+An outline of the protocol implementation is given by Algorithm~\ref{alg:DiLCO} which describes the execution of a period by a node (denoted by $s_j$ for a sensor node indexed by $j$). In the beginning, a node checks whether it has enough energy to stay active during the next sensing phase (i.e., the remaining energy $RE_j$ $\geq$ $E_{th}$ (the amount of energy required to be alive during one period)). If yes, it exchanges information with all the other nodes belonging to the same subregion: it collects from each node its position coordinates, remaining energy ($RE_j$), ID, and the number of one-hop neighbors still alive. Once the first phase is completed, the nodes of a subregion choose a leader to take the decision based on the criteria described in section \ref{ch4:sec:02:03:02}.
+%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{ch4:sec:03}) which provides a set of sensors planned to be active in the next sensing phase. As leader, it will send an ActiveSleep 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 ActiveSleep packet to know its state for the coming sensing phase.
+
+%Primary Points based
+\section{Coverage Problem Formulation}
+\label{ch4:sec:03}
+\indent Our model is based on the model proposed by \cite{ref156} where the
+objective is to find a maximum number of disjoint cover sets. To accomplish
+this goal, the 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 that the binary variable $X_{j}$ determines the activation of
+sensor $j$ in the sensing phase. We also consider primary points as targets.
+The set of primary points is denoted by $P$ and the set of sensors by $J$.
+
+\noindent Let $\alpha_{jp}$ denote the indicator function of whether the primary
+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 $p$ can then be
+computed by $\sum_{j \in J} \alpha_{jp} * X_{j}$ 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 (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}
