reading until the end of IV

[LiCO.git] / LiCO_Journal.tex

diff --git a/LiCO_Journal.tex b/LiCO_Journal.tex
index 8f1949b593bd4286643ec246cb8a1c267fcdb806..7e788d4f6524d030e7276e745ed2a82c4ba02b61 100644 (file)
--- a/LiCO_Journal.tex
+++ b/LiCO_Journal.tex
@@ -36,14 +36,16 @@
 \DeclareGraphicsRule{.ps}{pdf}{.pdf}{`ps2pdf -dEPSCrop -dNOSAFER #1 \noexpand\OutputFile}
 \begin{document}
 
 \DeclareGraphicsRule{.ps}{pdf}{.pdf}{`ps2pdf -dEPSCrop -dNOSAFER #1 \noexpand\OutputFile}
 \begin{document}
 

-\title{Lifetime Coverage Optimization Protocol \\
-  in Wireless Sensor Networks}  %LiCO Protocol 
+%\title{Lifetime Coverage Optimization Protocol \\
+%  in Wireless Sensor Networks}
+\title{Perimeter-based Coverage Optimization to Improve \\
+  Lifetime in Wireless Sensor Networks}

 
 \author{Ali Kadhum Idrees,~\IEEEmembership{}
         Karine Deschinkel,~\IEEEmembership{}
         Michel Salomon,~\IEEEmembership{}
         and~Rapha\"el Couturier ~\IEEEmembership{} 
 
 \author{Ali Kadhum Idrees,~\IEEEmembership{}
         Karine Deschinkel,~\IEEEmembership{}
         Michel Salomon,~\IEEEmembership{}
         and~Rapha\"el Couturier ~\IEEEmembership{} 

-        \thanks{The authors are with FEMTO-ST Institute, UMR 6174 CNRS, University of Franche-Comt\'e,
+        \thanks{The authors are with 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}}
 
           Belfort, France. Email: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel,
           michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr}}
 


@@ -53,24 +55,25 @@
 \maketitle
 
 \begin{abstract}
 \maketitle
 
 \begin{abstract}

-The most important problem in Wireless Sensor Networks (WSNs) is to optimize the
+The most important problem in a Wireless Sensor Network (WSN) is to optimize the

 use of its limited energy provision, so  that it can fulfill its monitoring task
 as  long as  possible. Among  known  available approaches  that can  be used  to
 improve  power  management,  lifetime coverage  optimization  provides  activity
 scheduling which ensures  sensing coverage while minimizing the  energy cost. In
 use of its limited energy provision, so  that it can fulfill its monitoring task
 as  long as  possible. Among  known  available approaches  that can  be used  to
 improve  power  management,  lifetime coverage  optimization  provides  activity
 scheduling which ensures  sensing coverage while minimizing the  energy cost. In

-this paper,  we propose  a such approach  called Lifetime  Coverage Optimization
+this paper,  we propose such an approach  called Lifetime  Coverage Optimization

 protocol (LiCO).   It is a  hybrid of  centralized and distributed  methods: the
 region of interest is first subdivided  into subregions and our protocol is then
 protocol (LiCO).   It is a  hybrid of  centralized and distributed  methods: the
 region of interest is first subdivided  into subregions and our protocol is then

-distributed among sensor nodes in each  subregion. A sensor node which runs LiCO
-protocol  repeats   periodically  four  stages:  information   exchange,  leader
-election, optimization decision, and sensing.  More precisely, the scheduling of
-nodes' activities (sleep/wake up duty cycles) is achieved in each subregion by a
-leader selected after  cooperation between nodes within the  same subregion. The
-novelty of  approach lies essentially in  the formulation of a  new mathematical
-optimization  model  based on  perimeter  coverage  level to  schedule  sensors'
-activities.  Extensive simulation experiments have been performed using OMNeT++,
-the  discrete event  simulator, to  demonstrate that  LiCO is  capable to  offer
-longer lifetime coverage for WSNs in comparison with some other protocols.
+distributed among sensor nodes in each  subregion.
+% A  sensor node  which runs  LiCO  protocol repeats  periodically four  stages:
+%information  exchange,  leader election,  optimization  decision, and  sensing.
+%More precisely, the scheduling of nodes' activities (sleep/wake up duty cycles)
+%is achieved  in each subregion by  a leader selected  after cooperation between
+%nodes within the same subregion.
+The  novelty of  our  approach lies  essentially  in the  formulation  of a  new
+mathematical optimization  model based on  perimeter coverage level  to schedule
+sensors' activities.  Extensive simulation experiments have been performed using
+OMNeT++, the  discrete event simulator, to  demonstrate that LiCO  is capable to
+offer longer lifetime coverage for WSNs in comparison with some other protocols.

 \end{abstract} 
 
 % Note that keywords are not normally used for peerreview papers.
 \end{abstract} 
 
 % Note that keywords are not normally used for peerreview papers.


@@ -92,7 +95,7 @@ communicating with one another through multi-hop radio communications. Each node
 can send the data  it collects in its environment, thanks to  its sensor, to the
 user by means of  sink nodes. The features of a WSN made  it suitable for a wide
 range of application  in areas such as business,  environment, health, industry,
 can send the data  it collects in its environment, thanks to  its sensor, to the
 user by means of  sink nodes. The features of a WSN made  it suitable for a wide
 range of application  in areas such as business,  environment, health, industry,

-military, and  son~\cite{yick2008wireless}.  Typically,  a sensor  node contains
+military, and so on~\cite{yick2008wireless}.   Typically, a sensor node contains

 three main components~\cite{anastasi2009energy}: a  sensing unit able to measure
 physical,  chemical, or  biological  phenomena observed  in  the environment;  a
 processing unit which will process and store the collected measurements; a radio
 three main components~\cite{anastasi2009energy}: a  sensing unit able to measure
 physical,  chemical, or  biological  phenomena observed  in  the environment;  a
 processing unit which will process and store the collected measurements; a radio


@@ -119,23 +122,23 @@ lifetime of the WSNs~\cite{rault2014energy}.
 
 This paper makes the following contributions.
 \begin{enumerate}
 
 This paper makes the following contributions.
 \begin{enumerate}

-\item We devise a framework to schedule nodes to be activated alternatively such
+\item We have devised a framework to schedule nodes to be activated alternatively such

   that the network lifetime is prolonged  while ensuring that a certain level of
   that the network lifetime is prolonged  while ensuring that a certain level of

-  coverage is preserved.  A  key idea in our framework is  to exploit spatial an
-  temporal subdivision.   On the one hand  the area of interest  if divided into
-  several smaller subregions and on the other hand the time line is divided into
+  coverage is preserved.  A key idea in  our framework is to exploit spatial and
+  temporal subdivision.   On the one hand,  the area of interest  if divided into
+  several smaller subregions and, on the other hand, the time line is divided into

   periods of equal length. In each subregion the sensor nodes will cooperatively
   choose a  leader which will schedule  nodes' activities, and this  grouping of
   sensors is similar to typical cluster architecture.
   periods of equal length. In each subregion the sensor nodes will cooperatively
   choose a  leader which will schedule  nodes' activities, and this  grouping of
   sensors is similar to typical cluster architecture.

-\item We  propose a new mathematical  optimization model.  Instead of  trying to
+\item We have proposed a new mathematical  optimization model.  Instead of  trying to

   cover a set of specified points/targets as  in most of the methods proposed in
   the literature, we formulate an integer program based on perimeter coverage of
   each sensor.  The  model involves integer variables to  capture the deviations
   cover a set of specified points/targets as  in most of the methods proposed in
   the literature, we formulate an integer program based on perimeter coverage of
   each sensor.  The  model involves integer variables to  capture the deviations

-  between  the actual  level of  coverage and  the required  level.  So  that an
+  between  the actual  level of  coverage and  the required  level.  Hence, an

   optimal scheduling  will be  obtained by  minimizing a  weighted sum  of these
   deviations.
   optimal scheduling  will be  obtained by  minimizing a  weighted sum  of these
   deviations.

-\item We  conducted extensive simulation  experiments, using the  discrete event
-  simulator OMNeT++, to demonstrate the  efficiency of our protocol. We compared
+\item We have conducted extensive simulation  experiments, using the  discrete event
+  simulator OMNeT++, to demonstrate the  efficiency of our protocol. We have compared

   our   LiCO   protocol   to   two   approaches   found   in   the   literature:
   DESK~\cite{ChinhVu} and  GAF~\cite{xu2001geography}, and also to  our previous
   work published in~\cite{Idrees2} which is  based on another optimization model
   our   LiCO   protocol   to   two   approaches   found   in   the   literature:
   DESK~\cite{ChinhVu} and  GAF~\cite{xu2001geography}, and also to  our previous
   work published in~\cite{Idrees2} which is  based on another optimization model


@@ -151,7 +154,7 @@ is devoted to the LiCO protocol  description and Section~\ref{cp} focuses on the
 coverage model  formulation which is used  to schedule the activation  of sensor
 nodes.  Section~\ref{sec:Simulation  Results and Analysis}  presents simulations
 results and discusses the comparison  with other approaches. Finally, concluding
 coverage model  formulation which is used  to schedule the activation  of sensor
 nodes.  Section~\ref{sec:Simulation  Results and Analysis}  presents simulations
 results and discusses the comparison  with other approaches. Finally, concluding

-remarks   are  drawn   and  some   suggestions   given  for   future  works   in
+remarks   are  drawn   and  some   suggestions are  given  for   future  works   in

 Section~\ref{sec:Conclusion and Future Works}.
 
 % that show that our protocol outperforms others protocols.
 Section~\ref{sec:Conclusion and Future Works}.
 
 % that show that our protocol outperforms others protocols.


@@ -165,7 +168,7 @@ the literature.
 The most  discussed coverage problems in  literature can be classified  in three
 categories~\cite{li2013survey}   according   to  their   respective   monitoring
 objective.  Hence,  area coverage \cite{Misra}  means that every point  inside a
 The most  discussed coverage problems in  literature can be classified  in three
 categories~\cite{li2013survey}   according   to  their   respective   monitoring
 objective.  Hence,  area coverage \cite{Misra}  means that every point  inside a

-fixed area  must be monitored, while  target coverage~\cite{yang2014novel} refer
+fixed area  must be monitored, while  target coverage~\cite{yang2014novel} refers

 to  the objective  of coverage  for a  finite number  of discrete  points called
 targets,  and  barrier coverage~\cite{HeShibo}\cite{kim2013maximum}  focuses  on
 preventing  intruders   from  entering   into  the   region  of   interest.   In
 to  the objective  of coverage  for a  finite number  of discrete  points called
 targets,  and  barrier coverage~\cite{HeShibo}\cite{kim2013maximum}  focuses  on
 preventing  intruders   from  entering   into  the   region  of   interest.   In


@@ -176,7 +179,7 @@ nodes    or   between    disk   of    sensor   nodes    and   boundaries.     In
 sensors are sufficiently  covered it will be  the case for the  whole area. They
 provide an algorithm in $O(nd~log~d)$  time to compute the perimeter-coverage of
 each  sensor,  where  $d$  denotes  the  maximum  number  of  sensors  that  are
 sensors are sufficiently  covered it will be  the case for the  whole area. They
 provide an algorithm in $O(nd~log~d)$  time to compute the perimeter-coverage of
 each  sensor,  where  $d$  denotes  the  maximum  number  of  sensors  that  are

-neighboring  to  a  sensor and  $n$  is  the  total  number of  sensors  in  the
+neighbors  to  a  sensor and  $n$  is  the  total  number of  sensors  in  the

 network. {\it In LiCO protocol, instead  of determining the level of coverage of
   a set  of discrete  points, our  optimization model is  based on  checking the
   perimeter-coverage of each sensor to activate a minimal number of sensors.}
 network. {\it In LiCO protocol, instead  of determining the level of coverage of
   a set  of discrete  points, our  optimization model is  based on  checking the
   perimeter-coverage of each sensor to activate a minimal number of sensors.}


@@ -201,28 +204,28 @@ each period.   {\it Motivated by  these works,  LiCO protocol works  in periods,
   decisions, followed by a sensing phase where one cover set is in charge of the
   sensing task.}
 
   decisions, followed by a sensing phase where one cover set is in charge of the
   sensing task.}
 

-Various centralized  and distributed approaches, or  even a mixing of  these two
+Various centralized  and distributed approaches, or  even a mixing  of these two

 concepts, have  been proposed  to extend the  network lifetime.   In distributed
 concepts, have  been proposed  to extend the  network lifetime.   In distributed

-algorithms~\cite{yangnovel,ChinhVu,qu2013distributed}  each  sensors decides  of
-its own  activity scheduling after  an information exchange with  its neighbors.
-The main interest of  a such approach is to avoid  long range communications and
-thus to reduce the energy dedicated to the communications.  Unfortunately, since
-each node has  only information on its immediate neighbors  (usually the one-hop
-ones)  it may  take a  bad  decision leading  to a  global suboptimal  solution.
-Conversely,                                                          centralized
-algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high}    always
+algorithms~\cite{yangnovel,ChinhVu,qu2013distributed} each sensor decides of its
+own activity scheduling  after an information exchange with  its neighbors.  The
+main interest of such an approach is to avoid long range communications and thus
+to reduce the energy dedicated to the communications.  Unfortunately, since each
+node has only information on  its immediate neighbors (usually the one-hop ones)
+it may take a bad decision leading to a global suboptimal solution.  Conversely,
+centralized
+algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high}     always

 provide nearly  or close to  optimal solution since  the algorithm has  a global
 view of the whole network. The disadvantage of a centralized method is obviously
 provide nearly  or close to  optimal solution since  the algorithm has  a global
 view of the whole network. The disadvantage of a centralized method is obviously

-its high cost  in communications needed to  transmit to a single  node, the base
-station which will globally schedule nodes'  activities, data from all the other
-sensor nodes in  the area.  The price  in communications can be  very huge since
-long range communications will be needed. In fact the larger the WNS, the higher
-the  communication and  thus energy  cost.   {\it In  order to  be suitable  for
-  large-scale networks,  in the LiCO  protocol the  area of interest  is divided
-  into several smaller subregions, and in each  one, a node called the leader is
-  in charge  for selecting the active  sensors for the current  period. Thus our
-  protocol  is  scalable  and  a  globally distributed  method,  whereas  it  is
-  centralized in each subregion.}
+its high  cost in communications needed to  transmit to a single  node, the base
+station which will globally schedule  nodes' activities, data from all the other
+sensor nodes  in the area.  The price  in communications can be  very huge since
+long range  communications will be  needed. In fact  the larger the WNS  is, the
+higher the  communication and  thus the energy  cost are.   {\it In order  to be
+  suitable for large-scale  networks, in the LiCO protocol,  the area of interest
+  is divided into several smaller subregions, and in each one, a node called the
+  leader  is  in  charge  of  selecting  the active  sensors  for  the  current
+  period.  Thus our  protocol is  scalable  and is a  globally distributed  method,
+  whereas it is centralized in each subregion.}

 
 Various  coverage scheduling  algorithms have  been developed  this last  years.
 Many of  them, dealing with  the maximization of the  number of cover  sets, are
 
 Various  coverage scheduling  algorithms have  been developed  this last  years.
 Many of  them, dealing with  the maximization of the  number of cover  sets, are


@@ -237,7 +240,7 @@ optimization  solver).  The  problem is  formulated as  an optimization  problem
 energy  constraints.   Column  generation   techniques,  well-known  and  widely
 practiced techniques for  solving linear programs with too  many variables, have
 also                                                                        been
 energy  constraints.   Column  generation   techniques,  well-known  and  widely
 practiced techniques for  solving linear programs with too  many variables, have
 also                                                                        been

-used~\cite{castano2013column,rossi2012exact,deschinkel2012column}. {\it  In LiCO
+used~\cite{castano2013column,rossi2012exact,deschinkel2012column}. {\it  In the LiCO

   protocol, each  leader, in charge  of a  subregion, solves an  integer program
   which has a twofold objective: minimize the overcoverage and the undercoverage
   of the perimeter of each sensor.}
   protocol, each  leader, in charge  of a  subregion, solves an  integer program
   which has a twofold objective: minimize the overcoverage and the undercoverage
   of the perimeter of each sensor.}


@@ -315,11 +318,11 @@ $R_c$ satisfies $R_c  \geq 2 \cdot R_s$. 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 active nodes.
 
 proved  that if  the  transmission  range fulfills  the  previous hypothesis,  a
 complete coverage of a convex area implies connectivity among active nodes.
 

-\indent  LiCO protocol  uses the  same perimeter-coverage  model than  Huang and
+The LiCO protocol  uses the  same perimeter-coverage  model as  Huang and

 Tseng in~\cite{huang2005coverage}. It  can be expressed as follows:  a sensor is
 said to be perimeter  covered if all the points on its  perimeter are covered by
 at least  one sensor  other than  itself.  They  proved that  a network  area is
 Tseng in~\cite{huang2005coverage}. It  can be expressed as follows:  a sensor is
 said to be perimeter  covered if all the points on its  perimeter are covered by
 at least  one sensor  other than  itself.  They  proved that  a network  area is

-$k$-covered if and only if each sensor in the network is $k$-perimeter-covered.
+$k$-covered if and only if each sensor in the network is $k$-perimeter-covered (perimeter covered by at least $k$ sensors).

 %According to this model, we named the intersections among the sensor nodes in the sensing field as intersection points. Instead of working with the coverage area, we consider for each sensor a set of intersection points which are determined by using perimeter-coverage model. 
 Figure~\ref{pcm2sensors}(a)  shows  the coverage  of  sensor  node~$0$. On  this
 figure, we can  see that sensor~$0$ has  nine neighbors and we  have reported on
 %According to this model, we named the intersections among the sensor nodes in the sensing field as intersection points. Instead of working with the coverage area, we consider for each sensor a set of intersection points which are determined by using perimeter-coverage model. 
 Figure~\ref{pcm2sensors}(a)  shows  the coverage  of  sensor  node~$0$. On  this
 figure, we can  see that sensor~$0$ has  nine neighbors and we  have reported on


@@ -333,8 +336,8 @@ arcs.
 \begin{figure}[ht!]
   \centering
   \begin{tabular}{@{}cr@{}}
 \begin{figure}[ht!]
   \centering
   \begin{tabular}{@{}cr@{}}

-    \includegraphics[width=75mm]{pcm.jpg} & \raisebox{3.25cm}{(a)}
-    \\ \includegraphics[width=75mm]{twosensors.jpg} & \raisebox{2.75cm}{(b)}
+    \includegraphics[width=75mm]{pcm.jpg} & \raisebox{3.25cm}{(a)} \\
+    \includegraphics[width=75mm]{twosensors.jpg} & \raisebox{2.75cm}{(b)}

   \end{tabular}
   \caption{(a) Perimeter  coverage of sensor node  0 and (b) finding  the arc of
     $u$'s perimeter covered by $v$.}
   \end{tabular}
   \caption{(a) Perimeter  coverage of sensor node  0 and (b) finding  the arc of
     $u$'s perimeter covered by $v$.}


@@ -343,29 +346,29 @@ arcs.
 
 Figure~\ref{pcm2sensors}(b) describes the geometric information used to find the
 locations of the  left and right points of  an arc on the perimeter  of a sensor
 
 Figure~\ref{pcm2sensors}(b) describes the geometric information used to find the
 locations of the  left and right points of  an arc on the perimeter  of a sensor

-node~$u$ covered by a sensor node~$v$. Node~$s$ is supposed to be located on the
+node~$u$ covered by a sensor node~$v$. Node~$v$ is supposed to be located on the

 west  side of  sensor~$u$,  with  the following  respective  coordinates in  the
 sensing area~: $(v_x,v_y)$ and $(u_x,u_y)$. From the previous coordinates we can
 compute the euclidean distance between nodes~$u$ and $v$: $Dist(u,v)=\sqrt{\vert
   u_x  - v_x  \vert^2 +  \vert u_y-v_y  \vert^2}$, while  the angle~$\alpha$  is
 west  side of  sensor~$u$,  with  the following  respective  coordinates in  the
 sensing area~: $(v_x,v_y)$ and $(u_x,u_y)$. From the previous coordinates we can
 compute the euclidean distance between nodes~$u$ and $v$: $Dist(u,v)=\sqrt{\vert
   u_x  - v_x  \vert^2 +  \vert u_y-v_y  \vert^2}$, while  the angle~$\alpha$  is

-obtained  through the  formula  $\alpha  = arccos  \left(\dfrac{Dist(u,v)}{2R_s}
-\right)$.   So, the  arc on  the perimeter  of node~$u$  defined by  the angular
-interval $[\pi - \alpha,\pi + \alpha]$ is said to be perimeter-covered by sensor
-node $v$.
+obtained through  the formula: $$\alpha =  \arccos \left(\dfrac{Dist(u,v)}{2R_s}
+\right).$$ The arc on the perimeter of~$u$ defined by the angular interval $[\pi
+  - \alpha,\pi + \alpha]$ is said to be perimeter-covered by sensor~$v$.

 
 Every couple of intersection points is placed on the angular interval $[0,2\pi]$
 in  a  counterclockwise manner,  leading  to  a  partitioning of  the  interval.
 Figure~\ref{pcm2sensors}(a)  illustrates  the arcs  for  the  nine neighbors  of
 
 Every couple of intersection points is placed on the angular interval $[0,2\pi]$
 in  a  counterclockwise manner,  leading  to  a  partitioning of  the  interval.
 Figure~\ref{pcm2sensors}(a)  illustrates  the arcs  for  the  nine neighbors  of

-sensor $0$ and  figure~\ref{expcm} gives the position of  the corresponding arcs
+sensor $0$ and  Figure~\ref{expcm} gives the position of  the corresponding arcs

 in  the interval  $[0,2\pi]$. More  precisely, we  can see  that the  points are
 ordered according  to the  measures of  the angles  defined by  their respective
 positions. The intersection points are  then visited one after another, starting
 in  the interval  $[0,2\pi]$. More  precisely, we  can see  that the  points are
 ordered according  to the  measures of  the angles  defined by  their respective
 positions. The intersection points are  then visited one after another, starting

-from  first  intersection point  after  point~zero,  and  the maximum  level  of
+from the first  intersection point  after  point~zero,  and  the maximum  level  of

 coverage is determined  for each interval defined by two  successive points. The
 maximum  level of  coverage is  equal to  the number  of overlapping  arcs.  For
 coverage is determined  for each interval defined by two  successive points. The
 maximum  level of  coverage is  equal to  the number  of overlapping  arcs.  For

-example, between~$5L$  and~$6L$ the maximum  level of  coverage is equal  to $3$
-(the value is highlighted in yellow  at the bottom of figure~\ref{expcm}), which
-means that at most 2~neighbors can cover  the perimeter in addition to node $0$.
+example, 
+between~$5L$  and~$6L$ the maximum  level of  coverage is equal  to $3$
+(the value is highlighted in yellow  at the bottom of Figure~\ref{expcm}), which
+means that at most 2~neighbors can cover  the perimeter in addition to node $0$. 

 Table~\ref{my-label} summarizes for each coverage  interval the maximum level of
 coverage and  the sensor  nodes covering the  perimeter.  The  example discussed
 above is thus given by the sixth line of the table.
 Table~\ref{my-label} summarizes for each coverage  interval the maximum level of
 coverage and  the sensor  nodes covering the  perimeter.  The  example discussed
 above is thus given by the sixth line of the table.


@@ -373,9 +376,9 @@ above is thus given by the sixth line of the table.
 %The points reported on the line segment $[0,2\pi]$ separates it in intervals as shown in figure~\ref{expcm}. For example, for each neighboring sensor of sensor 0, place the points  $\alpha^ 1_L$, $\alpha^ 1_R$, $\alpha^ 2_L$, $\alpha^ 2_R$, $\alpha^ 3_L$, $\alpha^ 3_R$, $\alpha^ 4_L$, $\alpha^ 4_R$, $\alpha^ 5_L$, $\alpha^ 5_R$, $\alpha^ 6_L$, $\alpha^ 6_R$, $\alpha^ 7_L$, $\alpha^ 7_R$, $\alpha^ 8_L$, $\alpha^ 8_R$, $\alpha^ 9_L$, and $\alpha^ 9_R$; on the line segment $[0,2\pi]$, and then sort all these points in an ascending order into a list.  Traverse the line segment $[0,2\pi]$ by visiting each point in the sorted list from left to right and determine the coverage level of each interval of the sensor 0 (see figure \ref{expcm}). For each interval, we sum up the number of parts of segments, and we deduce a level of coverage for each interval. For instance, the interval delimited by the points $5L$ and $6L$ contains three parts of segments. That means that this part of the perimeter of the sensor $0$ may be covered by three sensors, sensor $0$ itself and sensors $2$ and $5$. The level of coverage of this interval may reach $3$ if all previously mentioned sensors are active. Let say that sensors $0$, $2$ and $5$ are involved in the coverage of this interval. Table~\ref{my-label} summarizes the level of coverage for each interval and the sensors involved in for sensor node 0 in figure~\ref{pcm2sensors}(a). 
 % to determine the level of the perimeter coverage for each left and right point of a segment.
 
 %The points reported on the line segment $[0,2\pi]$ separates it in intervals as shown in figure~\ref{expcm}. For example, for each neighboring sensor of sensor 0, place the points  $\alpha^ 1_L$, $\alpha^ 1_R$, $\alpha^ 2_L$, $\alpha^ 2_R$, $\alpha^ 3_L$, $\alpha^ 3_R$, $\alpha^ 4_L$, $\alpha^ 4_R$, $\alpha^ 5_L$, $\alpha^ 5_R$, $\alpha^ 6_L$, $\alpha^ 6_R$, $\alpha^ 7_L$, $\alpha^ 7_R$, $\alpha^ 8_L$, $\alpha^ 8_R$, $\alpha^ 9_L$, and $\alpha^ 9_R$; on the line segment $[0,2\pi]$, and then sort all these points in an ascending order into a list.  Traverse the line segment $[0,2\pi]$ by visiting each point in the sorted list from left to right and determine the coverage level of each interval of the sensor 0 (see figure \ref{expcm}). For each interval, we sum up the number of parts of segments, and we deduce a level of coverage for each interval. For instance, the interval delimited by the points $5L$ and $6L$ contains three parts of segments. That means that this part of the perimeter of the sensor $0$ may be covered by three sensors, sensor $0$ itself and sensors $2$ and $5$. The level of coverage of this interval may reach $3$ if all previously mentioned sensors are active. Let say that sensors $0$, $2$ and $5$ are involved in the coverage of this interval. Table~\ref{my-label} summarizes the level of coverage for each interval and the sensors involved in for sensor node 0 in figure~\ref{pcm2sensors}(a). 
 % to determine the level of the perimeter coverage for each left and right point of a segment.
 

-\begin{figure*}[ht!]
+\begin{figure*}[t!]

 \centering
 \centering

-\includegraphics[width=137.5mm]{expcm.pdf}  
+\includegraphics[width=127.5mm]{expcm2.jpg}  

 \caption{Maximum coverage levels for perimeter of sensor node $0$.}
 \label{expcm}
 \end{figure*} 
 \caption{Maximum coverage levels for perimeter of sensor node $0$.}
 \label{expcm}
 \end{figure*} 


@@ -393,11 +396,11 @@ above is thus given by the sixth line of the table.
 
 \fi
 
 
 \fi
 

- \begin{table}[h]
+ \begin{table}[h!]

  \caption{Coverage intervals and contributing sensors for sensor node 0.}
 \begin{tabular}{|c|c|c|c|c|c|c|c|c|}
 \hline
  \caption{Coverage intervals and contributing sensors for sensor node 0.}
 \begin{tabular}{|c|c|c|c|c|c|c|c|c|}
 \hline

-\begin{tabular}[c]{@{}c@{}}Left \\ point \\ angle~$\alpha$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ left \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ right \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Maximum \\ coverage\\  level\end{tabular} & \multicolumn{5}{c|}{\begin{tabular}[c]{@{}c@{}}Set of sensors\\ involved \\ in interval coverage\end{tabular}} \\ \hline
+\begin{tabular}[c]{@{}c@{}}Left \\ point \\ angle~$\alpha$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ left \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ right \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Maximum \\ coverage\\  level\end{tabular} & \multicolumn{5}{c|}{\begin{tabular}[c]{@{}c@{}}Set of sensors\\ involved \\ in coverage interval\end{tabular}} \\ \hline

 0.0291    & 1L                                                                        & 2L                                                        & 4                                                                     & 0                     & 1                     & 3                    & 4                    &                      \\ \hline
 0.104     & 2L                                                                        & 3R                                                        & 5                                                                     & 0                     & 1                     & 3                    & 4                    & 2                    \\ \hline
 0.3168    & 3R                                                                        & 4R                                                        & 4                                                                     & 0                     & 1                     & 4                    & 2                    &                      \\ \hline
 0.0291    & 1L                                                                        & 2L                                                        & 4                                                                     & 0                     & 1                     & 3                    & 4                    &                      \\ \hline
 0.104     & 2L                                                                        & 3R                                                        & 5                                                                     & 0                     & 1                     & 3                    & 4                    & 2                    \\ \hline
 0.3168    & 3R                                                                        & 4R                                                        & 4                                                                     & 0                     & 1                     & 4                    & 2                    &                      \\ \hline


@@ -424,15 +427,15 @@ above is thus given by the sixth line of the table.
 
 %The optimization algorithm that used by LiCO protocol based on the perimeter coverage levels of the left and right points of the segments and worked to minimize the number of sensor nodes for each left or right point of the segments within each sensor node. The algorithm minimize the perimeter coverage level of the left and right points of the segments, while, it assures that every perimeter coverage level of the left and right points of the segments greater than or equal to 1.
 
 
 %The optimization algorithm that used by LiCO protocol based on the perimeter coverage levels of the left and right points of the segments and worked to minimize the number of sensor nodes for each left or right point of the segments within each sensor node. The algorithm minimize the perimeter coverage level of the left and right points of the segments, while, it assures that every perimeter coverage level of the left and right points of the segments greater than or equal to 1.
 

-In LiCO  protocol, scheduling of sensor  nodes' activities is formulated  with an
+In the LiCO  protocol, scheduling of sensor  nodes' activities is formulated  with an

 integer program  based on  coverage intervals. The  formulation of  the coverage
 optimization problem is  detailed in~section~\ref{cp}.  Note that  when a sensor
 node  has a  part of  its sensing  range outside  the WSN  sensing field,  as in
 integer program  based on  coverage intervals. The  formulation of  the coverage
 optimization problem is  detailed in~section~\ref{cp}.  Note that  when a sensor
 node  has a  part of  its sensing  range outside  the WSN  sensing field,  as in

-figure~\ref{ex4pcm}, the maximum coverage level for  this arc is set to $\infty$
+Figure~\ref{ex4pcm}, the maximum coverage level for  this arc is set to $\infty$

 and  the  corresponding  interval  will  not   be  taken  into  account  by  the
 optimization algorithm.
  
 and  the  corresponding  interval  will  not   be  taken  into  account  by  the
 optimization algorithm.
  

-\begin{figure}[t!]
+\begin{figure}[h!]

 \centering
 \includegraphics[width=62.5mm]{ex4pcm.jpg}  
 \caption{Sensing range outside the WSN's area of interest.}
 \centering
 \includegraphics[width=62.5mm]{ex4pcm.jpg}  
 \caption{Sensing range outside the WSN's area of interest.}


@@ -453,14 +456,14 @@ homogeneous subregions  using a divide-and-conquer  algorithm. In a  second step
 our  protocol  will  be  executed  in   a  distributed  way  in  each  subregion
 simultaneously to schedule nodes' activities for one sensing period.
 
 our  protocol  will  be  executed  in   a  distributed  way  in  each  subregion
 simultaneously to schedule nodes' activities for one sensing period.
 

-As  shown in  figure~\ref{fig2}, node  activity  scheduling is  produced by  our
+As  shown in  Figure~\ref{fig2}, node  activity  scheduling is  produced by  our

 protocol in a periodic manner. Each period is divided into 4 stages: Information
 (INFO)  Exchange,  Leader Election,  Decision  (the  result of  an  optimization
 problem),  and  Sensing.   For  each  period there  is  exactly  one  set  cover
 responsible for  the sensing task.  Protocols  based on a periodic  scheme, like
 LiCO, are more  robust against an unexpected  node failure. On the  one hand, if
 protocol in a periodic manner. Each period is divided into 4 stages: Information
 (INFO)  Exchange,  Leader Election,  Decision  (the  result of  an  optimization
 problem),  and  Sensing.   For  each  period there  is  exactly  one  set  cover
 responsible for  the sensing task.  Protocols  based on a periodic  scheme, like
 LiCO, are more  robust against an unexpected  node failure. On the  one hand, if

-node failure is discovered before  taking the decision, the corresponding sensor
-node will  not be considered  by the optimization  algorithm, and, on  the other
+a node failure is discovered before  taking the decision, the corresponding sensor
+node will  not be considered  by the optimization  algorithm. On  the other

 hand, if the sensor failure happens after  the decision, the sensing task of the
 network will be temporarily affected: only  during the period of sensing until a
 new period starts, since a new set cover will take charge of the sensing task in
 hand, if the sensor failure happens after  the decision, the sensing task of the
 network will be temporarily affected: only  during the period of sensing until a
 new period starts, since a new set cover will take charge of the sensing task in


@@ -474,7 +477,7 @@ the area.
 \begin{figure}[t!]
 \centering
 \includegraphics[width=80mm]{Model.pdf}  
 \begin{figure}[t!]
 \centering
 \includegraphics[width=80mm]{Model.pdf}  

-\caption{LiCO protocol}
+\caption{LiCO protocol.}

 \label{fig2}
 \end{figure} 
 
 \label{fig2}
 \end{figure} 
 


@@ -497,7 +500,7 @@ Five status are possible for a sensor node in the network:
   determine the activities scheduling;
 \item ACTIVE: node is sensing;
 \item SLEEP: node is turned off;
   determine the activities scheduling;
 \item ACTIVE: node is sensing;
 \item SLEEP: node is turned off;

-\item COMMUNICATION: transmits or recevives packets.
+\item COMMUNICATION: transmits or receives packets.

 \end{itemize}
 %\end{enumerate}
 %Below, we describe each phase in more details.
 \end{itemize}
 %\end{enumerate}
 %Below, we describe each phase in more details.


@@ -539,7 +542,7 @@ protocol applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
            }
       
         \emph{$s_k.status$ = COMMUNICATION}\;
            }
       
         \emph{$s_k.status$ = COMMUNICATION}\;

-        \emph{Send $ActiveSleep()$ to each node $l$ in subregion} \;
+        \emph{Send $ActiveSleep()$ to each node $l$ in subregion}\;

         \emph{Update $RE_k $}\;
       }          
       \Else{
         \emph{Update $RE_k $}\;
       }          
       \Else{


@@ -553,20 +556,19 @@ protocol applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
 \label{alg:LiCO}
 \end{algorithm}
 
 \label{alg:LiCO}
 \end{algorithm}
 

-In this  algorithm, K.CurrentSize and  K.PreviousSize refer to the  current size
-and the  previous size of  the subnetwork  in the subregion  respectively.  That
-means the  number of sensor nodes  which are still alive.  Initially, the sensor
-node checks its remaining energy $RE_k$,  which must be greater than a threshold
-$E_{th}$  in order  to  participate  in the  current  period.  Each sensor  node
-determines its  position and its subregion  using an embedded GPS  or a location
-discovery algorithm. After  that, all the sensors  collect position coordinates,
-remaining energy, sensor  node ID, and the number of  its one-hop live neighbors
-during the information  exchange. The sensors inside a same  region cooperate to
-elect a  leader. The selection  criteria for the  leader, in order  of priority,
-are: larger  number of neighbors, larger  remaining energy, and then  in case of
-equality,  larger  index.   Once  chosen, the  leader  collects  information  to
-formulate and  solve the integer  program which allows  to construct the  set of
-active sensors in the sensing stage.
+In this  algorithm, K.CurrentSize and K.PreviousSize  respectively represent the
+current number and  the previous number of alive nodes in  the subnetwork of the
+subregion.  Initially, the sensor node checks its remaining energy $RE_k$, which
+must be greater than a threshold $E_{th}$ in order to participate in the current
+period.  Each  sensor node  determines its position  and its subregion  using an
+embedded  GPS or a  location discovery  algorithm. After  that, all  the sensors
+collect position coordinates,  remaining energy, sensor node ID,  and the number
+of their  one-hop live  neighbors during the  information exchange.  The sensors
+inside a same region cooperate to elect a leader. The selection criteria for the
+leader, in order of priority,  are: larger number of neighbors, larger remaining
+energy, and  then in case  of equality, larger  index.  Once chosen,  the leader
+collects information to formulate and  solve the integer program which allows to
+construct the set of active sensors in the sensing stage.

 
 %After the cooperation among the sensor nodes in the same subregion, the leader will be elected in distributed way, where each sensor node and based on it's information decide who is the leader. The selection criteria for the leader in order  of priority  are: larger number of neighbors,  larger remaining  energy, and  then in  case of equality, larger index. Thereafter,  if the sensor node is leader, it will execute the perimeter-coverage model for each sensor in the subregion in order to determine the segment points which would be used in the next stage by the optimization algorithm of the LiCO protocol. Every sensor node is selected as a leader, it is executed the perimeter coverage model only one time during it's life in the network.
 
 
 %After the cooperation among the sensor nodes in the same subregion, the leader will be elected in distributed way, where each sensor node and based on it's information decide who is the leader. The selection criteria for the leader in order  of priority  are: larger number of neighbors,  larger remaining  energy, and  then in  case of equality, larger index. Thereafter,  if the sensor node is leader, it will execute the perimeter-coverage model for each sensor in the subregion in order to determine the segment points which would be used in the next stage by the optimization algorithm of the LiCO protocol. Every sensor node is selected as a leader, it is executed the perimeter coverage model only one time during it's life in the network.
 


@@ -588,7 +590,7 @@ First, we have the following sets:
 \end{itemize}
 $I_j$ refers to the set of  coverage intervals which have been defined according
 to the  method introduced in  subsection~\ref{CI}. For a coverage  interval $i$,
 \end{itemize}
 $I_j$ refers to the set of  coverage intervals which have been defined according
 to the  method introduced in  subsection~\ref{CI}. For a coverage  interval $i$,

-let $a^j_{ik}$ denote  the indicator function of whether  sensor~$k$ is involved
+let $a^j_{ik}$ denotes  the indicator function of whether  sensor~$k$ is involved

 in coverage interval~$i$ of sensor~$j$, that is:
 \begin{equation}
 a^j_{ik} = \left \{ 
 in coverage interval~$i$ of sensor~$j$, that is:
 \begin{equation}
 a^j_{ik} = \left \{ 


@@ -618,7 +620,7 @@ sensor $j$  is given by  $\sum_{k \in A} a^j_{ik}  X_k$.  To extend  the network
 lifetime,  the objective  is to  activate a  minimal number  of sensors  in each
 period to  ensure the  desired coverage  level. As the  number of  alive sensors
 decreases, it becomes impossible to reach  the desired level of coverage for all
 lifetime,  the objective  is to  activate a  minimal number  of sensors  in each
 period to  ensure the  desired coverage  level. As the  number of  alive sensors
 decreases, it becomes impossible to reach  the desired level of coverage for all

-coverage intervals. Therefore we uses variables $M^j_i$ and $V^j_i$ as a measure
+coverage intervals. Therefore we use variables  $M^j_i$ and $V^j_i$ as a measure

 of the  deviation between  the desired  number of active  sensors in  a coverage
 interval and  the effective  number. And  we try  to minimize  these deviations,
 first to  force the  activation of  a minimal  number of  sensors to  ensure the
 of the  deviation between  the desired  number of active  sensors in  a coverage
 interval and  the effective  number. And  we try  to minimize  these deviations,
 first to  force the  activation of  a minimal  number of  sensors to  ensure the


@@ -666,7 +668,7 @@ X_{k} \in \{0,1\}, \forall k \in A
 $\alpha^j_i$ and $\beta^j_i$  are nonnegative weights selected  according to the
 relative importance of satisfying the associated level of coverage. For example,
 weights associated with  coverage intervals of a specified part  of a region may
 $\alpha^j_i$ and $\beta^j_i$  are nonnegative weights selected  according to the
 relative importance of satisfying the associated level of coverage. For example,
 weights associated with  coverage intervals of a specified part  of a region may

-be  given a  relatively larger  magnitude than  weights associated  with another
+be  given by a  relatively larger  magnitude than  weights associated  with another

 region. This  kind of integer program  is inspired from the  model developed for
 brachytherapy    treatment   planning    for   optimizing    dose   distribution
 \cite{0031-9155-44-1-012}. The integer  program must be solved by  the leader in
 region. This  kind of integer program  is inspired from the  model developed for
 brachytherapy    treatment   planning    for   optimizing    dose   distribution
 \cite{0031-9155-44-1-012}. The integer  program must be solved by  the leader in


@@ -736,10 +738,10 @@ be active during at most 20 periods.
 The values  of $\alpha^j_i$ and  $\beta^j_i$ have been  chosen to ensure  a good
 network coverage and a longer WSN lifetime.  We have given a higher priority for
 the  undercoverage  (by  setting  the  $\alpha^j_i$ with  a  larger  value  than
 The values  of $\alpha^j_i$ and  $\beta^j_i$ have been  chosen to ensure  a good
 network coverage and a longer WSN lifetime.  We have given a higher priority for
 the  undercoverage  (by  setting  the  $\alpha^j_i$ with  a  larger  value  than

-$\beta^j_i$) so as to prevent the non-coverage  for the interval i of the sensor
-j. On the other hand, we have given  a little bit lower value for $\beta^j_i$ so
-as to  minimize the number of  active sensor nodes which  contribute in covering
-the interval.
+$\beta^j_i$)  so as  to prevent  the non-coverage  for the  interval~$i$ of  the
+sensor~$j$.  On the  other hand,  we have  given a  little bit  lower value  for
+$\beta^j_i$ so as to minimize the number of active sensor nodes which contribute
+in covering the interval.

 
 We introduce the following performance metrics to evaluate the efficiency of our
 approach.
 
 We introduce the following performance metrics to evaluate the efficiency of our
 approach.


@@ -751,7 +753,7 @@ approach.
   $Lifetime_{50}$  denote, respectively,  the  amount of  time  during which  is
   guaranteed a  level of coverage  greater than $95\%$  and $50\%$. The  WSN can
   fulfill the expected  monitoring task until all its nodes  have depleted their
   $Lifetime_{50}$  denote, respectively,  the  amount of  time  during which  is
   guaranteed a  level of coverage  greater than $95\%$  and $50\%$. The  WSN can
   fulfill the expected  monitoring task until all its nodes  have depleted their

-  energy or if the network is not more connected. This last condition is crucial
+  energy or if the network is no  more connected. This last condition is crucial

   because without  network connectivity a  sensor may not be  able to send  to a
   base station an event it has sensed.
 \item {\bf  Coverage Ratio (CR)} : it  measures how  well the  WSN is  able to
   because without  network connectivity a  sensor may not be  able to send  to a
   base station an event it has sensed.
 \item {\bf  Coverage Ratio (CR)} : it  measures how  well the  WSN is  able to


@@ -771,7 +773,7 @@ approach.
   follows:
   \begin{equation*}
     \scriptsize
   follows:
   \begin{equation*}
     \scriptsize

-    \mbox{ASR}(\%) =  \frac{\sum\limits_{r=1}^R \mbox{$|A_r|$}}{\mbox{$|S|$}} \times 100
+    \mbox{ASR}(\%) =  \frac{\sum\limits_{r=1}^R \mbox{$|A_r^p|$}}{\mbox{$|S|$}} \times 100

   \end{equation*}
   where $|A_r^p|$ is  the number of active  sensors in the subregion  $r$ in the
   current sensing period~$p$, $|S|$ is the number of sensors in the network, and
   \end{equation*}
   where $|A_r^p|$ is  the number of active  sensors in the subregion  $r$ in the
   current sensing period~$p$, $|S|$ is the number of sensors in the network, and


@@ -827,7 +829,7 @@ it corresponds to the configuration producing  the better results for DiLCO. The
 protocols are distinguished  from one another by the formulation  of the integer
 program providing the set of sensors which  have to be activated in each sensing
 phase. DiLCO protocol tries to satisfy the coverage of a set of primary points,
 protocols are distinguished  from one another by the formulation  of the integer
 program providing the set of sensors which  have to be activated in each sensing
 phase. DiLCO protocol tries to satisfy the coverage of a set of primary points,

-whereas LICO protocol objectif is to reach a desired level of coverage for each
+whereas LiCO protocol objective is to reach a desired level of coverage for each

 sensor perimeter. In our experimentations, we chose a level of coverage equal to
 one ($l=1$).
 
 sensor perimeter. In our experimentations, we chose a level of coverage equal to
 one ($l=1$).
 


@@ -868,7 +870,7 @@ rounds and  DiLCO and LiCO  protocols compete perfectly  with only 17.92  \% and
 20.16 \% active  nodes during the same  time interval. As the  number of periods
 increases, LiCO protocol  has a lower number of active  nodes in comparison with
 the three other approaches, while keeping a greater coverage ratio as shown in
 20.16 \% active  nodes during the same  time interval. As the  number of periods
 increases, LiCO protocol  has a lower number of active  nodes in comparison with
 the three other approaches, while keeping a greater coverage ratio as shown in

-figure \ref{fig333}.
+Figure \ref{fig333}.

 
 \begin{figure}[h!]
 \centering
 
 \begin{figure}[h!]
 \centering


@@ -881,7 +883,7 @@ figure \ref{fig333}.
 
 We study the effect of the energy  consumed by the WSN during the communication,
 computation, listening, active, and sleep status for different network densities
 
 We study the effect of the energy  consumed by the WSN during the communication,
 computation, listening, active, and sleep status for different network densities

-and  compare  it for  the  fours  approaches.  Figures~\ref{fig3EC}(a)  and  (b)
+and  compare  it for  the  four  approaches.  Figures~\ref{fig3EC}(a)  and  (b)

 illustrate  the  energy   consumption  for  different  network   sizes  and  for
 $Lifetime95$ and  $Lifetime50$. The results show  that our LiCO protocol  is the
 most competitive  from the energy  consumption point of  view. As shown  in both
 illustrate  the  energy   consumption  for  different  network   sizes  and  for
 $Lifetime95$ and  $Lifetime50$. The results show  that our LiCO protocol  is the
 most competitive  from the energy  consumption point of  view. As shown  in both


@@ -912,16 +914,17 @@ while keeping a good coverage level.
 \subsubsection{\bf Network Lifetime}
 
 We observe the superiority of LiCO and DiLCO protocols in comparison against the
 \subsubsection{\bf Network Lifetime}
 
 We observe the superiority of LiCO and DiLCO protocols in comparison against the

-two other approaches  in prolonging the network  WSN. In figures~\ref{fig3LT}(a)
-and (b),  $Lifetime95$ and $Lifetime50$  are shown for different  network sizes.
-As highlighted  by these figures,  the lifetime increases  with the size  of the
-network,  and it  is  clearly the  larger  for DiLCO  and  LiCO protocols.   For
-instance, for a network of 300 sensors  and coverage ratio greater than 50\%, we
-can see  on figure~\ref{fig3LT}(b) that the  lifetime is about two  times longer
-with LiCO compared to DESK protocol.  The performance difference is more obvious
-in  figure~\ref{fig3LT}(b)  than  in  figure~\ref{fig3LT}(a)  because  the  gain
-induced  by our  protocols increases  with  the time,  and the  lifetime with  a
-coverage of 50\% is far more longer than with 95\%.
+two    other   approaches    in    prolonging   the    network   lifetime.    In
+Figures~\ref{fig3LT}(a)  and (b),  $Lifetime95$ and  $Lifetime50$ are  shown for
+different  network  sizes.   As  highlighted  by  these  figures,  the  lifetime
+increases with the size  of the network, and it is clearly  the larger for DiLCO
+and LiCO  protocols.  For instance,  for a  network of 300~sensors  and coverage
+ratio greater than 50\%, we can  see on Figure~\ref{fig3LT}(b) that the lifetime
+is about two times longer with  LiCO compared to DESK protocol.  The performance
+difference    is    more    obvious   in    Figure~\ref{fig3LT}(b)    than    in
+Figure~\ref{fig3LT}(a) because the gain induced  by our protocols increases with
+the time, and the lifetime with a coverage  of 50\% is far more longer than with
+95\%.

 
 \begin{figure}[h!]
   \centering
 
 \begin{figure}[h!]
   \centering


@@ -945,8 +948,8 @@ that do not require a 100\% coverage of  the area to be monitored. LiCO might be
 an interesting  method since  it achieves  a good balance  between a  high level
 coverage ratio and network lifetime. LiCO always outperforms DiLCO for the three
 lower  coverage  ratios,  moreover  the   improvements  grow  with  the  network
 an interesting  method since  it achieves  a good balance  between a  high level
 coverage ratio and network lifetime. LiCO always outperforms DiLCO for the three
 lower  coverage  ratios,  moreover  the   improvements  grow  with  the  network

-size. DiLCO is better for coverage ratios near 100\%, but LiCO is not so bad for
-the smallest network sizes.
+size. DiLCO is better  for coverage ratios near 100\%, but in  that case LiCO is
+not so bad for the smallest network sizes.

 
 \begin{figure}[h!]
 \centering \includegraphics[scale=0.5]{R/LTa.eps}
 
 \begin{figure}[h!]
 \centering \includegraphics[scale=0.5]{R/LTa.eps}


@@ -962,7 +965,7 @@ the smallest network sizes.
 
 In this paper  we have studied the problem of  lifetime coverage optimization in
 WSNs. We designed  a new protocol, called Lifetime  Coverage Optimization, which
 
 In this paper  we have studied the problem of  lifetime coverage optimization in
 WSNs. We designed  a new protocol, called Lifetime  Coverage Optimization, which

-schedules  node' activities  (wake up and  sleep  stages) with  the objective  of
+schedules nodes'  activities (wake up  and sleep  stages) with the  objective of

 maintaining a  good coverage ratio  while maximizing the network  lifetime. This
 protocol is  applied in a distributed  way in regular subregions  obtained after
 partitioning the area of interest in a preliminary step. It works in periods and
 maintaining a  good coverage ratio  while maximizing the network  lifetime. This
 protocol is  applied in a distributed  way in regular subregions  obtained after
 partitioning the area of interest in a preliminary step. It works in periods and