-\noindent In this section, we summarize some related works regarding coverage problem , and distinguish our DiLCO protocol from the works presented in the literature.\\
-The most discussed coverage problems in literature
-can be classified into three types \cite{li2013survey}: area coverage (where
-every point inside an area is to be monitored), target coverage (where the main
-objective is to cover only a finite number of discrete points called targets),
-and barrier coverage (to prevent intruders from entering into the region of
-interest).
-{\it In DiLCO protocol, the area coverage, ie the coverage
-of every point in the sensing region, is transformed to the coverage of a fraction of points called primary points. }
-
-The major approach to extend network lifetime while preserving coverage is to divide/organize the sensors into a suitable number of set covers (disjoint or non-disjoint) where each set completely covers an interest region and to activate these set covers successively. The network activity can be planned in advance and scheduled for the entire network lifetime or organized in periods, and the set of
-active sensor nodes is decided at the beginning of each period.
-Active node selection is determined based on the problem
-requirements (e.g. area monitoring, connectivity, power
-efficiency). Different methods has been proposed in literature.
-
-{\it DiLCO protocol works in periods, each period contains a preliminary phase for information exchange and decisions, followed by a sensing phase where
-one cover set is in charge of the sensing task.}
-
-Various approaches, including centralised, distributed and localized algorithms, have been proposed to extend the network lifetime.
-%For instance, in order to hide the occurrence of faults, or the sudden unavailability of
-%sensor nodes, some distributed algorithms have been developed in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}.
-
-In distributed algorithms~\cite{yangnovel,ChinhVu,qu2013distributed}, information is disseminated throughout the network and sensors decide cooperatively by communicating with their neighbours which of them will remain in sleep mode for a certain period of time.
-The centralized algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high} always provide nearly
-or close to optimal solution since the algorithm has global view of the whole
-network, but such a method has the disadvantage of requiring
-high communication costs, since the node (located at the base station) making the decision needs information from all the sensor nodes in the area.
-
-A large variety of coverage scheduling algorithms have been proposed in the literature. Many of the existing algorithms, dealing with the maximisation of the number of cover sets, are heuristics. These heuristics involve the construction of a cover set by including in priority the sensor nodes which cover critical targets, that is to say targets that are covered by the smallest number of sensors. Other approaches are based on mathematical programming formulations and dedicated techniques (solving with a branch-and-bound algorithms available in optimization solver). The problem is formulated as an optimization problem (maximization of the lifetime, of the number of cover sets) under target coverage and energy constraints. Column generation techniques, well-known and widely practiced techniques for solving linear programs with too many variables, have been also used~\cite{castano2013column,rossi2012exact,deschinkel2012column}.
-
-Diongue and Thiare~\cite{diongue2013alarm} proposed an energy aware sleep scheduling algorithm for lifetime maximization in wireless sensor networks (ALARM). The proposed approach permits to schedule redundant nodes according to the weibull distribution. This work did not analyze the ALARM scheme under the coverage problem.
-
-Shi et al.~\cite{shi2009} modeled the Area Coverage Problem (ACP), which will be changed into a set coverage
-problem. By using this model, they are proposed an Energy-Efficient central-Scheduling greedy algorithm, which can reduces energy consumption and increases network lifetime, by selecting a appropriate subset of sensor nodes to support the networks periodically.
-
-In ~\cite{chenait2013distributed}, the authors presented a coverage-guaranteed distributed sleep/wake scheduling
-scheme so ass to prolong network lifetime while guaranteeing network coverage. This scheme mitigates scheduling process to be more stable by avoiding useless transitions between states without affecting the coverage level required by the application.
-
-The work in~\cite{cheng2014achieving} presented a unified sensing architecture for duty cycled sensor networks, called uSense, which comprises three ideas: Asymmetric Architecture, Generic Switching and Global Scheduling. The objective is to provide a flexible and efficient coverage in sensor networks.
-
-In~\cite{ling2009energy}, the lifetime of
-a sensor node is divided into epochs. At each epoch, the
-base station deduces the current sensing coverage requirement
-from application or user request. It then applies the heuristic algorithm in order to produce the set of active nodes which take the mission of sensing during the current epoch. After that, the produced schedule is sent to the sensor nodes in the network.
-
-{\it In DiLCO protocol, the area coverage is divided into several smaller subregions, and in each of which, a node called the leader is on charge for selecting the active sensors for the current period.}
-
-Yang et al.~\cite{yang2014energy} investigated full area coverage problem
-under the probabilistic sensing model in the sensor networks. They have studied the relationship between the
-coverage of two adjacent points mathematically and then convert the problem of full area coverage into point coverage problem. They proposed $\varepsilon$-full area coverage optimization (FCO) algorithm to select a subset
-of sensors to provide probabilistic area coverage dynamically so as to extend the network lifetime.
-
-The work in~\cite{cheng2014achieving} presented a unified sensing architecture for duty cycled sensor networks, called uSense, which comprises three ideas: Asymmetric Architecture, Generic Switching and Global Scheduling. The objective is to provide a flexible and efficient coverage in sensor networks.
-
-The work proposed by \cite{qu2013distributed} considers the coverage problem in WSNs where each sensor has variable sensing radius. The final objective is to maximize the network coverage lifetime in WSNs.
-
-{\it In DiLCO protocol, each leader, in each subregion, solves an integer program with a double objective consisting in minimizing the overcoverage and limiting the undercoverage. This program is inspired from the work of \cite{pedraza2006} where the objective is to maximize the number of cover sets.}
-
-
-\iffalse
-
-Some algorithms have been developed in ~\cite{yang2014energy,ChinhVu,vashistha2007energy,deschinkel2012column,shi2009,qu2013distributed,ling2009energy,xin2009area,cheng2014achieving,ling2009energy} to solve the area coverage problem so as to preserve coverage and prolong the network lifetime.
-
-
-Yang et al.~\cite{yang2014energy} investigated full area coverage problem
-under the probabilistic sensing model in the sensor networks. They have studied the relationship between the
-coverage of two adjacent points mathematically and then convert the problem of full area coverage into point coverage problem. They proposed $\varepsilon$-full area coverage optimization (FCO) algorithm to select a subset
-of sensors to provide probabilistic area coverage dynamically so as to extend the network lifetime.
-
-
-Vu et al.~\cite{ChinhVu} proposed a localized and distributed greedy algorithm named DESK for generating non-disjoint cover sets which provide the k-area coverage for the whole network.
-
-
-Qu et al.~\cite{qu2013distributed} developed a distributed algorithm using adjustable sensing sensors
-for maintaining the full coverage of such sensor networks. The
-algorithm contains two major parts: the first part aims at
-providing $100\%$ coverage and the second part aims at saving
-energy by decreasing the sensing radius.
-
-Shi et al.~\cite{shi2009} modeled the Area Coverage Problem (ACP), which will be changed into a set coverage
-problem. By using this model, they are proposed an Energy-Efficient central-Scheduling greedy algorithm, which can reduces energy consumption and increases network lifetime, by selecting a appropriate subset of sensor nodes to support the networks periodically.
-
-The work in~\cite{cheng2014achieving} presented a unified sensing architecture for duty cycled sensor networks, called uSense, which comprises three ideas: Asymmetric Architecture, Generic Switching and Global Scheduling. The objective is to provide a flexible and efficient coverage in sensor networks.
-
- In~\cite{ling2009energy}, the lifetime of
-a sensor node is divided into epochs. At each epoch, the
-base station deduces the current sensing coverage requirement
-from application or user request. It then applies the heuristic algorithm in order to produce the set of active nodes which take the mission of sensing during the current epoch. After that, the produced schedule is sent to the sensor nodes in the network.
-\fi
-
-\iffalse
-
-The work in ~\cite{vu2009delaunay} considered the area coverage problem for variable sensing radii in WSNs by improving the energy balancing heuristic proposed in ~\cite{wang2007energy} so that the area of interest can be full covered using Delaunay triangulation structure.
-
-Diongue and Thiare~\cite{diongue2013alarm} proposed an energy aware sleep scheduling algorithm for lifetime maximization in wireless sensor networks (ALARM). The proposed approach permits to schedule redundant nodes according to the weibull distribution. This work did not analyze the ALARM scheme under the coverage problem.
-
-
-In~\cite{xin2009area}, the authors proposed a circle intersection localized coverage algorithm
-to maintain connectivity based on loose connectivity critical condition
-. By using the connected coverage node set, it can maintain network
-connection in the case which loose condition is not meet.
-The authors in ~\cite{vashistha2007energy} addressed the full area coverage problem using information
-coverage. They are proposed a low-complexity heuristic algorithm to obtain full area information covers (FAIC), which they refer to as Grid Based FAIC (GB-FAIC) algorithm. Using these FAICs, they are obtained the optimal schedule for applying the sensing activity of sensor nodes in order to
-achieve increased sensing lifetime of the network.
-
-
-
-
-
-
-In \cite{xu2001geography}, Xu et al. proposed a Geographical Adaptive Fidelity (GAF) algorithm, 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.
-
-The main contributions of our DiLCO Protocol can be summarized as follows:
-(1) The distributed optimization over the subregions in the area of interest,
-(2) The distributed dynamic leader election at each period by each sensor node in the subregion,
-(3) The primary point coverage model to represent each sensor node in the network,
-(4) The activity scheduling based optimization on the subregion, which are based on the primary point coverage model to activate as less number as possible of sensor nodes to take the mission of the coverage in each subregion, and (5) The improved energy consumption model.
-\fi
-\iffalse
-The work presented in~\cite{luo2014parameterized,tian2014distributed} tries to solve the target coverage problem so as to extend the network lifetime since it is easy to verify the coverage status of discreet target.
-%Je ne comprends pas la phrase ci-dessus
-The work proposed in~\cite{kim2013maximum} considers the barrier-coverage problem in WSNs. The final goal is to maximize the network lifetime such that any penetration of the intruder is detected.
-%inutile de parler de ce papier car il concerne barrier coverage
-In \cite{ChinhVu}, the authors propose a localized and distributed greedy algorithm named DESK for generating non-disjoint cover sets which provide the k-coverage for the whole network.
-Our Work in~\cite{idrees2014coverage} proposes 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 are considered only distributing the coverage protocol over two subregions.
-
-The work presented in ~\cite{Zhang} focuses on a distributed clustering method, which aims to extend the network lifetime, while the coverage is ensured.
-
-The work proposed by \cite{qu2013distributed} considers the coverage problem in WSNs where each sensor has variable sensing radius. The final objective is to maximize the network coverage lifetime in WSNs.
-
-
