will explore how to exten our proposal to multicast. Finally, we inten to stuy how to cope with the mobility of the noes. 8. ACKNOWLEDGMENTS We thank Naouel Ben Salem for helpful iscussions an comments. 9. REFERENCES [] J. Chang an L. Tassiulas. Energy Conserving Routing in Wireless A-hoc Networks. In Proceeings of IEEE INFOCOM ACM, [2] A. Clementi, P. Crescenzi, P. Penna, G. Rossi, an P. Vocca. On the complexity of computing minimum energy consumption broacast subgraphs. In Proceeings of 8th Annual Symposium on Theoretical Aspects of Computer Science (STACS 200), 200. [3] T. Cormen, C. Leiserson, R. Rivest, an C. Stein. Introuction to Algorithms. Secon Eition. The MIT Press, 200. [4] D. Estrin, R. Govinan, J. Heiemann, an S. Kumar. Next century challenges: Scalable coorination in sensor networks. In Proceeings of the Fifth Annual International Conference on Mobile Computing an Networks (MOBICOM 999), Seattle, Washington, August 999. [5] O. Eğecioğlu an T. Gonzalez. Minimum-energy broacast in simple graphs with limite noe power. In Proceeings of IASTED International Conference on Parallel an Distribute Computing an Systems (PDCS 200), pages , Anaheim, CA, August 200. [6] A. Faragó an V. Syrotiuk. Algorithmic problems in power controlle a hoc networks. In Proceeings of the 4th International Conference on Parallel an