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\begin{document}

\title{Energy-Efficient Activity Scheduling in Heterogeneous Energy Wireless Sensor Networks}

% author names and affiliations
% use a multiple column layout for up to three different
% affiliations
\author{\IEEEauthorblockN{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon, and Rapha\"el Couturier }
\IEEEauthorblockA{FEMTO-ST Institute, UMR 6174 CNRS, University of Franche-Comte, Belfort, France \\
Email: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr}
%\email{\{ali.idness, karine.deschinkel, michel.salomon, raphael.couturier\}@univ-fcomte.fr}
%\and
%\IEEEauthorblockN{Homer Simpson}
%\IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France}
%\and
%\IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
%\IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France}
}

\maketitle

\begin{abstract}
One of  the fundamental challenges in Wireless  Sensor Networks (WSNs)
is  the coverage  preservation  and  the extension  of  the  network  lifetime
continuously  and  effectively  when  monitoring a  certain  area  (or
region) of interest. In this paper a coverage optimization protocol to
improve the lifetime in  heterogeneous energy wireless sensor networks
is proposed.   The area of  interest is first divided  into subregions
using a  divide-and-conquer method and then the scheduling  of sensor node
activity  is  planned for  each  subregion.   The proposed  scheduling
considers  rounds during  which  a small  number  of nodes,  remaining
active  for  sensing, is  selected  to  ensure  coverage.  Each  round
consists  of   four  phases:  (i)~Information   Exchange,  (ii)~Leader
Election, (iii)~Decision,  and (iv)~Sensing.  The  decision process is
carried  out  by  a  leader  node which  solves  an  integer  program.
Simulation  results show that  the proposed  approach can  prolong the
network lifetime and improve the coverage performance.
\end{abstract}

%\keywords{Area Coverage, Wireless Sensor Networks, lifetime Optimization, Distributed Protocol.}
 
\IEEEpeerreviewmaketitle

\section{Introduction}

\noindent Recent years have witnessed significant advances in wireless
communications and embedded micro-sensing MEMS technologies which have
led to the  emergence of wireless  sensor networks  as one  of the  most promising
technologies~\cite{asc02}.   In fact, they  present huge  potential in
several  domains ranging  from  health care  applications to  military
applications.  A sensor network is  composed of a large number of tiny
sensing  devices deployed  in a  region of  interest. Each  device has
processing  and wireless communication  capabilities, which  enable it to
sense its environment, to compute, to store information, and to deliver
report messages to a base station.
%These sensor nodes run on batteries with limited capacities. To achieve a long life of the network, it is important to conserve battery power. Therefore, lifetime optimisation is one of the most critical issues in wireless sensor networks.
One of the main design issues in Wireless Sensor Networks (WSNs) is to
prolong the  network lifetime,  while achieving acceptable  quality of
service for applications.  Indeed, sensor nodes have limited resources
in terms of memory, energy, and computational power.

Since sensor nodes have limited battery life and without being able to
replace batteries,  especially in remote and  hostile environments, it
is desirable that  a WSN should be deployed  with high density because
spatial redundancy can  then be exploited to increase  the lifetime of
the network. In such a high  density network, if all sensor nodes were
to be  activated at the same  time, the lifetime would  be reduced. To
extend the lifetime  of the network, the main idea  is to take advantage
of  the overlapping  sensing regions  of some  sensor nodes  to save
energy  by  turning  off  some  of  them  during  the  sensing  phase.
Obviously, the deactivation of nodes  is only relevant if the coverage
of the monitored area  is not affected.  Consequently, future softwares
may  need to  adapt  appropriately to  achieve  acceptable quality  of
service  for  applications.  In  this  paper  we  concentrate on  the area
coverage  problem,  with  the  objective  of  maximizing  the  network
lifetime  by using  an adaptive  scheduling. The  area of  interest is
divided into subregions and an activity scheduling for sensor nodes is
planned for  each subregion. 
 In fact, the nodes in a  subregion can be seen as a cluster where
  each node sends  sensing data to the cluster head  or the sink node.
  Furthermore, the activities in a subregion/cluster can continue even
  if another cluster stops due to too many node failures.
Our scheduling  scheme considers rounds,  where a round starts  with a
discovery  phase  to  exchange  information  between  sensors  of  the
subregion,  in order to  choose in  a suitable manner  a sensor  node to
carry  out a coverage  strategy. This  coverage strategy  involves the
solving of  an integer  program which provides  the activation  of the
sensors for the sensing phase of the current round.

The remainder of the paper is organized as follows.  The next section
% Section~\ref{rw}
reviews the related work in the field.  Section~\ref{pd} is devoted to
the    scheduling     strategy    for    energy-efficient    coverage.
Section~\ref{cp} gives the coverage model formulation which is used to
schedule  the  activation  of  sensors.  Section~\ref{exp}  shows  the
simulation  results obtained  using  the discrete  event simulator  on
OMNeT++  \cite{varga}. They  fully demonstrate  the usefulness  of the
proposed  approach.   Finally, we  give  concluding  remarks and  some
suggestions for future works in Section~\ref{sec:conclusion}.

\section{Related works}
\label{rw}

\noindent This section is dedicated to the various approaches proposed
in  the literature  for  the coverage  lifetime maximization  problem,
where the  objective is to  optimally schedule sensors'  activities in
order to  extend network lifetime  in a randomly deployed  network. As
this problem is subject to a wide range of interpretations, we have chosen
to recall the main definitions and assumptions related to our work.

%\begin{itemize}
%\item Area Coverage: The main objective is to cover an area. The area coverage requires
%that the sensing range of working Active nodes cover the whole targeting area, which means any
%point in target area can be covered~\cite{Mihaela02,Raymond03}.

%\item Target Coverage: The objective is to cover a set of targets. Target coverage means that the discrete target points can be covered in any time. The sensing range of working Active nodes only monitors a finite number of discrete points in targeting area~\cite{Mihaela02,Raymond03}. 

%\item Barrier Coverage An objective to determine the maximal support/breach paths that traverse a sensor field. Barrier coverage is expressed as finding one or more routes with starting position and ending position when the targets pass through the area deployed with sensor nodes~\cite{Santosh04,Ai05}.
%\end{itemize}
{\bf Coverage}

The most  discussed coverage problems in literature  can be classified
into two types \cite{ma10}: area coverage (also called full or blanket
coverage) and target coverage.  An  area coverage problem is to find a
minimum number of sensors to work, such that each physical point in the
area is within the sensing range  of at least one working sensor node.
Target coverage problem  is to cover only a  finite number of discrete
points  called targets.   This type  of coverage  has  mainly military
applications. Our work will concentrate on the area coverage by design
and implementation of a  strategy which efficiently selects the active
nodes   that  must   maintain  both   sensing  coverage   and  network
connectivity and at the same time improve the lifetime of the wireless
sensor  network.   But  requiring  that  all physical  points  of  the
considered region are covered may  be too strict, especially where the
sensor network is not dense.   Our approach represents an area covered
by a sensor as a set of primary points and tries to maximize the total
number  of  primary points  that  are  covered  in each  round,  while
minimizing  overcoverage (points  covered by  multiple  active sensors
simultaneously).

{\bf Lifetime}

Various   definitions   exist   for   the   lifetime   of   a   sensor
network~\cite{die09}.  The main definitions proposed in the literature are
related to the  remaining energy of the nodes or  to the coverage percentage. 
The lifetime of the  network is mainly defined as the amount
of  time during which  the network  can  satisfy its  coverage objective  (the
amount of  time that the network  can cover a given  percentage of its
area or targets of interest). In this work, we assume that the network
is alive  until all  nodes have  been drained of  their energy  or the
sensor network becomes disconnected, and we measure the coverage ratio
during the WSN lifetime.  Network connectivity is important because an
active sensor node without  connectivity towards a base station cannot
transmit information on an event in the area that it monitors.

{\bf Activity scheduling}

Activity scheduling is to  schedule the activation and deactivation of
sensor nodes.  The  basic objective is to decide  which sensors are in
what states (active or sleeping mode)  and for how long, so that the
application  coverage requirement  can be  guaranteed and  the network
lifetime can be  prolonged. Various approaches, including centralized,
distributed, and localized algorithms, have been proposed for activity
scheduling.  In  distributed algorithms, each node  in the network
autonomously makes decisions on whether  to turn on or turn off itself
only using  local neighbor information. In  centralized algorithms, a
central controller  (a node or  base station) informs every  sensors of
the time intervals to be activated.

{\bf Distributed approaches}

Some      distributed     algorithms      have      been     developed
in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}  to perform the
scheduling.   Distributed algorithms typically  operate in  rounds for
a predetermined  duration. At  the  beginning of  each  round, a  sensor
exchanges information with its neighbors and makes a decision to either
remain turned  on or to  go to sleep  for the round. This  decision is
basically made on simple greedy criteria like  the largest uncovered
area   \cite{Berman05efficientenergy},   maximum   uncovered   targets
\cite{1240799}.   In \cite{Tian02}, the  scheduling scheme  is divided
into rounds, where each round  has a self-scheduling phase followed by
a sensing phase.  Each sensor  broadcasts a message containing the node ID
and the node location  to its neighbors at the beginning  of each round. A
sensor determines  its status by  a rule named off-duty  eligible rule
which tells  him to  turn off if  its sensing  area is covered  by its
neighbors. A  back-off scheme is  introduced to let each  sensor delay
the decision process  with a random period of time,  in order to avoid
simultaneous conflicting decisions between nodes and  lack of coverage on any area.
\cite{Prasad:2007:DAL:1782174.1782218}  defines a model  for capturing
the dependencies  between different cover sets  and proposes localized
heuristic  based on this  dependency.  The  algorithm consists  of two
phases, an initial  setup phase during which each  sensor computes and
prioritizes the  covers and  a sensing phase  during which  each sensor
first decides  its on/off status, and  then remains on or  off for the
rest  of the  duration.  Authors  in \cite{chin2007}  propose  a novel
distributed  heuristic named  Distributed  Energy-efficient Scheduling
for k-coverage  (DESK) so  that the energy  consumption among  all the
sensors  is balanced,  and  network lifetime  is  maximized while  the
coverage requirement  is being  maintained.  This algorithm  works in
round, requires only  1-sensing-hop-neighbor information, and a sensor
decides  its status  (active/sleep)  based on  its perimeter  coverage
computed  through the k-Non-Unit-disk  coverage algorithm  proposed in
\cite{Huang:2003:CPW:941350.941367}.

Some other approaches do  not consider a synchronized and predetermined
period  of time  where the  sensors are  active or  not.  Indeed, each
sensor  maintains its  own timer  and its  wake-up time is randomized
\cite{Ye03} or regulated \cite{cardei05} over time.
%A ecrire \cite{Abrams:2004:SKA:984622.984684}p33

%The scheduling information is disseminated throughout the network and only sensors in the active state are responsible
%for monitoring all targets, while all other nodes are in a low-energy sleep mode. The nodes decide cooperatively which of them will remain in sleep mode for a certain
%period of time.

 %one way of increasing lifeteime is by turning off redundant nodes to sleep mode to conserve energy while active nodes provide essential coverage, which improves fault tolerance. 

%In this paper we focus on centralized algorithms because distributed algorithms are outside the scope of our work. Note that centralized coverage algorithms have the advantage of requiring very low processing power from the sensor nodes which have usually limited processing capabilities. Moreover, a recent study conducted in \cite{pc10} concludes that there is a threshold in terms of network size to switch from a localized to a centralized algorithm. Indeed the exchange of messages in large networks may consume  a considerable amount of energy in a localized approach compared to a centralized one. 

{\bf Centralized approaches}

Power  efficient  centralized  schemes  differ  according  to  several
criteria \cite{Cardei:2006:ECP:1646656.1646898},  such as the coverage
objective  (target coverage  or  area coverage),  the node  deployment
method (random or deterministic) and the heterogeneity of sensor nodes
(common sensing range, common battery lifetime). The major approach is
to divide/organize  the sensors into  a suitable number of  set covers
where each  set completely covers  an interest region and  to activate
these set covers successively.

The first algorithms  proposed in the  literature consider that  the cover
sets  are  disjoint: a  sensor  node appears  in  exactly  one of  the
generated  cover  sets.    For  instance,  Slijepcevic  and  Potkonjak
\cite{Slijepcevic01powerefficient}   propose    an   algorithm   which
allocates sensor nodes in mutually independent sets to monitor an area
divided into  several fields.  Their algorithm builds  a cover  set by
including in  priority the sensor  nodes which cover  critical fields,
that  is to  say fields  that are  covered by  the smallest  number of
sensors. The time complexity of  their heuristic is $O(n^2)$ where $n$
is the number of  sensors.  \cite{cardei02}~describes a graph coloring
technique  to achieve energy  savings by  organizing the  sensor nodes
into a maximum number of  disjoint dominating sets which are activated
successively. The dominating sets do not guarantee the coverage of the
whole        region        of        interest.        Abrams        et
al.~\cite{Abrams:2004:SKA:984622.984684}  design  three  approximation
algorithms  for a  variation of  the  set k-cover  problem, where  the
objective is to partition the sensors into covers such that the number
of covers that  includes an area, summed over  all areas, is maximized.
Their        work        builds        upon       previous        work
in~\cite{Slijepcevic01powerefficient} and the  generated cover sets do
not provide complete coverage of the monitoring zone.

%examine the target coverage problem by disjoint cover sets but relax the requirement that every  cover set monitor all the targets and try to maximize the number of times the targets are covered by the partition. They propose various algorithms and establish approximation ratio.

In~\cite{Cardei:2005:IWS:1160086.1160098},   the  authors   propose  a
heuristic  to compute  the  disjoint  set covers  (DSC).  In order  to
compute the maximum number of  covers, they first transform DSC into a
maximum-flow problem, which  is then formulated  as a  mixed integer
programming  problem (MIP).  Based on  the solution  of the  MIP, they
design a heuristic to compute  the final number of covers. The results
show  a slight  performance  improvement  in terms  of  the number  of
produced  DSC in comparison  to~\cite{Slijepcevic01powerefficient}, but
it incurs  higher execution  time due to  the complexity of  the mixed
integer      programming     solving.       %Cardei      and     Du
\cite{Cardei:2005:IWS:1160086.1160098} propose a method to efficiently
compute the maximum  number of disjoint set covers  such that each set
can  monitor all  targets. They  first  transform the  problem into  a
maximum  flow   problem  which  is  formulated  as   a  mixed  integer
programming (MIP). Then their heuristic  uses the output of the MIP to
compute  disjoint  set  covers.  Results  show  that  this  heuristic
provides  a   number  of  set  covers  slightly   larger  compared  to
\cite{Slijepcevic01powerefficient}  but with  a larger  execution time
due  to the complexity  of the  mixed integer  programming resolution.
Zorbas  et  al.  \cite{Zorbas2007}  present  B\{GOP\},  a  centralized
coverage   algorithm  introducing   sensor   candidate  categorization
depending on their  coverage status and the notion  of critical target
to  call  targets   that  are  associated  with  a   small  number  of
sensors. The total running time of their heuristic is $0(m n^2)$ where
$n$ is the number of sensors,  and $m$ the number of targets. Compared
to    algorithm's    results     of    Slijepcevic    and    Potkonjak
\cite{Slijepcevic01powerefficient},  their   heuristic  produces  more
cover sets with a slight growth rate in execution time.
%More recently Manju and Pujari\cite{Manju2011}

In the  case of non-disjoint algorithms  \cite{Manju2011}, sensors may
participate  in more  than  one cover  set.   In some  cases this  may
prolong  the lifetime  of the  network in  comparison to  the disjoint
cover set algorithms, but  designing algorithms for non-disjoint cover
sets  generally induces a  higher order  of complexity.   Moreover, in
case of a sensor's  failure, non-disjoint scheduling policies are less
resilient and less  reliable because a sensor may  be involved in more
than one  cover sets.  For instance,  Cardei et al.~\cite{cardei05bis}
present a  linear programming (LP)  solution and a greedy  approach to
extend the  sensor network lifetime  by organizing the sensors  into a
maximal  number of  non-disjoint cover  sets. Simulation  results show
that by allowing sensors to  participate in multiple sets, the network
lifetime         increases        compared         with        related
work~\cite{Cardei:2005:IWS:1160086.1160098}.   In~\cite{berman04}, the
authors  have formulated  the lifetime  problem and  suggested another
(LP)  technique to  solve this  problem. A  centralized  solution  based      on      the     Garg-K\"{o}nemann
algorithm~\cite{garg98}, probably near
the optimal solution,    is also proposed.

{\bf Our contribution}

There are  three main questions which  should be addressed  to build a
scheduling strategy. We  give a brief answer to  these three questions
to describe our  approach before going into details  in the subsequent
sections.
\begin{itemize}
\item {\bf How must the  phases for information exchange, decision and
  sensing be planned over time?}   Our algorithm divides the time line
  into a number  of rounds. Each round contains  4 phases: Information
  Exchange, Leader Election, Decision, and Sensing.

\item {\bf What are the rules to decide which node has to be turned on
  or off?}  Our algorithm tends to limit the overcoverage of points of
  interest  to avoid  turning on  too many sensors covering  the same
  areas  at the  same time,  and tries  to prevent  undercoverage. The
  decision  is  a  good   compromise  between  these  two  conflicting
  objectives.

\item {\bf Which  node should make such a  decision?}  As mentioned in
  \cite{pc10}, both centralized  and distributed algorithms have their
  own  advantages and  disadvantages. Centralized  coverage algorithms
  have the advantage  of requiring very low processing  power from the
  sensor  nodes which  have usually  limited  processing capabilities.
  Distributed  algorithms  are  very  adaptable  to  the  dynamic  and
  scalable nature of sensors network.  Authors in \cite{pc10} conclude
  that there is a threshold in  terms of network size to switch from a
  localized  to  a  centralized  algorithm.  Indeed  the  exchange  of
  messages  in large  networks may  consume a  considerable  amount of
  energy in a centralized approach  compared to a distributed one. Our
  work does not  consider only one leader to  compute and to broadcast
  the scheduling decision  to all the sensors.  When  the network size
  increases,  the network  is  divided into  many  subregions and  the
  decision is made by a leader in each subregion.
\end{itemize}

\section{Activity scheduling}
\label{pd}

We consider  a randomly and  uniformly deployed network  consisting of
static  wireless sensors. The  wireless sensors  are deployed  in high
density to ensure initially a full coverage of the interested area. We
assume that  all nodes are  homogeneous in terms of  communication and
processing capabilities and heterogeneous in term of energy provision.
The  location  information is  available  to  the  sensor node  either
through hardware  such as embedded  GPS or through  location discovery
algorithms.   The   area  of  interest   can  be  divided   using  the
divide-and-conquer strategy  into smaller areas  called subregions and
then  our coverage  protocol  will be  implemented  in each  subregion
simultaneously.   Our protocol  works in  rounds fashion  as  shown in
figure~\ref{fig1}.

%Given the interested Area $A$, the wireless sensor nodes set $S=\lbrace  s_1,\ldots,s_N \rbrace $ that are deployed randomly and uniformly in this area such that they are ensure a full coverage for A. The Area A is divided into regions $A=\lbrace A^1,\ldots,A^k,\ldots, A^{N_R} \rbrace$. We suppose that each sensor node $s_i$ know its location and its region. We will have a subset $SSET^k =\lbrace s_1,...,s_j,...,s_{N^k} \rbrace $ , where $s_N = s_{N^1} + s_{N^2} +,\ldots,+ s_{N^k} +,\ldots,+s_{N^R}$. Each sensor node $s_i$ has the same initial energy $IE_i$ in the first time and the current residual energy $RE_i$ equal to $IE_i$  in the first time for each $s_i$ in A. \\ 

\begin{figure}[ht!]
\centering
\includegraphics[width=85mm]{FirstModel.eps} % 70mm
\caption{Multi-round coverage protocol}
\label{fig1}
\end{figure} 

Each round  is divided  into 4 phases  : Information  (INFO) Exchange,
Leader  Election, Decision,  and  Sensing.  For  each  round there  is
exactly one set cover responsible  for the sensing task.  This protocol is
more reliable  against an unexpected node failure  because it works
in rounds.   On the  one hand,  if a node  failure is  detected before
making the decision, the node will not participate to this phase, and,
on the other hand, if the  node failure occurs after the decision, the
sensing task of the network  will be temporarily affected: only during
the period of sensing until a  new round starts, since a new set cover
will take  charge of the  sensing task in  the next round.  The energy
consumption  and  some other  constraints  can  easily  be taken  into
account  since  the  sensors   can  update  and  then  exchange  their
information (including their residual energy) at the beginning of each
round.  However,   the  pre-sensing  phases   (INFO  Exchange,  Leader
Election,  Decision) are energy  consuming for  some nodes,  even when
they do not  join the network to monitor the  area. Below, we describe
each phase in more details.

\subsection{Information exchange phase}

Each sensor node $j$ sends  its position, remaining energy $RE_j$, and
the number of local neighbors  $NBR_j$ to all wireless sensor nodes in
its subregion by using an INFO  packet and then listens to the packets
sent from  other nodes.  After that, each  node will  have information
about  all the  sensor  nodes in  the  subregion.  In  our model,  the
remaining energy corresponds to the time that a sensor can live in the
active mode.

%\subsection{\textbf Working Phase:}

%The working phase works in rounding fashion. Each round include 3 steps described as follow :

\subsection{Leader election phase}
This  step includes choosing  the Wireless  Sensor Node  Leader (WSNL)
which  will  be  responsible  for executing  the coverage  algorithm.  Each
subregion  in  the   area  of  interest  will  select   its  own  WSNL
independently  for each  round.  All the  sensor  nodes cooperate  to
select WSNL.  The nodes in the  same subregion will  select the leader
based on  the received  information from all  other nodes in  the same
subregion.  The selection criteria  in order  of priority  are: larger
number  of neighbors,  larger remaining  energy, and  then in  case of
equality, larger index.

\subsection{Decision phase}
The  WSNL will  solve an  integer  program (see  section~\ref{cp})  to
select which sensors will be  activated in the following sensing phase
to cover  the subregion.  WSNL will send  Active-Sleep packet  to each
sensor in the subregion based on the algorithm's results.
%The main goal in this step after choosing the WSNL is to produce the best representative active nodes set that will take the responsibility of covering the whole region $A^k$ with minimum number of sensor nodes to prolong the lifetime in the wireless sensor network. For our problem, in each round we need to select the minimum set of sensor nodes to improve the lifetime of the network and in the same time taking into account covering the region $A^k$ . We need an optimal solution with tradeoff between our two conflicting objectives.
%The above region coverage problem can be formulated as a Multi-objective optimization problem and we can use the Binary Particle Swarm Optimization technique to solve it. 

\subsection{Sensing phase}
Active  sensors  in the  round  will  execute  their sensing  task  to
preserve maximal  coverage in the  region of interest. We  will assume
that the cost  of keeping a node awake (or asleep)  for sensing task is
the same  for all wireless sensor  nodes in the  network.  Each sensor
will receive  an Active-Sleep  packet from WSNL  informing it  to stay
awake or to go to sleep  for a time  equal to  the period of  sensing until
starting a new round.

%\subsection{Sensing coverage model}
%\label{pd}

%\noindent We try to produce an adaptive scheduling which allows sensors to operate alternatively so as to prolong the network lifetime. For convenience, the notations and assumptions are described first.
%The wireless sensor node use the  binary disk sensing model by which each sensor node will has a certain sensing range is reserved within a circular disk called radius $R_s$.
\noindent We consider a boolean  disk coverage model which is the most
widely used sensor coverage model in the literature. Each sensor has a
constant sensing range $R_s$. All  space points within a disk centered
at  the sensor with  the radius  of the  sensing range  is said  to be
covered by this sensor. We also assume that the communication range is
at   least  twice    the size of the   sensing  range.   In  fact,   Zhang  and
Zhou~\cite{Zhang05} proved that if  the transmission range fulfills the
previous  hypothesis, a  complete coverage  of a  convex  area implies
connectivity among the working nodes in the active mode.
%To calculate the coverage ratio for the area of interest, we can propose the following coverage model which is called Wireless Sensor Node Area Coverage Model to ensure that all the area within each node sensing range are covered. We can calculate the positions of the points in the circle disc of the sensing range of wireless sensor node based on the Unit Circle in figure~\ref{fig:cluster1}:

%\begin{figure}[h!]
%\centering
%\begin{tabular}{cc}
%%\includegraphics[scale=0.25]{fig1.pdf}\\ %& \includegraphics[scale=0.10]{1.pdf} \\
%%(A) Figure 1 & (B) Figure 2
%\end{tabular}
%\caption{Unit Circle in radians. }
%\label{fig:cluster1}
%\end{figure}

%By using the Unit Circle in figure~\ref{fig:cluster1}, 
%We choose to representEach wireless sensor node will be represented into a selected number of primary points by which we can know if the sensor node is covered or not.
% Figure ~\ref{fig:cluster2} shows the selected primary points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.

\noindent Instead of working with the coverage area, we consider for each
sensor a set of points called  primary points. We also assume that the
sensing disk defined  by a sensor is covered if  all the primary points of
this sensor are covered.
%\begin{figure}[h!]
%\centering
%\begin{tabular}{cc}
%%\includegraphics[scale=0.25]{fig2.pdf}\\ %& \includegraphics[scale=0.10]{1.pdf} \\
%%(A) Figure 1 & (B) Figure 2
%\end{tabular}
%\caption{Wireless Sensor Node Area Coverage Model.}
%\label{fig:cluster2}
%\end{figure}
By  knowing the  position (point  center: ($p_x,p_y$))  of  a wireless
sensor node  and its $R_s$,  we calculate the primary  points directly
based on the proposed model. We  use these primary points (that can be
increased or decreased if necessary)  as references to ensure that the
monitored  region  of interest  is  covered  by  the selected  set  of
sensors, instead of using all the points in the area.

\noindent  We can  calculate  the positions  of  the selected  primary
points in  the circle disk of  the sensing range of  a wireless sensor
node (see figure~\ref{fig2}) as follows:\\
$(p_x,p_y)$ = point center of wireless sensor node\\  
$X_1=(p_x,p_y)$ \\ 
$X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           
$X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\
$X_4=( p_x + R_s * (0), p_y + R_s * (1) )$\\
$X_5=( p_x + R_s * (0), p_y + R_s * (-1 )) $\\
$X_6= ( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (0)) $\\
$X_7=( p_x + R_s *  (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
$X_8=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
$X_9=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
$X_{10}=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
$X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
$X_{12}=( p_x + R_s * (0), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $.

 \begin{figure}[h!]
%\centering
% \begin{multicols}{6}
\centering
%\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
%\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
\includegraphics[scale=0.25]{principles13.eps}
%\includegraphics[scale=0.10]{fig25.pdf}\\~ ~ ~(d)
%\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
%\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
%\end{multicols} 
\caption{Wireless sensor node represented by 13 primary points}
\label{fig2}
\end{figure}

\section{Coverage problem formulation}
\label{cp}
%We can formulate our optimization problem as energy cost minimization by minimize the number of active sensor nodes and maximizing the coverage rate at the same time in each $A^k$ . This optimization problem can be formulated as follow: Since that we use a homogeneous wireless sensor network, we will assume that the cost of keeping a node awake is the same for all wireless sensor nodes in the network. We can define the decision parameter  $X_j$ as in \eqref{eq11}:\\


%To satisfy the coverage requirement, the set of the principal points that will represent all the sensor nodes in the monitored region as $PSET= \lbrace P_1,\ldots ,P_p, \ldots , P_{N_P^k} \rbrace $, where $N_P^k = N_{sp} * N^k $ and according to the proposed model in figure ~\ref{fig:cluster2}. These points can be used by the wireless sensor node leader which will be chosen in each region in A to build a new parameter $\alpha_{jp}$  that represents the coverage possibility for each principal point $P_p$ of each wireless sensor node $s_j$ in $A^k$ as in \eqref{eq12}:\\

\noindent   Our   model   is   based   on  the   model   proposed   by
\cite{pedraza2006} where the objective is  to find a maximum number of
disjoint  cover sets.   To accomplish  this goal,  authors  proposed an
integer program which forces undercoverage and overcoverage of targets
to  become  minimal at  the  same  time.   They use  binary  variables
$x_{jl}$ to indicate  if sensor $j$ belongs to cover  set $l$.  In our
model,  we  consider  binary  variables $X_{j}$  which  determine  the
activation of  sensor $j$ in the  sensing phase of the  round. We also
consider  primary points  as targets.   The set  of primary  points is
denoted by $P$ and the set of sensors by $J$.

\noindent  For  a primary  point  $p$,  let  $\alpha_{jp}$ denote  the
indicator function of whether the point $p$ is covered, that is:
\begin{equation}
\alpha_{jp} = \left \{ 
\begin{array}{l l}
  1 & \mbox{if the primary point $p$ is covered} \\
 & \mbox{by sensor node $j$}, \\
  0 & \mbox{otherwise.}\\
\end{array} \right.
%\label{eq12} 
\end{equation}
The number of active sensors that cover the primary point $p$ is equal
to $\sum_{j \in J} \alpha_{jp} * X_{j}$ where:
\begin{equation}
X_{j} = \left \{ 
\begin{array}{l l}
  1& \mbox{if sensor $j$  is active,} \\
  0 &  \mbox{otherwise.}\\
\end{array} \right.
%\label{eq11} 
\end{equation}
We define the Overcoverage variable $\Theta_{p}$ as:
\begin{equation}
 \Theta_{p} = \left \{ 
\begin{array}{l l}
  0 & \mbox{if the primary point}\\
    & \mbox{$p$ is not covered,}\\
  \left( \sum_{j \in J} \alpha_{jp} * X_{j} \right)- 1 & \mbox{otherwise.}\\
\end{array} \right.
\label{eq13} 
\end{equation}
\noindent More precisely, $\Theta_{p}$ represents the number of active
sensor  nodes  minus  one  that  cover the  primary  point  $p$.\\
The Undercoverage variable $U_{p}$ of the primary point $p$ is defined
by:
\begin{equation}
U_{p} = \left \{ 
\begin{array}{l l}
  1 &\mbox{if the primary point $p$ is not covered,} \\
  0 & \mbox{otherwise.}\\
\end{array} \right.
\label{eq14} 
\end{equation}

\noindent Our coverage optimization problem can then be formulated as follows\\
\begin{equation} \label{eq:ip2r}
\left \{
\begin{array}{ll}
\min \sum_{p \in P} (w_{\theta} \Theta_{p} + w_{U} U_{p})&\\
\textrm{subject to :}&\\
\sum_{j \in J}  \alpha_{jp} X_{j} - \Theta_{p}+ U_{p} =1, &\forall p \in P\\
%\label{c1} 
%\sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &\forall j \in J \\
%\label{c2}
\Theta_{p}\in \mathbb{N} , &\forall p \in P\\
U_{p} \in \{0,1\}, &\forall p \in P \\
X_{j} \in \{0,1\}, &\forall j \in J
\end{array}
\right.
\end{equation}
\begin{itemize}
\item $X_{j}$  : indicates whether or  not the sensor  $j$ is actively
  sensing in the round (1 if yes and 0 if not);
\item $\Theta_{p}$  : {\it overcoverage}, the number  of sensors minus
  one that are covering the primary point $p$;
\item $U_{p}$  : {\it undercoverage},  indicates whether or  not the principal point
  $p$ is being covered (1 if not covered and 0 if covered).
\end{itemize}

The first group  of constraints indicates that some  primary point $p$
should be covered by at least one  sensor and, if it is not always the
case,  overcoverage  and  undercoverage  variables  help  balancing  the
restriction  equations by taking  positive values.  There are  two main         
objectives.  First we limit the overcoverage of primary points in order to
activate a minimum number of sensors.  Second we prevent the absence of monitoring on
 some parts of the subregion by  minimizing the undercoverage.   The
weights  $w_\theta$  and  $w_U$  must  be properly  chosen  so  as  to
guarantee that  the maximum number  of points are covered  during each
round.
 
%In equation \eqref{eq15}, there are two main objectives: the first one using  the Overcoverage parameter to minimize the number of active sensor nodes in the produced final solution vector $X$ which leads to improve the life time of wireless sensor network. The second goal by using the  Undercoverage parameter  to maximize the coverage in the region by means of covering each primary point in $SSET^k$.The two objectives are achieved at the same time. The constraint which represented in equation \eqref{eq16} refer to the coverage function for each primary point $P_p$ in $SSET^k$ , where each $P_p$ should be covered by
%at least one sensor node in $A^k$. The objective function in \eqref{eq15} involving two main objectives to be optimized simultaneously, where optimal decisions need to be taken in the presence of trade-offs between the two conflicting main objectives in \eqref{eq15} and this refer to that our coverage optimization problem is a multi-objective optimization problem and we can use the BPSO to solve it. The concept of Overcoverage and Undercoverage inspired from ~\cite{Fernan12} but we use it with our model as stated in subsection \ref{Sensing Coverage Model} with some modification to be applied later by BPSO.
%\subsection{Notations and assumptions}

%\begin{itemize}
%\item $m$ : the number of targets
%\item $n$ : the number of sensors
%\item $K$ : maximal number of cover sets
%\item $i$ : index of target ($i=1..m$)
%\item $j$ : index of sensor ($j=1..n$)
%\item $k$ : index of cover set ($k=1..K$)
%\item $T_0$ : initial set of targets
%\item $S_0$ : initial set of sensors
%\item $T $ : set of targets which are not covered by at least one cover set
%\item $S$ : set of available sensors
%\item $S_0(i)$ : set of sensors which cover the target $i$
%\item $T_0(j)$ : set of targets covered by sensor $j$
%\item $C_k$ : cover set of index $k$
%\item $T(C_k)$ : set of targets covered by the cover set $k$
%\item $NS(i)$ : set of  available sensors which cover the target $i$
%\item $NC(i)$ : set of cover sets which do not cover the target $i$
%\item $|.|$ : cardinality of the set

%\end{itemize}

\section{Simulation results}
\label{exp}

In this section, we conducted  a series of simulations to evaluate the
efficiency  and the relevance of  our approach,  using the  discrete event
simulator  OMNeT++  \cite{varga}. We  performed  simulations for  five
different densities varying from 50 to 250~nodes. Experimental results
were  obtained from  randomly generated  networks in  which  nodes are
deployed over a  $(50 \times 25)~m^2 $ sensing  field. 
More precisely, the deployment is controlled at a coarse scale in
  order to ensure that the  deployed nodes can fully cover the sensing
  field with the given sensing range.
10~simulation  runs  are performed  with
different  network  topologies for  each  node  density.  The  results
presented hereafter  are the  average of these  10 runs.  A simulation
ends  when  all the  nodes  are dead  or  the  sensor network  becomes
disconnected (some nodes may not be  able to send, to a base station, an
event they sense).

Our proposed coverage protocol uses the radio energy dissipation model
defined by~\cite{HeinzelmanCB02} as  energy consumption model for each
wireless  sensor node  when  transmitting or  receiving packets.   The
energy of  each node in a  network is initialized  randomly within the
range 24-60~joules, and each sensor node will consume 0.2 watts during
the sensing period which will last 60 seconds. Thus, an
active  node will  consume  12~joules during the sensing  phase, while  a
sleeping  node will  use  0.002  joules.  Each  sensor  node will  not
participate in the next round if its remaining energy is less than 12
joules.  In  all  experiments  the  parameters  are  set  as  follows:
$R_s=5~m$, $w_{\Theta}=1$, and $w_{U}=|P^2|$.

We  evaluate the  efficiency of  our approach by using  some performance
metrics such as: coverage ratio,  number of active nodes ratio, energy
saving  ratio, energy consumption,  network lifetime,  execution time,
and number of stopped simulation runs.  Our approach called strategy~2
(with two leaders)  works with two subregions, each  one having a size
of $(25 \times 25)~m^2$.  Our strategy will be compared with two other
approaches. The first one,  called strategy~1 (with one leader), works
as strategy~2, but considers only one region of $(50 \times 25)$ $m^2$
with only  one leader.  The  other approach, called  Simple Heuristic,
consists in uniformly dividing   the region into squares  of $(5 \times
5)~m^2$.   During the  decision phase,  in  each square,  a sensor  is
randomly  chosen, it  will remain  turned  on for  the coming  sensing
phase.

\subsection{The impact of the number of rounds on the coverage ratio} 

In this experiment, the coverage ratio measures how much the area of a
sensor field is  covered. In our case, the  coverage ratio is regarded
as the number  of primary points covered among the  set of all primary
points  within the field.  Figure~\ref{fig3} shows  the impact  of the
number of rounds on the  average coverage ratio for 150 deployed nodes
for the  three approaches.  It can be  seen that the  three approaches
give  similar  coverage  ratios  during  the first  rounds.  From  the
9th~round the  coverage ratio  decreases continuously with  the simple
heuristic, while the two other strategies provide superior coverage to
$90\%$ for five more rounds.  Coverage ratio decreases when the number
of rounds increases  due to dead nodes. Although  some nodes are dead,
thanks to  strategy~1 or~2,  other nodes are  preserved to  ensure the
coverage. Moreover, when  we have a dense sensor  network, it leads to
maintain the full coverage for a larger number of rounds. Strategy~2 is
slightly more efficient than strategy 1, because strategy~2 subdivides
the region into 2~subregions and  if one of the two subregions becomes
disconnected,  the coverage   may  be  still  ensured   in  the  remaining
subregion.

\parskip 0pt 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.55]{TheCoverageRatio150.eps} %\\~ ~ ~(a)
\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure} 

\subsection{The impact of the number of rounds on the active sensors ratio} 

It is important to have as few active nodes as possible in each round,
in  order to  minimize  the communication  overhead  and maximize  the
network lifetime.  This point is  assessed through the  Active Sensors
Ratio, which is defined as follows:
\begin{equation*}
\scriptsize
\mbox{ASR}(\%) = \frac{\mbox{Number of active sensors 
during the current sensing phase}}{\mbox{Total number of sensors in the network
for the region}} \times 100.
\end{equation*}
Figure~\ref{fig4} shows  the average active nodes ratio versus rounds
for 150 deployed nodes.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.55]{TheActiveSensorRatio150.eps} %\\~ ~ ~(a)
\caption{The impact of the number of rounds on the active sensors ratio for 150 deployed nodes }
\label{fig4}
\end{figure} 

The  results presented  in figure~\ref{fig4}  show the  superiority of
both proposed  strategies, the strategy  with two leaders and  the one
with a  single leader,  in comparison with  the simple  heuristic. The
strategy with one leader uses less active nodes than the strategy with
two leaders until the last  rounds, because it uses central control on
the whole sensing field.  The  advantage of the strategy~2 approach is
that even if a network is disconnected in one subregion, the other one
usually  continues  the optimization  process,  and  this extends  the
lifetime of the network.

\subsection{The impact of the number of rounds on the energy saving ratio} 

In this experiment, we consider a performance metric linked to energy.
This metric, called Energy Saving Ratio, is defined by:
\begin{equation*}
\scriptsize
\mbox{ESR}(\%) = \frac{\mbox{Number of alive sensors during this round}}
{\mbox{Total number of sensors in the network for the region}} \times 100.
\end{equation*}
The  longer the ratio  is,  the more  redundant sensor  nodes are
switched off, and consequently  the longer the  network may  live.
Figure~\ref{fig5} shows the average  Energy Saving Ratio versus rounds
for all three approaches and for 150 deployed nodes.

\begin{figure}[h!]
%\centering
% \begin{multicols}{6}
\centering
\includegraphics[scale=0.55]{TheEnergySavingRatio150.eps} %\\~ ~ ~(a)
\caption{The impact of the number of rounds on the energy saving ratio for 150 deployed nodes}
\label{fig5}
\end{figure} 

The simulation  results show that our strategies  allow to efficiently
save energy by  turning off some sensors during  the sensing phase. As
expected, the strategy with one leader is usually slightly better than
the second  strategy, because the  global optimization permits  to turn
off more  sensors. Indeed,  when there are  two subregions  more nodes
remain awake  near the border shared  by them. Note that  again as the
number of  rounds increases  the two leaders'  strategy becomes  the most
performing one, since it takes longer  to have the two subregion networks
simultaneously disconnected.

\subsection{The number of stopped simulation runs}

We  will now  study  the number  of  simulations which  stopped due  to
network  disconnections per round  for each  of the  three approaches.
Figure~\ref{fig6} illustrates the average number of stopped simulation
runs per  round for 150 deployed  nodes.  It can be  observed that the
simple heuristic is  the approach which  stops first because  the nodes
are   randomly chosen.   Among  the  two   proposed  strategies,  the
centralized  one  first  exhibits  network  disconnections.   Thus,  as
explained previously, in case  of the strategy with several subregions
the  optimization effectively  continues as  long  as a  network in  a
subregion   is  still   connected.   This   longer   partial  coverage
optimization participates in extending the network lifetime.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.55]{TheNumberofStoppedSimulationRuns150.eps} 
\caption{The number of stopped simulation runs compared to the number of rounds for 150 deployed nodes }
\label{fig6}
\end{figure} 

\subsection{The energy consumption}

In this experiment, we study the effect of the multi-hop communication
protocol  on the  performance of  the  strategy with  two leaders  and
compare  it  with  the  other  two  approaches.   The  average  energy
consumption  resulting  from  wireless  communications  is  calculated
by taking into account the  energy spent by  all the nodes when  transmitting and
receiving  packets during  the network  lifetime. This  average value,
which  is obtained  for 10~simulation  runs,  is then  divided by  the
average number of rounds to define a metric allowing a fair comparison
between networks having different densities.

Figure~\ref{fig7} illustrates the energy consumption for the different
network  sizes and  the three  approaches. The  results show  that the
strategy  with  two  leaders  is  the  most  competitive  from  the energy
consumption point  of view.  A  centralized method, like  the strategy
with  one  leader, has  a  high energy  consumption  due  to  many
communications.   In fact,  a distributed  method greatly  reduces the
number  of communications thanks  to the  partitioning of  the initial
network in several independent subnetworks. Let us notice that even if
a  centralized  method  consumes  far  more  energy  than  the  simple
heuristic, since the energy cost of communications during a round is a
small  part   of  the   energy  spent  in   the  sensing   phase,  the
communications have a small impact on the network lifetime.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.55]{TheEnergyConsumption.eps} 
\caption{The energy consumption}
\label{fig7}
\end{figure} 

\subsection{The impact of the number of sensors on execution time}

A  sensor  node has  limited  energy  resources  and computing  power,
therefore it is important that the proposed algorithm has the shortest
possible execution  time. The energy of  a sensor node  must be mainly
used   for  the  sensing   phase,  not   for  the   pre-sensing  ones.   
Table~\ref{table1} gives the average  execution times  in seconds
on a laptop of the decision phase (solving of the optimization problem)
during one  round.  They  are given for  the different  approaches and
various numbers of sensors.  The lack of any optimization explains why
the heuristic has very  low execution times.  Conversely, the strategy
with  one  leader which  requires  to  solve  an optimization  problem
considering  all  the  nodes  presents  redhibitory  execution  times.
Moreover, increasing the network size by 50~nodes   multiplies the time
by  almost a  factor of  10. The  strategy with  two leaders  has more
suitable times.  We  think that in distributed fashion  the solving of
the  optimization problem  in a  subregion  can be  tackled by  sensor
nodes.   Overall,  to  be  able to  deal  with  very  large  networks,  a
distributed method is clearly required.

\begin{table}[ht]
\caption{The execution time(s) vs the number of sensors}
% title of Table
\centering

% used for centering table
\begin{tabular}{|c|c|c|c|}
% centered columns (4 columns)
      \hline
%inserts double horizontal lines
Sensors number & Strategy~2 & Strategy~1  & Simple heuristic \\ [0.5ex]
 & (with two leaders) & (with one leader) & \\ [0.5ex]
%Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
% inserts table
%heading
\hline
% inserts single horizontal line
50 & 0.097 & 0.189 & 0.001 \\
% inserting body of the table
\hline
100 & 0.419 & 1.972 & 0.0032 \\
\hline
150 & 1.295 & 13.098 & 0.0032 \\
\hline
200 & 4.54 & 169.469 & 0.0046 \\
\hline
250 & 12.252 & 1581.163 & 0.0056 \\
% [1ex] adds vertical space
\hline
%inserts single line
\end{tabular}
\label{table1}
% is used to refer this table in the text
\end{table}

\subsection{The network lifetime}

Finally, we  have defined the network  lifetime as the  time until all
nodes  have  been drained  of  their  energy  or each  sensor  network
monitoring  an area has become  disconnected.  In  figure~\ref{fig8}, the
network  lifetime for different  network sizes  and for  both strategy
with two  leaders and the simple  heuristic is illustrated. 
  We do  not consider  anymore the  centralized strategy  with one
  leader, because, as shown above, this strategy results  in execution
  times that quickly become unsuitable for a sensor network.

\begin{figure}[h!]
%\centering
% \begin{multicols}{6}
\centering
\includegraphics[scale=0.5]{TheNetworkLifetime.eps} %\\~ ~ ~(a)
\caption{The network lifetime }
\label{fig8}
\end{figure} 

As  highlighted by figure~\ref{fig8},  the network  lifetime obviously
increases when  the size  of the network  increases, with  our approach
that leads to  the larger lifetime improvement.  By  choosing the  best 
suited nodes, for each round,  to cover the  region of interest  and by
letting the other ones sleep in order to be used later in next rounds,
our strategy efficiently prolonges the network lifetime. Comparison shows that
the larger  the sensor number  is, the more our  strategies outperform
the simple heuristic.  Strategy~2, which uses two leaders, is the best
one because it is robust to network disconnection in one subregion. It
also  means   that  distributing  the  algorithm  in   each  node  and
subdividing the sensing field  into many subregions, which are managed
independently and simultaneously, is the most relevant way to maximize
the lifetime of a network.

\section{Conclusion and future works}
\label{sec:conclusion}

In this paper, we have  addressed the problem of the coverage and the lifetime
optimization  in wireless  sensor networks.   This is  a key  issue as
sensor nodes  have limited  resources in terms  of memory,  energy and
computational power. To  cope with this problem, the  field of sensing
is   divided   into   smaller   subregions  using   the   concept   of
divide-and-conquer method,  and then a  multi-rounds coverage protocol
will optimize  coverage and  lifetime performances in  each subregion.
The  proposed  protocol  combines  two efficient  techniques:  network
leader election  and sensor activity scheduling,  where the challenges
include how to select the  most efficient leader in each subregion and
the best  representative active nodes that will  optimize the network lifetime
while  taking   the  responsibility  of   covering  the  corresponding
subregion.   The network lifetime  in each  subregion is  divided into
rounds, each round consists  of four phases: (i) Information Exchange,
(ii) Leader Election, (iii) an optimization-based Decision in order to
select  the  nodes remaining  active  for  the  last phase,  and  (iv)
Sensing.  The  simulations show the relevance  of the proposed
protocol in  terms of lifetime, coverage ratio,  active sensors ratio,
energy saving,  energy consumption, execution time, and  the number of
stopped simulation  runs due  to network disconnection.   Indeed, when
dealing with  large and dense wireless sensor  networks, a distributed
approach like the one we propose  allows to reduce the difficulty of a
single global optimization problem  by partitioning it in many smaller
problems, one per subregion, that can be solved more easily.

In  future work, we plan  to study  and propose  a coverage  protocol which
computes  all  active  sensor  schedules  in  a  single  round,  using
optimization  methods  such  as  swarms optimization  or  evolutionary
algorithms.  This single  round  will still  consists  of 4  phases, but  the
  decision phase will compute the schedules for several sensing phases
  which, aggregated together, define a kind of meta-sensing phase.
The computation of all cover sets in one round is far more
difficult, but will reduce the communication overhead.

% use section* for acknowledgement
%\section*{Acknowledgment}

\bibliographystyle{IEEEtran}
\bibliography{bare_conf}

% that's all folks
\end{document}