Energy-aware Broadcasting in Wireless Networks

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Energy-aware Broadcasting in Wireless Networks

Ioannis Papadimitriou, Leonidas Georgiadis

To cite this version:

Ioannis Papadimitriou, Leonidas Georgiadis. Energy-aware Broadcasting in Wireless Networks.

WiOpt’03: Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, Mar 2003, Sophia Antipolis, France. 11 p. �inria-00466720�

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Energy-aware Broadcasting in Wireless Networks

Ioannis Papadimitriou, Leonidas Georgiadis

Abstract—In this paper we address the problem of broadcasting in wireless networks, so that the power consumed by any node is as small as possible. This approach is motivated by the fact that nodes in such networks often use batteries and, hence, it is im- portant to conserve energy individually, so that they remain op- erational for a long time. We formulate the problem as alexi- cographicnode power optimization one. The problem is in gen- eral NP-complete. We provide an optimal algorithm which runs in polynomial time in certain cases. We also provide a heuristic algorithm whose performance relative to the optimal one is fairly satisfactory. We next show that these algorithms can also be used to solve the problem of broadcasting so that the remaining battery lifetime of any node is as large as possible. Finally, we discuss the issues of implementing the above algorithms distributively, as well as their multicast extensions.

Index Terms— Wireless Networks, Energy Conservation, Di- rected Spanning Tree, Lexicographic Optimization.

I. INTRODUCTION

Most wireless devices today are portable and operate on batteries withfinite amount of energy. Hence, it becomes impera- tive to take the issue of power management into account when designing a wireless network [1]. In this paper we address the problem of energy-aware broadcasting in wireless networks. In a wired environment, broadcasting is a well understood problem and polynomial algorithms exist for its solution [2], [3].

A common approach is to associate a cost with each link and determine a spanning tree whose sum of link costs is minimal.

However, as indicated in [4], broadcasting in a wireless environment where omnidirectional antennas are used, must take into account the fact that a node’s transmission can reach multiple neighbors at the same time. Hence, the power needed to reach a node’s set of neighbors is themaximumof the powers needed to reach each of the neighbors separately. Since the efficient management of battery power is sought, the objective then is to determine a set of retransmitting nodes and their corresponding transmission powers such that a measure of consumed node power is optimized. In [4] the optimization objective was considered to be the sum of consumed node powers needed to con- vey the information from a given source to all destination nodes.

The problem is NP-complete and heuristic algorithms were proposed and studied in [4], [5], [6]. In this paper we address the broadcast problem in wireless networks with the objective of ensuring that no node consumes excessive power, or no node’s battery lifetime is reduced significantly. This is a reasonable objective in an environment where each node has its own batteries and it is important to keep every node operational for as long

I. Papadimitriou and L. Georgiadis are with Electrical and Computer Engi- neering Dept., Aristotle University of Thessaloniki, Thessaloniki, GREECE.

E-mails: [email protected] , [email protected] .

I. Papadimitriou was fully supported for this work by the Public Benefit Foun- dation “ALEXANDER S. ONASSIS”, Athens, GREECE.

as possible. Minimizing the sum of node powers consumption in the network does not necessarily guarantee that each retransmitting node will consume a small amount of power.

Unlike much of previous work, where the wireless network is usually modeled as an undirected graph, the graphs we consider are directed. Our model does not necessarily imply the absence of bidirectional links between two nodes (although uni- directional links may exist in a real network [7]). It is rather used as a more realistic modelling of the fact that the power needed for communication between two nodes may not be the same in both directions due to noise or other signal propagation phenomena, and the heterogeneity in transmission hardware of nodes in the network. Wefirst formulate the problem as amin- maxnode power optimization one, where maximum consumed node power is sought to be minimized. Next we look at the stronger optimization criterion where, provided that we succeed in minimizing thei^thmaximum node power, we also seek to minimize the(i+ 1)^thmaximum. This objective is captured by the stronger lexicographicnode power optimization problem (see Section II-B). The general problem is NP-complete.

We present an optimal algorithm which is polynomial in certain cases. The algorithm runs in reasonable time for moderate size random networks but, as expected, its running time quickly deteriorates for larger networks. To deal with networks of larger sizes, we present a heuristic that is based on the steps of the optimal algorithm and avoids the most computing intensive op- erations. Numerical results show that the proposed heuristic has good running times and provides satisfactory performance relative to the optimal algorithm.

We also address the issue of lexicographic optimization under more general link cost functions. A special case of these functions can be used when the desirable performance metric is the node remaining lifetime. In this case it may be of interest to implement broadcasting in a manner that maximizes the minimum remaining node lifetime (max-min criterion), or maximizes lexicographically the remaining node lifetimes. To elucidate the difference between the two optimization criteria, consider that broadcasting from a given sourcertakes place until one of the nodes in the network runs out of batteries. Under both max-min and lexicographic criteria, thefirst timetwhere a node’s lifetime becomes zero (i.e., its battery is exhausted) is maximized. However, under the lexicographic criterion, the number of nodes whose lifetime is zero at timet, is also minimized. Moreover, if only one node’s battery is exhausted, then the minimum remaining lifetime of the rest of the nodes is maximized. Besides these two optimization criteria there are several others that can be proposed, e.g. maximization of the sum of node remaining lifetimes, maximization of convex combination of them, etc. The specific criterion to be used depends on the network requirements. An interesting issue is to determine and

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classify the network environments for which each of these optimization criteria is appropriate (see [8] for one such comparison). While we leave this issue as a subject for further study, we note that in applications where the max-min optimization of lifetime is important, the use of the lexicographic criterion provides solutions with additional desirable properties.

The objective of maximizing the minimum battery lifetime was considered in [9]. In the latter work in effect the reverse problem to ours was considered. That is, each node needs to transfer information to a specific gateway. The broadcast nature of each node’s transmission was not considered. Instead, a node was able to transmit to each of its neighbors with varying rate and the consumed node power for each “neighbor-to-neighbor”

transmission was proportional to the transmission rate. The Multipoint relaying algorithm, proposed in [10] and used in [11], has also some relevance to our work. An efficient tech- nique is proposed forflooding in wireless networks with limited information, so that the number of nodes that retransmit the information (relay nodes) is minimized. The latter setup can be considered as a special case of ours, by assuming that all nodes need unit power to reach their neighbors (see Section III).

The algorithms we develop assume knowledge of network topology and the powers needed by each node to reach its neighbors. Hence, they can be used in networks with infrequent topological changes and low mobility, such as computer termi- nals in a university campus or within a company. The issue of centralized versus distributed computation does not have a clear-cut answer [1]. As will be discussed in Section VI, in our setup at least partial optimization can be made in a distributed fashion by implementing distributively a basic algorithmic subroutine. In any case, our algorithms can be directly applied in network environments where at least partial information of network topology is proactively maintained at each node, as in OLSR [11] and ZRP [12]. Another important problem is energy-aware multicasting. The optimal algorithms presented in this paper can be used for the latter problem provided that a basic algorithmic subroutine is designed for multicasting. A simple algorithm for such a subroutine exists, but we believe that further improvements can be made.

The rest of the paper is organized as follows. Section II provides formal definitions and formulation of the min-max and lexicographic node power optimization problems. In Section III we indicate that the latter problem is NP-complete and provide an optimal algorithm which is polynomial in certain cases. We also provide a heuristic. In Section IV we consider generalized link cost functions and prove that the proposed algorithms can also be used to solve the broadcast problem so that the remaining battery lifetime of any node is as large as possible. Numer- ical results are presented in Section V. In Section VI we discuss the issues of implementing our algorithms in a distributed fashion and using them in the case of multicast communication.

Finally, Section VII summarizes the conclusions of our study.

All proofs of the lemmas are given in the Appendix.

II. DEFINITIONS ANDPROBLEMFORMULATION

Consider a directed graphG = (N, L), whereN is the set of nodes andLis the set of directed links. For a nodei ∈ N we denote byL^G_out(i)(L^G_in(i)) the set of links outgoing from

(incoming to)i. A nodej such that link(i, j)belongs toLis called a one-hop neighbor ofior simply a neighbor of i. A nodejis a two-hop neighbor ofiifjis not a neighbor ofiand for somek ∈ N,(i, k) ∈ L and(k, j) ∈ L. We denote by N₁^G(i)the set of (one-hop) neighbors and byN₂^G(i)the set of two-hop neighbors of nodei.

Given a root noder ∈ N, anr-rooted treeT = (N, L^T) spanningG(spanningtree for short) is a subgraph ofGhaving the following properties: a) there is a directed path from node rto every other node ofGusing only the links inL^T, and b)T has|N|−1links, where|N|is the cardinality of the setN. It follows from the definition ofT that: a) for every noden6=r there is exactly one link in the setL^T terminating atn, b) there is no link inL^T terminating at noder, and c) there are nodes in Tthat have no outgoing links inL^T. The latter nodes are called leaf nodes ofT. Hence, for a leaf nodeiwe haveL^T_out(i) =®. A. Wireless Communication Model

We model the wireless network as a directed graphG = (N, L).Nis the set of nodes in the network. If nodejcan suc- cessfully receive information transmitted by nodei, then link (i, j)belongs to the setLof links inG. The power needed for a successful transmission over linkl= (i, j)is denoted byc_l>0 and is also referred to as the link cost. Note that we allow for the possibility of asymmetric channels, i.e., it is possible that c_(i,j) 6=c_(j,i). Each node is equipped with an omnidirectional antenna. Hence, the following useful property holds:

Property1:If nodeitransmits with powerp_i, it can reach any nodejfor whichc_(i,j)≤p_i.

Note that during transmission, battery power is also consumed by the receiver of information. For simplicity in the discussion that follows we assume that the receive power is zero.

In Section IV we indicate how to incorporate it into the model.

Suppose that noder needs to broadcast information to all other nodes in the network. In this case, we have to determine a set of retransmitting nodes and their corresponding transmission powers, so that eventually all nodes receive the information. A way to achieve this, which will be useful in the sequel, is as follows. We define anr-rooted spanning treeT = (N, L^T) with the following interpretation:

1) Nodertransmits with powerp^T_r = max_l_∈_L^T

out(r){cl}. 2) Any nodenofT receiving the information, retransmits with powerp^T_n = max_l_∈_L^T

out(n){cl}, wheremax_l∈®{cl}= 0. We refer top^T_n as the power induced on nodenby treeT.

3) Step2is repeated until all non leaf nodes retransmit the information exactly once.

Note that, depending on the link costs, the same broadcast transmissions can be effected by more than one spanning trees.

Consider for example the network in Fig. 1 with root node A.

The spanning treesT₁andT₂, with links {(A,B),(B,C),(B,D)}

and {(A,B),(A,C),(B,D)} respectively, have the same leaf nodes. Hence, they both specify the same nodes for retransmission (node B in this case). Under link power set I, the node powers induced are the same for both trees,p^T_A¹ = p^T_A² = 2, p^T_B¹ = p^T_B² = 4,p^T_C¹ = p^T_D¹ =p^T_C² = p^T_D² = 0. Under link power set II, we havep^T_A¹ = 2,p^T_B¹ = 6,p^T_C¹ =p^T_D¹ = 0, and p^T_A² = 4,p^T_B² = 3,p^T_C² =p^T_D² = 0, and, hence, the node powers induced by the two trees are different.

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Fig. 1. Example of wireless broadcasting

B. Optimal Broadcast Trees

We are interested infinding a spanning tree for which the maximum induced node power is as small as possible. This re- quirement is captured by themin-maxnode power optimization problem defined as follows:

Problem 1: (Min-Max Node Power Optimization) Find a spanning tree T such that, for any spanning tree T of G, it holdsmax

n∈N{p^T_n}≤max

n∈N{p^T_n}.

Given a spanning treeT, letP^T = (p^T_n)_n_∈_N be the vector of induced node powers. A solution to Problem 1 minimizes the maximum consumed node power but does not specify how to treat nodes that consume power smaller than the maximum.

As discussed earlier, it may be desirable to also keep the consumed power of the latter nodes as small as possible, so that no node in the network consumes excessive power. This latter re- quirement is captured by the strongerlexicographicnode power optimization problem (referred to as min-max problem in [3]) stated below, after the following definition:

Definition 1: Given an n-dimensional real vector v=

(v1, v2, ..., vn), define byvbthen-dimensional vector whose co- ordinates are those of varranged in non-increasing order, i.e., bv = (bv1,bv2, ...,bvn) = (vi1, vi2, ..., vin),wherevi1 ≥ vi2 ≥ ... ≥v_i_n. Vectorvis called lexicographically smaller than or equal to vectoru, if eitherbv = buor there exists a numberl, 1≤l≤n, such thatbv_i=bu_ifor1≤i≤l−1andbv_l<bu_l. We writev¹lexuand, if in additionbv6=u,b v≺lexu.(For example, the vector(3,4,8)is lexicographically smaller than(5,8,2)).

Problem 2: (Lexicographic Node Power Optimization) Find a spanning tree T^∗ such that, for any spanning tree T of G, it holdsP^T^∗¹lexP^T.

While our main interest is in providing algorithms for Prob- lem 2, it will be seen in the sequel that Problem 1 is instrumental in the development of such algorithms. In the sequel, we will need the following notation. Letpe^T_i be thei^thdistinctmaximal node power induced by spanning treeT,Se_i^T the set of nodes that have to transmit with power ep^T_i, and ek_i^T = |Se_i^T|. That is,pe^T_i₋₁ > pe^T_i,i = 2, ...,Ie^T, andp^T_n = pe_i^T, for alln ∈ Se_i^T. According to this definition, pe^T_e

I^T = 0 andSe^T_e

I^T is the set of leaf nodes ofT. For example, ifP^T = (4,8,8,0,3,0,0), then e

p^T₁ = 8,Se₁^T ={2,3},pe^T₂ = 4,Se₂^T ={1},pe^T₃ = 3,Se₃^T ={5}, e

p^T₄ = 0,Se₄^T ={4,6,7}. It follows from the definition that any

Fig. 2. Example of graph reduction

lexicographically optimal (with respect to node powers) treeT^∗ has the following property (to avoid special cases, we define e

p^T₀ =∞andek₀^T = 0):

Lemma 1: For any spanning treeT of G, if for some l <

Ie^T^∗ it holdspe^T_i =pe^T_i^∗and ek_i^T =ek^T_i^∗for allisuch that0≤ i≤l, thenIe^T > l. Moreover, eitherpe^T_l+1^∗ <ep^T_l+1, orpe^T_l+1^∗ = e

p^T_l+1andek^T_l+1^∗ ≤ek_l+1^T .

It follows that all lexicographically optimal trees have the sameIe^T^∗,pe_i^T^∗, andek_i^T^∗, that is, ifT^∗is any lexicographically optimal tree, thenIe^T^∗ = Ie^∗, pe^T_i^∗ = pe^∗_i andek^T_i^∗ = ek^∗_i, for 1≤ i ≤Ie^∗. Finally, for the analysis in the next sections, we need to define the following transformation of a graphG(N, L):

Definition 2: Let L⁰ ⊆ L and p ≥ 0be given. Eliminate fromGall links inL−L⁰with power larger than or equal to p. Let L^R be the set of remaining links after the elimination.

Modify the link powers inL^Ras follows: c^R_l = 0, if l ∈ L⁰, andc^R_l =cl, otherwise. The resulting graph is called the “re- duction” ofGand is denoted byG^R(G, L⁰, p).

Fig. 2 shows an example of graph reduction, given that L⁰ ={(C,D),(D,E)} and p = 3 in the original graph G.

This example also shows the importance of Problem 2. As- sume that the root node is A and consider the spanning tree T of G with links {(A,B),(A,C),(C,D),(D,E)}. Tree T solves Problem 1 and induces a vector of node powers, (p^T_A, p^T_B, p^T_C, p^T_D, p^T_E) = (5,0,5,3,0), which is lexicographically larger than that induced by the spanning tree T^∗ with links {(A,C),(C,D),(D,E),(E,B)},(p^T_A^∗, p^T_B^∗, p_C^T^∗, p^T_D^∗, p^T_E^∗) = (2,0,5,3,1).

III. OPTIMIZATIONALGORITHMS

To avoid cluttering the discussion, in the following we make the assumption that there is anr-rooted spanning tree. There are well known algorithms that test for the existence of such a tree in a given network (see e.g. [13]). Let usfirst address Problem 1. We have

maxn∈N{p^T_n}= max

n∈N{ max

l∈L^T_out(n){cl}}= max

l∈L^T{cl}. (1) Hence,finding the spanning tree that minimizes the maximum induced node power is equivalent tofinding the tree that minimizes the maximum link power. For the latter problem, known

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also asbottleneckoptimization problem, polynomial time algorithms exist [14], [15], [16]. Note that since any lexicographically optimal treeT^∗ solves Problem 1, the value of any solution to Problem 1 ispe^T₁^∗=pe^∗₁.

We now address Problem 2. A special case of this problem is NP-complete. To see this, assume that all link costs ofGare equal, that is,cl=c >0, for anyl∈L. Then, since each node has power eithercor0, it follows from the definition of the lexicographically optimal point that the solution to the problem is a spanning treeT^∗with the following property; the number of non zero elements of the induced vector of node powersP^T^∗ (i.e., the number of non leaf nodes ofT^∗) is minimized. Hence, findingT^∗is equivalent tofinding the smallest set of nodes that will serve as relays to pass the information from the root noder to all other network nodes. This set is referred to as the optimal MultiPoint Relay (MPR) set in [10]. Consider now a graphG consisting of a noder∈N, the setN₁^G(r)(one-hop neighbors ofr), the setN₂^G(r)(two-hop neighbors ofr) and with bidirectional links. It is proved in [10] that for this graph,finding an optimal MPR set for noderis NP-complete.

According to the discussion above, an efficient algorithm for Problem 2 is very unlikely to exist. In the following, wefirst provide an optimal polynomial time algorithm under the condition that different nodes need different powers to reach their one-hop neighbors (see Condition 1 below for the exact state- ment). Next we will extend this algorithm to provide an optimal solution for the general case. In the latter case, as expected, the worst case running time of the algorithm is exponential in the size of the network. As will be seen in Section V, the derived algorithm has reasonable running times for moderate size random networks, where it can be employed to test the efficiency of proposed heuristics. Moreover, as will be discussed in Section III-C, this algorithm can be used as a basis for various heuristic algorithms that can be employed in larger networks.

A. Optimal Algorithm for a Special Case

We start with the optimal polynomial time algorithm. As- sume that the following condition holds in the network:

Condition 1: The powers of links outgoing from different nodes are different. That is,c_(i,n₁₎6=c_(j,n₂₎, for alli, j ∈N, i6=j,n₁∈N₁^G(i),n₂∈N₁^G(j).

Note that we allow for the possibility that links outgoing from the same node have equal power. As discussed above,pe^∗₁ can be obtained in polynomial time for any network. Accord- ing to Condition 1, there is a single node that needs powerpe^∗₁to reach some of its neighbors and can therefore be identified. In general, assume that we knowpe^∗_i for1 ≤ i ≤ m < Ie^∗−1, which, according to Condition 1, implies that we know Se_i^∗, 1 ≤ i ≤ m. Note that each Se_i^∗ contains a single node. We are interested infindingpe^∗_m+1. LetLe_ibe the set of links whose power is less than or equal tope^∗_i and emanate from the node in Se_i^∗. Define alsoLe^m=∪^mi=1Le_iandSe^m=∪^mi=1Se_i^∗. The search forpe^∗_m+1is based on the following lemma:

Lemma 2: Consider the solution to Problem1 on the re- duction of G, graphG^m+1 = G^R(G,Le^m,pe^∗_m). The value of this solution ispe^∗_m+1and the min-max optimal treeT^m+1has e

p^T_i^m+1=pe^∗_i for1≤i≤m+ 1.

Algorithm 1:

1.m= 0,G^m=G,Le^m=®^,pe^∗_m=∞^.

2.Form the reduction ofG^m,G^m+1=G^R(G^m,Le^m,pe^∗_m). 3. Solve the min-max optimization problem on G^m+1 and let T^m+1 be this solution. The maximum induced node power of T^m+1inG^m+1ispe^∗_m+1.

4.Ifpe^∗_m+1= 0, then ReturnT^m+1(T^∗is equal toT^m+1).

5. Else, identify the node inG^m+1that has at least one outgoing link with powerpe^∗_m+1. LetLem+1be the set of links whose power is less than or equal tope^∗_m+1and emanate from this node.

6.Le^m+1=Le^m∪Lem+1,m=m+ 1. Go to step2. Fig. 3. Optimal Algorithm for a Special Case

It follows from Lemma 2 that, if we knowpe^∗_i for 1≤ i ≤ m <Ie^∗−1, we canfindpe^∗_m+1by solving Problem 1 onG^m+1. At the same time, this solution provides a treeT^m+1for which we havep^T_n^m+1 =p^T_n^∗forn∈Se^m. If wefind thatpe^∗_m+1 = 0, then we know thatT^m+1 =T^∗. Otherwise, we can repeat the procedure tofind pe^∗_m+2. Also note that, as follows from the definitions, we haveG^m+1 = G^R(G^m,Le^m,pe^∗_m). Hence, we have the algorithm of Fig. 3 forfindingT^∗, under Condition 1.

Complexity of Algorithm1:Problem 1 at step3of the algorithm can be solved inO(|N|log|N|+|L|)time (see e.g.

[15]). The other steps take timeO(|L|)for one iteration and, since all steps of Algorithm1are invoked at most|N|times, its worst case running time isO(|N|²log|N|+|N||L|).

B. Optimal Algorithm for the General Case

Let us now remove Condition 1. According to (1), we can still solve Problem 1 andfind the valuepe^∗₁. Since in general there may be many nodes that can reach others with powerpe^∗₁, we must now identify a set of nodes that will transmit with powerpe^∗₁ when a lexicographically optimal treeT^∗ is imple- mented. Note that there may be multiple optimal trees and, therefore, multiple optimal node sets. However, according to Lemma 1, the number of nodes in all these sets will be the same, i.e.,ek^∗₁. The search forek^∗₁and the candidate optimal node sets is based on the two lemmas that follow. We denote byT^∗the set of all lexicographically optimal trees. Lemma 3 part 1 provides a method for testing whether a given node set isnotequal toSe₁^T^∗for anyT^∗ ∈ T^∗. Lemma 3 part 2 provides a method forfinding an upper bound forek^∗₁. Lemma 4 provides a method for testing whether a given node setS, such that|S| =ek₁^∗, is notequal toSe₁^T^∗for anyT^∗∈T^∗.

Lemma 3: Let S be a set of nodes with the property that each node inS has at least one outgoing link with powerpe^∗₁. LetL_S be the set of links emanating from some node inSand having power less than or equal tope^∗₁, i.e., L_S = {l ∈ L : l ∈ L^G_out(n), n ∈ S, c_l ≤ pe^∗₁}. Consider the reduced graph G¹=G^R(G, L_S,pe^∗₁).

1) If there is no spanning tree of G¹, thenS 6= Se₁^T^∗ for all T^∗∈T^∗.

2) Else, the value of the solution T¹ to Problem 1on G¹ is strictly smaller thanpe^∗₁,pe^T₁¹ =pe^∗₁andek^∗₁≤|S|.

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Lemma 4: LetS1,S2be sets of nodes with the property that each node inSi,i = 1,2, has at least one outgoing link with power ep^∗₁. Let LSi be the set of links emanating from some node inSiand having power less than or equal tope^∗₁. Consider the reductions of G, graphsG¹_i = G^R(G, LSi,pe^∗₁),i = 1,2.

Assume that there is at least one spanning tree of G¹_i and let v_i be the value of the solution to Problem1onG¹_i.If |S₁| =

|S2|=ek₁^∗andv1< v2<pe₁^∗, thenS26=Se₁^T^∗for allT^∗ ∈T^∗. Let us now describe how we can search for candidate optimal node sets based on the previous lemmas, once we have obtained e

p^∗₁. We test successively sets with cardinality 1first, then 2, 3 and so on, as follows. We pick a setS such that each node in S has at least one outgoing link with powerpe^∗₁ and generate the reduced graphG^R(G, L_S,pe^∗₁). If the latter has no spanning tree, then we know, from Lemma 3 part 1, thatS6=Se₁^T^∗for all T^∗ ∈ T^∗ and, therefore,S can be removed from further con- sideration. Else, from Lemma 3 part 2, we know thatek₁^∗≤|S|. Since we already know that there can be no optimal set with

|S|−1nodes, we haveek₁^∗=|S|. Hence, at this point we know ek^∗₁. However, we still have to identify the candidate optimal node sets. For this, we proceed as follows. We examine the sets with cardinalityek₁^∗and keep only those whose value of the solution to Problem 1 on the reduced graph is the smallest. From Lemma 4, it follows that only the latter sets are candidates for being equal toSe₁^T^∗for someT^∗ ∈T^∗. Moreover, the smallest value obtained should be equal tope^∗₂. Hence, at the end of this process we have the values ofpe^∗₁,ek^∗₁,pe^∗₂, and a set of candidate optimal node setsS1= {S_1,1, ..., S_1,s₁}that may be equal to Se₁^T^∗for someT^∗∈T^∗.

Unfortunately, with the information at hand we do not have any way to further isolate one of the sets in S¹ as be- longing to one of the lexicographically optimal trees. How- ever, the process described above can be generalized to allow further refinement of the search, that results in the itera- tive determination of sequences of candidate optimal node sets (S_1,m₁, S_2,m₂, ..., S_i,m_i)with the following property:

Property 2: It is possible that (S_1,m₁, S_2,m₂, ..., S_i,m_i) = (Se₁^T^∗,Se₂^T^∗, ...,Se_i^T^∗)for someT^∗∈T^∗,1≤i <Ie^T^∗.

The refinement process is based on direct generalization of Lemmas 3 and 4 and can be achieved as follows. At iteration iwe have a candidacy tree, whose levels and nodes have the following interpretation:

1) Level0is associated withpe^∗₁, obtained from the solution to Problem 1 on graphG. The root node at level0is associated with the empty set of nodes,®.

2) Levelj,j = 1, ..., i, is associated with the valuesek^∗_j,pe^∗_j+1 already obtained. Each nodemat levelj is associated with a candidate setSj,m. The cardinality of each of the sets at level j isek_j^∗; each node inS_j,m will transmit with powerpe^∗_j if in- deedS_j,m =Se^T_j^∗for someT^∗ ∈T^∗and the broadcast transmissions induced byT^∗are eventually chosen. We denote by Sj= ©

S_j,1, ..., S_j,s_jªthe set of candidate optimal node sets at levelj.

3) A path (S_1,m₁, S_2,m₂, ..., S_i,m_i)in the candidacy tree de- notes a sequence of candidate sets that may satisfy Property2.

Leaf nodes are located only at the highest level.

Algorithm 2:

1. For a candidate sequence (path from the root to a node at level i)Sⁱ = (S_1,m₁, S_2,m₂, ..., S_i,m_i), letL_Sⁱ =∪ⁱj=1L_S_j,mj and N_Sⁱ =∪ⁱj=1Sj,mj.

2.Test successively sets with cardinality 1first, then 2, 3 and so on, as follows. Pick a setSsuch that each node inShas at least one outgoing link with powerpe^∗_i+1andS⊆N−N_Sⁱ.

3. Generate the reduced graphG^R(G, L_S ∪L_Sⁱ,pe^∗_i+1). If the latter has no spanning tree, thenSis removed from further consid- eration. Else, it is known thatek_i+1^∗ =|S|^.

4.The setScorresponds to a node at leveli+1if the valuevof the solution to Problem 1 on the reduced graph is the smallest among the sets with cardinalityek_i+1^∗ . In this case, the node corresponding toSat leveli+ 1is connected to the node corresponding to set Si,miat leveli.

5. At the end of this procedure, there may be leaves in the candidacy tree that are located at level lower thani+ 1. The path that starts from the root and ends at one of these leaves corresponds to a sequence of node sets that cannot be a candidate satisfying Prop- erty2and should, therefore, be excluded from further considera- tion. Hence, we repeatedly eliminate all leaves that are not at level i+ 1of the candidacy tree, so that only leaves at leveli+ 1remain.

Fig. 4. Optimal Algorithm for the General Case

When we are at leveli,pe^∗_i+1 is known. Ifpe^∗_i+1 = 0, then any spanning tree corresponding to a candidate path up to level iis a lexicographically optimal tree. Else, leveli+ 1is cre- ated by adding and removing nodes in the candidacy tree as described in the algorithm of Fig. 4. Upon completion, Algorithm 2provides all lexicographically optimal trees and, therefore, all lexicographically optimal (with respect to node powers) sets of broadcast transmissions. Note that the number of nodes in the candidacy tree does not necessarily increase as the algorithm proceeds, since at step5the tree may be “pruned”.

Fig. 5 shows an example of candidacy tree for a graphG with root node A. After creation of level3, we observe that there is a node at level2of the candidacy tree, associated with node C ofG, which is a leaf node at level lower than 3and can, therefore, be eliminated. Then, the node at level1, associated with node A ofG, becomes such a leaf and is also eliminated. At level 4, we find that pe^∗₅ = 0 and the algorithm stops. In this example there are two optimal trees,T₁^∗,T₂^∗, with links {(A,B),(A,F),(A,G),(B,C),(B,D),(C,E),(F,H),(G,I)}

and {(A,B),(A,F),(A,G),(B,C),(B,D),(C,E),(F,H),(H,I)} respectively. T₁^∗ corresponds to the path B→C→{F,G}→A of the candidacy tree and T₂^∗ corresponds to the path B→C→{F,H}→A. The sets of node powers, corresponding to the trees, are easily derived by inspecting the candidacy tree.

For example, the power induced on node H byT₂^∗is found as follows. T₂^∗ corresponds to path B→C→{F,H}→A. For this path, node H is located at level3and, hence, its retransmission power should bepe^∗₃ = 3. The (lexicographically equal) node power vectors induced by the two trees are shown in Table I.

Algorithm2requires exponential number of computations in

|N| in the worst case. In a wireless network however, where nodes are placed at irregular locations, the cardinality of the set of nodes that have at least one outgoing link with powerpe^∗_i

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Fig. 5. Example of candidacy tree TABLE I

NODEPOWERS INDUCED BY THEOPTIMALTREES

A B C D E F G H I

T₁^∗ 2 5 4 0 0 3 3 0 0 T₂^∗ 2 5 4 0 0 3 0 3 0

may not be very large. Moreover, in such an environment, the condition for a branching to occur in the candidacy tree does not seem very likely. Note also that, even if a branching does occur, it is likely to be eliminated in the next few iterations, according to step5of the algorithm. Hence, for moderate size random networks, Algorithm2may not require many computations. However, as the simulation results show, its running time rapidly deteriorates as the number of nodes increases. In the next subsection we provide a heuristic that can be used for larger networks as well. As we will see, the various steps of Algorithm2are useful for the development of the heuristic.

C. Heuristic Algorithm

The main causes of explosion in the number of computations of Algorithm2are the following:

1) The cardinality of the setA_i, the set of nodes in the graph that have at least one outgoing link with powerpe^∗_i. In general, subsets of these nodes have to be tested to create the nodes of the candidacy tree at leveliand, hence, the worst case number of computations required is proportional to2^|^Aⁱ^|.

2) The branchings of the candidacy tree. A branching will occur if two or more of the node sets being tested result in the smallest value of the solution to Problem 1 on the corresponding reduced networks.

Simulation results show that branchings do not occur very frequently. On the other hand, as will be seen in Section V,

|A_i|does increase as the number of nodes within the same geo- graphical area increases. Therefore, in order to reduce the computations required to obtain a good tree, we must develop efficient algorithms for selecting “good” subsetsSe_i^T. Motivated by these observations, we modify Algorithm2as follows.

We solve Problem 1 onG. Letp^H₁ = pe^∗₁ be the value of this solution andA^H₁ the set of nodes inG with at least one outgoing link with powerp^H₁. According to Algorithm2, we have to select a subsetS₁^H ⊆A^H₁ of minimal cardinality, such that the reduced graphG^R(G, L_S^H

1 , p^H₁), whereL_S^H

1 = {l ∈ L:l∈L^G_out(n), n∈S₁^H, c_l ≤p^H₁}, has at least one spanning tree, and solve Problem 1 on that graph.

To avoid excessive computations, we pickS^H₁ as follows. We eliminate fromGall links with power larger thanp^H₁ . LetG⁰ be the resulting graph. For each noden ∈ A^H₁ we form the vectorvn = (cl)_l_∈_L^G0

out(n). The maximal element ofvn is by definition ofG⁰ equal top^H₁, but the rest of its elements may have any values smaller than or equal top^H₁. Any noden∈S₁^H will have to transmit with powerp^H₁ and, according to Prop- erty1, will also transfer the required information to all nodes inN₁^G⁰(n). Hence, it is preferable to include nodes inS₁^Hwith large values of elements invn and exclude those with small values. To achieve this, we arrange the nodes inA^H₁ in non- decreasing lexicographic order of the vectorsvn. Starting from thefirst nodenin the lexicographic order, we test whether by eliminating fromG⁰the outgoing links ofnwith powerp^H₁,G⁰ still has at least one spanning tree. If this is the case, we eliminate fromG⁰all outgoing links ofnwith power equal top^H₁ and examine the next node in the lexicographic order. Else, we add nodentoS₁^H and consider the next node in the lexicographic order. Note that, according to the selection procedure above, a node with lexicographically small vectorvn is not included inS^H₁ , unless it is necessary to maintain network connectivity.

Hence,S₁^Hwill generally contain nodes with lexicographically large vectorsvn.

At the end of this procedure, the reduced graph G¹ = G^R(G, L_SH

1 , p^H₁) is formed and Problem 1 is solved on that graph to obtainp^H₂ . Note thatp^H₂ may not be equal toep^∗₂. All steps above are then applied toG¹, instead ofG, and the process is repeated untilp^H_i = 0for somei >1. The last spanning tree obtained is returned as the solutionT^H of the algorithm.

The node power vector(p^T_n^H)n∈N, induced byT^Hin the original graphG, defines the set of broadcast transmissions of the proposed heuristic.

Note that in the previous algorithm, branchings in the candidacy tree are eliminated. In effect, we now have a single path at each step of the iteration. Some further optimization can be performed by observing that if a noden∈A^H₁ contains a link l = (n, m)such thatc_l =p^H₁, andl is the only incoming link to nodemin graphG⁰, then nodenmust transmit with power p^H₁ and, therefore, is included in the setS₁^H. Thefinal heuristic algorithm is presented in Fig. 6.

As an example, we apply the heuristic algorithm to graphG of Fig. 5. Sincep^H₁ = 5, we have thatA^H₁ = {A,B,C}and S₁^H = {B}. After solving Problem 1 on the reduced graph G^R(G, L_{B},5) we obtainp^H₂ = 4, A^H₂ = S^H₂ = {C}and, consecutively,p^H₃ = 3,A^H₃ = {F,G,H},S₃^H ={F,H},p^H₄ = 2, A^H₄ = S₄^H = {A}and p^H₅ = 0. The last spanning tree obtained is the solutionT^Hof the algorithm. For this particular graph,T^H=T₂^∗, whereT₂^∗is one of the two lexicographically optimal trees ofGfound in the previous subsection.

Complexity of Algorithm 3: We will first provide the

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Algorithm 3:

1.i= 0,Gⁱ=G.

2.Letp^H_i+1be the value of the solutionTⁱ⁺¹to Problem 1 onGⁱ. 3. Ifp^H_i+1 = 0, then ReturnTⁱ⁺¹(T^H is equal toTⁱ⁺¹). Else, setS_i+1^H =®and identifyA^H_i+1, the set of nodes with at least one outgoing link with powerp^H_i+1inGⁱ.

4.Eliminate fromGⁱall links with power larger thanp^H_i+1and let G⁰be the resulting graph.

5.For each noden∈A^H_i+1, if there is a linkl= (n, m)such that cl=p^H_i+1, andlis the only incoming link to nodemin graphG⁰, then addntoS_i+1^H .

6. If the reduced graphG^R(Gⁱ, L_S^H

i+1, p^H_i+1)has at least one spanning tree, then go to step9. Else, for each noden∈A^H_i+1− S_i+1^H form the vectorvn = (cl)_l_∈_L^G0

out(n)and arrange these nodes in non-decreasing lexicographic order of the vectorsvn.

7. Starting from thefirst nodenin the lexicographic order, test whether by eliminating fromG⁰the outgoing links ofnwith power p^H_i+1,G⁰still has at least one spanning tree.

8.If this is the case, eliminate fromG⁰all outgoing links ofnwith power equal top^H_i+1and examine the next node in the lexicographic order. Else, add nodentoS_i+1^H and consider the next node in the lexicographic order.

9.Gⁱ⁺¹=G^R(Gⁱ, L_SH

i+1, p^H_i+1),i=i+ 1. Go to step2. Fig. 6. Heuristic Algorithm

complexity for one iteration of the algorithm. Step 2 takes O(|N|log|N|+|L|)time. Steps3-5takeO(|L|)time. Step 6 requires the arrangement of at most |N| nodes in non- decreasing lexicographic order of the vectorsvn. We can assume that the coordinates of these vectors are already sorted in non-increasing order (this can be made during initializa- tion inO(P

n∈Nx_nlogx_n) = O(|L|log|L|)time, wherex_n is the number of elements in v_n). To evaluate the complexity of the above arrangement, let us assume that we implement bubble sort [13] - as will be seen, because of step 7, more efficient sorting does not improve worst-case complexity. Lexicographic comparison of vectors vn andvn+1 takes timeO(min{xn, xn+1})and a singlepassof the vectorsvn, n ∈ N, takes timeO(P_|N|−1

n=1 min{x_n, x_n+1}), which is at mostO(P|N|

n=1xn) =O(|L|). Since at most|N|passes of the vectorsvn,n∈N, can occur, the complexity of the above arrangement with bubble sort isO(|N||L|). A spanning tree can be obtained inO(|L|)time [13] and, therefore, step7of the algorithm requiresO(|N||L|)time in the worst case. Hence, one iteration of steps1-9of the algorithm requiresO(|N||L|)time and, since at most|N|such iterations may occur, the worst case running time of Algorithm3isO(|N|²|L|).

IV. MOREGENERALCOSTFUNCTIONS

Assume that with each noden∈ N there is a cost function f_n(p), which expresses the cost incurred at nodenif it trans- mits with powerp. We assume thatf_n(p)is strictly increasing inpand nonnegative. For reasons that will become apparent in the sequel, we allow for the possibilityf_n(0)>0. As in Sec- tion II-A, given anr-rooted spanning treeT, we define for any

noden ∈ N,φ^T_n = max_l_∈_L^T

out(n){fn(cl)}, ifL^T_out(n) 6= ∅, andφ^T_n = fn(0), ifL^T_out(n) = ∅. The objective now is to find the spanning tree for which the vector(φ^T_n)n∈N is lexicographically minimal. The case fn(p) = pfor all n ∈ N corresponds to the problem studied in Section II. In fact, if we use the quantitiesf_n(c_l)as link cost functions, then the main difference between the current setup and the one examined up to now, is that the “powerφ^T_n” of a leaf nodenmay be non zero in the general case. However, we will show that the same algorithms can be used in this case as well, by modifyingG(N, L) to a new networkG(N , L) as follows. Gcontains all nodes and links ofG. For each noden∈ N, a new nodenis added to the setN of nodes inGand a new link(n, n)is added to the setLof links inG. The link powers in Lare defined as c_(n,m)=fn(c_(n,m)), ifm6=n, andc_(n,n)=fn(0). The node powers for a treeTinGare defined asθ^T_n = max

l∈L^T_out(n){c_l}, ifL^T_out(n)6=∅, andθ^T_n = 0, ifL^T_out(n) =∅. Note that for all outgoing links of noden, the same functionf_nis used to determine the corresponding generalized link costs. Sincef_n(p)is strictly increasing inp, Property1holds in networkGas well (by replacing “powerp” with “powerf_n(p)”) and the problem of minimizing lexicographically the vector(θ^T_n)_n_∈_N inGcan be solved using the previously presented algorithms. According to the above definition, each spanning treeT ofGcorresponds to a spanning treeT of Gand vice versa. TreeT is obtained fromT by eliminating every link(n, n) ∈ L. Based on this observation, the following lemma holds:

Lemma 5: LetT^∗be a tree that minimizes lexicographically the vector(θ^T_n)_n_∈_NinG. Then, the corresponding treeT^∗ in G, minimizes lexicographically the vector(φ^T_n)n∈N.

Application I: Taking Into Account Node Receive Power Consumption.

Assume that nodenconsumes powerq_nduring the reception of information. Since all nodes in a broadcast treeT receive the information transmitted by the root noder, the total power consumed by noden6=ris nowp^T_n +qn. Therefore, if we are interested in minimizing lexicographically the total consumed node power, we can setfn(p) =p+qnifn6=r, andfr(p) =p.

Note that in this case we havefn(0)6= 0ifn6=r.

Application II: Maximizing Lexicographically the Mini- mum Remaining Node Battery Lifetime.

Assume we know the duration of the broadcast transmission, t, and the energyE_n of the battery at noden ∈ N at the be- ginning of the transmission, which we call “battery lifetime”.

Assume also for simplicity that the receive powerqn is zero.

Then, if nodentransmits with powerp, at the end of the broadcast transmission, its remaining battery lifetime will beEn−pt.

In an environment where spare batteries are not easily avail- able, it is reasonable to attempt to maintain the remaining battery lifetime at each node as high as possible. Given anr-rooted spanning treeT, the broadcast transmissions defined byT result in remaining battery lifetime at noden∈N,

E_n^T =En−p^T_nt=En− max

l∈L^T_out(n){cl}t

=− max

l∈L^T_out(n){c_lt−E_n+E}+E, whereE= max

n∈N{E_n}.