based on the Garg-K\"{o}nemann algorithm~\cite{garg98}, provably near the
optimal solution, is also proposed.
+In~\cite{yang2014maximum}, The authors are proposed a linear programming approach for selecting the minimum number of sensor nodes in working station so as to preserve a maximum coverage and extend lifetime of the network. Cheng et al.~\cite{cheng2014energy} are proposed a heuristic algorithm called Cover Sets Balance (CSB) algorithm to choose a set of active nodes using the tuple (data coverage range, residual energy). Then, they are introduced a new Correlated Node Set Computing (CNSC) algorithm to find the correlated node set for a given node. After that, they are proposed a High Residual Energy First (HREF) node selection algorithm to minimize the number of active nodes so as to prolong the network lifetime.
+In~\cite{castano2013column,rossi2012exact,deschinkel2012column}, The authors are proposed a centralized methods based on column generation approach to extend lifetime in wireless sensor networks while coverage preservation.
+
+
\subsection{Distributed approaches}
%{\bf Distributed approaches}
In distributed and localized coverage algorithms, the required computation to
% is used to refer this table in the text
\end{table}
-Our protocol is declined into four versions: MuDiLCO-1, MuDiLCO-3, MuDiLCO-5,
+Our protocol is declined into four versions: MuDiLCO-1, MuDiLCO-3, MuDiLCO-5,
and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$ ($T$ the number of
rounds in one sensing period). In the following, the general case will be
-denoted by MuDiLCO-T. We compare MuDiLCO-T with two other methods. The first
+denoted by MuDiLCO-T. We are studied the impact of dividing the sensing feild on the performance of our MuDiLCO-T protocol with different network sizes using Divide and Conquer method, and we are found that as the number of subregions increase, the network lifetime increase and the MuDiLCO-T protocol become more powerful against the network disconnection.
+This subdivision should be stopped when there is no benefit from the optimization, therefore Our MuDiLCO-T protocol is distributed over 16 rather than 32 subregions because there is a balance between the benefit from the optimization and the execution time is needed to sove it. We compare MuDiLCO-T with two other methods. The first
method, called DESK and proposed by \cite{ChinhVu} is a full distributed
coverage algorithm. The second method, called GAF~\cite{xu2001geography},
consists in dividing the region into fixed squares. During the decision phase,
\end{equation*}
where $n^t$ is the number of covered grid points by the active sensors of all
subregions during round $t$ in the current sensing phase and $N$ is total number
-of grid points in the sensing field of the network.
+of grid points in the sensing field of the network. In our simulation $N = 51 \times 26 = 1326$ grid points.
%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 $51 \times 26~ = ~ 1326$ points in total.
%In future work, we plan to study and propose adjustable sensing range coverage optimization protocol, which computes all active sensor schedules in one time, by using
%optimization methods. This protocol can prolong the network lifetime by minimizing the number of the active sensor nodes near the borders by optimizing the sensing range of sensor nodes.
% use section* for acknowledgement
-%\section*{Acknowledgment}
+
+\section*{Acknowledgment}
+As a Ph.D. student, Ali Kadhum IDREES would like to gratefully acknowledge the University of Babylon - IRAQ for the financial support and in the same time would like to acknowledge Campus France (The French national agency for the promotion of higher education, international student services, and international mobility) and University of Franche-Comt\'e - FRANCE for all the support in FRANCE.
%% \linenumbers
