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Applied Soft Computing 11 (2011) 5684–5694

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l ho me p age: www.elsev ier .com/ l ocate /asoc

inimum energy broadcasting in wireless sensor networks: An ant colonyptimization approach for a realistic antenna model�

ugo Hernández, Christian Blum ∗

LBCOM Research Group, Universitat Politècnica de Catalunya, Barcelona, Spain

r t i c l e i n f o

rticle history:eceived 31 August 2010eceived in revised form 20 January 2011ccepted 23 March 2011vailable online 8 April 2011

a b s t r a c t

The classical minimum energy broadcast (MEB) problem in wireless adhoc networks, which is well-studied in the scientific literature, considers an antenna model that allows the adjustment of thetransmission power to any desired real value from zero up to the maximum transmission power. How-ever, when specifically considering sensor networks, a look at the currently available hardware shows thatthis antenna model is not very realistic. A first contribution of this work is therefore the re-formulation

eywords:inimum energy broadcastingireless sensor networks

ealistic antenna modelnt colony optimization

of the MEB problem for an antenna model that is realistic for sensor networks. In this antenna modeltransmission power levels are chosen from a finite set of possible ones. The second contribution con-cerns the adaptation of ant colony optimization, a current state-of-the-art algorithm for the classicalMEB problem, to the more realistic problem version. The obtained results show that the advantage ofant colony optimization over classical heuristics even grows when the number of possible transmissionpower levels decreases.

. Introduction

Sensor networks are wireless adhoc networks that are beingsed in practical scenarios such as the monitoring of certain events1]. Sensor nodes are generally equipped with omni-directionalntennas for sending and receiving information. They have aacket-forwarding capability in order to communicate via sharednd limited radio channels. In order to transmit information, aender node must adjust its transmission power in order to reachhe desired receiver node. As network lifetime is limited by bat-eries, energy saving is critical. A fundamental problem in sensoretworks arises when one node is required to transmit data to allther nodes of the network. This scenario is known as broadcasting.bviously, for broadcasting to be energy-efficient, the transmis-

ion powers of the sensor nodes should be adjusted such that the

um of the energy spent by all nodes is minimized. This prob-em is know as the minimum energy broadcast (MEB) problem inhe literature [2]. To our knowledge, all existing works from the

� This work was supported by grant TIN2007-66523 (FORMALISM) of the Spanishovernment, and by the EU project FRONTS (FP7-ICT-2007-1). In addition, C. Blumcknowledges support from the Ramón y Cajal program of the Spanish Government,nd H. Hernández acknowledges support from the Comissionat per a Universitats

Recerca del Departament d’Innovació, Universitats i Empresa de la Generalitat deatalunya and from the European Social Fund.∗ Corresponding author.

E-mail addresses: hhernandez@lsi.upc.edu (H. Hernández), cblum@lsi.upc.eduC. Blum).

568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.asoc.2011.03.023

© 2011 Elsevier B.V. All rights reserved.

literature use an antenna model which allows the adjustment ofthe transmission powers to any real value between zero and themaximum transmission power. However, available hardware suchas SunSPOTs (see http://www.sunspotworld.com/) or iSense sen-sor nodes (see http://www.coalesenses.com/) are equipped withantennas that offer a limited set of different transmission powerlevels; 201 in the case of SunSPOTs and seven in the case of theiSense hardware. Note that SunSPOTs are among the most widelyused sensor hardware, while iSense nodes are used by two of thecurrently largest European projects on sensor networks, FRONTSand WISEBED.1

1.1. Related work

The classical minimum energy broadcast (MEB) problem, assum-ing omni-directional antennas whose transmission power can beadjusted to any desired real value between zero and the maximumtransmission power, has mainly been tackled with centralizedheuristics. Only very few distributed approaches can be found inthe literature (see [3,4]). Due to the fact that the ACO approach

proposed in this paper is centralized, we focus in the following onother centralized proposals.

Centralized heuristics include the ones presented in [2,5,6]. Themost popular constructive technique is the broadcast incremen-

1 Concerning FRONTS, see http://fronts.cti.gr; concerning WISEBED, seehttp://www.wisebed.eu.

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H. Hernández, C. Blum / Applied

al power (BIP) algorithm proposed in [2]. Moreover, local searchethods including tree-based methods such as [7,8] and power-

ased methods such as [9] have been developed. More recentlyhe MEB problem was also tackled by metaheuristics [10–13]. Theatest approaches are an ant colony optimization (ACO) approachroposed in [14] and the hybrid genetic algorithm presented in [15].ven though no direct comparison is available between ACO and theybrid genetic algorithm, a rough comparison can be made basedn the results obtained by the two algorithms for a commonly usedet of benchmark instances from the literature. In particular, forenchmark instances with 50 nodes, ACO finds on average for each

nstance an optimal solution in 29.7 out of 30 trials. On the con-rary, the hybrid genetic algorithm on average only finds in 22.9rials an optimal solution for the same problem instances. More-ver, the results of ACO are achieved in less computational time.herefore, the ACO algorithm from [14] can currently be regardeds a state-of-the-art approach among the centralized techniques.

A comprehensive survey of existing work for the MEB problemnd other related problems can be found in [16].

.2. Our contribution

This work offers two contributions. The first one concerns a re-ormulation of the classical MEB problem for an antenna modelhat is frequently encountered, for example, in antennas of sensorodes such as SunSPOTs and iSense nodes. While the classical MEBroblem is formulated for antennas where the transmission poweran be adjusted to any desired real-value between 0 and the maxi-um transmission power, our re-formulation considers an antennaodel where transmission powers must be chosen from a finite

et of discrete values, as it is the case for the antennas found inunSPOTs and iSense sensor nodes.

The second contribution of this article concerns the adaptationf the current state-of-the-art algorithm for the classical MEB prob-em to the re-formulated problem. The obtained results show thathe advantage of this algorithm over classical heuristics even growshen the number of possible transmission power levels decreases.

.3. Organization of the paper

Section 2 is devoted to the description of the minimum energyroadcast problem with realistic antennas. Then, in Sections 3 and

are described the adapted BIP algorithm, respectively the adaptednt colony optimization algorithm. Finally, in Section 5 we providen experimental evaluation of the presented algorithms, whereasection 6 will offer conclusions and an outlook to future work.

. Minimum energy broadcasting with realistic antennas

Given a set of sensor nodes V, we assume that each node i ∈ V canhoose a transmission power level pi such that pi ∈ P = {tp1, . . ., tpm},here P is a finite set of m different transmission powers such that

p1 = 0 and tpl < tpl+1 for all l = 1, . . ., m − 1. Moreover, we assumehat signal power diminishes at a rate proportional to r−˛, where

is the distance to the signal source, and is a parameter that,epending on the environment, takes typically values between 2nd 4. In our work we choose = 2, as in most other works (see,or example, [2]). A sender node i is able to successfully transmit

signal to a receiver node j if pi ≥ k · d(i, j)˛, where d(i, j) is theuclidean distance between i and j, and k is the receiving node’sower threshold for signal detection which is usually normalizedo 1.

The minimum energy broadcast problem with realistic anten-as (MEBRA), as introduced in the following, is NP-hard. This easily

ollows from being a generalization of the standard MEB problems defined in the literature [17]. It can be stated as follows. Given

mputing 11 (2011) 5684–5694 5685

is a set V of nodes with fixed positions in a 2-dimensional area.Introducing a directed link (i, j) between all (ordered) pairs i /= j ofnodes such that d(i, j)˛ ≤ tpm, where d(i, j) is the Euclidean distancebetween i and j, induces a directed network G = (V, E). Given a sourcenode s ∈ V, one must find transmission powers for all nodes suchthat a broadcast from s to all other nodes is possible, and such thatthe sum of all transmission powers is minimal. This correspondsto finding a directed spanning tree T = (V, ET) with root node s in Gsuch that function f(), whose definition is given in the following, isminimized:

f (T) :=∑

i ∈ V

max(i,j) ∈ ETpij (1)

where pij is the minimum transmission power level with whichnode j can receive the signal sent by node i. Technically, pij can bedefined as follows:

pij := min{

tpl ∈ P|l ∈ {2, . . . , m}, tpl ≥ d(i, j)˛, tpl−1 < d(i, j)˛}

(2)

Note that a solution T is easily converted to a transmission powerlevel pi for each node i ∈ V as follows. For all leaf nodes of T we choosepi = 0, and for all other nodes pi := max(i,j) ∈ ET

pij . An example of theMEBRA problem is given in Fig. 1.

3. BIP-algorithm for the MEBRA problem

As mentioned already in Section 1.1, the classical heuristic forthe original MEB problem is the broadcast incremental power (BIP)algorithm proposed in [2]. In the following we outline the adap-tation of this algorithm for the MEBRA problem, for two reasons.First, the way of constructing solutions that is employed by the BIPalgorithm is also used by the ACO algorithm that will be outlinedin Section 4. Second, the adapted BIP algorithm will be used as abenchmark method for comparison in Section 5.

The BIP algorithm starts off with a partial solution T = (VT, ET),where VT : = {s} and ET : = ∅. In other words, T only contains thesource node s. Henceforth we denote by VT the set of sensor nodeswhich are not included in the current partial solution, that is,VT := V \ VT . At each construction step, an additional link is addedto the current partial solution. The set NT of potential links that canbe added to T is defined as follows:

NT := {(i, j) ∈ E|i ∈ VT , j ∈ VT }, (3)

where E is the edge set of the directed network G as defined inSection 2. More specifically, NT consists of those links whose sourcenode is in T whereas the goal node is not in T. The choice of a linkfrom NT is done by means of a greedy function �() that assigns avalue to each e ∈ NT . The BIP algorithm uses the following greedyfunction:

�(e) := f (T ′) − f (T), (4)

where T′ is defined as T with link e already added.2 In other words,the greedy function accounts for the increase of transmission powercaused by adding a link e. At each step of the BIP algorithm, the linke ∈ NT with minimal greedy value is chosen.

After choosing a link e = (i, j) for the expansion of the currentpartial solution T, all the other links from NT (if any) that can beadded to T without any further increase of transmission powersare also added to T (in addition to e). This concerns all links e′ =(i, k) ∈ NT with d(i, k) ≤ pij. Note that the solution construction stops

2 Remember that f(·) denotes the objective function of the MEBRA problem asdefined in Equation 1.

5686 H. Hernández, C. Blum / Applied Soft Computing 11 (2011) 5684–5694

Fig. 1. An example for the MEBRA problem and different solutions. We assume that each sensor node has an antenna as shown in (a), that is, with four transmission powerl tp2, tl ance w2 al sol

4

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evels. The first one, tp1, is always zero. The three other transmission power levels—et us assume that 2 · tp2 < tp4. (b) and (c) each show a solution for a problem inst

· tp2 < tp4, the solution in (c) is better than the one from (b). In fact, (c) is the optim

. The ant colony optimization algorithm

lgorithm 1. ACO for the MEBRA problem.1: INPUT: the network G = (V, E) and a source node s ∈ V2: Tbs := NULL, Trb := NULL, cf := 0, bs update := FALSE3: forall e ∈ Edo �e := 0.5end forall4: while termination conditions not satisfied do5: for j = 1 to nado6: Tj := ConstructBroadcastTree(G, s)7: Tj := Local Search(Tj)8: end for9: Tib := argmin{f (T1), . . . , f (Tna )}10: Update(Tib, Trb, Tbs)11: ApplyPheromoneValueUpdate(cf, bs update, T, Tib, Trb, Tbs)12: cf := ComputeConvergenceFactor(T, Trb, Tbs)13: if cf ≥ 0.99then14: if bs update = TRUE then15: forall e ∈ Edo �e := 0.5end forall16: Trb := NULL, bs update := FALSE17: else18: bs update := TRUE19: end if20: end if21: end while22: OUTPUT: Tbs

In the following we present the adaptation of the ACO algorithmroposed for the classical MEB problem [14] to the MEBRA prob-

em. This algorithm is a so-called MAX–MIN Ant System (MMAS)n the Hyper-Cube Framework [18], which is an iterative algorithmhat works roughly as follows. First, at each iteration na solutionso the problem are constructed in a probabilistic way based onheromone information and heuristic information. Remember thatolutions in this context are spanning trees rooted in source node s.econd, improvement procedures are applied to each of these trees.he pheromone model T used by our ACO algorithm consists of aheromone value �e for each link e ∈ E. After the initialization of theariables Tbs (i.e., the best-so-far solution), Trb (i.e., the restart-bestolution), and cf (i.e., the convergence factor), all the pheromonealues are set to 0.5. At each iteration, after the generation of thea solutions, some of them are used for updating the pheromonealues. The details of the algorithmic framework shown in Algo-ithm 1 are explained in the following. ConstructBroadcastTree(G, s):he construction of a solution in ACO is basically done in the sameay as explained in the context of the BIP algorithm in Section 3.owever, there are two exceptions. First, note that NT may contain

lot of bad-quality links, especially at the beginning of the solu-

ion construction process. Therefore, we considered the followingestriction of NT :

rT := {(∗, l) ∈ NT }, (5)

p3, and tp4—are shown as concentric circles of the area they cover. In this exampleith three sensor nodes: the source node s and nodes 1 and 2. Note that because

ution for this small example.

where l is the best node in VT , which is defined as follows. A node lis called the best node in VT if and only if exists a link e = (k, l) ∈ NT

such that �(e) < �(e′) for all e′ ∈ NT . Hereby, �(·) is the greedy func-tion as defined in Equation 4. After this reduction, set Nr

T may stillcontain quite a lot of links. Therefore, we additionally applied a so-called candidate list strategy, that is, Nr

T is restricted to c elementswith the greatest heuristic information values. After tuning by handwe decided for a setting of c = 8 for all experiments.

The second aspect of the solution construction that is differentto the BIP algorithm is the fact that a link from Nr

T is chosen in aprobabilistic way, rather than deterministically. More in detail, alink from Nr

T is chosen according to the following probabilities:

p(e) := �e · �(e)−1

∑

e′ ∈NrT

�e′ · �(e′)−1, e ∈ Nr

T (6)

LocalSearch(Tj): Note that solutions constructed by the ants maycontain nodes whose transmission power level can be reducedwithout destroying the broadcast property of the solution. There-fore, we first apply the so-called SWEEP procedure (see [2]) in orderto detect and fix these cases. Afterwards the variable neighborhooddescent (VND) algorithm [19] outlined in Algorithm 2 is appliedto further improve the given solution. VND is based on the localsearch procedure r-shrink, which was developed by Das et al. in[9] for the original MEB problem. The only difference between ourimplementation and the one by Das et al. is the re-definition of thetransmission power pij necessary to reach node j from node i. Whilefor the original MEB problem, pij is defined as d(i, j)˛, in the con-text of the MEBRA problem this transmission power is defined asin Section 2. After tuning by hand, parameter rmax from Algorithm2 was set to | V |−1.

Algorithm 2. Variable neighborhood descent (VND).1: INPUT: the network G = (V, E), a source node s ∈ V , a solution T = (V, ET ), aparameter rmax

2: r := 13: while r ≤ rmaxdo4: T ′ := r-shrink(T)5: if f (T ′) < f (T)then T := T ′ and r := 1else r := r + 16: end while7: OUTPUT: a (possibly) improved solution T

Update(Tib, Trb, Tbs): In this procedure Trb and Tbs are set to Tib

(i.e., the iteration-best solution), if f(Tib) < f(Trb) and f(Tib) < f(Tbs).

ApplyPheromoneUpdate(cf, bs update, T, Tib, Trb, Tbs): As it is the

case for all MAX–MIN Ant Systems implemented in the Hyper-Cube Framework, the proposed ACO algorithm may use threedifferent solutions for updating the pheromone values: (i) the

H. Hernández, C. Blum / Applied Soft Co

Table 1The schedule used for values �ib , �rb and �bs depending on cf (the convergence factor)and the Boolean control variable bs update.

bs update = FALSE bs update = TRUE

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teration-best solution Tib, (ii) the restart-best solution Trb and, (iii)he best-so-far solution Tbs. The influence of each of these threeolutions depends on the so-called convergence factor cf, which pro-ides an estimate on the state of convergence of the algorithm. Theheromone update is then performed as follows. First, an updatealue �e for each link e ∈ E is computed:

e := �ib · ı(Tib, e) + �rb · ı(Trb, e) + �bs · ı(Tbs, e), (7)

here �ib is the weight of solution Tib, �rb is the weight of Trb, and �bss the weight of Tbs. Hereby, the three parameters must be chosenuch that �ib + �rb + �bs = 1.0. Moreover, in Eq. (7) the ı-function ishe characteristic function of the set of links in a tree T, that is, ı(T,) = 1 if e ∈ ET, and ı(T, e) = 0 otherwise. Given the �e values for all

∈ E, the following update rule is applied to all pheromone valuese:

e := min{

max{�min, �e + � ·(

�e − �e

)}, �max

}, (8)

here � ∈ (0, 1] is the learning rate, which we have set to 0.1.he upper and lower bounds �max = 0.99 and �min = 0.01 keep theheromone values always in the range (�min, �max), thus prevent-

ng the algorithm from converging to a solution. After tuning, thealues for �ib, �rb and �bs are chosen as shown in Table 1.

ComputeConvergenceFactor(T, Trb, Tbs): This function computes,t each iteration, the convergence factor as

f : =

∑

e ∈ E(Trb)

�e

(|E(Trb)|−1) · �max,respectively cf : =

∑

e ∈ E(Tbs)

�e

(|E(Tbs)|−1) · �max,

(9)

f bs update = FALSE, respectively if bs update = TRUE. Here, �max ishe upper limit for the pheromone values. The convergence factorf can therefore only assume values between 0 and 1. The closer cfs to 1, the higher is the probability to produce the solution Trb (orbs analogously).

. Experimental evaluation

The adapted BIP algorithm from Section 3 and the proposed ACOlgorithm from Section 4 were implemented in ANSI C++. Moreover,CC 3.2.2 was used for compiling the software. All experimentsere executed on a PC with an AMD64X2 4400 processor and 4 GB

f memory. The parameter setting of the ACO algorithm is summa-ized in Table 2. Note that the values of parameters na, �, �min andmax are the standard values chosen for most MAX–MIN Ant Sys-

able 2ummary of the parameters of ACO with their values.

Parameter Value Role of the parameter

na 10 Number of ants per iterationc 8 Number of elements in the candidate list� 0.1 Learning rate (controls the speed of convergence)�min 0.01 Lower limit of the pheromone values�max 0.99 Upper limit of the pheromone valuesrmax | V |−2 Upper limit of r within the r-shrink algorithm used by VND

mputing 11 (2011) 5684–5694 5687

tems implemented in the Hyper-Cube Framework. As mentionedbefore, the values for c and rmax were chosen after tuning by hand.

5.1. Experimental setup

Two sets of problem instances were used for the experimentalevaluation. The first set, consisting of 30 problem instances witheach 50 nodes was introduced in [12,13] for the original MEB prob-lem. The second set, which was introduced in [14], contains 30problem instances with 100 nodes each.

As an example, we decided to work with antennas whose max-imum transmission power level reaches a distance of 500 spaceunits. Moreover, we assume that antennas offer x transmissionpower levels that are equidistantly distributed in [0, 500]. Forexample, when x = 3 the distances reached with the three differenttransmission power levels are {0, 250, 500}. With the motivationof testing two antennas that are used in practice, we performedtests with x = 7 (corresponding to the antennas of iSense nodes),and x = 201 (corresponding to the antennas of SunSPOTs).3

Three algorithms were applied to all problem instances: (1) theadapted BIP algorithm, (2) BIP+VND, which is the BIP algorithmwith the subsequent application of VND as outlined in Section 4,and (3) the ACO algorithm. All algorithms were applied consid-ering antennas with seven transmission power levels, and alsoconsidering antennas with 201 transmission power levels. As BIPand BIP+VND are deterministic algorithms they were applied toeach problem instance exactly once (per antenna version). On theother side, ACO was applied 30 times to each problem instance (perantenna version). Hereby, the following computation time limitswere used. Concerning the instances with 50 nodes, a computationtime limit of 20 seconds per run was used, whereas 100 secondswere used as a computation time limit for the larger instances with100 nodes.

5.2. Results

Detailed numerical results are shown in Tables 3–6 (seeAppendix A). The structure of these tables is as follows. The firstcolumn of each table provides the instance name. Afterwardsthe results of the three algorithms are provided. Concerning BIP,which is a deterministic constructive heuristic, the result and thecomputation time are provided. Note that a computation time of0.00 means that the algorithm was faster than 0.01 seconds. Thesame holds for BIP + VND. In addition, the percentage improvement(labelled deviation) over the results of BIP are provided. Note thatnegative percentages indicate an improvement over BIP. Finally,the results of ACO are given in four columns. More specifically, theprovided information consists of the best result obtained within30 runs (see column best), the average result over 30 runs (seecolumn average), and the average computation time for reachingthe best solution of each run (see column time (s)). As in the caseof BIP+VND, the percentage improvement of ACO over BIP is alsoshown (see column deviation). Note that the calculation of thisdeviation was based on the average results of ACO, rather than onthe best result. Finally, the last row of each table provides aver-ages of all measures over all instances considered in the respectivetable. In order to improve the readability of the results, the per-

centage deviations of BIP + VND and ACO over BIP are presented ingraphical form in Fig. 2. Hereby, each of the four subfigures of Fig. 2corresponds to one of the four tables.

3 Note that x = 7 means that the antennas of iSense nodes(http://www.coalesenses.com/) have six transmission power levels greaterthan zero. The same for SunSPOTs (http://www.sunspotworld.com/): x = 201 meansthat SunSPOT antennas have 200 transmission power levels greater than zero.

5688 H. Hernández, C. Blum / Applied Soft Computing 11 (2011) 5684–5694

F ances,i

serat

ig. 2. Graphical presentation of the results. The x-axes range over the problem instndicate an improvement over BIP.

The results show, first, that both BIP + VND and ACO are—for eachingle instance and antenna version—better than BIP. Moreover,

xcept for one single case where both algorithms achieve the sameesult, ACO always improves quite substantially over BIP + VND. Theverage percentages of deviation, which are to be found in the lastable rows, are summarized in Fig. 3a and b. In addition to the results

whereas the y-axes show the percent deviation from BIP. Note that negative values

of BIP + VND and ACO for the antenna versions with seven, respec-tively 201, transmission power levels, these tables also provide the

results of BIP + VND and ACO for the original MEB problem (as takenfrom [14]). There are several interesting observations that we wantto point out. First, the results of BIP + VND and ACO are very similarfor the original MEB problem and the MEBRA problem with anten-

H. Hernández, C. Blum / Applied Soft Co

Problem version BIP + VND ACO ACOMEBRA (7 le vels) -14. 61 % -31.8 8 % -31.7 1 %MEBRA (201 le vels ) -10. 67 % -21.1 7 % -20.5 7 %Original MEB -10. 61 % -21.4 7 % -21.2 4 %

(a) 50 node instance s, advantage over BIP

Problem version BIP + VND ACO ACOMEBRA (7 le vels) -15. 07 % -36.9 5 % -35.3 7 %MEBRA (201 le vels ) -10. 56 % -20.1 3 % -17.5 5 %Original MEB -10.31 % -19.6 9 % -18.4 0 %

(b) 100 node instance s, advantage over BIP

Problem versio n BIP + VND ACO ACOMEBRA (7 le vels ) 0.0 6 sec 0.7 6 sec 4.0 7 secMEBRA (201 le vels ) 0.0 5 sec 0.9 2 sec 7.5 7 secOriginal MEB 0.0 7 sec 1.5 1 sec 8.4 2 sec

(c) 50 node instances , computatio n tim e

Problem version BIP + VND ACO ACOMEBRA (7 le vels) 0.45 sec 26. 74 sec 41. 42 secMEBRA (201 le vels ) 0.40 sec 29. 11 sec 61. 02 secOriginal MEB 0.47 sec 28. 70 sec 63. 83 sec

nfngtTaBMmi

F(

(d) 100 node instances , computatio n tim e

Fig. 3. Summarized algorithm behaviour for different problem versions.

as of 201 transmission power levels. This can be explained by theact that the MEBRA problem becomes more similar to the origi-al MEB problem when the number of transmission power levelsrows. Second, the advantage of BIP + VND and ACO over BIP seemso grow when the number of transmission power levels decreases.his is indicted by the average percentages of deviation from BIPs summarized in Fig. 3c and d. The advantage of BIP+VND over

IP improves from being at around −10% (concerning the originalEB problem and the MEBRA problem with antennas of 201 trans-ission power levels) to an advantage of around −15%, both for

nstances of 50 nodes and instances of 100 nodes. In the case of ACO,

Number of

Dev

iatio

n (in

per

cent

)

981716151413121117

−15

−20

−25

−30

−35

−40

(a) Instan

Number of

Dev

iatio

n (in

per

cent

)

981716151413121117

−15

−20

−25

−30

−35

−40

(b) Instan

ig. 4. Deviation of the average solution quality of ACO from the solution quality obtainedb) shows the same information concerning the instances with 100 nodes. The x-axis ran

mputing 11 (2011) 5684–5694 5689

this improvement over BIP is even more pronounced. The advan-tage increases from around −20% to more than −30%. In order tostudy this aspect in more detail we repeated all the experiments forantennas with a number of transmission power levels inbetweenseven and 201. More precisely, experiments were conducted forall numbers of transmission power levels in {11 + 10 · i | 0 ≤ i ≤ 18}.The results are shown by means of bar-plots in Fig. 4. These resultsconfirm, indeed, that the deviation of the average solution qualityof ACO from the solution quality obtained by BIP grows when thenumber of transmission power levels decreases.

5.3. Study of the structure of solutions

In order to study possible reasons for the increasing advantageof ACO over BIP when the number of transmission power levelsdecreases, we exemplary provide the solutions of BIP and ACO forproblem instance p50.25 (one of the problem instances with 50nodes) in graphical form; see Fig. 5 for seven transmission powerlevels, and Fig. 6 for 201 transmission power levels. In the followingwe first focus on the solutions for antennas with 201 transmis-sion power levels. In this case the disadvantage of BIP seems to bebased on the fact that many nodes use an intermediate transmis-sion power level (neither small nor large), whereas in the solutionof ACO only one node uses a large transmission power level and veryfew additional nodes use rather small transmission power levels.With this in mind let us have a look now at the solutions of BIP andACO for antennas with only seven transmission power levels (seeFig. 5). The solution of BIP seems to suffer from the same disad-vantage as in the case of 201 transmission power levels. However,

we can also observe an additional disadvantage. The rings paintedin light-gray indicate wasted transmission power, which occurswhen the furthest neighbor that is reached by the chosen transmis-sion power level of a node is quite far away from the transmission

transmission power levels

2011911811711611511411311211111011

ce s with 50 nodes

transmission power levels

2011911811711611511411311211111011

ces with 100 nodes

by BIP. (a) The average deviation concerning the 30 instances with 50 nodes, whileges over different numbers of transmission power levels between seven and 201.

5690 H. Hernández, C. Blum / Applied Soft Computing 11 (2011) 5684–5694

0

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

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ig. 5. Solutions for problem instance p50.25 using seven transmission power levelsensor nodes. Gray shaded rings show the wasted energy that is due to transmitting

ower range of this level. The BIP solution seems to suffer quite aot from this phenomenon, which may explain the growing advan-age of ACO over BIP when the number of transmission power levelsecreases.

.4. Effect of VND on the performance of ACO

Finally, we were interested in the contribution of VND to the per-ormance of ACO. Therefore, all experiments concerning antennasith seven and 201 transmission power levels were repeated forCO without the application of VND. The corresponding algorithm

s henceforth denoted by ACO�. The average deviation of the resultsbtained by ACO� from the results obtained by BIP is shown in the

ast column of the tables of Fig. 3a and b. The corresponding aver-ge computation times are given in the tables of Fig. 3c and d. Theseesults show that the performance of ACO� is only slightly worsehan the performance of ACO. However, ACO needs significantly

472222.2 2value:functionion

ce node is node 46. The circles show the transmission power levels of the individualmore power than necessary in order reach the furthest neighbor node.

less computation time than ACO�. This is because the applicationof VND helps the ACO algorithm in quickly reaching high qualityareas of the search space.

5.5. Considerations concerning computation time

Concerning computation time we observe that the type of prob-lem (MEB versus MEBRA with different transmission power levels)does not seem to play an important role for the computation times,that is, they stay in the same order of magnitude. The averagecomputation times of BIP + VND and ACO are given in the second,respectively third, column of Table 3 (for instances with 50 nodes)and Table 3 (for instances with 100 nodes). In this context note

that both the algorithms for the original MEB problem from [14]and the adapted algorithms from this paper have been run with thesame computation time limits on the same processors. Therefore,the computation times can be directly compared.

H. Hernández, C. Blum / Applied Soft Computing 11 (2011) 5684–5694 5691

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each the furthest neighbor node.

. Conclusions

In this work we have first introduced a generalization of thelassical minimum energy broadcast problem in wireless adhocetworks such as sensor networks. The introduced generalizationoncerns the fact that in many radio antennas used in real sen-or network hardware, the choice of a transmission power level is

imited to a discrete set of values, rather than being unlimited. As

second contribution we adapted ant colony optimization, a cur-ent state-of-the-art algorithm for the classical minimum energyroadcast problem, to the problem generalization. The experimen-

asted energy that is due to transmitting with more power than necessary in order

tal results have shown that the advantage of this algorithm overclassical heuristics even grows when the number of available trans-mission power levels decreases.

In the future we plan to develop a distributed version of the antcolony optimization algorithm for minimum energy broadcasting.Note that the use of a distributed algorithm is always necessarywhen the structure of the deployed network is unknown, which

might be the case in quite a few applications of sensor networks.In this context, the development of suitable heuristic informationfor solution construction as well as the development of distributedsolution improvement procedures will be the main challenges.

5692 H. Hernández, C. Blum / Applied Soft Computing 11 (2011) 5684–5694

Table 3Results for the 30 instances with 50 nodes when using antennas with seven transmission power levels.

Instance BIP BIP + VND ACO

Obj. fun Time (s) Deviation Obj. fun Time (s) Best Deviation Average Time (s)

p50.00 736,111.11 0.00 −16.98% 611,111.11 0.06 506,944.44 −31.13% 506,944.44 0.14p50.01 611,111.11 0.00 −25.00% 458,333.33 0.06 458,333.33 −25.00% 458,333.33 0.18p50.02 805,555.56 0.00 −12.07% 708,333.33 0.06 520,833.33 −35.34% 520,833.33 0.62p50.03 590,277.78 0.00 −9.41% 534,722.22 0.05 423,611.11 −28.24% 423,611.11 0.68p50.04 701,388.89 0.00 −17.82% 576,388.89 0.06 430,555.56 −38.61% 430,555.56 0.18p50.05 611,111.11 0.00 −5.68% 576,388.89 0.06 486,111.11 −20.45% 486,111.11 0.44p50.06 729,166.67 0.00 −20.95% 576,388.89 0.06 479,166.67 −34.29% 479,166.67 0.15p50.07 722,222.22 0.00 −14.42% 618,055.56 0.05 527,777.78 −26.92% 527,777.78 0.44p50.08 722,222.22 0.00 −7.69% 666,666.67 0.05 402,777.78 −44.23% 402,777.78 0.43p50.09 680,555.56 0.00 −10.20% 611,111.11 0.06 493,055.56 −27.55% 493,055.56 1.79p50.10 743,055.56 0.00 −13.08% 645,833.33 0.05 534,722.22 −28.04% 534,722.22 0.33p50.11 708,333.33 0.00 −10.78% 631,944.44 0.06 444,444.44 −37.25% 444,444.44 0.08p50.12 659,722.22 0.00 −7.37% 611,111.11 0.06 493,055.56 −25.26% 493,055.56 1.55p50.13 750,000.00 0.00 −12.04% 659,722.22 0.05 520,833.33 −30.56% 520,833.33 0.66p50.14 812,500.00 0.00 −24.79% 611,111.11 0.08 513,888.89 −36.75% 513,888.89 0.25p50.15 694,444.44 0.00 −11.00% 618,055.56 0.06 493,055.56 −29.00% 493,055.56 1.99p50.16 729,166.67 0.00 −15.24% 618,055.56 0.06 541,666.67 −25.71% 541,666.67 1.22p50.17 631,944.44 0.00 −18.68% 513,888.89 0.05 500,000.00 −20.88% 500,000.00 0.17p50.18 763,888.89 0.00 −16.36% 638,888.89 0.05 479,166.67 −37.27% 479,166.67 0.21p50.19 680,555.56 0.00 −27.55% 493,055.56 0.06 430,555.56 −36.16% 434,490.74 3.11p50.20 777,777.78 0.00 −21.43% 611,111.11 0.06 493,055.56 −36.61% 493,055.56 0.11p50.21 736,111.11 0.00 −4.72% 701,388.89 0.05 472,222.22 −35.85% 472,222.22 0.43p50.22 645,833.33 0.00 −10.75% 576,388.89 0.06 437,500.00 −32.26% 437,500.00 0.09p50.23 729,166.67 0.00 −8.57% 666,666.67 0.05 527,777.78 −27.62% 527,777.78 0.60p50.24 736,111.11 0.00 −10.38% 659,722.22 0.06 500,000.00 −32.08% 500,000.00 0.50p50.25 791,666.67 0.00 −17.54% 652,777.78 0.05 472,222.22 −40.35% 472,222.22 0.28p50.26 763,888.89 0.00 −11.82% 673,611.11 0.06 527,777.78 −30.82% 528,472.22 0.62p50.27 805,555.56 0.00 −10.34% 722,222.22 0.06 555,555.56 −31.03% 555,555.56 0.82p50.28 826,388.89 0.00 −27.73% 597,222.22 0.08 548,611.11 −33.53% 549,305.56 4.05p50.29 736,111.11 0.00 −17.92% 604,166.67 0.05 458,333.33 −37.74% 458,333.33 0.59

721,064.82 0.00 −14.61% 614,814.82 0.06 489,120.37 −31.88% 489,297.84 0.76

Table 4Results for the 30 instances with 50 nodes when using antennas with 201 transmission power levels.

Instance BIP BIP + VND ACO

Obj. fun Time (s) Deviation Obj. fun Time (s) Best Deviation Average Time (s)

p50.00 523,075.00 0.00 −16.82% 435,093.75 0.08 405,175.00 −22.52% 405,272.08 0.66p50.01 526,025.00 0.00 −11.67% 464,662.50 0.06 378,718.75 −28.00% 378,718.75 2.10p50.02 544,862.50 0.00 −11.52% 482,112.50 0.05 398,787.50 −26.81% 398,787.50 1.01p50.03 412,675.00 0.00 −7.40% 382,137.50 0.06 321,968.75 −21.98% 321,968.75 0.41p50.04 429,493.75 0.00 −9.73% 387,712.50 0.05 331,293.75 −22.86% 331,293.75 1.60p50.05 471,518.75 0.00 −11.06% 419,375.00 0.06 392,512.50 −16.76% 392,512.50 0.36p50.06 507,443.75 0.00 −10.50% 454,175.00 0.05 392,218.75 −22.71% 392,218.75 0.30p50.07 502,312.50 0.00 −9.90% 452,568.75 0.06 408,406.25 −18.69% 408,406.25 0.48p50.08 433,562.50 0.00 −12.53% 379,250.00 0.05 347,325.00 −19.89% 347,325.00 0.96p50.09 452,750.00 0.00 −10.12% 406,925.00 0.05 353,212.50 −21.99% 353,212.50 1.11p50.10 499,337.50 0.00 −3.42% 482,281.25 0.06 423,068.75 −15.27% 423,068.75 0.49p50.11 442,775.00 0.00 −6.84% 412,481.25 0.05 374,225.00 −15.48% 374,225.00 3.42p50.12 465,137.50 0.00 −8.30% 426,518.75 0.06 398,731.25 −14.28% 398,731.25 0.29p50.13 516,387.50 0.00 −4.27% 494,343.75 0.04 406,487.50 −21.28% 406,487.50 0.71p50.14 566,643.75 0.00 −5.95% 532,931.25 0.04 420,356.25 −25.82% 420,356.25 0.39p50.15 438,243.75 0.00 −4.60% 418,093.75 0.05 374,937.50 −14.45% 374,937.50 0.74p50.16 520,862.50 0.00 −13.37% 451,212.50 0.06 420,850.00 −19.20% 420,850.00 0.33p50.17 425,031.25 0.00 −10.36% 381,000.00 0.06 361,987.50 −14.83% 361,987.50 0.47p50.18 458,956.25 0.00 −9.98% 413,137.50 0.05 383,562.50 −16.43% 383,562.50 0.29p50.19 497,393.75 0.00 −7.75% 458,837.50 0.04 340,006.25 −31.64% 340,006.25 0.41p50.20 538,018.75 0.00 −13.95% 462,962.50 0.06 419,450.00 −22.04% 419,450.00 0.47p50.21 463,031.25 0.00 −10.22% 415,712.50 0.05 367,487.50 −20.63% 367,487.50 0.23p50.22 477,400.00 0.00 −11.69% 421,600.00 0.05 351,343.75 −26.40% 351,343.75 3.49p50.23 492,225.00 0.00 −17.92% 404,031.25 0.06 389,487.50 −20.87% 389,487.50 0.52p50.24 534,443.75 0.00 −15.62% 450,975.00 0.06 409,250.00 −23.43% 409,250.00 0.59p50.25 533,143.75 0.00 −13.80% 459,568.75 0.06 391,675.00 −26.53% 391,675.00 3.19p50.26 528,281.25 0.00 −12.02% 464,756.25 0.05 411,450.00 −22.12% 411,450.00 0.80p50.27 557,493.75 0.00 −10.20% 500,656.25 0.06 458,293.75 −17.79% 458,293.75 1.06p50.28 529,150.00 0.00 −13.11% 459,762.50 0.04 422,187.50 −20.21% 422,187.50 0.44p50.29 525,537.50 0.00 −15.55% 443,825.00 0.05 398,131.25 −24.24% 398,131.25 0.14

493,773.75 0.00 −10.67% 440,623.33 0.05 388,419.58 −21.17% 388,422.82 0.92

H. Hernández, C. Blum / Applied Soft Computing 11 (2011) 5684–5694 5693

Table 5Results for the 30 instances with 100 nodes when using antennas with seven transmission power levels.

Instance BIP BIP + VND ACO

Obj. fun Time (s) Deviation Obj. fun Time (s) Best Deviation Average Time (s)

p100.00 722,222.22 0.00 −15.38% 611,111.11 0.44 437,500.00 −39.42% 437,500.00 13.32p100.01 729,166.67 0.00 −7.62% 673,611.11 0.42 465,277.78 −35.81% 468,055.56 32.55p100.02 805,555.56 0.00 −22.41% 625,000.00 0.48 486,111.11 −39.66% 486,111.11 21.01p100.03 694,444.44 0.00 −12.00% 611,111.11 0.42 472,222.22 −31.67% 474,537.04 35.26p100.04 763,888.89 0.00 −13.64% 659,722.22 0.44 500,000.00 −33.82% 505,555.56 41.86p100.05 840,277.78 0.00 −11.57% 743,055.56 0.42 541,666.67 −34.05% 554,166.67 37.91p100.06 847,222.22 0.00 −19.67% 680,555.56 0.46 493,055.56 −41.80% 493,055.56 23.68p100.07 770,833.33 0.00 −16.22% 645,833.33 0.46 472,222.22 −37.96% 478,240.74 12.10p100.08 763,888.89 0.00 −16.36% 638,888.89 0.52 506,944.44 −32.85% 512,962.96 36.60p100.09 729,166.67 0.00 −6.67% 680,555.56 0.42 506,944.44 −29.97% 510,648.15 31.61p100.10 687,500.00 0.00 −15.15% 583,333.33 0.43 409,722.22 −38.75% 421,064.81 35.02p100.11 708,333.33 0.00 −9.80% 638,888.89 0.46 472,222.22 −33.33% 472,222.22 25.26p100.12 826,388.89 0.00 −21.85% 645,833.33 0.44 500,000.00 −39.19% 502,546.30 25.60p100.13 833,333.33 0.00 −20.83% 659,722.22 0.51 444,444.44 −46.67% 444,444.44 7.44p100.14 694,444.44 0.00 −10.00% 625,000.00 0.42 465,277.78 −32.10% 471,527.78 30.60p100.15 680,555.56 0.00 −12.24% 597,222.22 0.44 451,388.89 −33.61% 451,851.85 17.56p100.16 798,611.11 0.00 −18.26% 652,777.78 0.46 437,500.00 −44.32% 444,675.93 35.96p100.17 763,888.89 0.00 −15.45% 645,833.33 0.43 472,222.22 −38.15% 472,453.70 27.16p100.18 645,833.33 0.00 −10.75% 576,388.89 0.44 458,333.33 −28.03% 464,814.81 34.36p100.19 833,333.33 0.00 −20.83% 659,722.22 0.43 465,277.78 −43.92% 467,361.11 19.28p100.20 715,277.78 0.00 −16.50% 597,222.22 0.42 458,333.33 −35.83% 459,027.78 28.54p100.21 847,222.22 0.00 −19.67% 680,555.56 0.46 465,277.78 −44.95% 466,435.19 34.01p100.22 784,722.22 0.00 −15.93% 659,722.22 0.49 506,944.44 −34.54% 513,657.41 35.27p100.23 826,388.89 0.00 −18.49% 673,611.11 0.41 527,777.78 −36.13% 527,777.78 18.59p100.24 715,277.78 0.00 −17.48% 590,277.78 0.41 479,166.67 −32.78% 480,787.04 31.25p100.25 708,333.33 0.00 −13.73% 611,111.11 0.47 472,222.22 −33.33% 472,222.22 9.55p100.26 770,833.33 0.00 −18.92% 625,000.00 0.44 444,444.44 −42.13% 446,064.81 28.26p100.27 805,555.56 0.00 −15.52% 680,555.56 0.45 486,111.11 −39.14% 490,277.78 27.12p100.28 750,000.00 0.00 −12.96% 652,777.78 0.47 465,277.78 −37.96% 465,277.78 12.58p100.29 770,833.33 0.00 −6.31% 722,222.22 0.42 486,111.11 −36.70% 487,962.96 32.92

761,111.11 0.00 −15.07% 644,907.41 0.45 475,000.00 −36.95% 478,109.57 26.74

Table 6Results for the 30 instances with 100 nodes when using antennas with 201 transmission power levels.

Instance BIP BIP + VND ACO

Obj. fun Time (s) Deviation Obj. fun Time (s) Best Deviation Average Time (s)

p100.00 459,212.50 0.00 −12.04% 403,943.75 0.40 345,231.25 −24.82% 34,5231.25 11.33p100.01 441,043.75 0.00 −10.29% 395,675.00 0.37 361,331.25 −18.01% 361,623.75 40.57p100.02 474,512.50 0.00 −9.01% 431,768.75 0.36 385,406.25 −18.78% 385,411.67 9.36p100.03 458,281.25 0.00 −15.01% 389,487.50 0.42 365,556.25 −20.12% 366,090.62 23.86p100.04 479,650.00 0.00 −9.08% 436,093.75 0.40 393,343.75 −17.89% 393,829.58 15.68p100.05 531,850.00 0.00 −11.72% 469,525.00 0.48 424,700.00 −20.15% 424,700.00 18.91p100.06 491,356.25 0.00 −14.97% 417,781.25 0.36 385,681.25 −20.89% 388,715.83 39.89p100.07 463,506.25 0.00 −10.25% 415,993.75 0.36 350,681.25 −24.24% 351,135.42 14.84p100.08 472,868.75 0.00 −9.71% 426,931.25 0.38 380,725.00 −19.41% 381,064.58 43.65p100.09 480,850.00 0.00 −7.26% 445,956.25 0.38 374,731.25 −22.07% 374,731.25 4.91p100.10 424,231.25 0.00 −11.23% 376,575.00 0.38 342,006.25 −19.38% 342,006.25 3.86p100.11 471,656.25 0.00 −11.29% 418,425.00 0.41 362,737.50 −23.09% 362,737.50 9.21p100.12 500,500.00 0.00 −10.30% 448,943.75 0.38 401,350.00 −19.70% 401,879.17 43.62p100.13 451,593.75 0.00 −11.31% 400,525.00 0.45 338,781.25 −24.98% 338,783.96 39.31p100.14 429,406.25 0.00 −11.30% 380,875.00 0.41 351,075.00 −18.17% 351,387.92 42.73p100.15 432,450.00 0.00 −9.02% 393,431.25 0.38 358,181.25 −17.17% 358,181.25 26.72p100.16 424,568.75 0.00 −11.37% 376,300.00 0.36 344,987.50 −18.56% 345,751.67 32.89p100.17 458,856.25 0.00 −8.91% 417,956.25 0.38 378,987.50 −15.69% 386,868.33 50.71p100.18 408,706.25 0.00 −9.82% 368,581.25 0.38 338,337.50 −16.09% 342,941.04 25.93p100.19 479,712.50 0.00 −8.98% 436,618.75 0.44 370,693.75 −22.73% 370,693.75 10.19p100.20 454,231.25 0.00 −9.47% 411,225.00 0.40 360,156.25 −20.42% 361,480.62 49.89p100.21 450,000.00 0.00 −9.33% 408,031.25 0.41 369,750.00 −17.71% 370,295.62 22.70p100.22 454,412.50 0.00 −6.26% 425,981.25 0.32 373,293.75 −17.84% 373,324.58 24.32p100.23 537,300.00 0.00 −12.88% 468,118.75 0.40 417,356.25 −22.25% 417,749.38 48.98p100.24 442,118.75 0.00 −9.51% 400,068.75 0.43 364,725.00 −17.41% 365,165.00 47.77p100.25 445,268.75 0.00 −11.48% 394,168.75 0.46 365,462.50 −17.82% 365,938.96 44.36p100.26 465,362.50 0.00 −10.09% 418,418.75 0.42 357,131.25 −23.26% 357,131.25 17.85p100.27 459,843.75 0.00 −8.68% 419,950.00 0.37 377,900.00 −17.55% 379,123.96 32.73p100.28 473,806.25 0.00 −16.98% 393,337.50 0.45 356,293.75 −24.74% 356,566.46 49.76p100.29 470,712.50 0.00 −9.19% 427,462.50 0.37 363,006.25 −22.88% 363,006.25 26.78

462,928.96 0.00 −10.56% 413,938.33 0.40 368653.33 −20.13% 369,451.56 29.11

5 Soft Co

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694 H. Hernández, C. Blum / Applied

ppendix A

See Tables 3–6.

eferences

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