-Some distributed algorithms have been developed
-in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02, yardibi2010distributed, prasad2007distributed,Misra}
-to perform the scheduling so as to preserve coverage. 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} or maximum uncovered targets
-\cite{lu2003coverage}. The authors in \cite{yardibi2010distributed} have 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 have 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 one-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}.
-
-%Our Work, which is presented in~\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 coverage-aware,
-distributed energy-efficient, and distributed clustering methods respectively,
-which aims to extend the network lifetime, while the coverage is ensured. More recently, Shibo et al. \cite{Shibo} have 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. In
-\cite{xu2001geography}, Xu et al. have proposed an algorithm, called
-Geographical Adaptive Fidelity (GAF), which uses geographic location information
-to divide the area of interest into fixed square grids. Within each grid, it
-keeps only one node staying awake to take the responsibility of sensing and
-communication.
-
-Some other approaches (outside the scope of our work) 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 MuDiLCO protocol (for Multiround Distributed Lifetime Coverage Optimization
-protocol) presented in this paper is an extension of the approach introduced
-in~\cite{idrees2014coverage}. In~\cite{idrees2014coverage}, the protocol is
-deployed over only two subregions. Simulation results have shown that it was
-more interesting to divide the area into several subregions, given the
-computation complexity. Compared to our previous paper, in this one we study the
-possibility of dividing the sensing phase into multiple rounds and we also add
-an improved model of energy consumption to assess the efficiency of our
-approach.
-
-
-
-
-
-\iffalse
-
-\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. Note that centralized algorithms have the advantage of requiring very
-low processing power from the sensor nodes, which usually have limited
-processing capabilities. The main drawback of this kind of approach is its
-higher cost in communications, since the node that will take the decision needs
-information from all the sensor nodes. Moreover, centralized approaches usually
-suffer from the scalability problem, making them less competitive as the network
-size increases.
-
-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} have
-proposed 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} have designed 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 include 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.
-
-In \cite{cardei2005improving}, the authors have proposed 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 centralized 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 algorithms, where CCF means that
-they use a cost function called Critical Control Factor, 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.
-
-In~\cite{yang2014maximum}, the authors have proposed a linear programming
-approach for selecting the minimum number of working sensor nodes, in order to
-as to preserve a maximum coverage and extend lifetime of the network. Cheng et
-al.~\cite{cheng2014energy} have defined a heuristic algorithm called Cover Sets
-Balance (CSB), which choose a set of active nodes using the tuple (data coverage
-range, residual energy). Then, they have introduced a new Correlated Node Set
-Computing (CNSC) algorithm to find the correlated node set for a given node.
-After that, they proposed a High Residual Energy First (HREF) node selection
-algorithm to minimize the number of active nodes so as to prolong the network
-lifetime. Various centralized methods based on column generation approaches have
-also been proposed~\cite{castano2013column,rossi2012exact,deschinkel2012column}.
-
-
-
-\subsection{Distributed approaches}
-%{\bf Distributed approaches}
