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Using dynamic programming to solve the WirelessSensor Network Configuration Problem

Ada Gogu, Dritan Nace, Enrico Natalizio, Yacine Challal

To cite this version:Ada Gogu, Dritan Nace, Enrico Natalizio, Yacine Challal. Using dynamic programming to solve theWireless Sensor Network Configuration Problem. Journal of Network and Computer Applications(JNCA), Elsevier, 2017, 83, pp.140-154. �10.1016/j.jnca.2017.01.022�. �hal-01469355�

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Using dynamic programming to solve theWireless Sensor Network Configuration

ProblemAda Gogu, Dritan Nace, Enrico Natalizio and Yacine Challal

Abstract—This work studies the problem of networkconfiguration for Wireless Sensor Networks (WSN), con-sisting of two interdependent problems: sensor placementand topology control, by taking into consideration boththe traffic pattern and the transmission range assignment.The design objectives are i) reducing the overall energyconsumption and ii) ensuring node energy consumptionfairness between the sensors. First, the problem of placingthe sensors in the optimal positions is studied and thena power control scheme is put in place to manage thetopology of the network. For both the two sub-problems,the linear network case is treated and then extended to thetwo-dimensional case. The two sub-problems are treatedwithin a unifying mathematical framework based on dy-namic programming, in order to guarantee the optimalityof the solution. The method can easily be adapted to solvethe problem for discrete values of transmission range. Themethod presented in this work shows a low computationalcomplexity in comparison to other methods, and, due toits implementation simplicity, it will be of great helpto network designers in the planning phase of WSNdeployment.

Index Terms—WSN, Dynamic programming, SensorPlacement, Network configuration;

I. Introduction

Wireless Sensor Networks (WSN) is a disruptivetechnology with potentially a wide range spectrumof applications such as remote health surveillance,precision agriculture, air and water pollution detectionand containment, home automation, supply chain mon-itoring, etc. [? ]. WSN development is the culminationof recent advances in microelectromechanical systems(MEMS) and wireless communication technologies. Atypical WSN is composed of a set of tiny motes thatsense the environment and transmit sensed informa-tion, hop-by-hop, to a processing workstation throughthe Base Station (BS). A simple WSN is presented inFig. 1. Each mote comprises sensors, a processing unit,memory, radio transceiver and a battery.

Faced with stringent constraints that affect energy,bandwidth and memory use among others, WSN

Ada Gogu is with Polytechnic University of Tirana, MotherTeresa Square, 4, Tirana, Albania. D. Nace, E. Natalizio and Y.Challal are with the Heudiasyc Laboratory (UMR CNRS 7253),Universite de Technologie de Compiegne, Compiegne 60205, France.Y. Challal is also with Laboratoire de Methodes de Conceptionde Systemes at Ecole nationale Superieure d Informatique, Alge-ria E-mail: ada.goguhds.utc.fr, dnacehds.utc.fr, enataliziohds.utc.fr,ychallalhds.utc.fr

Processing and Decision

Work StationSink

Sensed Field

WSN

Interrogation

Through Internet

Motes

Figure 1: Wireless sensor network

technology needs to be carefully managed in order tomeet the requirements of applications. Optimizationtechniques and strategies are applied at the physical,access control, network and application layers toimprove their performance. A primary concern in WSNis the energy constraint. A carefully designed networkcan be a very effective means to conserve energy andtherefore extend the lifetime of the network.

In this work, we focus on the network configurationproblem, which consists in assessing two main aspectsof WSN:

1) Transmission structure: The transmission struc-ture determines the organization between thesensors such that their data flows may reach theBase Station (BS) through a many-to-one trafficparadigm. For an energy constrained technology,the design of a data transmission structure isstrategic. The design process can be even moredifficult if other application-dependent issues likedata aggregation need to be taken into account.

2) Deployment and range assignment: A many-to-one traffic pattern tends to overuse the nodesclosest to the BS for relaying the traffic generatedby other nodes, thus causing an unbalanceddistribution of the traffic load throughout thenetwork known as the energy hole problem.As remarked in different works [? ? ? ] thesenodes will be more prone to failures, and theirfailure may culminate in disconnections, networkpartitioning, and even complete paralysis of thenetwork.Two possible ways of overcoming the issues

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raised by the mentioned aspects are1:• Energy-aware deployment. This aims at find-

ing a deployment scheme that distributesload fairly over all the sensors in the network.

• Transmission range assignment. An effectivetransmission range assignment will balancethe energy consumption between the sensorsin the network. Obviously, the transmissionrange assignment should take into accountthe corresponding sensor load. Moreover,the transmission range is closely related tothe distance from the BS which comes totake into consideration the number of hopsneeded to reach it. Trade-offs between theload distribution and the number of hopscomplicate the problem of transmission rangeassignment under energy constraints.

In this study, we focus on the design of optimalstrategies in terms of a given objective (i.e.: energyefficiency) for WSN. We examine two sub-problems,which are derived by the aforementioned main aspectsof the network configuration problem, namely: i) sen-sor placement and ii) topology control based on powercontrol. Both problems seek to extend the lifetime ofthe network by finding the best combination betweena feasible transmission structure and a particularscheme (a deployment scheme in the case of thesensor placement problem and a range assignmentscheme in the case of the topology control problem).Nonetheless, the context of these two problems, relatedto the sensor deployment assumption, is not the same.Sensor placement might be seen as the pre-deploymentenergy-aware network configuration problem, whereastopology control corresponds to the post-deploymentenergy-aware network configuration problem. For eachof the problems we begin our investigation with asimple case, i.e. a linear network, and then extend itto the two-dimensional network case.

The contribution of this paper is twofold and canbe summarized as follows:• We solve the sensor placement problem and the

topology control one at the optimality. Further-more, we provide a complete study as we considerseveral scenarios involving traffic aggregationand equitable expenditure of energy, and providecomparisons between discrete and continuouspower assignment scenarios.

• We propose a unifying mathematical framework todeal with the two sub-problems, which is based ondynamic programming. Our approach has a lowcomplexity compared with nonlinear program-ming approaches proposed in the literature, andwe believe it provides a powerful tool to deal withthe high combinatorics of such problems.

The rest of the paper is organized as follows.

1Load balancing at the maintenance phase is beyond the scope ofthis work.

Section II presents a brief state of the art for differentvariants of the problem and position this work inthe body of existing literature. In Section III, weexamine the sensor placement problem and proposedynamic programming approaches for both linearand two-dimensional networks. Section IV is devotedto the network configuration problem. Apart thesolution approaches for several scenarios, we providecomparative computational tests. Finally, section Vconcludes the paper.

II. Problem description and related work

This section gives an overview of works related tothe network configuration problem. Specifically, weinvestigate the state of the art of the two sub-problemstreated in this paper: sensor placement and topologycontrol based on power control. In literature, only onework has investigated these two sub-problems withina unifying framework [? ]. However, the authors of thementioned work propose heuristic solutions for boththe sub-problems, and they validate the results throughsimulations, without ensuring the optimality of theirsolution. Our contribution aims at building an analysisframework based on a more rigorous mathematicalapproach. As, at the best of our knowledge, no otherworks have investigated the two sub-problems by usinga unifying solution methodology, we will present theliterature split into two parts.

A. Sensor placementIn WSN literature the sensor placement problem may

group different optimization problems depending onthe constraints and objective function. For an extendedstate of art of the sensor placement problem, the readeris referred to [? ]. Some formulations of this problemaims at deterministically placing the sensors in orderto meet some requirement such as minimizing energyconsumption [? ? ], fault tolerance [? ], or maximizinglifetime under physical attacks [? ]. Other works [? ? ]consider dynamic environments in which the objectivesand tasks can change over the time and the sensorplacement is reconfigured frequently. In [? ? ] theauthors determine the optimal placement of sensornodes along the straight line between a source and adestination node. They model the energy consumedfor packet transmission in a mono-directional/bi-directional flow between source and destination withina mathematical framework using non-linear program-ming. The proposed mathematical model allows theauthors to find the optimal positions for the nodes ina closed formula. In this work, we use a more generalframework for the sensor placement problem, whichgeneralize the mentioned works, as we also considertraffic aggregation at the intermediate relays. Theproblem discussed in [? ] consists in deciding how toplace N sensors in a linear network topology, such thatall the sensors expend the same amount of energy. This

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Table I: Related works

Scheme Sensor placement Topology control Objective(s) Methodology[? ] l m Energy consumption Multi-variable nonlinear programming

[? ? ] l m Energy consumption Nonlinear programming[? ] l m Energy consumption Dynamic Programming[? ] l m Fault tolerance in terms of connectivity Heuristic based on Steiner Tree (STP-MSP)[? ] l m Resistance to physical attacks Heuristic based on binary search[? ] l m Dynamic Distributed heuristic[? ] l m Dynamic Heuristic based on Dijkstra[? ] l m Multiobjective Genetic algorithm[? ] m l Multiobjective Genetic algorithm[? ] m l Load balancing Distributed annealing algorithm[? ] m l Energy consumption Heuristic based on spanning tree[? ] m l Energy consumption Analytical method[? ] m l Energy consumption Heuristic based on clustering&energy harvesting[? ] m l Coverage&Energy consumption Heuristics based on clustering[? ] l l Energy consumption Heuristic based on virtual tree

work assumes a linear transmission structure whereeach node has to relay all the collected and generateddata towards the immediate one-hop neighbor. Theproblem is formulated as a multi-variable nonlinearprogramming problem and solved to optimality, butwithout considering the idle and receiving energy. Amore general formulation is given by [? ], where theproblem is to find the optimal number of sensors andtheir respective positions. The optimization problem issolved in two steps. First, the sensor position problemfor a given number of sensors N is formulated asa nonlinear programming problem similar to that of[? ]. In the second step, the goal is to optimize thenumber of sensors such that the ratio of networklifetime to the number of sensors N is maximized.Notice that the network lifetime is highly dependenton the number of sensors N calculated in the firststep. This ratio represents a new metric for evaluatingtrade-offs between network lifetime and the cost ofsensors. In contrast to sensor placement, the densitycontrol problem seeks to distribute load uniformlyover the monitoring area so that traffic per node isbalanced over the whole area. Density strategies [?? ] are defined as part of the pre-deployment phaseand provide a non-uniform random deployment inwhich the density increases closer to the BS. In thisstudy we do not consider this problem but the densitychanges may be extrapolated from the results of thesensor placement problem. Another solution approachused in literature for the sensor placement problem isthe multi-objective optimization. In fact, the sensorplacement can serve simultaneously to reduce theenergy consumption, increase the connectivity and thecoverage. In [? ], the authors propose a multi-objectiveevolutionary algorithm that uses a decompositionapproach for converting the problem of approximationof the Pareto fronts (PF) into a number of single-objective optimization problems. Even if the multi-objective approach seems promising, in this work, wefocus on the energy efficiency of the WSN, as our maincontribution is the unified solution methodology for

the two sub-problems. However, our work could beeasily extended to treat multiple objectives.

B. Topology control based on power controlThe topology control problem is a very important

problem in WSN as it includes several sub-problems,which have been grouped in literature according totheir main objective and to the approach used toreach this objective. Specifically, In [? ] and in [? ] theauthors classify topology control schemes accordingto different attainable objectives: network coverage,network connectivity, network lifetime extension. Foreach of these categories they present protocols andtechniques with a focus on blanket coverage, barriercoverage, sweep coverage, power management, andpower control. According to the classification proposedin these papers, our work would fall in the categorythat concerns power control approaches for networklifetime extension. Originally, this approach has beenproposed by Santi et al [? ]. In this paper, we takeinto account only the topology control based on powercontrol, and we redefine the problem as it follows. Wedivide the network into coronas or grids of differentlengths such that the energy is balanced in each ofthem. This problem assumes a uniform node distribu-tion and hence may be defined in a post-deploymentphase. To take account of the energy hole problem, acorona-based model is usually employed, the readeris referred to [? ] for an extensive survey on corona-based deployment strategies. In this model the networkarea is assumed to be circular, and it is divided intoconcentric circles forming coronas. Just like in a linearnetwork topology, nodes belonging to a corona willforward the data generated by the nodes themselvestogether with data generated by higher-level coronas.In the case of linear or rectangular networks, a grid isgenerally employed, corresponding to a unit divisionof the network. The network configuration problemis discussed in [? ], which proposes a method fordividing the network into coronas, then the coronasinto subcoronas and finally the subcoronas into zones

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such that the load in every zone is balanced. Inthis model, there is always a mapping between thesensors of a given zone belonging to a corona, andthe sensors of another zone, belonging to an upstreamcorona. The problem of finding the optimal numberof coronas is modeled as an optimization problemand a simulated annealing algorithm is proposed.The problem of dividing the coronas into subcoronasand zones is solved iteratively. This network divisionscheme assumes that the coronas and the subcoronashave the same width and that the number of zonesis the same for each of them. In [? ], on the otherhand, the number of coronas and their respectivelengths are assumed to be known, and the transmissionrange must be assigned for each corona so as tomaximize network lifetime. The transmission rangewill actually determine the next hop corona. Thealgorithm proposed for this problem builds a graph inwhich the nodes represent the coronas and the edgesrepresent the feasible wireless communication linksbetween any two coronas. The algorithm builds a setof spanning trees for each node, and those havingthe best performances in terms of network lifetimeare selected. Among the spanning trees that includethe most distant corona and the BS, the one withthe longest lifetime is retained as the solution to theproblem. In [? ] the authors discuss the same problem,however their transmission structure is different thanours. According to [? ] sensors in a corona transmittheir data to any of sensors in the downstream corona.In our model, the sensors are grouped in clustersand it is only the cluster head that transmit the data.By considering only the transmitting energy, in [? ],the authors prove that the total energy consumptionminimization is achieved when the coronas have thesame length. They also propose an iterative algorithmto consider the uneven energy depletion of energyand avoid the energy holes. Even for the topologycontrol based on power control there are some worksthat deal with multiple objectives. For example, in [?] the authors assume eight predefined transmissionrange levels for each sensor, and propose a geneticalgorithm for optimizing coverage, connectivity andlifetime of the sensor network. As already mentionedfor the sensor placement sub-problem, our approachfocuses on a single objective in order to offer a moreperforming solution.

In order to simplify the further reading of the paper,we present below the table of notations.

III. Sensor placement problem

In this section the goal is to find the networkdimensioning where the sensors consume the leastenergy in transmitting their data to the destination.Given a number of source nodes at a distance dfrom the BS, the sensor placement problem is todetermine the number of sensors and their respec-tive positions from the BS such that the total energy

B a s e S t a t i o n B a s e S t a t i o n

a ) b )

S o u r c e S o u r c e

1

2

3

n

2

3

n

d

1x

x

x

x

t a g

t a g 1

S c e n a r i o S c e n a r i o

Figure 2: The multihop transmission for linear network

consumption is minimized. In Section III-A we solvethe problem for a linear topology. This type of topologyhas applications in border surveillance, highway trafficmonitoring and oil pipelines. Subsequently, in SectionIII-B we generalize the problem to the two dimensionalnetwork in which a given number of source nodes areplaced in the surveying area.

A. Linear network

Here we assume that the sensors will be deployedin a linear network as in Fig. (2).

The energy model that we use to estimate the energyconsumed by them is proposed in [? ]. ETX denotesthe energy used for transmitting and ERX the energyused for receiving, as in equation (1).{

ETX = (Eelec + Eampdγ) · BERX = Erec · B

(1)

In these equation, d corresponds to the distance ofcommunication. However, practically the sensor rangeof communication is upper limited by Txmax. Besidesthe energy spent for receiving and transmitting thedata, sensors consume energy also in the idle mode.At this stage, we assume that the MAC protocolsguarantee perfect synchronization and successful trans-missions between sensors, therefore we do not takethe idle energy into consideration. For this problemwe examine two scenarios: (a) the source sensor nodehas to send data to the BS and all the intermediatesensor nodes act simply as relaying nodes and (b)the intermediate sensors add their own informationbefore relaying. Fig. 2 illustrates these two casesfor the linear network. The solution for scenario(a) places the sensors equidistantly (see [? ]). Whileobtaining an optimal solution for this first scenariois fairly straightforward, solving the second is morecomplex. The problem corresponding to scenario (b)is formulated as follows:

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Table II: Table of notations

Type Parameter Description

Networkd Network length.

R Network radius.

x Network length from the source.

ki = xid Ratio between the sensors’ distances.

n Number of hops, deployment problem.

Number of cells, network configuration problem.

γ0 The angle between two nodes in the uppermost circle.

r Ratio between the nodes in two consecutive circles.

li Transmission range for nodes in the ith circle.

NiNumber of nodes placed in the ith circle, deployment problem.

Number of zones in the ith corona, network configurationproblem.

Applicationβ Data generation rate in bps

α Data traffic per node in Erlang

ε A very small constant

m Compression ratio

c Coefficient c = α · βfn(d) The total energy function for a network with a length (d) and

n hops.

δy = 1 Distance discretion

Energy

Eelec Energy/bit consumed by the transmitter electronics.

Eamp Energy/bit consumed by the amplifier.

Erec Energy/bit consumption of the receiving circuitry

Eidle Energy consumption during the idle mode.

B Number of bits

Txmax Maximal transmission range

γ Path loss exponent [2− 6]

L = {l1, l2, · · · lm} Discrete distance transmission of a sensor

E = {e1 , e2 , · · · em} Discrete energy transmission of a sensor

Problem Definition P1.Input : A source node at distance d from the BShas to transmit its own information to the BSacross n intermediate nodes such that:• every intermediate node (sensor) transmits

to the downstream neighbor its own infor-mation as well as the information receivedfrom the upstream neighbor;

• the information is not aggregated.Objective : Minimize total energy consumption.Output : The number n of intermediate nodesand their respective distances di 1 ≤ i ≤ n fromthe source node.

We remark that the total energy consumption func-tion is convex (the proof is omitted on this paper),therefore it exists a global minimum of energy. Incontrast to the first scenario, in this problem eachintermediate node adds a given amount of traffic

(see Fig. 2 scenario (b)). Two parameters have to becalculated, namely the number of relaying sensornodes and the distances between them, in order toattain the minimum of the associated energy function.The difficulty and the combinatorics of this problemlie in the interdependence of two terms: the numberof intermediate nodes and the respective distancesbetween them. Nonetheless, there is a nice propertyregarding these parameters.

Let us suppose that xn is the optimal location of thelast relay node in a linear network of length d where nrelay nodes are employed. Given the energy formula(1), it can be shown that the value kn = xn/d does notdepend on the distance d. The following propositionholds:

Proposition 1. ki values depend only on the number nand not on the distance d.

Proof: This result can be proved by mathematical

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induction on the number n. Let us take n = 1. Clearly,the energy function associated with the transmissionthrough an intermediate sensor placed at x1 is at aminimum when the derivative is 0. The energy functionis of the form

α1(x1)γ + α2(d− x1)γ + ζ = α1dγ(k1)γ + α2dγ(1− k1)γ + ζ(2)

where α, β and ζ are constants. The derivative givesα1dγγ(k1)γ−1 − α2dγγ(1− k1)γ−1. Simple calculationsshow that this function attains 0 for some k1 =

1(α1/α2)1−γ+1

that depends only on the values of α1, α2

and γ. The same reasoning can be applied to anyhigher n, provided that the result holds for n − 1.Then, from the recurrence assumption we can deducethat also for some n the energy function is similarto the energy function given in (2) (apart from usingother constants). Let explain this in details below. Wesuppose that n sensors are placed between the sourceand the BS and xi give the distance of sensor i fromsource. Then, the energy formula has the followingform:

(3)α1(x1)γ + α2(x2 − x1)γ + α3(x3 − x2)γ

+ · · · + αn−1(xn−1 − xn−2)γ

+ αn(xn − xn−1)γ + βn(d − xn)γ + ζn

As we are looking for minimum energy formula,the placement of x1 to xn−1 is done optimally andthe recurrence hypothesis applies. Hence, we havexn−1 = kn−1 · x, xn−2 = kn−2 · kn−1 · x, ..., x1 = k1 ·k2 · · · kn−2 · kn−1 · x. Replacing xi in formula (3), weobtain an energy formula similar to (2):

α′(xn)γ + β′(d− xn)γ + ζn = α′dγ(kn)γ + β′dγ(1− kn)γ + ζn(4)

Then we can therefore apply the same schema as forn = 1 which ends the proof.

We will consider in the following the case with γ = 2.The objective function representing the overall energyconsumption of the network is given by equation(5). It is obtained by summing the energy expendedby all sensors according to equation (1). Regardingthe sensors, we assume that n sensors are placedrespectively at distances d1, d2, · · · , dn and each ofthem generates one unit traffic.

(5)E(n,d1 ,d2 ,···dn) = (Eelec + Eampd21) + Erec + 2(Eelec + Eampd2

2)

+ · · · + nErec + (n + 1)(Eelec + Eampd2n)

We solve this problem using a dynamic program-ming approach. Let us first briefly recall the principleof Dynamic Programming (DP). DP is a sequentialapproach, proposed by Bellman [? ] for optimizing agiven objective function. The problem is thus brokendown into stages and the aim at every stage is toselect the optimal decision so that the objective isoptimized over the current number of stages. Hence,in each stage we solve only once the corresponding

subproblem. The results of each stage are stored andlater used to backtrack the optimal values. The mosttypical example of a DP implementation is the shortestpath. If the path (A− > B− > C) is the shortest pathbetween the points A and C that passes through B,then AB is evidently the shortest path between thepoints A and B.

More particularly, our problem is separated intostages determined by the number of intermediatesensor nodes (or by the number of hops). At somestage n we determine the minimum energy necessaryfor transmitting the information if n sensor nodes areplaced between the source node and the base station,using the energy values obtained from the previousstage. The optimal solution for an instance of a problemwith n hops and a given distance d can be seen asequivalent to the optimal solution for some distancex < d and n− 1 hops, plus a final hop from this point(x) to the BS. Let fn(d) denote the minimum energy fortransmitting from the source node to the BS where theinformation passes through n intermediate relayingnodes that add their own information to the flow (Figure 2 scenario (b)). Applying this principle to ourproblem we obtain the following energy recurrenceformula:

(6)fn(d) = min

0≤x≤d(d−x)≤Txmax

{fn−1(x) + (n − 1)Erec + n

·(

Eelec + Eamp(d − x)2)}

where fn−1(x) is the minimum energy used over adistance x from the source, passing through n − 1intermediate sensors, and the remainder of the formularepresents the energy added by the nth sensor placedat x. We note that at each step n we only need toknow the minimum value for the energy functionfn−1(x), 0 ≤ x ≤ d. Moreover, the distance d shouldbe updated (d = d − Txmax) if the distance (d − x)becomes bigger than Txmax. In the following, we willanalyze the problem for a distance d and assumingthat in each step the the transmission range will notoverpass the maximal transmission range. Hence, westart with { f1(x)|0 ≤ x ≤ d}, and build the energyfunction recursively for the upper levels. In our casethe problem can be solved analytically. We begin byexpressing the function fn−1(x) included in the right-hand term of equation (6) through fn−2(x), and soon. Then, we replace fn−1(x) by the obtained recursiveexpression in (6) and derive the right-hand term, whichallows us to find fn(d). Let us assume that we have0 ≤ xn−1 ≤ x such that:

fn−1(x) = fn−2(xn−1) + n · Eamp(x− xn−1)2

+nEelec + (n− 1)Erec .

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Let kn−1 denote xn−1x . We have:

fn(d) = min0≤x≤d { fn−2(x · kn−1)

+nEamp(1− kn−1)2x2 + nEelec + (n− 1)Erec

+(n + 1)Eamp(d− x)2 + (n + 1)Eelec + nErec}.

By developing further we obtain the following recur-sive formula for fn(d):

fn(d) = min0≤x≤d {∑n−1l=0 ((l + 1)Eamp(1− kl)2

·∏n−1p=l+1 k2

p · x2) + (n + 1)Eamp(d− x)2

+ ∑nk=1 ((k + 1)Eelec + kErec)}.

(7)

Considering the derivative function of (7) withrespect to x and using the result of Proposition 1,which proves the independence of kp with respect tox, we obtain:

n−1

∑l=0

(l + 1) · (1− kl)2

n−1

∏p=l+1

k2p

· x + (n + 1)x = (n + 1)d. (8)

From (8) we find that the minimum of the right handterm of (7) is attained for the value of x = kn · d where

kn =n + 1

∑n−1l=0 (l + 1) · (1− kl)2(∏n−1

p=l+1 k2p) + n + 1

(9)

Hence, we deduce:

fn(d) = ∑n−1l=0 (Eamp(l + 1)(1− kl)2 ·∏n−1

p=l+1 k2p · (knd)2)

+(n + 1)(1− kn)2d2 + Eelec(n + 1)(n + 2)/2

+Erecn(n + 1)/2

= Eampd2 ∑nl=0(l + 1)(1− kl)2(∏n

p=l+1 k2p)

+Eelec(n + 1)(n + 2)/2 + Erecn(n + 1)/2.(10)

This formula generalizes the work of [? ] aimed atfinding the most energy efficient position for a sensor.Values kp for any p do not depend on the distanced (Prop. 1) and can be computed beforehand like informula (11).

kp =j + 1

∑j−1l=0 (l + 1) · (1− kl)2(∏

j−1j=l+1 k2

j ) + j + 1(11)

For scenario (b), these values are k0 = 0, k1 = 2/3 =0.66, k2 = 9/11 = 0.82, k3 = 0.88, and so on. Then, giventhe distance d between the source node and the BSwe can compute the distance from the source node foreach intermediate node (see also Fig. 1, scenario (b)).Finally, from Formula (10), we deduce the minimumamount of energy used for problem P1. Given theabove, the following proposition holds:

Proposition 2. For any distance d and number of interme-diate nodes n for problem P1, the total energy consumptionis minimized when each intermediate node i is placed atdistance xi, calculated by formula xi = ∏n

p=i kpd, withcoefficients kp computed according to (11).

We simply need to calculate fk(d) iteratively for k > 0and stop for n such that fn+1(d) ≥ fn(d).

In Fig. 3 we show the behavior of the energy functionfor d = 100 and 250m when the number of intermediatesensors is made to vary. In this work, the numericalresults are calculated using Matlab 7.11.0. In Fig. 4 the

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8x 10

−6

Intermediare sensor number

Energ

y (

J)

d=100 Scenario b)

d=250 Scenario b)

d=100 Scenario a)

d=250 Scenario a)

Figure 3: Energy versus the number of relay sensors for d = 100and d = 250 meters

graphics present the optimal number of sensors versusdistance for the two scenarios (a) and (b). The number

0 50 100 150 200 250 300 3500

1

2

3

4

5

6

7

8

9

Distance (m)

Num

ber

of sensors

Scenario b)

Scenario a)

Figure 4: Optimal number of sensors

of hops obtained for scenario (a) is larger than for (b).This is because each additional hop for scenario (b)increases the traffic to be transmitted and hence theenergy function. In other words, in contrast to scenario(a), decreasing the number of hops for scenario (b) maylead to energy conservation at a certain level. The maininterest of scenario (b) is that for minimum energyconsumption and a given number of hops n we knowthe distances between the relaying nodes exactly, asshown in Fig. 1.b). Furthermore, the distances from thebase station are proportional to a constant sequenceof n values determined by k1, k2, ...kn.

B. Two-dimensional network

In this section, we extend the study to a two-dimensional network. Here, we assume that the nodeswill be uniformly placed in circles. The general prob-lem is defined as follows:

8

−60 −40 −20 0 20 40 60−60

−40

−20

0

20

40

60

R

d

Base Station

l

k

Sensor node

Figure 5: Sensor placement for the case with traffic aggregation

Problem Definition P2.Input : We have N0 sensor nodes uniformly de-ployed according to a circle with radius R from theBS, which need to transmit their information to theBS. We assume that:• the information is transmitted through several

relaying nodes placed on intermediate circles.Each intermediate node transmits the trafficreceived from the upstream neighbor;

• all sensors placed in the same circle will trans-mit with the same transmission range;

Objective : Minimize the total energy consumption.Output : The distance di from the BS, for allintermediate circles i and the number of nodes ineach circle.

For the above problem, we use an aggregation modelproposed in [? ], which has the form y = mx + c wherex and y are the node input and output informationquantities respectively, m varies in [0, 1] and c is aconstant. For the sake of simplicity, we assume thateach node will generate an unit traffic.

The total energy consumption for a scheme withN0 sensor nodes placed in the outer circle and N1other nodes placed in an intermediate circle is givenin equation (12).

E = Eouter + Einner with :

Eouter = N0(Eelec + Eamp · l2)

Einner = N0Erec + N1(Eelec + Eampd2) (12)

where l is the transmission range of the sensorsbelonging to the outer circle (path loss exponent isequal to 2). Hence, Eouter is the energy expended byN0 nodes of the outer circle, and Einner is the energyexpended by N1 = N0

r nodes placed in the inner circle.To calculate the transmission range l (Fig. 5) we applythe cosine formula (13) on the triangle delimited bysides of length R, d and l:

l2 =(

d2 + R2 − 2Rd cos(r− 1

2γ0))

(13)

in which γ0 gives the angle between two successivesensors in the outer circle and the BS (i.e. γ0 = (2 ·π)/N0). As it can be observed, we have r nodes of theouter circle that will transmit to a node in the inner one.This is why the variable r has to take integer values.The equation (13) is true for every even or odd valueof r. Moreover, as we assume that the transmissionrange of the nodes in the outer circle is the same for allthe nodes, the distance l is computed as the maximaldistance between a sensor in the outer circle and hisrespective receiver in the inner circle.

Using the same reasoning as in the linear case, andconsidering the energy function in (12), we find thatthe minimum energy consumption is attained for somer such that:

r2 sin(r− 1

2γ0) =

Eelec + Eampd2

dREampγ0.

Unlike the linear network case, the problem withtraffic aggregation is difficult to solve analyticallyby the DP method. We opt for a method based onparameter discretization of the distance and the numberof sensor nodes associated with the distance from the BSrespectively, and apply the DP principle. Hence, fora given number of sensors N placed uniformly atdistance d from the base station, with n intermediatelevels up to the BS, we compute the energy functionusing the following recursive formula:

End,N = minx≤d,P≤N {( m·N

P + 1) · En−1x,P + NEelec + NErec

+NEamp(d2 + x2 − 2dx cos(NP −1

2 γ0))}(14)

Here the variables x and P correspond respectively tothe distance and the the number of sensors in the leveln− 1. Based on the DP algorithm, the energy of then− 1 level, En−1

x,P , is optimal. We add the energy addedby the nth hop which consists of the transmitting andreceiving energy of N sensors placed at the distanced to P sensors in the distance x. The coefficient mdetermines the aggregation data ratio, where the twoextreme cases are with aggregation (m = 0) andwithout aggregation (m = 1). In each step (the stepis determined by the number of hops) we build amatrix containing the energy values for an interval ofdistances x ≤ d and number of sensor nodes P ≤ N.Hence, we begin by computing the first matrix E1

containing values of {E1d,N |d ≤ R, N ≤ N0}. This

corresponds to the case when nodes transmit directlyto the BS. In the next step, from the recursive formula(14) for which n = 2 (that is, a single intermediatecircle), we calculate all {E2

d,N |d ≤ R, N ≤ N0} values.We continue like this with {En

d,N |d ≤ R, N ≤ N0} forsubsequent values of parameter n. We stop the calcula-tions when we reach some n such that En−1

R,N0≤ En

R,N0,

as shown in Algorithm 1.At the end of the computations we obtain the

number of intermediate levels (n), or the number

9

Algorithm 1: 2D networkInput: The number of nodes N0, Radius R, Max

transmission range Txmax;Output: Number of intermediate circles n,

Minimal energy E;Initialization : {E1

d,N |d≤R,N≤N0}, n=1;repeat

Calculate {En+1d,N |d≤R,N≤N0} matrix:

En+1d,N =minx≤d,P≤N {( m·N

P +1)·Enx,P+NEelec+NErec

+NEamp(d2+x2−2dx cos(NP −1

2 γ0))}

Calculate the transmission range lk accordingto 15;

if (lk>Txmax) thenlk=Txmax

n←n+1;until En−1

R,N0≤En

R,N0;

display n,E.

of hops necessary to achieve a minimum energyscheme. From this we can derive as well the numberof nodes (Nk) and the distance (dk) from the BS foreach k intermediate level. These values are stored inthe matrix Ek

d,N of the Algorithm 1. The associatedtransmission range can now be calculated for nodesin each k intermediate circle using formula (15):

lk =

√d2

k + d2k−1 − 2dkdk−1 cos(

NkNk−1− 1

2γ0) (15)

The following proposition holds:

Proposition 3. For any given number of sensors N0placed at a distance d from the BS, problem P2 can besolved through dynamic programming using the recurrenceformula (14).

The case in which the nodes act only as relayingnodes2 is easier. The nodes will transmit to the BSvia the shortest path. The number of nodes on eachintermediate circle will be the same as for the outercircle (N0) and the distances from the BS are the sameas with scenario (a) for the linear network.

1) Numerical results for the 2D Network: Numericalresults are provided in table III to illustrate theapplication for both algorithms with a radius R = 100mand the number of nodes N0 = 50 and N0 = 60.

Regarding the first algorithm, we stopped at thefourth iteration for the scenario with N0 = 50. Weremark that in this case only four hops are necessaryto transmit the information to the BS with minimumenergy, whereas for N0 = 60, one additional hop isneeded. On the other hand there are less hops forAlgorithm 2 applied to the same scenarios. This is

2The sensor will neither generate nor aggregate the data trafficbut simply forward it.

Table III: Energy values (in 10−4 J) for each n

With Aggregation Without AggregationN0 = 50 N0 = 60 N0 = 50 N0 = 60

n = 1 0.5250 0.6300 0.5250 0.6300n = 2 0.1641 0.1855 0.3669 0.4351n = 3 0.1280 0.1414 0.3895 0.4587n = 4 0.1249 0.1334 - -n = 5 0.1253 0.1326 - -n = 6 - 0.1337 - -

obvious as the intermediate nodes increase the trafficload.

However, the above model remains essentially theo-retical, because in real world applications a node mayreceive and transmit a certain quantity of traffic basedon a network traffic pattern or the nodes can be alreadydeployed and one needs to take decisions only for therange assignment. Nevertheless, it may be useful tocalculate the lower bounds of energy consumption.This work gives a solid basis for tackling the topologycontrol problem, which is discussed in the followingsection.

IV. Topology control problem based on power

control

In some WSN applications the sensors may bealready deployed in the monitoring area, the require-ment being to transmit the data as energy-efficientlyas possible. Here the problem is how to configure thenetwork so as to meet this objective. Configuring alinear network involves creating a feasible division ofthe network into cells and determining their respectivesizes (see Fig. 6). In the case of a two-dimensionalnetwork, the problem involves not only dividing thenetwork into coronas of different lengths, but alsosubdividing these coronas into different zones (see Fig.9). Regarding these objectives, we consider two strate-gies: (i) minimizing the overall energy consumption inthe network (which we shall call Strategy 1) and (ii)guaranteeing a fair allocation of energy consumptionbetween the sensors (which we shall call Strategy 2).

In this section we adapt the mathematical modelused for the sensor placement problem to solve thetopology control problem. We propose a method foroptimally solving this problem under different condi-tions of traffic load. Finally, we show how to adaptthe method to different energy consumption modelswithout increasing the computational complexity ofthe solution.

A. Linear networkWe consider a linear network with a length d from

the BS, where the nodes are uniformly distributed,as in Fig. 6. The network will be divided into cells. Thetransmission structure is quite similar to the cluster-based scheme. For any cell there will be only one node,

10

Figure 6: Network model

known as the cluster head, which receives the informa-tion from the other nodes in the same cell and fromthe cluster head of the upstream cell. The transmissionrange of a sensor corresponds to the length of therespective cell. Communication between sensors canbe guaranteed as the sensors deployment is dense andthe length of each cell can not exceed the maximaltransmission range Txmax. The received data are thentransmitted to the cluster head of the downstreamcell without performing any type of aggregation. Thenetwork model considered here is proposed in [? ]. Weassume that each node has α Erlang of traffic, and theradio data rate is β bps. Regarding the data aggregationmodel (see sectionIII-B), we take c = α · β and m = 1.The nodes are uniformly distributed with a densitynd. The ith cell in the network (Figure 6) will consumea quantity of energy equal to ETX when transmittingthe information (see equation (16)).

ETX(i) = (Eelec + Eamp · (xi − xi−1)γ) · xindαβ (16)

where xi and xi−1 are the distances of the ith and(i − 1)th cells respectively and γ as before presentsthe path loss exponent. As regards the reception ofinformation, the energy consumed by the ith cell whenreceiving is expressed by equation (17).

ERX(i) = Erec · xi−1ndαβ (17)

Besides the energy dissipation for receiving andtransmitting, we take into account the energy spentduring the idle state noted as Eidle. Based on thenetwork model presented in [? ], the idle energy for acluster head of the ith cell is calculated as follows:

EIdle(i) = Eidle · (1− 2xi−1ndα) · β (18)

Here it is assumed that each node requires a timewindow equal with (δt) to transmit its own data. Forreceiving all the data, a cluster head of the ith cellneeds a time3 equal to (xi−1ndδt). If T is the time ofa round for the data collection, then the time thecluster head passes in the idle mode is (T − 2xi−1ndδt)and the division with T leads to (1− 2xi−1ndα). Thenetwork configuration problem consists in determiningthe optimal number of cells and the length of each cellsuch that (i) the total energy consumption is minimized

3The cluster head will spend the same time either for receivingor transmitting the data.

(Strategy 1) and (ii) energy fairness is obtained byensuring that every cell in the network consumes thesame amount of energy (Strategy 2). So far we haveassumed that the node transmission energy dependson the communication distance. Since real sensorshave discrete levels of power transmission we focus onminimizing the total energy consumption of nodes inthe linear network using only discrete energy values.More specifically, given a linear network, we needto find the optimal number of cells, together withtheir respective distances, that guarantees not only alower bound of energy consumption, but also energybalancing between cells or the minimizing of energyconsumption for discrete energy levels (in IV-A1). Aswe can see, for any of these problems the number ofpossible combinations is exponential, and therefore wepropose a method based on dynamic programming.

1) Optimization criteria for the linear network:a) Total energy minimization: Here the objective

is to minimize the total energy consumption for thelinear network presented in IV-A. The total energyconsumption function is convex, therefore it exists aglobal minimum of energy. We have proved the con-vexity of the function but for reason of space we haveomitted it in this paper. As in the previous scenarios,the problem can be separated into a number of stagesdepending on the number of cells. At some stage nwe determine, using the energy values obtained fromthe previous stage, the minimum energy necessary totransmit the data. Hence, the optimal solution for aninstance of a problem with n cells and a given distanced can be seen as equivalent to the optimal solution forsome distance x < d and n− 1 hops, plus a final hopfrom this point (x) to the BS. In the nth iteration, thedistance (d− x) represents the length of the nth cell.The Algorithm 2 begins with n = 1 which assumesthat all the nodes transmit with a range d. Next, nis incremented and for each distance y (1 ≤ y ≤ d)the best division is depicted by x. Importantly, whenthe optimal n is found out, the x value is identifiedas well. We backtrack all the xi values for each stageni to construct the whole solution and the difference(xi− xi−1) represents the length of each cell. The energyadded by the last hop E(y− x) is calculated accordingto formula (19).

(19)E(y − x) = e1 + e2 + e3 + e4

where:e1 = Erec · xndαβ

is the energy spent for receiving the data of theupstream cell,

e2 = (Eelec + Eamp(y − x)γ) · yndαβ

gives the energy dissipated by the cluster head totransmit all the collected data,

e3 = (Eelec + Eamp(y − x)γ + Erec) · (y − x)ndαβ

11

Algorithm 2: Strategy 1

Input: Distance d, node density nd;Output: Optimal number of network division n,

Minimal energy E;Initialization : {E1

y |y≤d}, n = 1;

repeatfor y←1 to d do

En+1y =∞;

for x←1 to y−δy doEn+1

y ←min{En+1y , En

x +E(y−x)}

n←n+1;until En−1

d ≤End ;

display n,E.

corresponds to the energy consumed for transmittingand receiving the data of the other sensors deployedin the cluster,

e4 = Eidle · (1− 2xndα) · β

and finally the e4 is the energy spent during the idlestate. Note that En

y represents the total energy con-sumption where the network length is y and dividedinto n cells. The distances x and y are discretized usinga distance discretization step δy. The initializationphase of the algorithm requires O(y/δy) computationsto construct the vector En

y . The calculation procedurecosts at most O

((y/δy)2) and it will be repeated

n times. Finally, the computation complexity of thealgorithm is O(n · (y/δy)2).

Remark 1. Energy minimization with discrete transmis-sion levels

In the above strategy while using the energy modeldescribed in (1), we estimate the sensor transmissionenergy according to the exact distance of transmission.However, the transmission power of sensor nodes isdiscrete and therefore it takes only some given values.In this section, we provide Algorithm 3, a modifiedversion of Algorithm 2, which addresses this problem.

In the Algorithm 3, the term (E dy− xe) is com-puted using equation (19) while the formula of thetransmission energy is modified. Hence, (dy− xe) ismatched with the closest element4 of the list L. Weassociate to each distance li from the list l a powervale ei belonging to list e. Then, the terms e2 and e3 ofthe equation (19) will be computed using the ei powervalues as following:

e2 = ei · yndαβ

ande3 = (ei + Erec) · (y − x)ndαβ

4We use the ceiling value of the distance (y− x) to ensure thatthe closest matched element in the list L is not less than (y-x).

Algorithm 3: Energy minimization with discretetransmission levels

Input: Distance d, node density nd, the list ofenergy power levels e={e1 ,e2 ,...en}, thecorresponding list of the respectivedistances L={l1 ,l2 ,..lm};

Output: Optimal number of network division n,Optimal cell sizes x, Minimal energy E;

Initialization : {E1y |y≤d}, n = 1;

repeatfor y←1 to d do

En+1y ← min 0<x<y

dy−xe∈L{En

x + E dy− xe }

n←n+1;

until En−1d ≤En

d ;

display n,E.

b) Energy fairness: In order to avoid the energyhole problem, we need to guarantee that each nodein the network will consume the same amount ofenergy, independently of the quantity of received andgenerated data. This is the motivation behind oursecond strategy which seeks to ensure the equitableexpenditure of energy in the network. One way tosolve this problem is to divide the network into k cellsand write a system of k equations, where for each cellthe total energy consumption Ei is given by equation(20).

Ei = (Eelec + Eamp(xi − xi−1)γ) · xindαβ

+(Eelec + Eamp(xi − xi−1)γ) · (xi − xi−1)ndαβ

+Erec · ndαβxi−1 + Eidle · (1− 2xi−1αnd) · β(20)

The equation ∑ki=1(xi − xi−1) = d enables us to have

k equations and to find all the variables xi for alli ∈ 1 . . . k and E. However, this method proposed in [?] needs to solve a system of k nonlinear equations fork cells. Instead, we propose Algorithm 4, also basedon the dynamic programming method.

Here Eny represents the total energy consumption

whileEn

yy·nd

is the average energy consumption of asensor in a cell5 when the network of length y isdivided into n cells. E(y− x) is the energy added onlyby the last hop (see equation (19)). The ratio E(y−x)

(y−x)·ndcalculates the sensor node energy consumption for thelast cell. The IF condition in the algorithm identifiesthe solutions which guarantee a fairness dissipation ofenergy with a given threshold. The algorithm worksas follows: it finds the network division (xi values) fora given n such that energy is balanced. Next, it repeatsthis procedure till it reaches the n which minimizesthe total energy consumption. At the end of execution,

5The sensors in all cells will consume an identical amount ofenergy.

12

Algorithm 4: Strategy 2

Input: Distance d, Node density nd, Coefficient ε;Output: Optimal number of network division n,

Minimal energy E;Initialization : {E1

y |y≤d}, n = 1;repeat

for y←1 to d doEn+1

y =∞;for x←1 to y−δy do

if ( Enx

x·nd− E(y−x)

(y−x)·nd<ε· E(y−x)

(y−x)·ndthen

En+1y ←min{En+1

y , Enx +E(y−x)};

n←n+1;until En−1

d ≤End ;

display n,E.

the algorithm displays the number of cells and thetotal energy consumption while the length of each cellis found by backtracking the xi values.

2) Numerical results: We tested our algorithms withrespect to optimization strategies 1 and 2 and com-pared their results with the best case of uniform celldivision. Uniform cell division means that the networkis divided into n equal size cells, and the best casecorresponds to the n that minimizes the energy value.The results of total energy consumption for the threecases with respect to the distance are shown in Fig 7.For this simulation we set the simulation parametersas shown in Table IV.

Table IV: Simulation parameters for the linear network

Type Parameter Value

NetworkNetwork Length 200 ∼ 2000m

Node distribution Uniformly

Node density 0.4

ApplicationData generation rate β 485 bps

Data traffic per node α 0.003 Erlang

ε 0.07

Compression ratio m = 1 | c = α · β

Energy

Eelec 50nJ/bit

Eamp 100pJ/bit/m2

Erec 50nJ/bit

Eidle 40nJ/bit

γ 2

L = {l1 , l2 , · · · lm} [20, 40, 60, 80] in meters

E = {e1 , e2 , · · · em} [0.09, ..., 0.69] · 10−6J/bit

Method Distance discretization δy = 1

We notice that strategies 1 and 2 outperform the bestcase of uniform cell division considering the overallconsumption energy. If we zoom out the results for

200 400 600 800 1000 1200 1400 1600 1800 20000

1

2

3

4

5

6

7x 10

−3

Network length in (/m)

Energ

y

Overall energy consumption

Strategy 1

Uniform

Strategy 2

Figure 7: Energy consumption

the extremity points taken in our simulation (200mand 2000m), the strategy 1 consumes 15% (resp. 36.2%)less energy than uniform cell division for 200m (resp.2000m). On the other hand, strategies 1 and 2 dividethe network such that the energy consumption is morebalanced compared to the uniform one. Strategy 1gives a lower bound for total energy consumption,but remains comparable to Strategy 2. For a networkwith a distance of 200m (resp. 2000m) the differenceis around 4% (resp. 4.5%) in energy saving comparedto Strategy 2.

Table V gives the cell divisions, beginning with thecell closest to the Base Station, generated by strategies1 and 2. We notice that these strategies yield differentnetwork scenarios, even where they both have the sameoptimal number of cells. As the distance increases,the configuration scenarios different more obviouslyfrom each other. The radio parameters also stronglyinfluence this behavior.

Network length Network Division

(250 m ) Strategy 1 Strategy 2 Uniform

cell 1 35 36 41.66

cell 2 37 37 41.66

cell 3 39 40 41.66

cell 4 42 42 41.66

cell 5 47 45 41.66

cell 6 50 50 41.66

Table V: Linear network configuration

As expected, for the energy discretization problemthe sensors consume more energy. For this simulationwe assume that each node can achieve the distancesL = l1, l2, . . . , lm using the respective energies E =e1, e2, . . . , em. Fig. 8 shows that the continuous energyvalue still remains an acceptable approximation ofenergy behavior (the gap between the discrete and the

13

continuous case reaches 6% for 400m). Energy doesnot diverge between the discrete and continuous cases,regardless of the distance, given that they both use thesame energy model.

100 150 200 250 300 350 4000.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−4

Network length in (/m)

Ene

rgy

Energy discretization − Strategy 1

Discrete Case

Continuous Case

Figure 8: Energy discretization

B. Two-dimensional sensor network

The second problem considered here is the two-dimensional sensor network configuration problem.We assume that the sensors are uniformly distributedin a circular monitoring area. There is only one sink,positioned at the center of the zone.

The sensors will transmit their information accord-ing to a many-to-one traffic pattern using a multihopscheme. Total energy consumption will be minimizedby optimally dividing the area into concentric coronasC1, C2, ...Cn centered at the sink and then subdividingeach corona into a given number of zones. Fig. 9gives an example of network division with 3 coronascontaining respectively 1, 4 and 8 zones. Moreover,the length of every corona can be varied. For eachzone there is only one sensor, the cluster head, thatis able to receive information from other sensors inthe zone, aggregate this information and transmit it tothe downstream zone closer to the BS. We assumethat any sensor in the zone is a potential clusterhead. Notice that we can easily shift from this modelto a specific-sensor placement model in which thecluster head position must be explicitly determinedby adapting the radio transmission range allocated toeach zone. In accordance with the aggregation modelintroduced in Section III-B, here we analyze the casewith m = 0 in which a node will totally aggregatethe information and transmit only a fixed amountof information. If we assume that no aggregation ofsensor information will be performed by the clusterheads then the problem is reduced to the linearnetwork case, because sensors will choose the shortestpath when transmitting information. Finally, for thenetwork model as described above, the questions tobe answered are:

−60 −40 −20 0 20 40 60−50

−40

−30

−20

−10

0

10

20

30

40

50

x

y

BS

P zones

Q zones

Figure 9: Two-dimensional network

• What is the optimal number of coronas thatminimizes energy across the whole network?What are their respective radii?

• What is the optimal number of zones that a coronais divided in?Here we assume that each zone has a clusterhead that will aggregate the traffic and transmit ittowards the cluster head of the downstream zonecloser to the BS. For the sake of the simplicity, theidle energy is not considered for this problem.

1) Optimization criteria:a) Minimal energy consumption - Strategy 1: The

problem for a two-dimensional network with trafficaggregation is more complex than the linear case. Weopt once again for a method based on parameterdiscretization of the radius and the number of zonesfor each corona respectively, and apply the dynamicprogramming principle. We assume that the sensornetwork has an uniform node distribution nd and eachsensor generates a constant traffic α · β. Hence, for asuch network with radius R and number of zones N,we apply Algorithm 5.

In the initialization phase, the algorithm computesthe energy spent in a simple cell (N = 1) in which allthe sensors transmit their information directly to theBS using the same transmission range y. The energyvalues for the first vector E1

y,1 are computed accordingto formula (21).

E1y,1 = (Eelec + Eamp · yγ) · ndπy2αβ (21)

In the next step, the number of coronas is two (n = 2)and the algorithm computes the matrix {E2

y,Q|(y ≤R, Q ≤ N)}. For each n coronas’ number, a new matrix{En

y,Q|(y ≤ R, Q ≤ N)} is generated where N givesthe maximal number of the zone divisions of the lastcorona. The elements of each matrix for a given nare computed based on the dynamic programmingprinciple where the last hop term is represented byE(y− x, Q− P) and computed according to formula(22). Here (y− x) gives the length of the last coronaand Q (resp. P) gives the number of zones in the last

14

Algorithm 5: Network area configuration - Strategy1

Input: Node density nd, Network area radius R,Maximal number of zone divisions N,Generated traffic per node αβ;

Output: Minimal energy E, Optimal number ofcoronas n;

Initialization :{

E1y,1|y≤R

}, n=1;

repeatfor ((y←1 to R) and (Q←2 to N)) do

En+1y,Q =∞;

for ((x←1 to y) and (P←2 to Q)) do

En+1y,Q←min{En+1

y,Q , Enx,P+E(y−x,Q−P)}

n←n+1;En

y←minQ≤N{Eny,Q};

until En−1R ≤En

R;

display n,E.

(resp. last but one) corona. A simple illustration isgiven in Fig. 9.

E(y− x, Q− P) = Q · f1(y, x, P) + f2(y, x, Q) (22)

where:

f1(y, x, P) =(Eelec + Eamp · (Rt(y, x, P))γ + Erec

)· αβ (23)

with

Rt(y, x, P) =(

y2 + x2 − 2yxcos(2π

P)) 1

2

(24)

Equation (23) gives the energy used by the clusterhead of each zone in the nth corona to transmit theinformation towards the next cluster head closer tothe BS while using a transmission range defined bythe Rt(y, x, P) function. As in the sensor placementproblem, the transmission range is calculated basedon the cosine formula. However, in this problem wedo not know the cluster head placement, thereforewe consider the maximal angle for computing thetransmission range which depends only on the numberof zones P of the precedent level n− 1.

f2(y, x, Q) =(

Eelec + Eamp(

Rt(y, x, Q))γ + Erec

)· ndπ(y2

− x2)αβ

(25)

Equation (25) gives the energy consumed by allthe sensors in a corona for performing intra-zonetransmissions6. Regarding algorithm 5, we calculate thematrix {En

y,Q|(y ≤ R, Q ≤ N)} for subsequent valuesof parameter n. Next, we find the En

R which is theminimal value of the last row of the matrix {En

R,N}. Westop the calculations when we reach some n such thatEn

R ≤ En+1R . During the execution of the algorithm we

6Intra-zone transmissions include the receiving and transmittingoperations inside a zone between the cluster head and the othersensors in the zone.

keep track of those values (xi , Pi) that allow to achievethe minimum of energy at the current iteration. Thesolution can then easily be built by backtracking thesestored values. At the end of the algorithm we obtainthe optimal number of coronas for minimum energyconsumption, as well as the number of zones for eachintermediate corona and the corresponding radius.

For the simulations, the nodes are uniformly de-ployed in the circular area and generate a constantamount of traffic. The cluster head of each zone aggre-gates all the information that it receives and transmitsa constant amount of traffic to the closest cluster head.Table VI lists the system configuration parameters indetail whereas the values of the energy model are givenin Table IV. For these parameters, some experimentalresults for Algorithm 5 are presented in Table VII. Aswe can observe, the optimal number of coronas is 9,and the algorithm determines the radius of each coronaand the number of zones, both of which decreaseuniformly.

Table VI: Simulation parameters

Type Parameter Value

NetworkArea radius 50m - 300m

Node distribution Uniformly

Node density 0.02

ApplicationData generation rate β 100 bps

Data traffic per node α 0.003 Erlang

ε 0.07

Compression ratio m = 0 | c = α · β

Cell sizes Algorithm results

Radius of each Number of zonescorona from the BS

corona 9 100 27

corona 8 90 24

corona 7 80 21

corona 6 70 18

corona 5 60 15

corona 4 50 12

corona 3 40 9

corona 2 30 6

corona 1 22 1

Table VII: Optimal configuration for overall energyconsumption, network radius R = 100m

The energy consumptions of the whole networkfor different network radii and node densities arepresented in Fig. 10.

b) Energy fairness - Strategy 2: Assuring energyfairness between the network clusters is an alternativeobjective to minimizing total energy consumption.

15

50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−4

Area Radius (/m)

Ene

rgy (

/J)

Node density = 0.04 m2

= 0.028 m2

= 0.02m2

Figure 10: Energy versus network area radius

Here the optimal solution to the problem gives anetwork configuration such that differences in theenergy consumed by the sensors of different cellsand the overall energy consumed are minimized. Tosolve this problem, we propose the Algorithm 6. Here

Algorithm 6: Network area configuration - Strategy2Input: Node density nd, Network area radius R,

Maximal number of zone divisions N,Generated traffic per node αβ, Coefficient ε;

Output: Minimal energy E, Optimal number ofcoronas n;

Initialization :{

E1y,1|y≤R

}, n=1 ;

repeatfor ((y←1 to R) and (Q←2 to N)) do

En+1y,Q = ∞;

for ((x←1 to y) and (P←2 to Q)) do

if (En

x,PN1 −( f1(y,x,P)+ f2(y,x,Q)

N2 )<ε·En

x,PN1 ) then

En+1y,Q←min

{En+1

y,Q , Enx,P+E(y−x,Q−P)

}

n←n+1;En

y←minQ≤N{Eny,Q};

until En−1R ≤En

R;

display n,E.

Eny,Q represents the total energy consumption while

Eny,Q

N1 is the average energy consumed by a sensorwhen the network’s radii is y and its nth corona isdivided into Q zones. The term N1 is the numberof the sensors deployed in the area with a radiiy. In the initialization phase, matrix E1

y,1 is com-puted according to equation (21). The E(y− x, Q− P),f1(y, x, P), Rt(y, x, P) and f2(y, x, Q) functions are givenby the corresponding equations (22), (23), (24) and

(25). The ratio f1(y,x,P)+ f2(y,x,Q)N2 gives the sensor’s energy

in the respective zone while N2 = (y2−x2)·2π·ndQ is the

corresponding number of the sensors in this zone.At first, the algorithm finds the network division

(xi , Pi values) for a given n which guarantees a fairnessconsumption of energy for each sensor in the networkwith a coefficient ε. Second, by iterating this procedurefor different n, it seeks the minimal value of totalenergy consumption. This value is reached for some nwhich satisfies the inequality En−1

R ≤ EnR. Algorithm 6

is implemented for the scenario described in SectionIV-B. The application and parameter values are definedin Table VI wheres the energy parameters are givenin Table IV. The results are shown in Table VIII. Forapplication needs, the number of zones in one coronashould be a multiple of this in the downstream corona.However, this can be easily taken into account byassuming that the number Q of zones, in algorithm 6,is a multiple of P.

Cell sizes Algorithm results

Radius of each Number of zonescorona from the BS

corona 8 100 30

corona 7 86 27

corona 6 72 21

corona 5 60 15

corona 4 48 12

corona 3 36 9

corona 2 24 6

corona 1 12 1

Table VIII: Optimal configuration for energy fairness,network radius R = 100m

2) Numerical results: We remark that Strategy 1builds a multi-hop tree rooted at the BS which mini-mizes the total energy consumption for the communi-cations. One basic assumption of this strategy is fullaggregation, meaning that each node will transmitonly one packet to its parent regardless of the numberof received data packets. The work in [? ] considersthe same constraints and objective function to build amulti-hop tree and propose a protocol PEDAP(PowerEfficient Data Gathering and Aggregation Protocol).The algorithm behind the PEDAP protocol builds anear optimum spanning tree and is designed for datagathering in sensor networks. One basic assumptionof this algorithm is full aggregation, meaning thateach node will transmit only one packet to its parentregardless of the number of received data packets.In both cases, we assume that nodes are uniformlydistributed in a circular area. To calculate the energyconsumed by PEDAP we construct a spanning treethat includes all the nodes. In the spanning tree wedifferentiate between terminal and in-network nodes.

16

40 60 80 100 120 140 160 180 200 2200

1

2

3

4

5

6

7

8x 10

−4

Area Radius (/m)

Ene

rgy (

/J)

Energy consumption

Strategy 1

PEDAP

nd=0.1

nd=0.04

Node density (nd) node/m2

Figure 11: PEDAP and Strategy 1

50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5x 10

−4

Network Area Radius (/m)

Energ

y (

/J)

Energy consumption

Strategy 1

Stategy 2

Figure 12: Energy consumption

The terminal nodes use energy only for transmitting,while the others use energy for transmitting andreceiving, in accordance with equations (1). In bothscenarios each node has α Erlang of traffic, and theradio data rate is β bps. The results of the comparisonbetween Strategy 1 and PEDAP are given in Fig. 11.Our algorithm outperforms PEDAP, and more so inthe case of large dense networks, which shows that itis quite scalable. The overall energy consumption forStrategies 1 and 2 is given in Fig. 12.

Through simulation we also remarked that thebehavior of energy function toward the node densitychanges is similar for both strategies.

For the topology control problem we have looked attwo strategies which seek respectively to i) minimizethe energy consumption and ii) ensure a fair energydistribution while minimizing the total energy con-sumption. The trade-offs between the two strategies arein terms of global energy, fairness and complexity. Theresults show that Strategy 1 presents a lower boundof network energy consumption, but also that Strategy2 is very close to this bound (see Fig. 12).

V. Conclusions

In this paper we analyzed the network configura-tion problem for a many-to-one WSN. For a given

network where the nodes are uniformly distributed,the problem is to determine the optimal numberof sensors, their positions and their transmissionrange such that some energy objectives are met. Weshowed that an appropriate configuration of the WSN,addressing the sensor deployment and the respectivepower assignments, leads to considerable savings inenergy. Our algorithms, whose aims are to minimizethe total energy consumption and to ensure fairness inthe energy consumption by different nodes, are basedon the dynamic programming method. We showedhow this method can be adapted even to the discretecase of the energy consumption minimization problem.Then, we extended the linear network case to the caseof a two-dimensional network. Our overall observationis that dynamic programming is an effective methodfor handling trade-offs between the parameters for anumber of variants of the network configuration prob-lem. It provides an optimal solution to the problemfor a given objective function and reduces the compu-tational complexity in comparison with other methodsproposed in the literature. Moreover, the method canbe adapted to different energy consumption modelswithout increasing the computational complexity of thesolution. In the future we shall be looking at the effectsof combining different routing protocols with solutionsobtained for the network configuration problem. Wethink that this sort of combination will be of interestat the maintenance phase and for implementation inreal world applications.