completed, the nodes of a subregion choose a leader to take the decision based
on the following criteria with decreasing importance: larger number of
neighbors, larger remaining energy, and then in case of equality, larger index.
-After that, if the sensor node is leader, it will execute the integer program
-algorithm (see Section~\ref{cp}) which provides a set of sensors planned to be
-active in the next sensing phase. As leader, it will send an Active-Sleep packet
+After that, if the sensor node is leader, it will solve an integer program
+(see Section~\ref{cp}). \textcolor{blue}{This integer program contains boolean variables $X_j$ where ($X_j=1$) means that sensor $j$ will be active in the next sensing phase. Only sensors with enough remaining energy are involved in the integer program ($J$ is the set of all sensors involved). As the leader consumes energy (computation energy, denoted by $E^{comp}$) to solve the optimization problem, it will be included in the integer program only if it has enough energy to achieve the computation and to stay alive during the next sensing phase, that is to say if $RE_j > E^{comp}+E_{th}$. Once the optimization problem is solved, each leader will send an Active-Sleep packet
to each sensor in the same subregion to indicate it if it has to be active or
-not. Alternately, if the sensor is not the leader, it will wait for the
-Active-Sleep packet to know its state for the coming sensing phase.
+not. Otherwise, if the sensor is not the leader, it will wait for the
+Active-Sleep packet to know its state for the coming sensing phase.}
+%which provides a set of sensors planned to be
+%active in the next sensing phase.
+
\begin{algorithm}[h!]
The objective function consists in minimizing the undercoverage and the overcoverage of the area as suggested in \cite{pedraza2006}.
The area coverage problem is expressed as the coverage of a fraction of points called primary points.
Details on the choice and the number of primary points can be found in \cite{idrees2014coverage}. The set of primary points is denoted by $P$
-and the set of sensors by $J$. As we consider a boolean disk coverage model, we use the boolean indicator $\alpha_{jp}$ which is equal to 1 if the primary point $p$ is in the sensing range of the sensor $j$. The binary variable $X_j$ represents the activation or not of the sensor $j$. So we can express the number of active sensors that cover the primary point $p$ by $\sum_{j \in J} \alpha_{jp} * X_{j}$. We deduce the overcoverage denoted by $\Theta_p$ of the primary point $p$ :
+and the set of alive sensors by $J$. As we consider a boolean disk coverage model, we use the boolean indicator $\alpha_{jp}$ which is equal to 1 if the primary point $p$ is in the sensing range of the sensor $j$. The binary variable $X_j$ represents the activation or not of the sensor $j$. So we can express the number of active sensors that cover the primary point $p$ by $\sum_{j \in J} \alpha_{jp} * X_{j}$. We deduce the overcoverage denoted by $\Theta_p$ of the primary point $p$ :
\begin{equation}
\Theta_{p} = \left \{
\begin{array}{l l}
\end{array}
\right.
\end{equation}
-The objective function is a weighted sum of overcoverage and undercoverage. The goal is to limit the overcoverage in order to activate a minimal number of sensors while simultaneously preventing undercoverage. Both weights $w_\theta$ and $w_U$ must be carefully chosen in
-order to guarantee that the maximum number of points are covered during each
-period.
+The objective function is a weighted sum of overcoverage and undercoverage. The goal is to limit the overcoverage in order to activate a minimal number of sensors while simultaneously preventing undercoverage. \textcolor{blue}{ By
+ choosing $w_{U}$ much larger than $w_{\theta}$, the coverage of a
+ maximum of primary points is ensured. Then for the same number of covered
+ primary points, the solution with a minimal number of active sensors is
+ preferred. }
+%Both weights $w_\theta$ and $w_U$ must be carefully chosen in
+%order to guarantee that the maximum number of points are covered during each
+%period.
% MODIF - END
