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\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{Energy-Efficient Distributed Multirounds Coverage Optimization Protocol to Prolong  the Lifetime in Wireless Sensor Networks}

\author{Ali Kadhum Idrees,~\IEEEmembership{}
        Karine Deschinkel,~\IEEEmembership{}
        Michel Salomon,~\IEEEmembership{}
        and~Rapha\"el Couturier ~\IEEEmembership{}% <-this % stops a space
%\thanks{are members in the AND team - DISC department - FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France.
% e-mail: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}% <-this % stops a space
%\thanks{}% <-this % stops a space
\thanks{Manuscript received April 19, 2005; revised January 11, 2007.}}



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\markboth{Journal of \LaTeX\ Class Files,~Vol.~6, No.~1, January~2007}%
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\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 Energy-Efficient Distributed 
Multirounds Coverage Optimization Protocol (EDMCOP)
to  improve  the  lifetime  in wireless  sensor
networks  is proposed. The area  of interest  is first  divided into
subregions using  a divide-and-conquer method and then the  EDMCOP protocol is distributed on the sensor nodes in each subregion. The EDMCOP combines two efficient techniques: Leader election for each subregion after that activity scheduling based optimization is planned for each subregion. The proposed
EDMCOP  works into rounds  during which a small number  of nodes,
remaining active for sensing, is  selected to ensure coverage so as to maximize the lifetime of wireless sensor network. 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. Compared with some existing
protocols, simulation  results show that the proposed protocol can  prolong the
network lifetime and improve the coverage performance effectively.
\end{abstract}

\begin{IEEEkeywords}
Wireless   Sensor   Networks,   Area   Coverage,   Network   lifetime,
Optimization, Scheduling.
\end{IEEEkeywords}
%\keywords{Area Coverage, Network lifetime, Optimization, Distributed Protocol}
 
\IEEEpeerreviewmaketitle

\section{Introduction}

\indent  The fast developments  in the  low-cost sensor  devices and
wireless  communications  have allowed  the  emergence  the WSNs.  WSN
includes  a large  number of  small, limited-power sensors that can
sense, process and transmit data over a wireless communication. They
communicate with each other by using multi-hop wireless communications, cooperate together to monitor the area of interest, 
and the measured data can be reported to a monitoring center called sink  
for analysis it~\cite{Sudip03}. There are several applications used the
WSN including health, home, environmental, military, and industrial
applications~\cite{Akyildiz02}. One of the major scientific research challenges in WSNs, which are addressed by a large number of literature during the last few years is to design energy efficient approches for coverage and connectivity in WSNs~\cite{conti2014mobile}. The coverage problem is one  of the
fundamental challenges in WSNs~\cite{Nayak04} that consists in monitoring efficiently and continuously
the area of interest. Thelimited energy  of sensors represents  the main challenge in  the WSNs
design~\cite{Sudip03}, where it is difficult to replace and/or recharge their batteries because the the area of interest nature (such
as hostile environments) and the cost. So, it is necessary that a WSN
deployed  with high  density because  spatial redundancy  can  then be
exploited to increase  the lifetime of the network. However, turn on
all the sensor nodes, which monitor the same region at the same time
leads to decrease the lifetime of the network. 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~\cite{Misra05}. WSNs require
energy-efficient solutions to improve the network lifetime that is
constrained by the limited power of each sensor node ~\cite{Akyildiz02}. 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. In section~\ref{Pr}, the problem definition and some background are described. Section~\ref{pd} is devoted to
the EDMCOP Protocol Description. 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 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}

\indent 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 WSNs. Cardei and Wu \cite{cardei2006energy} provide a taxonomy for coverage algorithms in WSNs according to several design choices:
\begin{itemize}
\item Sensors scheduling Algorithms, i.e. centralized or distributed/localized algorithms.
\item The objective of sensor coverage, i.e. to maximize the network lifetime
or to minimize the number of sensors during the sensing period.
\item The homogeneous or heterogeneous nature of the
nodes, in terms of sensing or communication capabilities.
\item The node deployment method, which may be random or deterministic.
\item Additional requirements for energy-efficient
coverage and connected coverage.
\end{itemize}

 The independency in the cover set (i.e. whether the cover sets are disjoint or non-disjoint) \cite{zorbas2010solving} is another design choice that can be added to the above
list.
   
\subsection{Centralized Approaches}
%{\bf Centralized approaches}
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 centralized algorithms always provide nearly or close to optimal solution since the algorithm has global view of the whole network. However, its advantage of
this type of algorithms is that it requires very low processing power from the sensor nodes, which usually have
limited processing capabilities where the schdule of selected sensor nodes will be computed on the base stations and then sent it to the sensor nodes to apply it to monitor the area of interest.

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. Abrams et al.~\cite{abrams2004set}  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.
\cite{cardei2005improving} 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{zorbas2010solving}  presented a centralised greedy
algorithm for the efficient production of both node disjoint
and non-disjoint cover sets. Compared to algorithm's results  of  Slijepcevic and Potkonjak
\cite{Slijepcevic01powerefficient},  their   heuristic  produces  more
disjoint cover sets with a slight growth rate in execution time. When producing non-disjoint cover sets, both Static-CCF and Dynamic-CCF provide cover sets offering longer network lifetime than those produced by
\cite{cardei2005energy}. Also, they require a smaller number of node participations in order to
achieve these results.

In the  case of non-disjoint algorithms  \cite{pujari2011high}, 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{cardei2005energy}
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{cardei2005improving}.   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}, provably near
the optimal solution,    is also proposed.

\subsection{Distributed approaches}
%{\bf Distributed approaches}
In distributed $\&$ localized coverage algorithms, the required computation to schedule the activity of sensor nodes will be done by the cooperation among the neighbours nodes. These algorithms may require more computation power  for the processing by the cooperated sensor nodes but they are more scaleable for large WSNs. Normally, the localized and distributed algorithms result in non-disjoint set covers. 

Some      distributed     algorithms      have      been     developed
in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02, yardibi2010distributed}  to perform the
scheduling so as to coverage preservation.   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{lu2003coverage}.   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{prasad2007distributed}  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. 

The authors in \cite{yardibi2010distributed}, are developed a distributed adaptive sleep scheduling algorithm (DASSA) for WSNs with partial coverage. DASSA does not require location information of sensors while maintaining connectivity and satisfying a user defined coverage target. In DASSA, nodes use the residual energy levels and feedback from the sink for scheduling the activity of their neighbors. This feedback mechanism reduces the randomness in scheduling that would otherwise occur due to the absence of location information. 

In \cite{ChinhVu},  the author proposed a novel  distributed heuristic, called
Distributed Energy-efficient  Scheduling for k-coverage  (DESK), which
ensures that the energy consumption  among the sensors is balanced and
the lifetime  maximized while the coverage  requirement is maintained.
This  heuristic   works  in  rounds,  requires   only  1-hop  neighbor
information,  and each  sensor decides  its status  (active  or sleep)
based    on    the    perimeter    coverage    model    proposed    in
\cite{Huang:2003:CPW:941350.941367}.
Idrees et al. ~\cite{idrees2014coverage} proposed a coverage optimization protocol to improve the lifetime in
heterogeneous energy wireless sensor networks. In this work, the coverage protocol distributed in each sensor node in the subregion but the optimization take place over the the whole subregion. We consider only distributing the coverage protocol over two subregions. 

The works presented in \cite{Bang, Zhixin, Zhang}  focuses on a Coverage-Aware, Distributed  Energy- Efficient and distributed clustering methods respectively, which aims to extend the network lifetime, while the coverage is ensured.
S.  Misra  et al.  \cite{Misra}  proposed  a  localized algorithm  for
coverage in  sensor networks. The algorithm conserve  the energy while
ensuring  the network coverage  by activating  the subset  of sensors,
with  the  minimum  overlap  area.The proposed  method  preserves  the
network connectivity by formation of the network backbone. 
More    recently,   Shibo   et
al. \cite{Shibo}  expressed the coverage  problem as a  minimum weight
submodular  set cover  problem  and proposed  a Distributed  Truncated
Greedy Algorithm  (DTGA) to  solve it. They  take advantage  from both
temporal  and spatial  correlations between  data sensed  by different
sensors,  and   leverage  prediction,  to  improve   the  lifetime. 


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{cardei2005maximum} over time.

The main contributions of our approach can be summarized as follows: (1) The high coverage ratio, (2) The reduced number of active nodes. 3) The optimization distributed over the subregions in the area of interest. 4) The dynamic leader election at each round based on some priority factors that will apply activity scheduling based optimization on the subregion to activate as less number as possible of sensor nodes  to take the mission of the coverage in each subregion(5) The very low energy consumption. 6) The higher network lifetime. 7) primary point coverage model to represent the sensing range of the sensor node. All this came from addresses three main questions to build a scheduling strategy by our EDMCOP Protocol :

 {\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.

 {\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.

 {\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.
 

\section{Preliminaries}
\label{Pr}

\subsection{Coverage Problem}
The most discussed coverage problems in literature can be classified
into three types \cite{ghosh2008coverage}\cite{wang2011coverage}: area coverage \cite{mulligan2010coverage}(also called full or blanket
coverage), target coverage \cite{yang2014novel}, and barrier coverage \cite{HeShibo}.  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. The problem of preventing an intruder from entering a region of interest is referred to as the barrier coverage . 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).


\subsection{Network 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.

\subsection{Activity Scheduling }
Activity scheduling is to schedule the activation and deac-
tivation 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. There are many sensor node scheduling methods are proposed in \cite{wang2010clique}, where they are grouped into two main categories:round-based sensor node scheduling in which, sensor nodes will execute the scheduling
algorithm during the initialization of each round and group-based sensor node scheduling in which, each node will performs the scheduling algorithm only once after its deployment and after
the execution of scheduling algorithm, all nodes will be allocated into different groups.



\section{ The EDMCOP Protocol Description}
\label{pd}

In this section, we introduce Energy-Efficient Distributed 
Multirounds Coverage Optimization Protocol named EDMCOP, which is distributed on the subregions for the area of interest. It is based on two efficient techniques: network
leader election and sensor activity scheduling for coverage preservation and energy conservation continuously and efficiently to maximize the lifetime in the network.  
The main features of our EDMCOP protocol: i)It divides the area of interest into subregions. ii)It requires only the information of the nodes within the subregion. iii) it divides the network lifetime into rounds. iv)It based on the autonomous distributed decision by the nodes in the subregion to elect the Leader, which will apply the activity scheduling based optimization on the subregion. v)  it achieves an energy consumption balancing among the nodes in the subregion  by selecting the nodes with maximum energy as leader in each round. vi) It uses the optimization to select the best representative set of sensors in the subregion by optimize the coverage and the lifetime over the area of interest.
vii)It uses our developed primary point coverage model, which represent the sensing range of the sensor as a set of points, which are used by the our optimization algorithm. viii) It uses a simple energy model that takes communication, sensing and computation energy consumptions into account to evaluate the performance of our Protocol.


\subsection{ Assumptions and Models}
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.   
\indent 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 $R_c \geq 2R_s$.
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.



%\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.

\indent 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.

\indent  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{fig1}) 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{fig1}
\end{figure}

\subsection{The Main Idea}
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 EDMCOP protocol works in rounds fashion as shown in figure~\ref{fig2}.
\begin{figure}[ht!]
\centering
\includegraphics[width=85mm]{FirstModel.eps} % 70mm
\caption{EDMCOP protocol}
\label{fig2}
\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. 
We define two types of packets to be used by our EDMCOP protocol.
\begin{enumerate}[(a)] 
\item INFO packet: sent by each sensor node to all the nodes of it's subregion for information exchange.
\item ActiveSleep packet: sent by the leader to all the nodes in the same of it's subregion to inform them to be Active or Sleep during the sensing phase.
\end{enumerate}

There are four status for each sensor node in the network
\begin{enumerate}[(a)] 
\item LISTENING: Sensor has not yet decided.
\item ACTIVE: Sensor is active.
\item SLEEP: Sensor decides to turn off.
\item COMMUNICATION: Sensor is Transmitting or Receiving packet.
\end{enumerate}

Below, we describe each phase in more details.

\subsubsection{Information Exchange Phase}

Each sensor node $j$ sends its position, remaining energy $RE_j$, and
the number of neighbours  $NBR_j$ 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 :

\subsubsection{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 neighbours,  larger remaining  energy, and  then in  case of
equality, larger index. The pseudo-code for leader election phase is provided in Algorithm~1.
\begin{algorithm}                
  \KwIn{all the parameters related to information exchange}
  \KwOut{$node-id$ (: the id of the winner sensor node, which is the leader of current round)}
  \BlankLine
  \emph{Select the node(s) with higher $NBR_j$ and $ RE_j \geqslant E_{th}$} \; 
  
  \If{ there are more than two nodes with the same maximum $NBR_j$ }{
         \If{ there are more than two nodes with the same maximum $NBR_j$ and the same $RE_j$}{
                \emph{ Select the node with higher id} \;
         }       
         \Else{
             \emph{Select the node with maximum $RE_j$} \;
         }
   }
   \Else{
            \emph{ Select the node with higher $NBR_j$ } \;
    }
  
  \emph{return node-id} \;
\caption{LEADER ELECTION}
\label{alg:LEADER}

\end{algorithm}

Where $E_{th}$ is the minimum energy needed to stay active during the sensing phase. As shown in Algorithm~1, the more priority selection factor is the number of $1-hop$ neighbours, $NBR j$, which can  minimize the energy consumption during the communication Significantly.  


\subsubsection{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.


\subsubsection{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{EDMCOP protocol Algorithm}
we first show the pseudo-code of EDMCOP protocol, which is executed by each sensor in the subregion and then describe it in more detail. 

\begin{algorithm}                
 % \KwIn{all the parameters related to information exchange}
%  \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
  \BlankLine
  \emph{Initialize the sensor node and determine it's position and it's subregion} \; 
  
  \If{ $RE_j \geq E_{th}$ }{
     \emph{ Send and Receive INFO Packet to and from other nodes in the subregion}\;
      \emph{ Collect information and construct the list L for all nodes in the subregion}\;
      \emph{ $s_j.status$ = LISTENING}\;
       \If{ the received INFO Packet = number of nodes in it's subregion -1  }{
         \emph{ LeaderID $\leftarrow$ \bf Algorithm~\ref{alg:LEADER}}\;
         \If{ $ s_j.ID = LeaderID $}{ 
                \emph{Execute Integer Program Algorithm (Gbest) }\;
                \For{$k\leftarrow 1$ \KwTo No. of nodes in subregion}{
                    \If{$ s_j.ID \neq L_k$ }{
                         \If{$ Gbest_k = 1$ }{
                                   \emph{ Send ActiveSleep() Packet with status = ACTIVE  }\;
                          }
                          \Else{\emph{Send ActiveSleep() Packet with status = SLEEP}\;}
                          \Else{
                                        \If{$ Gbest_k = 1$ }{
                                           \emph{ $s_j.status$ = ACTIVE}\; 
                                           \emph{UPDATE Remaining Energy $RE_j $}\;
                                        }
                                        \Else{ 
                                              \emph{ $s_j.status$ = SLEEP}\;
                                              \emph{UPDATE Remaining Energy $RE_j $}\;
                                         }
                          }
                }         
            }     
           
         }
         \Else{ 
                \emph{Wait ActiveSleep() Packet from the Leader}\;
                 \If{received ActiveSleep().status = ACTIVE  }{ 
                  \emph{ $s_j.status$ = ACTIVE}\; 
                  \emph{UPDATE Remaining Energy $RE_j $}\;
                 }
                 \Else{
                  \emph{ $s_j.status$ = SLEEP}\;
                  \emph{UPDATE Remaining Energy $RE_j $}\;
                 }
         }  
       }

   }
   \Else { Exclude me from entering in the current round}
   
 %   \emph{return X} \;
\caption{EDMCOP($s_j$)}
\label{alg:DMRCLOP}

\end{algorithm}

The EDMCOP protocol work in rounds and executed at each sensor node in the network , each sensor node can still sense data while being in
LISTENING mode. Thus, by entering the LISTENING mode at the beginning of each round,
sensor nodes still executing sensing task while participating in the leader election and decision phases. More specifically, The EDMCOP protocol algorithm works as follow: 
Initially, the sensor node check it's remaining energy in order to participate in the current round. Each sensor node determines it's position and it's subregion based Embedded GPS  or Location Discovery Algorithm. After that, All the sensors collect position coordinates, current remaining energy, sensor node id, and the number of its one-hop live neighbors during the information exchange. It stores this information into a list L.
The sensor node enter in listening mode waiting to receive ActiveSleep packet from the leader to take the decision. Each sensor node will execute the Algorithm~1 to know who is the leader. After that, if the sensor node is leader, It will execute the integer program algorithm ( see section~\ref{cp}) to optimize the coverage and the lifetime in it's subregion. After the decision, the optimization approach will select the set of sensor nodes to take the mission of coverage during the sensing phase. The leader will send ActiveSleep packet to each sensor node in the subregion to inform him to it's status during the period of sensing, either Active or sleep until the starting of next round. Based on the decision, the leader as other nodes in subregion, either go to be active or go to be sleep during current sensing phase. the other nodes in the same subregion will stay in listening mode waiting the ActiveSleep packet from the leader. After finishing the time period for sensing, all the sensor nodes in the same subregion will start new round by executing the EDMCOP protocol and the lifetime in the subregion will continue until all the sensor nodes are died or the network becomes disconnected in the subregion.

\section{Coverage problem formulation}
\label{cp}


\indent   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 primary 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.
 

\section{Simulation Results and Analysis}
\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}. The simulation parameters are summarized in
Table~\ref{table3}

\begin{table}[ht]
\caption{Relevant parameters for network initializing.}
% title of Table
\centering
% used for centering table
\begin{tabular}{c|c}
% centered columns (4 columns)
      \hline
%inserts double horizontal lines
Parameter & Value  \\ [0.5ex]
   
%Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
% inserts table
%heading
\hline
% inserts single horizontal line
Sensing  Field  & $(50 \times 25)~m^2 $   \\
% inserting body of the table
%\hline
Nodes Number &  50, 100, 150, 200 and 250~nodes   \\
%\hline
Initial Energy  & 50-75~joules  \\  
%\hline
Sensing Period & 20 Minutes \\
$E_{thr}$ & 12.2472 \\
$R_s$ & 5~m   \\     
%\hline
$w_{\Theta}$ & 1   \\
% [1ex] adds vertical space
%\hline
$w_{U}$ & $|P^2|$
%inserts single line
\end{tabular}
\label{table3}
% is used to refer this table in the text
\end{table}

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 a simple energy model
defined by~\cite{ChinhVu} that based on ~\cite{raghunathan2002energy} with some modification as  energy consumption model for each wireless  sensor node in the network and for all the simulations.

The modification is to add the energy consumption for receiving the packets as well as we ignore the part that related to the sensing range because we used fixed sensing range. The new energy consumption model will take inro account the energy consumption for communication (packet transmission/reception), data sensing and computational energy. 
There are four subsystems in each sensor node that consume energy: the micro-controller
unit (MCU) subsystem which is capable of computation, communication subsystem which is responsible for
transmitting/receiving messages, sensing subsystem that collects data, and the powe suply which supplies power to the complete sensor node ~\cite{raghunathan2002energy}. In our model, we will concentrate on first three main subsystems and each subsystem can be turned on or off depending on the current status of the sensor which is summarized in Table~\ref{table4}.

\begin{table}[ht]
\caption{The Energy Consumption Model}
% title of Table
\centering
% used for centering table
\begin{tabular}{|c|c|c|c|c|}
% centered columns (4 columns)
      \hline
%inserts double horizontal lines
Sensor mode & MCU   & Radio & Sensing & Power (mW) \\ [0.5ex]
\hline
% inserts single horizontal line
Listening & On & On & On & 20.05 \\
% inserting body of the table
\hline
Active & On & Off & On & 9.72 \\
\hline
Sleep & Off & Off & Off & 0.02 \\
\hline
 \multicolumn{4}{|c|}{Energy needed to send/receive a 1-bit} & 0.2575\\
 \hline
\end{tabular}

\label{table4}
% is used to refer this table in the text
\end{table}

For the simplicity, we ignore the energy needed to turn on the
radio, to start up the sensor node, the transition from mode to another, etc. We also do not consider the need of collecting sensing data. Thus, when a sensor becomes active (i.e., it already decides it status), it can turn its radio off to save battery. Since our couverage optimization protocol uses two types of the packets, the size of the INFO-Packet and Status-Packet are 112 bits and 16 bits respectively. The value of energy spent to send a message shown in Table~\ref{table4} is obtained by using the equation in ~\cite{raghunathan2002energy} to calculate the energy cost for transmitting messages and we propose the same value for receiving the packets.


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.
The energy of  each node in a  network is initialized  randomly within the
range 50-75~joules.  Each  sensor  node will  not participate in the next round if its remaining energy is less than $E_{thr}$, the minimum energy needed for the node to stay alive during one round. 

In the simulations, we introduce the following performance metrics to
evaluate the  efficiency of our approach: 

(i) Coverage Ratio (CR): the coverage ratio measures how much the area of a sensor field is  covered. In our case, we treated the sensing fields as a grid, and used each grid point as a sample point
for calculating the coverage. The coverage ratio can be calculated by:
\begin{equation*}
\scriptsize
\mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100.
\end{equation*}
Where: $n$ is the Number of Covered Grid points by the Active Sensors of the all subregions of the network during the current sensing phase and $N$ is total number of grid points in the sensing field of the network.
The accuracy of this method depends on the distance between grids. In our
simulations, the sensing field has been divided into 50 by 25 grid points, which means
there are $51~x~26~ = ~ 1326$ points in total. Therefore, for our simulations, the error in the coverage calculation is less than ~ 1 $\% $.

(ii) Number of Active Sensors Ratio(ASR): 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.The Active Sensors Ratio is defined as follows:
\begin{equation*}
\scriptsize
\mbox{ASR}(\%) = \sum\limits_{r=1}^R \left( \frac{\mbox{$A_r$}}{\mbox{$S$}} \times 100 \right) .
\end{equation*}
Where: $A_r$ is the number of active sensors in the subregion $r$ during the current sensing phase, $S$ is the total number of sensors in the network, and $R$ is the total number of the subregions in the network.

(iii) Energy Saving  Ratio(ESR):is defined by:
\begin{equation*}
\scriptsize
\mbox{ESR}(\%) = \sum\limits_{r=1}^R \left( \frac{\mbox{${ES}_r$}}{\mbox{$S$}} \times 100 \right) .
\end{equation*}
Where: ${ES}_r$ is the number of alive sensors in subregion $r$ during this round. The  longer the ratio  is,  the more  redundant sensor  nodes are switched off, and consequently  the longer the  network may  live.

(iv) Energy Consumption:

 Energy Consumption (EC) can be seen as the total energy consumed by the sensors during the lifetime of the network divided by the total number of rounds. The EC can be computed as follow: \\
 \begin{equation*}
\scriptsize
\mbox{EC} =  \frac{\mbox{$\sum\limits_{d=1}^D \left( E^c_d + E^l_d + E^a_d + E^s_d \right)$ }}{\mbox{$D$}}   .
\end{equation*}
%\begin{equation*}
%\scriptsize
%\mbox{EC} =  \frac{\mbox{$\sum\limits_{d=1}^D E^c_d$}}{\mbox{$D$}} + \frac{\mbox{$\sum\limits_{d=1}^D %E^l_d$}}{\mbox{$D$}} + \frac{\mbox{$\sum\limits_{d=1}^D E^a_d$}}{\mbox{$D$}} + %\frac{\mbox{$\sum\limits_{d=1}^D E^s_d$}}{\mbox{$D$}}.
%\end{equation*}

Where: D is the total number of rounds. 
The total energy consumed by the sensors (EC) comes through taking into consideration four main energy factors, which are $E^c_d$, $E^l_d$, $E^a_d$, and $E^s_d$.
The factor $E^c_d$ represents the  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 round $d$. The $E^l_d$ represents the energy consumed by all the sensors during the listening mode before taking the decision to go Active or Sleep in round $d$. The $E^a_d$ and $E^s_d$  are refered to enegy consumed by the turned on and turned off sensors in the period of sensing during the round $d$. 

(v) Network Lifetime: 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.


(vi) 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.   
  
(vii)The number of stopped simulation runs: we  will study the percentage of simulations, which are stopped due to network disconnections per round.



\subsection{Performance Comparison for differnet subregions}
\label{sub1}
In this subsection, we will study the performance of our approach for a different number of regions (Leaders).
10~simulation  runs  are performed  with different  network  topologies for  each  node  density.  The  results presented hereafter  are the  average of these  10 runs.  
Our approach are called strategy 1 ( With One Leader), strategy 2 ( With Two Leaders), strategy 3 ( With Four Leaders), and strategy 4 ( With Eight Leaders). The strategy 1 ( With One Leader) is a centralized approach on all the area of the interest, while  strategy 2 ( With Two Leaders), strategy 3 ( With Four Leaders), and strategy 4 ( With Eight Leaders) are distributed on two, four, and eight subregions respectively.  
\subsubsection{The impact of the number of rounds on the coverage ratio} 
In this experiment, Figure~\ref{fig3} shows  the impact  of the
number of rounds on the  average coverage ratio for 150 deployed nodes
for the  four strategies.  
\parskip 0pt    
\begin{figure}[h!]
\centering
 \includegraphics[scale=0.43] {R1/CR.eps} 
\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure} 

It can be  seen that the four strategies
give  nearly similar  coverage  ratios  during  the first  three rounds.  
As shown in  the figure ~\ref{fig3}, when we increase the number of sub-regions, It will leads to cover the area of interest for a larger number of rounds. Coverage ratio decreases when the number of rounds increases  due to dead nodes. Although  some nodes are dead,
thanks to  strategy~4,  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~4 is
slightly more efficient than other strategies, because strategy~4 subdivides
the Area of interest into 8~subregions and  if one of the eight subregions becomes
disconnected,  the coverage   may  be  still  ensured   in  the  remaining
subregions.

\subsubsection{The impact of the number of rounds on the active sensors ratio} 
 Figure~\ref{fig4} shows the average active nodes ratio versus the number of rounds
for 150 deployed nodes.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/ASR.eps}  
\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
the proposed  strategy 4, in comparison with the other strategies. The
strategy with less number of leaders uses less active nodes than the other strategies, which uses a more number of leaders until the last rounds, because it uses central control on
the larger area of the sensing field. The  advantage of the strategy~4 approach is
that even if a network is disconnected in one subregion, the other ones
usually  continues  the optimization  process,  and  this extends  the lifetime of the network.

\subsubsection{The impact of the number of rounds on the energy saving ratio} 
In this experiment, we consider a performance metric linked to energy. Figure~\ref{fig5} shows the average Energy Saving Ratio versus number of rounds for all four strategies and for 150 deployed nodes.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/ESR.eps}  
\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 4 is usually slightly better than
the other strategies, because the distributed optimization on larger number of subregions permits to minimize the energy needed for communication and It led to save more energy obviously. Indeed,  when there are more than one subregion more nodes
remain awake near the border shared by them but the energy consumed by these nodes have no effect in comparison with the energy consumed by the communication. Note that again as the number of  rounds increases  the strategy 4 becomes the most
performing one, since it takes longer to have the eight subregion networks
simultaneously disconnected.

\subsubsection{The percentage of stopped simulation runs}
Figure~\ref{fig6} illustrates the percentage of stopped simulation
runs per  round for 150 deployed  nodes. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.43]{R1/SR.eps} 
\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }
\label{fig6}
\end{figure} 
It can be  observed that the
strategy 1 is the approach which  stops first because it apply the
centralized  control on all the area of interest  That is why it is  first  exhibits  network  disconnections.   Thus,  as explained previously, in case  of the strategy 4 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.

\subsubsection{The Energy Consumption}
In this experiment, we study the effect of the energy consumed by the sensors during the communication , listening, active, and sleep modes for different network densities. Figure~\ref{fig7} illustrates the energy consumption for the different
network  sizes and for the four proposed stratgies. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/EC.eps} 
\caption{The Energy Consumption}
\label{fig7}
\end{figure} 

The  results show  that the strategy  with  eight  leaders  is  the  most  competitive  from  the energy
consumption point of view.  The other strategies have a  high energy  consumption  due  to  many
communications as well as the energy consumed during the listening before taking the decision. In fact,  a distributed  method on the subregions greatly reduces the number  of communications and the time of listening so thanks to the  partitioning of the initial
network in several independent subnetworks. 


\subsubsection{The impact of the number of sensors on execution time}
In this experiment, we study the the impact of the size of the network on the excution time of the our distributed optimization approach. Table~\ref{table1} gives the average  execution times in seconds for the decision phase (solving of the optimization problem)
during one  round.  They  are given for  the different  approaches and
various numbers of sensors. We can see from Table~\ref{table1}, that the strategy 4 has very low execution times in comparison with other strategies, because it distributed on larger number of small subregions.  Conversely, the strategy 1 which  requires  to  solve  an optimization  problem considering  all  the  nodes  presents  high  execution  times.
Moreover, increasing the network size by 50~nodes  multiplies the time
by  almost a  factor of  10. The  strategy 4 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|c|}
% centered columns (4 columns)
      \hline
%inserts double horizontal lines
Sensors number & Strategy~1 & Strategy~2  & Strategy~3  & Strategy~4 \\ [0.5ex]
\hline
% inserts single horizontal line
50      & 0.1848396     & 0.04655324 &  0.0117669418    & 0.0033982841  \\
% inserting body of the table
\hline
100     & 1.895704      & 0.2190363     & 0.0445454425  & 0.0111192651 \\
\hline
150     & 12.211936     & 0.6322759     & 0.095189135   & 0.0228475025 \\
\hline
200     & 152.25806     & 2.285319      & 0.184868      &  0.0412940988 \\
\hline
250     & 1542.5396     & 5.656081      & 0.3147647     & 0.0609174088 \\
% [1ex] adds vertical space
\hline
%inserts single line
\end{tabular}
\label{table1}
% is used to refer this table in the text
\end{table}

\subsubsection{The Network Lifetime}
Finally, in  figure~\ref{fig8}, the
network  lifetime for different  network sizes  and for the four strategies is illustrated. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/LT.eps}  
\caption{The Network Lifetime }
\label{fig8}
\end{figure} 
We see that the strategy 1 results in execution times that quickly become unsuitable for a sensor network as well as the energy consumed during the communication seems to be huge because it used a centralised control on the all the area of interest.

As  highlighted by figure~\ref{fig8},  the network  lifetime obviously
increases when  the size  of the network  increases, with  our approach strategy~4
that leads to  the larger lifetime improvement.  By  choosing the  best 
suited nodes, for each round,  to cover the Area of interest and by
letting the other ones sleep in order to be used later in next rounds,
our strategy~4 efficiently prolonges the network lifetime. Comparison shows that
the Strategy~4, which uses eight leaders, is the best
one because it is robust to network disconnection during the network lifetime. 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.


\subsection{Performance Comparison for Differnet Primary Point Models}
\label{sub2}
Based on the results, which are conducted in subsection~\ref{sub1}, we will study the performance of the Strategy~4 approach for a different primary point models.
50~simulation runs are performed  with different  network  topologies for each node density. The  results presented hereafter are the average of these 50 runs.  
In this comparisons, our approaches are called Model~1( With 5 Primary Points), Model~2 ( With 13 Primary Points), Model~3 ( With 17 Primary Points), and Model~4 ( With 21 Primary Points). The simulation will applied with strategy~4 by subdividing the area of interest into eight subregions and distribute our strategy~4 approach on the all subregions. 
\subsubsection{The impact of the number of rounds on the coverage ratio} 
In this experiment, we Figure~\ref{fig33} shows  the impact  of the
number of rounds on the  average coverage ratio for 150 deployed nodes
for the  four strategies.  
\parskip 0pt    
\begin{figure}[h!]
\centering
 \includegraphics[scale=0.5] {R2/CR.eps} 
\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}
\label{fig33}
\end{figure} 
It is shown that Model~4 and Model~3 provides a
better coverage ratio with 99.5\% and 99.4\% against 99.2\% and 98.1\% produced by
model~2 and Model~1 for the first ninth rounds, because they are used a larger number of primary points to represent the sensing range of the sensor during optimization process. Moreover, when the number of rounds increases, coverage
ratio produced by Model~4 and Model~3 decreases in comparison with Model~1 and Model~2 due to the high energy consumption during the listening to take the decision after finishing optimization process for larger number of primary points. As shown in  the figure ~\ref{fig33}, Coverage ratio decreases when the number of rounds increases  due to dead nodes. Although  some nodes are dead,
thanks to  Model~2, which is slightly more efficient than other Models, because Model~2 balances between the number of rounds and the better coverage ratio in comparison with other Models.

\subsubsection{The impact of the number of rounds on the active sensors ratio} 
 Figure~\ref{fig44} shows the average active nodes ratio versus the number of rounds
for 150 deployed nodes.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/ASR.eps}  
\caption{The impact of the number of rounds on the active sensors ratio for 150 deployed nodes }
\label{fig44}
\end{figure} 

The  results presented  in figure~\ref{fig44}  show the  superiority of
the proposed  Model 1, in comparison with the other Models. The
model with less number of primary points uses less active nodes than the other models, which uses a more number of primary points to represent the area of the sensor. According to the results that presented in figure~\ref{fig33}, we observe that although the Model~1 continue to a larger number of rounds, but it has less coverage ratio compared with other models.The advantage of the Model~2 approach is to use less number of active nodes for each round compared with Model~3 and Model~4, and this led to continue for a larger number of rounds with extending the network lifetime. Model~2 has a better coverage ratio compared to Model~1 and acceptable number of rounds.

\subsubsection{The impact of the number of rounds on the energy saving ratio} 
In this experiment, we study the effect of increasing primary points on the energy conservation in the wireless sensor network. Figure~\ref{fig55} shows the average Energy Saving Ratio versus number of rounds for all four Models and for 150 deployed nodes.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/ESR.eps}  
\caption{The impact of the number of rounds on the energy saving ratio for 150 deployed nodes}
\label{fig55}
\end{figure} 
The simulation results show that our Models allow to efficiently
save energy by turning off the redundant  sensors during the sensing phase.
 As expected, the Model 1 is usually slightly better than the other Models, because it turn on a less number of nodes during the sensing phase in comparison with other models and according to the results, which are observed in figure ~\ref{fig33}, and It led to save more energy obviously.
 Indeed,  when there are more primary points to represent the area of the sensor leads to activate more nodes to cover them and in the same time ensuring more coverage ratio. From the previous presented results, we see it is preferable to choose the model that balance between the coverage ratio and the number of rounds. The Model~2 becomes the most performing one, since it could apply this requirement where, It can coverage the area of interest with a good coverage ratio and for a larger number of rounds prolonging the lifetime of the wireless sensor network.

\subsubsection{The percentage of stopped simulation runs}
In this study, we want to show the effect of increasing the primary points on the number of stopped simulation runs for each round. Figure~\ref{fig66} illustrates the percentage of stopped simulation
runs per round for 150 deployed nodes. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/SR.eps} 
\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }
\label{fig66}
\end{figure} 
As shown in Figure~\ref{fig66}, when the number of primary points increase leads to increase the percentage of the stopped simulation runs per rounds and starting from round 11 until the the network is died. The reason behind the increase is the increase in the sensors dead When the primary points increases. We can observe that the Model~1 is a better than other models because it conserve more energy by turn on less number of sensors during the sensing phase, but in the same time it preserve the coverage with a less coverage ratio in comparison with other models. Model~2 seems to be more suitable to be used in sensor networks.


\subsubsection{The Energy Consumption}
In this experiment, we study the effect of increasing the primary points to represent the area of the sensor on the the energy consumed by the wireless sensor network for different network densities. Figure~\ref{fig77} illustrates the energy consumption for the different
network sizes and for the four proposed Models. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/EC.eps} 
\caption{The Energy Consumption}
\label{fig77}
\end{figure} 

We see from the results presented in Figure~\ref{fig77}, The energy consumed by the network for each round increases when the primary points increases, because the decision for optimization process will takes more time leads to consume more energy during the listening mode. The  results show  that the Model~1 is  the  most  competitive  from  the energy
consumption point of view but the worst one from coverage ratio point of view.  The other Models  have a  high energy  consumption  due  to  the increase in the primary points, which are led to increase the energy consumption during the listening mode before taking the optimization decision. In fact, we see that the Model~2 is a good candidate to be used the wireless sensor network because I have a good coverage ratio and a suitable energy consumption in comparison with other models. 


\subsubsection{The impact of the number of sensors on execution time}
In this experiment, we study the the impact of the increase in primary points on the excution time of the our distributed optimization approach. Table~\ref{table11} gives the average  execution times in seconds for the decision phase (solving of the optimization problem) during one round.  
\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|c|}
% centered columns (4 columns)
      \hline
%inserts double horizontal lines
Sensors number & Model~1 & Model~2  & Model~3  & Model~4 \\ [0.5ex]
\hline
% inserts single horizontal line
50      & 0.0024106288  & 0.0034540527  & 0.0045273282  & 0.0064475788  \\
% inserting body of the table
\hline
100     & 0.0045475502  & 0.0104693717  & 0.0144968192  & 0.0198072788 \\
\hline
150     & 0.0090504642  & 0.0224529648  & 0.031136629   & 0.0442327285 \\
\hline
200     & 0.0156788154  & 0.0412170018  & 0.0566654188  & 0.0785000165 \\
\hline
250     & 0.023145026   & 0.0618075108  & 0.0853965538  & 0.1195402003 \\
% [1ex] adds vertical space
\hline
%inserts single line
\end{tabular}
\label{table11}
% is used to refer this table in the text
\end{table}

They  are given for  the different primary point models and
various numbers of sensors. We can see from Table~\ref{table11}, that the Model~1 has  lower execution time in comparison with other Models, because it used smaller number of primary points to represent the area of the sensor.  Conversely, the other primary point models  presents  higher  execution  times.
Moreover, the Model~2 has more suitable times, coverage ratio, and saving energy ratio    leads to continue for a larger number of rounds extending the network lifetime. We  think that a good primary point model, this one that balances between the coverage ratio and the number of rounds during the lifetime of the network.

\subsubsection{The Network Lifetime}
Finally, we will study the effect of increasing the primary points on the lifetime of the network. In figure~\ref{fig88}, the network  lifetime for different network sizes and for the four proposed models is illustrated. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/LT.eps}  
\caption{The Network Lifetime }
\label{fig88}
\end{figure} 
As  highlighted by figure~\ref{fig88},  the network  lifetime obviously
increases when  the size  of the network  increases, with  our approach Model~1
that leads to  the larger lifetime improvement.
Comparison shows that the Model~1, which uses less number of primary points , is the best one because it is less energy consumption during the network lifetime. It is also  the worst one from the point of view of coverage ratio. Our proposed Model~2 efficiently prolongs the network lifetime with a good coverage ratio in comparison with other models.
 
\subsection{Performance Comparison for differnet Approaches}
Based on the results, which are conducted from previous two subsections, ~\ref{sub1} and \ref{sub2}, we found that the Strategy~4 with Model~2 is the best candidate to be compared with other two approches. The first approach, called DESK that proposed by ~\cite{ChinhVu}, which is a full distributed coverage algorithm. The second 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. In this subsection, 50 simulation runs are
performed with different network topologies. The results
presented hereafter are the average of these 50 runs.

\subsubsection{The impact of the number of rounds on the coverage ratio} 
In this experiment, Figure~\ref{fig333} shows  the impact  of the
number of rounds on the  average coverage ratio for 150 deployed nodes
for the  three approaches.  
\parskip 0pt    
\begin{figure}[h!]
\centering
 \includegraphics[scale=0.45] {R3/CR.eps} 
\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}
\label{fig333}
\end{figure} 

It is shown that DESK provides a
better coverage ratio with 99.9 \% against 99.2 \% produced by
our approach Strategy~4 for the lowest number of rounds. This is due to
the fact that our protocol Strategy~4 put in sleep mode
redundant sensors using optimization (which lightly decreases the coverage
ratio) while there are more nodes are active in the case of DESK.
Moreover, when the number of rounds increases, coverage
ratio produced by DESK protocol decreases. This is due to dead nodes. However, Our protocol Strategy~4 maintains almost full
coverage. This is because it optimize the coverage and the lifetime in wireless sensor network by selecting the best representative sensor nodes to take the reponsibilty of coverage during the sensing phase and this will leads to continue for a larger number of rounds and prolonging the network lifetime; although some nodes are dead, sensor activity scheduling of our protocol chooses other nodes to ensure the coverage of the area of interest. It can be seen that the  Simple Heuristic approach gives similar coverage ratios  99.8 \% during the first four rounds. From the 
5th~round the coverage ratio decreases continuously and quickly with the simple heuristic until the 8th~round, the network is died.

\subsubsection{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. Figure~\ref{fig444} shows the average active nodes ratio versus the number of rounds for 150 deployed nodes. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R3/ASR.eps}  
\caption{The impact of the number of rounds on the active sensors ratio for 150 deployed nodes }
\label{fig444}
\end{figure} 

The  results presented  in figure~\ref{fig444}  show the  superiority of
the proposed protocol with Strategy~4, in comparison with the other approaches.  We can observe that DESK has 37.5 \%  active nodes and our protocol with Strategy~4 
competes perfectly with only 22.6 \%  active nodes for the first four rounds. Then as the number of rounds increases our protocol with Strategy~4  generates less active nodes than DESK until the $10^{th}$ round, we see that the DESK has less number of active nodes because there are many nodes are died due to the high energy consumption by the redundant nodes during the sensing phase. The Simple Heuristic turn on 33.3 \% for the first four rounds after that the number of active nodes decreased in the next rounds due to the died nodes until the died of the network in the $8^{th}$ round.  

\subsubsection{The impact of the number of rounds on the energy saving ratio} 
In this experiment, we will perform a comparison study for the performance of our protocol with Strategy~4 with two other approaches from the point of view of energy conservation. Figure~\ref{fig555} shows the average Energy Saving Ratio versus number of rounds for all three approaches and for 150 deployed nodes.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R3/ESR.eps}  
\caption{The impact of the number of rounds on the energy saving ratio for 150 deployed nodes}
\label{fig555}
\end{figure} 
The simulation results show that our protocol with Strategy~4 allow to efficiently
save energy by turning off the redundant sensors during the sensing phase. As
expected, our protocol with Strategy~4 is usually slightly better than
the other approaches, because the distributed optimization on the subregions permits to minimize the energy needed for communication as well as turn off all the redundant sensor nodes, which are led to save more energy obviously. Note that again as the number of rounds increases, our protocol with Strategy~4 becomes the most performing one, since it is distributed the optimization process on the eight subregion networks simultaneously so as to optimize the coverage and the lifetime in the network.

\subsubsection{The percentage of stopped simulation runs}
The results presented in this experiment, is to show the comparison of our protocol with Strategy~4 with other two approaches from the point of view the stopped simulation runs per round.
Figure~\ref{fig666} illustrates the percentage of stopped simulation
runs per round for 150 deployed nodes. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.4]{R3/SR.eps} 
\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }
\label{fig666}
\end{figure} 
It can be  observed that the Simple Heuristic is the approach, which stops first because it consumes more energy for communication as well as it turn on a large number of redundant nodes during the sensing phase. Our protocol with Strategy~4 has less stopped simulation runs in comparison with DESK because it distributed the optimization on several subregions in order to optimize the coverage and the lifetime of the network by activating a less number of nodes during the sensing phase leading to extend the network lifetime and coverage preservation.The optimization effectively continues as long as a network in a subregion is still connected.

\subsubsection{The Energy Consumption}
In this experiment, we study the effect of the energy consumed by the wireless sensor network during the communication , listening, active, and sleep modes for different network densities and compare it with other approaches. Figure~\ref{fig777} illustrates the energy consumption for the different
network sizes and for the three approaches. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R3/EC.eps} 
\caption{The Energy Consumption}
\label{fig777}
\end{figure} 
The  results show  that our protocol with Strategy~4  is  the  most  competitive  from  the energy consumption point of view.  The other approaches have a  high energy  consumption  due  to  activating a larger number of redundant nodes as well as the energy consumed for communication, active and listening modes. In fact,  a distributed  method on the subregions greatly reduces the number  of communications and the time of listening so thanks to the  partitioning of the initial network into several independent subnetworks. 
\subsubsection{The Network Lifetime}
Finally, In this experiment, we will show the superiority of our protocol with Strategy~4 against other two approaches in prolonging the network lifetime. In Figure~\ref{fig888}, the
network  lifetime for different  network sizes  and for the four strategies is illustrated. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R3/LT.eps}  
\caption{The Network Lifetime }
\label{fig888}
\end{figure} 

As highlighted by figure~\ref{fig888}, the network lifetime obviously
increases when the size of the network increases, with our protocol with Strategy~4
that leads to maximize the lifetime of the network compared with other approaches. 
By choosing the best suited nodes, for each round, by optimizing the coverage and lifetime of the network to cover the area of interest and by
letting the other ones sleep in order to be used later in next rounds, our protocol with Strategy~4 efficiently prolonges the network lifetime. 
Comparison shows that our protocol with Strategy~4, which uses distributed optimization on the subregions, is the best
one because it is robust to network disconnection during the network lifetime as well as it consume less energy in comparison with other approaches. 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  one time,  using
optimization  methods  such  as  swarms optimization  or  evolutionary
algorithms.  The round  will still  consist 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 time is far more
difficult, but will reduce the communication overhead.
% use section* for acknowledgement
%\section*{Acknowledgment}




\bibliographystyle{IEEEtran}
\bibliography{biblio}


%\appendices
%\section{Proof of the First Zonklar Equation}
%Appendix one text goes here.

% you can choose not to have a title for an appendix
% if you want by leaving the argument blank
%\section{}
%Appendix two text goes here.


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

We would like to thank , and also thank anonymous reviewers for their constructive comments which helped us to
improve the quality of this paper.



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\ifCLASSOPTIONcaptionsoff
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\end{document}