-
\documentclass[conference]{IEEEtran}
-
\ifCLASSINFOpdf
\else
\hyphenation{op-tical net-works semi-conduc-tor}
-\usepackage{float}
+\usepackage{float}
\usepackage{epsfig}
-\usepackage{subfigure}
\usepackage{calc}
\usepackage{times,amssymb,amsmath,latexsym}
\usepackage{graphics}
\usepackage{graphicx}
\usepackage{amsmath}
-\usepackage{txfonts}
+%\usepackage{txfonts}
\usepackage{algorithmic}
\usepackage[T1]{fontenc}
\usepackage{tikz}
\usepackage{epsfig}
\usepackage{caption}
\usepackage{multicol}
-
+\usepackage{times}
+\usepackage{graphicx,epstopdf}
+\epstopdfsetup{suffix=}
+\DeclareGraphicsExtensions{.ps}
+\DeclareGraphicsRule{.ps}{pdf}{.pdf}{`ps2pdf -dEPSCrop -dNOSAFER #1 \noexpand\OutputFile}
\begin{document}
-\title{Distributed Coverage Optimization Protocol to Improve the Lifetime in Heterogeneous Energy Wireless Sensor Networks}
-
+%\title{ Coverage and Lifetime Optimization in Heterogeneous Energy Wireless Sensor Networks}
+\title{Coverage and Lifetime Optimization in Heterogeneous Energy Wireless Sensor Networks}
+%Activity Scheduling for Coverage and Lifetime Optimization in Wireless Sensor Networks}
% author names and affiliations
% use a multiple column layout for up to three different
% affiliations
-\author{\IEEEauthorblockN{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon and Raphael Couturier }
-\IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France \\
-Email:$\lbrace$ali.idness, karine.deschinkel, michel.salomon,raphael.couturier$\rbrace$@edu.univ-fcomte.fr
-}
+\author{\IEEEauthorblockN{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon, and Rapha\"el Couturier}
+\IEEEauthorblockA{FEMTO-ST Institute, UMR 6174 CNRS \\
+University of Franche-Comt\'e, Belfort, France \\
+Email: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr}
%\email{\{ali.idness, karine.deschinkel, michel.salomon, raphael.couturier\}@univ-fcomte.fr}
%\and
%\IEEEauthorblockN{Homer Simpson}
%\IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France}
}
-
\maketitle
-
\begin{abstract}
-%\boldmath
-One of the fundamental challenges in Wireless Sensor Networks (WSNs) is Coverage preservation and extension of network lifetime continuously and effectively during monitoring a certain geographical area.In this paper
-a distributed coverage optimization protocol to improve the lifetime in in Heterogeneous Energy Wireless Sensor Networks is proposed. The area of interest is divided into subregions using Divide-and-conquer method and an activity scheduling for sensor nodes is planned for each subregion.Our protocol is distributed in each subregion. It divides the network lifetime into activity rounds. In each round a small
-number of active nodes is selected to ensure coverage.Each round includes four phases: INFO Exchange, Leader election, decision and sensing.Simulation results show that the proposed protocol can prolong the network
-lifetime and improve network coverage effectively.
-
-
+One of the fundamental challenges in Wireless Sensor Networks (WSNs)
+is the coverage preservation and the extension of the network lifetime
+continuously and effectively when monitoring a certain area (or
+region) of interest. In this paper, a coverage optimization protocol to
+improve the lifetime in heterogeneous energy wireless sensor networks
+is proposed. The area of interest is first divided into subregions
+using a divide-and-conquer method and then the scheduling of sensor node
+activity is planned for each subregion. The proposed scheduling
+considers rounds during which a small number of nodes, remaining
+active for sensing, is selected to ensure coverage. Each round
+consists of four phases: (i)~Information Exchange, (ii)~Leader
+Election, (iii)~Decision, and (iv)~Sensing. The decision process is
+carried out by a leader node, which solves an integer program.
+Simulation results show that the proposed approach can prolong the
+network lifetime and improve the coverage performance.
\end{abstract}
- %\keywords{Area Coverage, Wireless Sensor Networks, lifetime Optimization, Distributed Protocol.}
-
+\begin{IEEEkeywords}
+Area Coverage, Network lifetime, Optimization, Scheduling, Distributed Protocol.
+\end{IEEEkeywords}
+%\keywords{Area Coverage, Network lifetime, Optimization, Distributed Protocol}
- \IEEEpeerreviewmaketitle
-
+\IEEEpeerreviewmaketitle
\section{Introduction}
-\noindent Recent years have witnessed significant advances in wireless sensor
-networks which emerge as one of the most promising technologies for
-the 21st century~\cite{asc02}. In fact, they present huge potential in
-several domains ranging from health care applications to military
-applications.
-A sensor network is composed of a large number of tiny sensing devices deployed in a region of interest. Each device has processing and wireless communication capabilities, which enable to sense its environment, to compute, to store information and to deliver report messages to a base station.
-%These sensor nodes run on batteries with limited capacities. To achieve a long life of the network, it is important to conserve battery power. Therefore, lifetime optimisation is one of the most critical issues in wireless sensor networks.
-One of the main design challenges in Wireless Sensor Networks (WSN) is to prolong the system lifetime, while achieving acceptable quality of service for applications. Indeed, sensor nodes
-have limited resources in terms of memory, energy and computational powers.
-
-%\medskip
-Since sensor nodes have limited battery life and without being able to replace
-batteries, especially in remote and hostile environments,
-it is desirable that a WSN should be deployed with
-high density and thus redundancy can be exploited to increase
-the lifetime of the network. In such a high density network, if all sensor nodes
-were to be activated at the same time, the lifetime would be reduced. Consequently,
-future software may need to adapt appropriately to achieve acceptable quality of service for applications.
-In this paper we concentrate on area coverage problem, with the objective of maximizing the network lifetime by using an adaptive scheduling. Area of interest is divided into subregions and an activity scheduling for sensor nodes is planned for each subregion.
-Our scheduling scheme works in period which includes a discovery phase to exchange information between sensors of the subregion, then a sensor is chosen in suitable manner to carry out a coverage strategy. This coverage strategy involves the resolution of an integer program which provides the activation of the sensors for the $t$ next round.
-
-
-The remainder of the paper is organized as follows.
-Section~\ref{rw} reviews the related work in the field.
-Section \ref{pd} is devoted to the scheduling strategy for energy-efficient coverage.
-Section \ref{cp} gives the coverage model formulation which is used to schedule the activation of sensors.
-Section \ref{exp} shows the simulation results conducted on OMNET++, that fully demonstrate the usefulness of the proposed approach. Finally, we give concluding remarks in Section~\ref{sec:conclusion}.
-
-\section{\uppercase{Related work}}
+
+\noindent The fast developments in the low-cost sensor devices and wireless communications have allowed the emergence the WSNs. WSN includes a large number of small , limited-power sensors that can sense, process and transmit
+ data over a wireless communication . They communicate with each other by using multi-hop wireless communications , cooperate together to monitor the area of interest, and the measured data can be reported
+ to a monitoring center
+called, sink, for analysis it~\cite{Ammari01, Sudip03}. There are several applications used the WSN including health, home, environmental, military,and industrial applications~\cite{Akyildiz02}.
+The coverage problem is one of the fundamental challenges in WSNs~\cite{Nayak04} that consists in monitoring efficiently and continuously the area of interest. The limited energy of sensors represents the main challenge in the WSNs design~\cite{Ammari01}, where it is difficult to replace and/or
+ recharge their batteries because the the area of interest nature (such as hostile environments) and the cost. So, it is necessary that a WSN deployed with high density because spatial redundancy can then be exploited to increase the lifetime of the network . However, turn on all the sensor nodes, which monitor the same region at the same time leads to decrease the lifetime of the network. To extend the lifetime of the network, the main idea is to take advantage of the overlapping sensing regions of some sensor nodes to save energy by turning off some of them during the sensing phase~\cite{Misra05}. WSNs require energy-efficient solutions to improve the network lifetime that is constrained by the limited power of each sensor node ~\cite{Akyildiz02}.
+In this paper, we concentrate on the area
+coverage problem, with the objective of maximizing the network
+lifetime by using an adaptive scheduling. The area of interest is
+divided into subregions and an activity scheduling for sensor nodes is
+planned for each subregion.
+ In fact, the nodes in a subregion can be seen as a cluster where
+ each node sends sensing data to the cluster head or the sink node.
+ Furthermore, the activities in a subregion/cluster can continue even
+ if another cluster stops due to too many node failures.
+Our scheduling scheme considers rounds, where a round starts with a
+discovery phase to exchange information between sensors of the
+subregion, in order to choose in a suitable manner a sensor node to
+carry out a coverage strategy. This coverage strategy involves the
+solving of an integer program, which provides the activation of the
+sensors for the sensing phase of the current round.
+
+The remainder of the paper is organized as follows. The next section
+% Section~\ref{rw}
+reviews the related work in the field. Section~\ref{pd} is devoted to
+the scheduling strategy for energy-efficient coverage.
+Section~\ref{cp} gives the coverage model formulation, which is used to
+schedule the activation of sensors. Section~\ref{exp} shows the
+simulation results obtained using the discrete event simulator OMNeT++ \cite{varga}. They fully demonstrate the usefulness of the
+proposed approach. Finally, we give concluding remarks and some
+suggestions for future works in Section~\ref{sec:conclusion}.
+
+\section{Related works}
\label{rw}
-\noindent
-This section is dedicated to the various approaches proposed in the literature for the coverage lifetime maximization problem where the objective is to optimally schedule sensors'activities in order to extend network lifetime in a randomly deployed network. As this problem is subject to a wide range of interpretations, we suggest to recall main definitions and assumptions related to our work.
-{\bf Coverage}
+\noindent This section is dedicated to the various approaches proposed
+in the literature for the coverage lifetime maximization problem,
+where the objective is to optimally schedule sensors' activities in
+order to extend network lifetime in a randomly deployed network. As
+this problem is subject to a wide range of interpretations, we have chosen
+to recall the main definitions and assumptions related to our work.
+
%\begin{itemize}
%\item Area Coverage: The main objective is to cover an area. The area coverage requires
%that the sensing range of working Active nodes cover the whole targeting area, which means any
%\item Barrier Coverage An objective to determine the maximal support/breach paths that traverse a sensor field. Barrier coverage is expressed as finding one or more routes with starting position and ending position when the targets pass through the area deployed with sensor nodes~\cite{Santosh04,Ai05}.
%\end{itemize}
-
-The most discussed coverage problems in literature can be classified into two types \cite{} : area coverage and targets coverage. An area coverage problem is to find a minimum number of sensors to work such that each physical point in the area is monitored by at least a working sensor. Target coverage problem is to cover only a finite number of discrete points called targets.
- Our work will concentrate on the area coverage by design and implement a strategy which efficiently select the active nodes that must maintain both sensing coverage and network connectivity and in the same time improve the lifetime of the wireless sensor network. But requiring that all physical points are covered may be too strict, specially where the sensor network is not dense.
-Our approach represents an area covered by a sensor as a set of principle points and tries to maximize the total number of principles points that are covered in each round, while minimizing overcoverage (points covered by multiple active sensors simultaneously).\\
-{\bf Lifetime}\\
-Various definitions exist for the lifetime of a sensor network. Main definitions proposed in the literature are related to the remaining energy of the nodes \cite{} or to the percentage of coverage \cite{}. The lifetime of the network is mainly defined as the amount of time that the network can satisfy its coverage objective (the amount of time that the network can cover a given percentage of its area or targets of interest) . In our simulation we assume that the network is alive until all sensor nodes are died and we measure the coverage ratio during the process.
-
-{\bf Activity scheduling}\\
-Activity scheduling is to schedule the activation and deactivation of nodes 'sensor units. The basic objective is to decide which sensors are in which states (active or sleeping mode) and for how long a time such that the application coverage requirement can be guaranteed and network lifetime can be prolonged. Various approaches, including centralized, distributed and localized algorithms, have been proposed for activity scheduling. In the distributed algorithms, each node in the network autonomously makes decisions on whether to turn on or turn off itself only using local neighbor information. In centralized algorithms, a central controller (node or base station) informs every sensor of the time intervals to be activated.
-
-{\bf Distributed approaches}
-
-Some distributed algorithms have been developed in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}. Distributed algorithms typically operate in roundsf predetermined duration. At the beginning of each round, a sensor exchange information with its neighbors and makes a decision to either turn on or go to sleep for the round. This decision is basically based on simple greedy criteria like the largest uncovered area \cite{Berman05efficientenergy}, maximum uncovered targets \cite{1240799}.
-In \cite{Tian02}, the sheduling scheme is divided into rounds, where each round has a self-scheduling phase followed by a sensing phase. Each sensor broadcasts a message to its neighbors containing node ID and node location at the beginning of each round. Sensor determines its status by a rule named off-duty eligible rule which tells him to turn off if its sensing area is covered by its neighbors. A back-off scheme is introduced to let each sensor delay the decision process with a random period of time, in order to avoid that nodes make conflicting decisions simultaneously and that a part of the area is no longer covered.
-\cite{Prasad:2007:DAL:1782174.1782218} propose a model for capturing the dependencies between different cover sets and propose localized heuristic based on this dependency. The algorithm consists of two phases, an initial setup phase during which each sensor calculates and prioritize the covers and a sensing phase during which each sensor first decides its on/off status and then remains on or off for the rest of the duration.
-Authors in \cite{chin2007} propose a novel distributed heuristic named distributed Energy-efficient Scheduling for k-coverage (DESK) so that the energy consumption among all the sensors is balanced, and network lifetime is maximized while the coverage requirements being maintained. This algorithm works in round, requires only 1-sensing-hop-neigbor information, and a sensor decides its status (active/sleep) based on its perimeter coverage computed through the k-Non-Unit-disk coverage algorithm proposed in \cite{Huang:2003:CPW:941350.941367}.\\
-
-Some others approaches do not consider synchronized and predetermined period of time where the sensors are active or not. Each sensor maintains its own timer and its time wake-up is randomized \cite{Ye03} or regulated \cite{cardei05} over time.
+\subsection{Coverage}
+%{\bf Coverage}
+
+The most discussed coverage problems in literature can be classified
+into two types \cite{ma10}: area coverage (also called full or blanket
+coverage) and target coverage. An area coverage problem is to find a
+minimum number of sensors to work, such that each physical point in the
+area is within the sensing range of at least one working sensor node.
+Target coverage problem is to cover only a finite number of discrete
+points called targets. This type of coverage has mainly military
+applications. Our work will concentrate on the area coverage by design
+and implementation of a strategy, which efficiently selects the active
+nodes that must maintain both sensing coverage and network
+connectivity and at the same time improve the lifetime of the wireless
+sensor network. But, requiring that all physical points of the
+considered region are covered may be too strict, especially where the
+sensor network is not dense. Our approach represents an area covered
+by a sensor as a set of primary points and tries to maximize the total
+number of primary points that are covered in each round, while
+minimizing overcoverage (points covered by multiple active sensors
+simultaneously).
+
+\subsection{Lifetime}
+%{\bf Lifetime}
+
+Various definitions exist for the lifetime of a sensor
+network~\cite{die09}. The main definitions proposed in the literature are
+related to the remaining energy of the nodes or to the coverage percentage.
+The lifetime of the network is mainly defined as the amount
+of time during which the network can satisfy its coverage objective (the
+amount of time that the network can cover a given percentage of its
+area or targets of interest). In this work, we assume that the network
+is alive until all nodes have been drained of their energy or the
+sensor network becomes disconnected, and we measure the coverage ratio
+during the WSN lifetime. Network connectivity is important because an
+active sensor node without connectivity towards a base station cannot
+transmit information on an event in the area that it monitors.
+
+\subsection{Activity scheduling}
+%{\bf Activity scheduling}
+
+Activity scheduling is to schedule the activation and deactivation of
+sensor nodes. The basic objective is to decide which sensors are in
+what states (active or sleeping mode) and for how long, so that the
+application coverage requirement can be guaranteed and the network
+lifetime can be prolonged. Various approaches, including centralized,
+distributed, and localized algorithms, have been proposed for activity
+scheduling. In distributed algorithms, each node in the network
+autonomously makes decisions on whether to turn on or turn off itself
+only using local neighbor information. In centralized algorithms, a
+central controller (a node or base station) informs every sensors of
+the time intervals to be activated.
+
+\subsection{Distributed approaches}
+%{\bf Distributed approaches}
+
+Some distributed algorithms have been developed
+in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02} to perform the
+scheduling. 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}, maximum uncovered targets
+\cite{1240799}. In \cite{Tian02}, the scheduling scheme is divided
+into rounds, where each round has a self-scheduling phase followed by
+a sensing phase. Each sensor broadcasts a message containing the node ID
+and the node location to its neighbors at the beginning of each round. A
+sensor determines its status by a rule named off-duty eligible rule,
+which tells him to turn off if its sensing area is covered by its
+neighbors. A back-off scheme is introduced to let each sensor delay
+the decision process with a random period of time, in order to avoid
+simultaneous conflicting decisions between nodes and lack of coverage on any area.
+\cite{Prasad:2007:DAL:1782174.1782218} defines a model for capturing
+the dependencies between different cover sets and proposes localized
+heuristic based on this dependency. The algorithm consists of two
+phases, an initial setup phase during which each sensor computes and
+prioritizes the covers and a sensing phase during which each sensor
+first decides its on/off status, and then remains on or off for the
+rest of the duration. Authors in \cite{chin2007} propose a novel
+distributed heuristic named Distributed Energy-efficient Scheduling
+for k-coverage (DESK) so that the energy consumption among all the
+sensors is balanced, and network lifetime is maximized while the
+coverage requirement is being maintained. This algorithm works in
+round, requires only 1-sensing-hop-neighbor information, and a sensor
+decides its status (active/sleep) based on its perimeter coverage
+computed through the k-Non-Unit-disk coverage algorithm proposed in
+\cite{Huang:2003:CPW:941350.941367}.
+
+Some other approaches 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{cardei05} over time.
%A ecrire \cite{Abrams:2004:SKA:984622.984684}p33
-
%The scheduling information is disseminated throughout the network and only sensors in the active state are responsible
%for monitoring all targets, while all other nodes are in a low-energy sleep mode. The nodes decide cooperatively which of them will remain in sleep mode for a certain
%period of time.
%one way of increasing lifeteime is by turning off redundant nodes to sleep mode to conserve energy while active nodes provide essential coverage, which improves fault tolerance.
%In this paper we focus on centralized algorithms because distributed algorithms are outside the scope of our work. Note that centralized coverage algorithms have the advantage of requiring very low processing power from the sensor nodes which have usually limited processing capabilities. Moreover, a recent study conducted in \cite{pc10} concludes that there is a threshold in terms of network size to switch from a localized to a centralized algorithm. Indeed the exchange of messages in large networks may consume a considerable amount of energy in a localized approach compared to a centralized one.
-{\bf Centralized approaches}\\
-Power efficient centralized schemes differ according to several criteria \cite{Cardei:2006:ECP:1646656.1646898}, such as the coverage objective (target coverage or area coverage), the node deployment method (random or deterministic) and the heterogeneity of sensor nodes (common sensing range, common battery lifetime). 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.
-
-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} propose an algorithm which allocates sensor nodes in mutually independent sets to monitor an area divided into several fields. Their algorithm constructs 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. ~\cite{cardei02} present
-a graph coloring technique to achieve energy savings
-by organizing the sensor nodes into a maximum number of disjoint
-dominating sets which are activated successively. The dominating
-sets do not guarantee the coverage of the whole region of interest.
-Abrams et al.\cite{Abrams:2004:SKA:984622.984684} design 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.
+\subsection{Centralized approaches}
+%{\bf Centralized approaches}
+
+Power efficient centralized schemes differ according to several
+criteria \cite{Cardei:2006:ECP:1646656.1646898}, such as the coverage
+objective (target coverage or area coverage), the node deployment
+method (random or deterministic) and the heterogeneity of sensor nodes
+(common sensing range, common battery lifetime). 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 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} propose 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. In~\cite{cardei02}, a graph coloring
+technique is described to achieve energy savings by organizing the sensor nodes
+into a maximum number of disjoint dominating sets, which are activated
+successively. The dominating sets do not guarantee the coverage of the
+whole region of interest. Abrams et
+al.~\cite{Abrams:2004:SKA:984622.984684} design 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 includes 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.
%examine the target coverage problem by disjoint cover sets but relax the requirement that every cover set monitor all the targets and try to maximize the number of times the targets are covered by the partition. They propose various algorithms and establish approximation ratio.
-In~\cite{Cardei:2005:IWS:1160086.1160098}, the authors propose a heuristic to
-compute the disjoint set covers (DSC). In order to compute the maximum number of covers, they
-first transform DSC into a maximum-flow problem ,
-which is then formulated as a mixed integer programming problem
-(MIP). Based on the solution of the MIP, they design a heuristic
-to compute the final number of covers. The results show a slight performance
-improvement in terms of the number of produced DSC in comparison to~\cite{Slijepcevic01powerefficient} but it incurs
-higher execution time due to the complexity of the mixed integer programming resolution.
- %Cardei and Du \cite{Cardei:2005:IWS:1160086.1160098} propose 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 these 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{Zorbas2007} present B\{GOP\}, a centralized coverage algorithm introducing sensor candidate categorisation depending on their coverage status and the notion of critical target to call targets that are associated with a small number of sensors. The total running time of their heuristic is $0(m n^2)$ where $n$ is the number of sensors, and $m$ the number of targets. Compared to algorithm's results of Slijepcevic and Potkonjak \cite{Slijepcevic01powerefficient}, their heuristic produces more cover sets with a slight growth rate in execution time.
+In~\cite{Cardei:2005:IWS:1160086.1160098}, the authors propose a
+heuristic to compute the disjoint set covers (DSC). In order to
+compute the maximum number of covers, they first transform DSC into a
+maximum-flow problem, which is then formulated as a mixed integer
+programming problem (MIP). Based on the solution of the MIP, they
+design a heuristic to compute the final number of covers. The results
+show a slight performance improvement in terms of the number of
+produced DSC in comparison to~\cite{Slijepcevic01powerefficient}, but
+it incurs higher execution time due to the complexity of the mixed
+integer programming solving. %Cardei and Du
+\cite{Cardei:2005:IWS:1160086.1160098} propose 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{Zorbas2007} present B\{GOP\}, a centralized
+coverage algorithm introducing sensor candidate categorization
+depending on their coverage status and the notion of critical target
+to call targets that are associated with a small number of
+sensors. The total running time of their heuristic is $0(m n^2)$ where
+$n$ is the number of sensors, and $m$ the number of targets. Compared
+to algorithm's results of Slijepcevic and Potkonjak
+\cite{Slijepcevic01powerefficient}, their heuristic produces more
+cover sets with a slight growth rate in execution time.
%More recently Manju and Pujari\cite{Manju2011}
-In the case of non-disjoint algorithms \cite{Manju2011}, 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 incurs 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{cardei05bis} 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{Cardei:2005:IWS:1160086.1160098}. In~\cite{berman04}, the authors have formulated the lifetime problem and suggested another (LP) technique to solve this problem. A centralized provably near
-optimal solution based on the Garg-K\"{o}nemann algorithm~\cite{garg98} is also proposed.
-
-{\bf Our contribution}
-%{decoupage de la region en sous region, selection de noeud leader, formulation %et resolution du probleme de couverture, planification périodique
-There are three main questions which should be answered to build a scheduling strategy. We give a brief answer to these three questions to describe our approach before going into details in the subsequent sections.
+In the case of non-disjoint algorithms \cite{Manju2011}, 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{cardei05bis}
+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{Cardei:2005:IWS:1160086.1160098}. 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.
+
+\subsection{Our contribution}
+%{\bf Our contribution}
+
+There are three main questions, which should be addressed to build a
+scheduling strategy. We give a brief answer to these three questions
+to describe our approach before going into details in the subsequent
+sections.
\begin{itemize}
-\item {\bf How must be planned the
-phases for information exchange, decision and sensing over time?}
-Our algorithm partitions the time line into a number of periods. Each period contains 4 phases : information Exchange, Leader Election, Decision, and Sensing. Our work further divides sensing phase into a number of rounds of predetermined length.
-\item {\bf What are the rules to decide which node has to turn on or off?}
-Our algorithm tends to limit the overcoverage of points of interest to avoid turning on too much sensors covering the same areas at the same time, and tries to prevent undercoverage. The decision is a good compromise between these two conflicting objectives and is made for the next $T$ rounds of sensing. In our experimentations we will check which value of $T$ is the most appropriate.
-\item {\bf Which node should make such decision ?}
-As mentioned in \cite{pc10}, both centralized and distributed algorithms have their own advantages and disadvantages. Centralized coverage algorithms have the advantage of requiring very low processing power from the sensor nodes which have usually limited processing capabilities. Distributed algorithms are very adaptable to the dynamic and scalable nature of sensors network. Authors in \cite{pc10} concludes that there is a threshold in terms of network size to switch from a localized to a centralized algorithm. Indeed the exchange of messages in large networks may consume a considerable amount of energy in a localized approach compared to a centralized one. Our work does not consider only one leader to compute and to broadcast the schedule decision to all the sensors. When the size of network increases, the network is divided in many subregions and the decision is made by a leader in each subregion.
+\item {\bf How must the phases for information exchange, decision and
+ sensing be planned over time?} Our algorithm divides the time line
+ into a number of rounds. Each round contains 4 phases: Information
+ Exchange, Leader Election, Decision, and Sensing.
+
+\item {\bf What are the rules to decide which node has to be turned on
+ or off?} Our algorithm tends to limit the overcoverage of points of
+ interest to avoid turning on too many sensors covering the same
+ areas at the same time, and tries to prevent undercoverage. The
+ decision is a good compromise between these two conflicting
+ objectives.
+
+\item {\bf Which node should make such a decision?} As mentioned in
+ \cite{pc10}, both centralized and distributed algorithms have their
+ own advantages and disadvantages. Centralized coverage algorithms
+ have the advantage of requiring very low processing power from the
+ sensor nodes, which have usually limited processing capabilities.
+ Distributed algorithms are very adaptable to the dynamic and
+ scalable nature of sensors network. Authors in \cite{pc10} conclude
+ that there is a threshold in terms of network size to switch from a
+ localized to a centralized algorithm. Indeed, the exchange of
+ messages in large networks may consume a considerable amount of
+ energy in a centralized approach compared to a distributed one. Our
+ work does not consider only one leader to compute and to broadcast
+ the scheduling decision to all the sensors. When the network size
+ increases, the network is divided into many subregions and the
+ decision is made by a leader in each subregion.
\end{itemize}
+\section{Activity scheduling}
+\label{pd}
+We consider a randomly and uniformly deployed network consisting of
+static wireless sensors. The wireless sensors are deployed in high
+density to ensure initially a full coverage of the interested area. We
+assume that all nodes are homogeneous in terms of communication and
+processing capabilities and heterogeneous in term of energy provision.
+The location information is available to the sensor node either
+through hardware such as embedded GPS or through location discovery
+algorithms. The area of interest can be divided using the
+divide-and-conquer strategy into smaller areas called subregions and
+then our coverage protocol will be implemented in each subregion
+simultaneously. Our protocol works in rounds fashion as shown in
+figure~\ref{fig1}.
- \section{\uppercase{Distributed coverage model}}
-\label{pd}
-We consider a randomly and uniformly deployed network consisting of static wireless sensors. The wireless sensors are deployed in high density to ensure initially a full coverage of the interested area. We assume that all nodes are homogeneous in terms of communication and processing capabilities and heterogeneous in term of energy. The location information is available to the sensor node either through hardware such as embedded GPS or through location discovery algorithms.
-The area of interest can be divided using the divide-and-conquer strategy into smaller area called subregions and then our coverage protocol will be implemented in each subregion simultaneously. Our protocol works in rounds fashion as in figure \ref{fig:4}.
%Given the interested Area $A$, the wireless sensor nodes set $S=\lbrace s_1,\ldots,s_N \rbrace $ that are deployed randomly and uniformly in this area such that they are ensure a full coverage for A. The Area A is divided into regions $A=\lbrace A^1,\ldots,A^k,\ldots, A^{N_R} \rbrace$. We suppose that each sensor node $s_i$ know its location and its region. We will have a subset $SSET^k =\lbrace s_1,...,s_j,...,s_{N^k} \rbrace $ , where $s_N = s_{N^1} + s_{N^2} +,\ldots,+ s_{N^k} +,\ldots,+s_{N^R}$. Each sensor node $s_i$ has the same initial energy $IE_i$ in the first time and the current residual energy $RE_i$ equal to $IE_i$ in the first time for each $s_i$ in A. \\
\begin{figure}[ht!]
\centering
-\includegraphics [width=70mm]{FirstModel.eps}
-\caption{Multi-Round Coverage Protocol}
-\label{fig:4}
+\includegraphics[width=85mm]{FirstModel.eps} % 70mm
+\caption{Multi-round coverage protocol}
+\label{fig1}
\end{figure}
-Each round is divided into 4 phases : INFO Exchange, Leader Election, Decision, and Sensing. For each round there is exactly one set cover responsible for sensing task. This protocol is more reliable against the unexpectedly node failure because it works into rounds,and if the node failure detected before taking the decision, the node will not participate in decision and if the the node failure obtain after the decision the sensing task of the network will be affected temporarily only during the period of sensing until starting new round, since a new set cover will take charge of the sensing task in the next round. The energy consumption and some other constraints can easily be taken into account since the sensors can update and then exchange the information (including their residual energy) at the beginning of each round. However, the preprocessing phase (INFO Exchange, leader Election, Decision) are energy consuming for some nodes even when they not join the network to monitor the area. We describe each phase in more detail.
-
-\subsection{\textbf INFO Exchange Phase}
-
-Each sensor node $j$ sends its position, remaining energy $RE_j$, number of local neighbours $NBR_j$ to all wireless sensor nodes in its subregion by using INFO packet and listen to the packets sent from other nodes. After that, each node will have information about all the sensor nodes in the subregion. In our model.
-
-% the remaining energy corresponds to the time that a sensor can live in the active mode.
-
+Each round is divided into 4 phases : Information (INFO) Exchange,
+Leader Election, Decision, and Sensing. For each round there is
+exactly one set cover responsible for the sensing task. This protocol is
+more reliable against an unexpected node failure because it works
+in rounds. On the one hand, if a node failure is detected before
+making the decision, the node will not participate to this phase, and,
+on the other hand, if the node failure occurs after the decision, the
+sensing task of the network will be temporarily affected: only during
+the period of sensing until a new round starts, since a new set cover
+will take charge of the sensing task in the next round. The energy
+consumption and some other constraints can easily be taken into
+account since the sensors can update and then exchange their
+information (including their residual energy) at the beginning of each
+round. However, the pre-sensing phases (INFO Exchange, Leader
+Election, Decision) are energy consuming for some nodes, even when
+they do not join the network to monitor the area. Below, we describe
+each phase in more details.
+
+\subsection{Information exchange phase}
+
+Each sensor node $j$ sends its position, remaining energy $RE_j$, and
+the number of local neighbours $NBR_j$ to all wireless sensor nodes in
+its subregion by using an INFO packet and then listens to the packets
+sent from other nodes. After that, each node will have information
+about all the sensor nodes in the subregion. In our model, the
+remaining energy corresponds to the time that a sensor can live in the
+active mode.
%\subsection{\textbf Working Phase:}
%The working phase works in rounding fashion. Each round include 3 steps described as follow :
-\subsection{\textbf Leader Election Phase}
-This step includes choosing the Wireless Sensor Node Leader (WSNL) which will be responsible of executing coverage algorithm to choose the list of active sensor nodes that contribute in covering the subregion.
-% The sensors in the same region are capable to communicate with each others using a routing protocol provided by the simulator OMNET++ in order to provide multi-hop communication protocol.
-The WSNL will be chosen based on the number of local neighbours $NBR_j$ of sensor node $s_j$ and it's remaining energy $RE_j$.
-If we have more than one node has the same $NBR_j$ and $RE_j$, this leads to choose WSNL based on the largest index among them. Each subregion in the area of interest will select its WSNL independently for each round.
-
-
-\subsection{\textbf Decision Phase}
-The WSNL will execute the GLPK algorithm to select which sensors will be activated in the next rounds to cover the subregion. WSNL will send Active-Sleep packet to each sensor in the subregion based on algorithm's results.
+\subsection{Leader election phase}
+This step includes choosing the Wireless Sensor Node Leader (WSNL),
+which will be responsible for executing the coverage algorithm. Each
+subregion in the area of interest will select its own WSNL
+independently for each round. All the sensor nodes cooperate to
+select WSNL. The nodes in the same subregion will select the leader
+based on the received information from all other nodes in the same
+subregion. The selection criteria in order of priority are: larger
+number of neighbours, larger remaining energy, and then in case of
+equality, larger index.
+
+\subsection{Decision phase}
+The WSNL will solve an integer program (see section~\ref{cp}) to
+select which sensors will be activated in the following sensing phase
+to cover the subregion. WSNL will send Active-Sleep packet to each
+sensor in the subregion based on the algorithm's results.
%The main goal in this step after choosing the WSNL is to produce the best representative active nodes set that will take the responsibility of covering the whole region $A^k$ with minimum number of sensor nodes to prolong the lifetime in the wireless sensor network. For our problem, in each round we need to select the minimum set of sensor nodes to improve the lifetime of the network and in the same time taking into account covering the region $A^k$ . We need an optimal solution with tradeoff between our two conflicting objectives.
%The above region coverage problem can be formulated as a Multi-objective optimization problem and we can use the Binary Particle Swarm Optimization technique to solve it.
-\\
-
-\subsection{\textbf Sensing Phase}
- The algorithm will produce the best representative set of the active nodes that will take the mission of coverage preservation in the subregion during the Sensing phase. Since that we use a homogeneous wireless sensor network, we will assume that the cost of keeping a node awake (or sleep) for sensing task is the same for all wireless sensor nodes in the network.
-
+\subsection{Sensing phase}
+Active sensors in the round will execute their sensing task to
+preserve maximal coverage in the region of interest. We will assume
+that the cost of keeping a node awake (or asleep) for sensing task is
+the same for all wireless sensor nodes in the network. Each sensor
+will receive an Active-Sleep packet from WSNL informing it to stay
+awake or to go to sleep for a time equal to the period of sensing until
+starting a new round.
%\subsection{Sensing coverage model}
%\label{pd}
%\noindent We try to produce an adaptive scheduling which allows sensors to operate alternatively so as to prolong the network lifetime. For convenience, the notations and assumptions are described first.
%The wireless sensor node use the binary disk sensing model by which each sensor node will has a certain sensing range is reserved within a circular disk called radius $R_s$.
-\noindent We consider a boolean disk coverage model which is the most widely used sensor coverage model in the literature. Each sensor has a constant sensing range $R_s$. All space points within a disk centered at the sensor with the radius of the sensing range is said to be covered by this sensor. We also assume that the communication range is at least twice of the sening range. In fact, Zhang and Zhou ~\cite{Zhang05} prove that if the tranmission range is at least twice of the sensing range, a complete coverage of a convex area implies connectivity amnong the working nodes in the active mode.
+\indent We consider a boolean disk coverage model which is the most
+widely used sensor coverage model in the literature. Each sensor has a
+constant sensing range $R_s$. All space points within a disk centered
+at the sensor with the radius of the sensing range is said to be
+covered by this sensor. We also assume that the communication range is
+at least twice the size of the sensing range. In fact, Zhang and
+Zhou~\cite{Zhang05} proved that if the transmission range fulfills the
+previous hypothesis, a complete coverage of a convex area implies
+connectivity among the working nodes in the active mode.
%To calculate the coverage ratio for the area of interest, we can propose the following coverage model which is called Wireless Sensor Node Area Coverage Model to ensure that all the area within each node sensing range are covered. We can calculate the positions of the points in the circle disc of the sensing range of wireless sensor node based on the Unit Circle in figure~\ref{fig:cluster1}:
%\begin{figure}[h!]
%\end{figure}
%By using the Unit Circle in figure~\ref{fig:cluster1},
-%We choose to representEach wireless sensor node will be represented into a selected number of principle points by which we can know if the sensor node is covered or not.
-% Figure ~\ref{fig:cluster2} shows the selected principle points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.
-
-\noindent Instead of working with area coverage, we consider for each sensor a set of points called principal points. And we assume the sensing disk defined by a sensor is covered if all principal points of this sensor are covered.
+%We choose to representEach wireless sensor node will be represented into a selected number of primary points by which we can know if the sensor node is covered or not.
+% Figure ~\ref{fig:cluster2} shows the selected primary points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.
+\indent Instead of working with the coverage area, we consider for each
+sensor a set of points called primary points. We also assume that the
+sensing disk defined by a sensor is covered if all the primary points of
+this sensor are covered.
%\begin{figure}[h!]
%\centering
%\begin{tabular}{cc}
%\caption{Wireless Sensor Node Area Coverage Model.}
%\label{fig:cluster2}
%\end{figure}
-
-
-
-\noindent By knowing the position (point center :($p_x,p_y$) of the Wireless sensor node and its $R_s$ , we calculate the principle points directly based on proposed model. We use these principle points (that can be increased or decreased as if it is necessary) as references to ensure that the monitoring area of the region is covered by the selected set of sensors instead of using the all points in the area.
-
- \begin{figure}[h!]
-%\centering
-% \begin{multicols}{6}
-\centering
-%\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
-%\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
-\includegraphics[scale=0.2]{principles13.eps}
-%\includegraphics[scale=0.10]{fig25.pdf}\\~ ~ ~(d)
-%\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
-%\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
-%\end{multicols}
-\caption{Wireless Sensor node represented by 13 principle points }
-\label{fig3}
-\end{figure}
-
-\noindent We can calculate the positions of the selected principle points in the circle disk of the sensing range of wireless sensor node in figure ~\ref{fig3} as follow:\\
-$p_x,p_y$ = point center of wireless sensor node. \\
+By knowing the position (point center: ($p_x,p_y$)) of a wireless
+sensor node and its $R_s$, we calculate the primary points directly
+based on the proposed model. We use these primary points (that can be
+increased or decreased if necessary) as references to ensure that the
+monitored region of interest is covered by the selected set of
+sensors, instead of using all the points in the area.
+
+\indent We can calculate the positions of the selected primary
+points in the circle disk of the sensing range of a wireless sensor
+node (see figure~\ref{fig2}) as follows:\\
+$(p_x,p_y)$ = point center of wireless sensor node\\
$X_1=(p_x,p_y)$ \\
$X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\
$X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\
$X_{10}=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
$X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
$X_{12}=( p_x + R_s * (0), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
-$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
-
-
+$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $.
+ \begin{figure}[h!]
+%\centering
+% \begin{multicols}{6}
+\centering
+%\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
+%\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
+\includegraphics[scale=0.25]{principles13.eps}
+%\includegraphics[scale=0.10]{fig25.pdf}\\~ ~ ~(d)
+%\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
+%\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
+%\end{multicols}
+\caption{Wireless sensor node represented by 13 primary points}
+\label{fig2}
+\end{figure}
-\section{\uppercase{Coverage problem formulation}}
+\section{Coverage problem formulation}
\label{cp}
%We can formulate our optimization problem as energy cost minimization by minimize the number of active sensor nodes and maximizing the coverage rate at the same time in each $A^k$ . This optimization problem can be formulated as follow: Since that we use a homogeneous wireless sensor network, we will assume that the cost of keeping a node awake is the same for all wireless sensor nodes in the network. We can define the decision parameter $X_j$ as in \eqref{eq11}:\\
%To satisfy the coverage requirement, the set of the principal points that will represent all the sensor nodes in the monitored region as $PSET= \lbrace P_1,\ldots ,P_p, \ldots , P_{N_P^k} \rbrace $, where $N_P^k = N_{sp} * N^k $ and according to the proposed model in figure ~\ref{fig:cluster2}. These points can be used by the wireless sensor node leader which will be chosen in each region in A to build a new parameter $\alpha_{jp}$ that represents the coverage possibility for each principal point $P_p$ of each wireless sensor node $s_j$ in $A^k$ as in \eqref{eq12}:\\
-
-\noindent Our model is based on the model proposed by \cite{pedraza2006} where the objective is to find a maximum number of disjoint cover sets. To accomplish this goal, authors propose a integer program which forces undercoverage and overcoverage of targets to become minimal at the same time. They use variables $x_{s,l}$ to indicate if the sensor $s$ belongs to cover set $l$. In our model, we consider binary variables $X_{j,t}$ which determine the activation of sensor $j$ in round $t$. We replace the constraint guarantying that each sensor is a member of only one cover of the entire set of disjoint covers by a constraint specifying that the sum of energy consumed by the activation of sensor during several rounds is less than or equal to the remaining energy of the sensor. We also consider principle points as targets. \\
-\noindent For a principle point $p$, let $\alpha_{jp}$ denote the indicator function of whether the point $p$ is covered, that is, \\
+\indent Our model is based on the model proposed by
+\cite{pedraza2006} where the objective is to find a maximum number of
+disjoint cover sets. To accomplish this goal, 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 binary variables $X_{j}$, which determine the
+activation of sensor $j$ in the sensing phase of the round. We also
+consider primary points as targets. The set of primary points is
+denoted by $P$ and the set of sensors by $J$.
+
+\noindent For a primary point $p$, let $\alpha_{jp}$ denote the
+indicator function of whether the point $p$ is covered, that is:
\begin{equation}
\alpha_{jp} = \left \{
\begin{array}{l l}
- 1 & \mbox{if the principal point $p$ is covered} \\
- & \mbox{by active sensor node $j$} \\
- 0 & \mbox{Otherwise}\\
+ 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 sensors that are covering point $p$ during a round $t$ is equal to $\sum_{j \in J} \alpha_{jp} * X_{j,t}$ where :
+The number of active sensors that cover the primary point $p$ is equal
+to $\sum_{j \in J} \alpha_{jp} * X_{j}$ where:
\begin{equation}
-X_{j,t} = \left \{
+X_{j} = \left \{
\begin{array}{l l}
- 1& \mbox{if sensor $s_j$ is active during round } t\\
- 0 & \mbox{Otherwise}\\
+ 1& \mbox{if sensor $j$ is active,} \\
+ 0 & \mbox{otherwise.}\\
\end{array} \right.
%\label{eq11}
\end{equation}
-We define the Overcoverage variable $\Theta_{p,t}$ .\\
-
+We define the Overcoverage variable $\Theta_{p}$ as:
\begin{equation}
- \Theta_{p,t} = \left \{
+ \Theta_{p} = \left \{
\begin{array}{l l}
- 0 & \mbox{if point p is not }\\
-&\mbox{covered during round } t\\
- \left( \sum_{j \in J} \alpha_{jp} * X_{j,t} \right)- 1 & \mbox{Otherwise}\\
+ 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$\Theta_{p}$ represents the number of active sensor nodes minus one that cover the principle point $p$.\\
-The Undercoverage variable $U_{p,t}$ of the principle point $p$ is defined as follow :\\
-
+\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,t} = \left \{
+U_{p} = \left \{
\begin{array}{l l}
- 1 &\mbox{if point } $p$ \mbox{ is not covered during round } $t$\\
- 0 & \mbox{Otherwise}\\
+ 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 be formulated as follow.\\
+\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,t} \Theta_{p,t} + w_{u,t} U_{p,t})&\\
+\min \sum_{p \in P} (w_{\theta} \Theta_{p} + w_{U} U_{p})&\\
\textrm{subject to :}&\\
-\sum_{j \in J} \alpha_{jp} X_{j,t} - \Theta_{p,t}+ U_{p,t} =1, &\forall p \in P, \forall t \in T\\
+\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 \\
+%\sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &\forall j \in J \\
%\label{c2}
-\Theta_{p,t}\in \mathbb{N} , &\forall p \in P, \forall t \in T\\
-U_{p,t} \in \{0,1\}, &\forall p \in P, \forall t \in T \\
-X_{j,t} \in \{0,1\}, &\forall j \in J, \forall t \in T
+\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 $X_{j,t}$ : indicating whether or not sensor $j$ is active in round $t$(1 if yes and 0 if not)
-\item $\Theta_{p,t}$ : {\it overcoverage}, the number of sensors minus one that are covering point $p$ in round $t$
-\item $U_{p,t}$ : {\it undercoverage}, indicating whether or not point $p$ is being covered (1 if not covered and 0 if covered) in round $t$
+\item $X_{j}$ : indicates whether or not the sensor $j$ is actively
+ sensing in the round (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}
-The first group of constraints indicates that some point $p$ should be covered by at least one sensor in every round $t$ and, if it is not always the case, overcoverage and undercoverage variables help balance the restriction equation by taking positive values. Second group of contraints ensures for each sensor that the amount of energy consumed during its activation periods will be less than or equal to its remaining energy.
-There are two main objectives. We limit overcoverage of principle points in order to activate a minimum number of sensors and we prevent that parts of the subregion are not monitored by minimizing undercoverage. The weights $w_{\theta,t}$ and $w_{u,t}$ must be properly chosen so as to guarantee that the maximum number of points are covered during each round.
+
+The first group of constraints indicates that some primary point $p$
+should be covered by at least one sensor and, if it is not always the
+case, overcoverage and undercoverage variables help balancing the
+restriction equations by taking positive values. There are two main
+objectives. First, we limit the overcoverage of primary points in order to
+activate a minimum number of sensors. Second we prevent the absence of monitoring on
+ some parts of the subregion by minimizing the undercoverage. The
+weights $w_\theta$ and $w_U$ must be properly chosen so as to
+guarantee that the maximum number of points are covered during each
+round.
-%In equation \eqref{eq15}, there are two main objectives: the first one using the Overcoverage parameter to minimize the number of active sensor nodes in the produced final solution vector $X$ which leads to improve the life time of wireless sensor network. The second goal by using the Undercoverage parameter to maximize the coverage in the region by means of covering each principle point in $SSET^k$.The two objectives are achieved at the same time. The constraint which represented in equation \eqref{eq16} refer to the coverage function for each principle point $P_p$ in $SSET^k$ , where each $P_p$ should be covered by
+%In equation \eqref{eq15}, there are two main objectives: the first one using the Overcoverage parameter to minimize the number of active sensor nodes in the produced final solution vector $X$ which leads to improve the life time of wireless sensor network. The second goal by using the Undercoverage parameter to maximize the coverage in the region by means of covering each primary point in $SSET^k$.The two objectives are achieved at the same time. The constraint which represented in equation \eqref{eq16} refer to the coverage function for each primary point $P_p$ in $SSET^k$ , where each $P_p$ should be covered by
%at least one sensor node in $A^k$. The objective function in \eqref{eq15} involving two main objectives to be optimized simultaneously, where optimal decisions need to be taken in the presence of trade-offs between the two conflicting main objectives in \eqref{eq15} and this refer to that our coverage optimization problem is a multi-objective optimization problem and we can use the BPSO to solve it. The concept of Overcoverage and Undercoverage inspired from ~\cite{Fernan12} but we use it with our model as stated in subsection \ref{Sensing Coverage Model} with some modification to be applied later by BPSO.
%\subsection{Notations and assumptions}
%\item $|.|$ : cardinality of the set
%\end{itemize}
-\section{\uppercase{Simulation Results}}
-\label{exp}
-In this section, we conducted a series of simulations to evaluate the efficiency of our approach
-based on the discrete event simulator OMNeT++ (http://www.omnetpp.org/).we conduct simulations for six
-different densities varying from 50 to 300 nodes. Experimental results were obtained from randomly generated
-networks in which nodes are deployed over a $ 50\times25(m2) $sensing field. For each network deployment, we
-assume that the deployed nodes can fully cover the sensing field with the given sensing range. 100 simulation runs are performed with different network topologies. The results presented hereafter are the average of these 100 runs.Simulation ends when there is at least one active node has no connectivity with the network.Our proposed coverage protocol use the Radio energy dissipation model that defined by~\cite{HeinzelmanCB02} as energy consumption model by each wireless sensor node for transmitting and receiving the packets in the network.The energy of each node in the network is initialized randomly within the range 24-60 joules, and each sensor will consumes 0.2 watts during the period of sensing which it is 60 seconds.Each active node will consumes 12 joules during sensing phase and each sleep node will consumes 0.002 joules.Each sensor node will not participate in the next round if it's remaining energy less than 12 joules. In all experiments the parameters are given by $R_s = 5m $ , $ W_{\Theta} =1$ and $W_{\Psi} = P^2$.
-We evaluate the efficiency of our approach using some performance metrics such as:coverage ratio, number of
-active nodes ratio, energy saving ratio, number of rounds, network lifetime and execution time of our approach.Coverage ratio measures how much area of a sensor field is covered. In our case, the coverage ratio is regarded as the number of principle points covered among the set of all principle points within the field.In our simulation the sensing field is sub divided into two subregions each one equal to $ 25\times25(m2) $ of the sensing field.
-\subsection{The impact of the Number of Rounds on Coverage Ratio:}
-In this experiment, we study the impact of the number of rounds on the coverage ratio and for different sizes for sensor network.For each Sensor network size we will take the average of coverage ratio per round and for 100 simulation.Fig. 3 show the impact of the number of rounds on coverage ratio for different network sizes and for two subregions.
+\section{Simulation results}
+\label{exp}
- \begin{figure}[h!]
-%\centering
-% \begin{multicols}{6}
+In this section, we conducted a series of simulations to evaluate the
+efficiency and the relevance of our approach, using the discrete event
+simulator OMNeT++ \cite{varga}. We performed simulations for five
+different densities varying from 50 to 250~nodes. Experimental results
+were obtained from randomly generated networks in which nodes are
+deployed over a $(50 \times 25)~m^2 $ sensing field.
+More precisely, the deployment is controlled at a coarse scale in
+ order to ensure that the deployed nodes can fully cover the sensing
+ field with the given sensing range.
+10~simulation runs are performed with
+different network topologies for each node density. The results
+presented hereafter are the average of these 10 runs. A simulation
+ends when all the nodes are dead or the sensor network becomes
+disconnected (some nodes may not be able to send, to a base station, an
+event they sense).
+
+Our proposed coverage protocol uses the radio energy dissipation model
+defined by~\cite{HeinzelmanCB02} as energy consumption model for each
+wireless sensor node when transmitting or receiving packets. The
+energy of each node in a network is initialized randomly within the
+range 24-60~joules, and each sensor node will consume 0.2 watts during
+the sensing period, which will last 60 seconds. Thus, an
+active node will consume 12~joules during the sensing phase, while a
+sleeping node will use 0.002 joules. Each sensor node will not
+participate in the next round if its remaining energy is less than 12
+joules. In all experiments, the parameters are set as follows:
+$R_s=5~m$, $w_{\Theta}=1$, and $w_{U}=|P^2|$.
+
+We evaluate the efficiency of our approach by using some performance
+metrics such as: coverage ratio, number of active nodes ratio, energy
+saving ratio, energy consumption, network lifetime, execution time,
+and number of stopped simulation runs. Our approach called strategy~2
+(with two leaders) works with two subregions, each one having a size
+of $(25 \times 25)~m^2$. Our strategy will be compared with two other
+approaches. The first one, called strategy~1 (with one leader), works
+as strategy~2, but considers only one region of $(50 \times 25)$ $m^2$
+with only one leader. The other approach, called Simple Heuristic,
+consists in uniformly dividing the region into squares of $(5 \times
+5)~m^2$. During the decision phase, in each square, a sensor is
+randomly chosen, it will remain turned on for the coming sensing
+phase.
+
+\subsection{The impact of the number of rounds on the coverage ratio}
+
+In this experiment, the coverage ratio measures how much the area of a
+sensor field is covered. In our case, the coverage ratio is regarded
+as the number of primary points covered among the set of all primary
+points within the field. Figure~\ref{fig3} shows the impact of the
+number of rounds on the average coverage ratio for 150 deployed nodes
+for the three approaches. It can be seen that the three approaches
+give similar coverage ratios during the first rounds. From the
+9th~round the coverage ratio decreases continuously with the simple
+heuristic, while the two other strategies provide superior coverage to
+$90\%$ for five more rounds. Coverage ratio decreases when the number
+of rounds increases due to dead nodes. Although some nodes are dead,
+thanks to strategy~1 or~2, other nodes are preserved to ensure the
+coverage. Moreover, when we have a dense sensor network, it leads to
+maintain the full coverage for a larger number of rounds. Strategy~2 is
+slightly more efficient than strategy 1, because strategy~2 subdivides
+the region into 2~subregions and if one of the two subregions becomes
+disconnected, the coverage may be still ensured in the remaining
+subregion.
+
+\parskip 0pt
+\begin{figure}[h!]
\centering
-%\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
-%\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
-\includegraphics[scale=0.5]{CR2R2L_1.eps}\\~ ~ ~(a)
-\includegraphics[scale=0.5]{CR2R2L_2.eps}\\~ ~ ~(b)
-%\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
-%\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
-%\end{multicols}
-\caption{The impact of the Number of Rounds on Coverage Ratio.(a):subregion 1. (b): subregion 2 }
+\includegraphics[scale=0.5]{TheCoverageRatio150g.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure}
-As shown Fig. 3 (a) and (b) our protocol can give a full average coverage ratio in the first rounds and then it decreases when the number of rounds increases due to dead nodes.Although some nodes are dead, sensor activity scheduling choose other nodes to ensure the coverage of interest area. Moreover, when we have a dense sensor network, it leads to maintain the full coverage for larger number of rounds.
-
-\subsection{The impact of the Number of Rounds on Energy Saving Ratio:}
-
-\subsection{The impact of the Number of Rounds on Active Sensor Ratio:}
+\subsection{The impact of the number of rounds on the active sensors ratio}
+
+It is important to have as few active nodes as possible in each round,
+in order to minimize the communication overhead and maximize the
+network lifetime. This point is assessed through the Active Sensors
+Ratio (ASR), which is defined as follows:
+\begin{equation*}
+\scriptsize
+\mbox{ASR}(\%) = \frac{\mbox{Number of active sensors
+during the current sensing phase}}{\mbox{Total number of sensors in the network
+for the region}} \times 100.
+\end{equation*}
+Figure~\ref{fig4} shows the average active nodes ratio versus rounds
+for 150 deployed nodes.
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.5]{TheActiveSensorRatio150g.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the active sensors ratio for 150 deployed nodes }
+\label{fig4}
+\end{figure}
-\subsection{The impact of Number of Sensors on Number of Rounds:}
+The results presented in figure~\ref{fig4} show the superiority of
+both proposed strategies, the strategy with two leaders and the one
+with a single leader, in comparison with the simple heuristic. The
+strategy with one leader uses less active nodes than the strategy with
+two leaders until the last rounds, because it uses central control on
+the whole sensing field. The advantage of the strategy~2 approach is
+that even if a network is disconnected in one subregion, the other one
+usually continues the optimization process, and this extends the
+lifetime of the network.
+
+\subsection{The impact of the number of rounds on the energy saving ratio}
+
+In this experiment, we consider a performance metric linked to energy.
+This metric, called Energy Saving Ratio (ESR), is defined by:
+\begin{equation*}
+\scriptsize
+\mbox{ESR}(\%) = \frac{\mbox{Number of alive sensors during this round}}
+{\mbox{Total number of sensors in the network for the region}} \times 100.
+\end{equation*}
+The longer the ratio is, the more redundant sensor nodes are
+switched off, and consequently the longer the network may live.
+Figure~\ref{fig5} shows the average Energy Saving Ratio versus rounds
+for all three approaches and for 150 deployed nodes.
+
+\begin{figure}[h!]
+%\centering
+% \begin{multicols}{6}
+\centering
+\includegraphics[scale=0.5]{TheEnergySavingRatio150g.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the energy saving ratio for 150 deployed nodes}
+\label{fig5}
+\end{figure}
-\subsection{The impact of Number of Sensors on Network Lifetime:}
+The simulation results show that our strategies allow to efficiently
+save energy by turning off some sensors during the sensing phase. As
+expected, the strategy with one leader is usually slightly better than
+the second strategy, because the global optimization permits to turn
+off more sensors. Indeed, when there are two subregions more nodes
+remain awake near the border shared by them. Note that again as the
+number of rounds increases the two leaders' strategy becomes the most
+performing one, since it takes longer to have the two subregion networks
+simultaneously disconnected.
+
+\subsection{The percentage of stopped simulation runs}
+
+We will now study the percentage of simulations, which stopped due to
+network disconnections per round for each of the three approaches.
+Figure~\ref{fig6} illustrates the percentage of stopped simulation
+runs per round for 150 deployed nodes. It can be observed that the
+simple heuristic is the approach, which stops first because the nodes
+are randomly chosen. Among the two proposed strategies, the
+centralized one first exhibits network disconnections. Thus, as
+explained previously, in case of the strategy with several subregions
+the optimization effectively continues as long as a network in a
+subregion is still connected. This longer partial coverage
+optimization participates in extending the network lifetime.
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.5]{TheNumberofStoppedSimulationRuns150g.eps}
+\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }
+\label{fig6}
+\end{figure}
-\subsection{The impact of Number of Sensors on Execution Time:}
+\subsection{The energy consumption}
+
+In this experiment, we study the effect of the multi-hop communication
+protocol on the performance of the strategy with two leaders and
+compare it with the other two approaches. The average energy
+consumption resulting from wireless communications is calculated
+by taking into account the energy spent by all the nodes when transmitting and
+receiving packets during the network lifetime. This average value,
+which is obtained for 10~simulation runs, is then divided by the
+average number of rounds to define a metric allowing a fair comparison
+between networks having different densities.
+
+Figure~\ref{fig7} illustrates the energy consumption for the different
+network sizes and the three approaches. The results show that the
+strategy with two leaders is the most competitive from the energy
+consumption point of view. A centralized method, like the strategy
+with one leader, has a high energy consumption due to many
+communications. In fact, a distributed method greatly reduces the
+number of communications thanks to the partitioning of the initial
+network in several independent subnetworks. Let us notice that even if
+a centralized method consumes far more energy than the simple
+heuristic, since the energy cost of communications during a round is a
+small part of the energy spent in the sensing phase, the
+communications have a small impact on the network lifetime.
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.5]{TheEnergyConsumptiong.eps}
+\caption{The energy consumption}
+\label{fig7}
+\end{figure}
-\subsection{Performance Comparison:}
-\label{Simulation Results}
+\subsection{The impact of the number of sensors on execution time}
+
+A sensor node has limited energy resources and computing power,
+therefore it is important that the proposed algorithm has the shortest
+possible execution time. The energy of a sensor node must be mainly
+used for the sensing phase, not for the pre-sensing ones.
+Table~\ref{table1} gives the average execution times in seconds
+on a laptop of the decision phase (solving of the optimization problem)
+during one round. They are given for the different approaches and
+various numbers of sensors. The lack of any optimization explains why
+the heuristic has very low execution times. Conversely, the strategy
+with one leader, which requires to solve an optimization problem
+considering all the nodes presents redhibitory execution times.
+Moreover, increasing the network size by 50~nodes multiplies the time
+by almost a factor of 10. The strategy with two leaders has more
+suitable times. We think that in distributed fashion the solving of
+the optimization problem in a subregion can be tackled by sensor
+nodes. Overall, to be able to deal with very large networks, a
+distributed method is clearly required.
+
+\begin{table}[ht]
+\caption{THE EXECUTION TIME(S) VS THE NUMBER OF SENSORS}
+% title of Table
+\centering
+% used for centering table
+\begin{tabular}{|c|c|c|c|}
+% centered columns (4 columns)
+ \hline
+%inserts double horizontal lines
+Sensors number & Strategy~2 & Strategy~1 & Simple heuristic \\ [0.5ex]
+ & (with two leaders) & (with one leader) & \\ [0.5ex]
+%Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
+% inserts table
+%heading
+\hline
+% inserts single horizontal line
+50 & 0.097 & 0.189 & 0.001 \\
+% inserting body of the table
+\hline
+100 & 0.419 & 1.972 & 0.0032 \\
+\hline
+150 & 1.295 & 13.098 & 0.0032 \\
+\hline
+200 & 4.54 & 169.469 & 0.0046 \\
+\hline
+250 & 12.252 & 1581.163 & 0.0056 \\
+% [1ex] adds vertical space
+\hline
+%inserts single line
+\end{tabular}
+\label{table1}
+% is used to refer this table in the text
+\end{table}
+
+\subsection{The network lifetime}
+
+Finally, we have defined the network lifetime as the time until all
+nodes have been drained of their energy or each sensor network
+monitoring an area has become disconnected. In figure~\ref{fig8}, the
+network lifetime for different network sizes and for both strategy
+with two leaders and the simple heuristic is illustrated.
+ We do not consider anymore the centralized strategy with one
+ leader, because, as shown above, this strategy results in execution
+ times that quickly become unsuitable for a sensor network.
+
+\begin{figure}[h!]
+%\centering
+% \begin{multicols}{6}
+\centering
+\includegraphics[scale=0.5]{TheNetworkLifetimeg.eps} %\\~ ~ ~(a)
+\caption{The network lifetime }
+\label{fig8}
+\end{figure}
-\section{\uppercase{Conclusions}}
+As highlighted by figure~\ref{fig8}, the network lifetime obviously
+increases when the size of the network increases, with our approach
+that leads to the larger lifetime improvement. By choosing the best
+suited nodes, for each round, to cover the region of interest and by
+letting the other ones sleep in order to be used later in next rounds,
+our strategy efficiently prolonges the network lifetime. Comparison shows that
+the larger the sensor number is, the more our strategies outperform
+the simple heuristic. Strategy~2, which uses two leaders, is the best
+one because it is robust to network disconnection in one subregion. It
+also means that distributing the algorithm in each node and
+subdividing the sensing field into many subregions, which are managed
+independently and simultaneously, is the most relevant way to maximize
+the lifetime of a network.
+
+\section{Conclusion and future works}
\label{sec:conclusion}
-In this paper, we have addressed the problem of lifetime optimization in wireless sensor networks. This is a very
-natural and important problem, as sensor nodes
-have limited resources in terms of memory, energy and computational power.
-%energy-efficiency is crucial in power-limited wireless sensor network.
-To cope with this problem,
-%an efficient centralized energy-aware algorithm is presented and analyzed. Our algorithm seeks to
-%Energy-efficiency is crucial in power-limited wireless sensor network, since sensors have significant power constraints (battery life). In this paper we have investigated the problem of
-
-
-
-
+In this paper, we have addressed the problem of the coverage and the lifetime
+optimization in wireless sensor networks. This is a key issue as
+sensor nodes have limited resources in terms of memory, energy and
+computational power. To cope with this problem, the field of sensing
+is divided into smaller subregions using the concept of
+divide-and-conquer method, and then a multi-rounds coverage protocol
+will optimize coverage and lifetime performances in each subregion.
+The proposed protocol combines two efficient techniques: network
+leader election and sensor activity scheduling, where the challenges
+include how to select the most efficient leader in each subregion and
+the best representative active nodes that will optimize the network lifetime
+while taking the responsibility of covering the corresponding
+subregion. The network lifetime in each subregion is divided into
+rounds, each round consists of four phases: (i) Information Exchange,
+(ii) Leader Election, (iii) an optimization-based Decision in order to
+select the nodes remaining active for the last phase, and (iv)
+Sensing. The simulations show the relevance of the proposed
+protocol in terms of lifetime, coverage ratio, active sensors ratio,
+energy saving, energy consumption, execution time, and the number of
+stopped simulation runs due to network disconnection. Indeed, when
+dealing with large and dense wireless sensor networks, a distributed
+approach like the one we propose allows to reduce the difficulty of a
+single global optimization problem by partitioning it in many smaller
+problems, one per subregion, that can be solved more easily.
+
+In future work, we plan to study and propose a coverage protocol, which
+computes all active sensor schedules in one time, using
+optimization methods such as swarms optimization or evolutionary
+algorithms. The round will still consist of 4 phases, but the
+ decision phase will compute the schedules for several sensing phases,
+ which aggregated together, define a kind of meta-sensing phase.
+The computation of all cover sets in one time is far more
+difficult, but will reduce the communication overhead.
% use section* for acknowledgement
-\section*{Acknowledgment}
-
-
-
-
-\begin{thebibliography}{1}
-
-
-\bibitem{wns07}
- J. Wang, C. Niu, and R. Shen,~\emph{Randomized approach for target coverage scheduling in directional sensor network},~ICESS2007,~pp. 379-390,~2007.
-
-
-
-\bibitem{dw60}
- G.~B. Dantzig and P.~ Wolfe,~\emph {Decomposition principle for linear programs},~Operations Research,~pp. 101-111,~ 1960.
-
-
-\bibitem{asc02}
- I.~ Akyildiz ,~ W.~ Su ,~ Y.~Sankarasubramniam and E. Cayirci,~ emph{A survey on sensor networks},~
- IEEE Comm. Magazine,~pp. 102-114,~2002.
-
-
-\bibitem{pc10}
- T.~V.~ Padmavathy and M. Chitra,~emph{Extending the Network Lifetime of Wireless Sensor Networks Using Residual Energy Extraction—Hybrid Scheduling Algorithm},~Int. J. of Communications, Network and System Sciences ,~ VOL.~3,No. ~1,pp.~98-106,~2010.
-
-
-\bibitem{Slijepcevic01powerefficient}
- S.~ Slijepcevic and M.~ Potkonjak,~emph{Power Efficient Organization of Wireless Sensor Networks},~
- IEEE International conference on Communications,~pp.~472--476,~2001.
-
-\bibitem{Cardei:2005:IWS:1160086.1160098}
- M.~Cardei, and D.~ Du, ~emph {Improving wireless sensor network lifetime through power aware organization},~
- Wireless Networks,~ vol.~11,~ issue~3,~pp.~333--340,~May~2005.
-
-
-
-\bibitem{Cardei05energy-efficienttarget}
- M.~ Cardei ,~ M.~ T.~ Thai,~ Y.~ Li and W.~ Wu,~ emph{Energy-efficient target coverage in wireless sensor networks},IEEE Infocom,~pp.~1976--1984,~2005.
-
-
-\bibitem{b04}
- P.~ Berman,~ G.~ Calinescu ,~ C.~ Shah and A.~ Zelikovsky,~emph{Power efficient monitoring management in sensor networks},~Wireless Communications and Networking Conference,~ WCNC. ~2004,~2004.
-
-\bibitem{Berman05efficientenergy}
- P.~ Berman ,~ G.~ Calinescu ,~ C.~ Shah and A. Zelikovsky,~emph {Efficient energy management in sensor networks},~ Ad Hoc and Sensor Networks. Nova Science Publishers,~Nova Science Publisher,~2005.
-
-\bibitem{1240799}
- J.~ Lu and T.~ Suda,~ emph{Coverage-aware self-scheduling in sensor networks},~ Proceedings. 2003 IEEE 18th Annual Workshop on Computer Communications CCW 2003,~pp.~117-123,~2003.
-
-
-\bibitem{Prasad:2007:DAL:1782174.1782218}
- S.~K.~Prasad, and A.~Dhawan,~emph{Distributed algorithms for lifetime of wireless sensor networks based on dependencies among cover sets},~Proceedings of the 14th international conference on High performance computing ,~HiPC'07,pp.~381--392,~2007.
-
-
-
-\bibitem{Zorbas:2010:SCP:1715927.1715956}
- D.~ Zorbas ,~ D. Glynos ,~ P.~ Kotzanikolaou and C.~ Douligeris,~emph {Solving coverage problems in wireless sensor networks using cover sets},~ Ad Hoc Netw.,~ vol.~8,issue~4,~pp.~400--415,June~ 2010.
-
-
-\bibitem{Garey:1990:CIG:574848}
- M.~R.~ Garey and D.~S.~ Johnson,~ emph {Computers and Intractability; A Guide to the Theory of NP-Completeness},~ W. H. Freeman Co.,~1990.
-
-
-\bibitem{Caprara98algorithmsfor}
- A.~ Caprara ,~ M.~ Fischetti and P.~ Toth ,~ emph{Algorithms for the Set Covering Problem},~ Annals of Operations Research,~ vol.~98,~1998.
-
-
-\bibitem{DBLP:journals/eor/AlfieriBBC07}
- A.~Alfieri ,~ A.~ Bianco,~ P.~ Brandimarte and C. Chiasserini,~emph {Maximizing system lifetime in wireless sensor networks},~European Journal of Operational Research,~ vol.~181,~ No.~1,~pp.390-402,~2007.
-
-
-\bibitem{Gu:2009:QTC:1673018.1673025}
- Y.~ Gu,~ Y.~Ji,~ J. ~Li and B.~ Zhao,~emph {QoS-aware target coverage in wireless sensor networks},~Wirel. Commun. Mob. Comput.,~ vol.~9,~issue~12,~pp.~1645--1659,December~ 2009.
-
-
-\bibitem{springerlink:10.1007/BF01594937}
- Y.~ Ye,~emph{An O($n^3L$)) potential reduction algorithm for linear programming},~Mathematical Programming,~vol.~50,~issue~1,~ pp.~239-258,~ 1991.
-
-
-
-\bibitem{glpk}
- A.~ Mahkorin,~emph {GNU Linear Programming Kit, Reference Manual},~2010.
-
-\bibitem{Garg98fasterand}
- N.~ Garg and J.~ K\"{o}nemann,~emph {Faster and simpler algorithms for multicommodity flow and other fractional packing problems},~In Proceedings of the 39th Annual Symposium on Foundations of Computer Science,pp.~300--309,~ 1998.
-
-
-\bibitem{Kuhn55}
- H.~ W.~ Kuhn,~emph{The Hungarian Method for the assignment problem},~Naval Research Logistics Quarterly,~vol.~2,~pp.~83--97,~1955.
-
-\bibitem{Cardei:2006:ECP:1646656.1646898}
-M.~Cardei and J.~ Wu,~emph {Energy-efficient coverage problems in wireless ad-hoc sensor networks},~Comput. Commun.,~ vol.~29,~ no.~4,~pp.~413--420,~February 2006.
-
-
-\bibitem{Manju2011}
- M.~ Chaudhary and A.~ K. ~ Pujari,~emph{High-Energy-First (HEF) Heuristic for Energy-Efficient Target Coverage Problem},~International Journal of Ad hoc, Sensor and Uniquitous computing (IJASUC),~ vol.~2,
- ~no. ~1,~ 2011.
-
-
-
-\bibitem{Abrams:2004:SKA:984622.984684}
-Z.~ Abrams,~ A. ~ Goel and S.~ Plotkin, ~ emph{Set k-cover algorithms for energy efficient monitoring in wireless sensor networks},~ Proceedings of the 3rd international symposium on Information processing in sensor networks,~pp.~ 424--432,~ 2004.
-
-
-
-\bibitem{Zorbas2007}
-D.~ Zorbas ,~ D.~ Glynos ,~ P.~ Kotzanikolaou and C.~ Douligeris,~emph {B\{GOP\}: an adaptive coverage algorithm for wireless sensor networks},~Proceedings of the 13th European Wireless Conference,~Paris,~ France,~2007.
-
-\bibitem{cardei02}
- M.~ Cardei ,~D.~ MacCallum ,~M.~ X.~ Cheng ,~M.~ Min ,~X.~ Jia ,~D.~ Li and
- D.~ Du,~emph {Wireless Sensor Networks with Energy Efficient Organization},~Journal of Interconnection Networks,~ vol.~3,~ no.~3-4,pp.~213-229,~2002.
-
-
-
-\bibitem{cardei05}
- M.~ Cardei ,~ J.~ Wu, N.~ Lu and M.~O.~ Pervaiz,~emph{Maximum Network Lifetime with Adjustable Range},~WiMob,~2005.
-
-
-\bibitem{cardei05bis}
- M.~ Cardei ,~ M.~ T.~ Thai ,~Y.~ Li and W.~ Wu,~emph {Energy-efficient target coverage in wireless sensor networks},~INFOCOM,~pp.~1976-1984,~2005.
-
-
-
-\bibitem{berman04}
- P.~ Berman and G.~ Calinescu,~emph {Power Efficient Monitoring Management in Sensor Networks},~Proceedings of IEEE Wireless Communication and Networking Conference (WCNC'04),~pp.~2329--2334,~2004.
-
-
-\bibitem{garg98}
- N.~ Garg and J.~ Koenemann,~emph {Faster and Simpler Algorithms for Multicommodity Flow and other Fractional Packing Problems.},~Proceedings of the 39th Annual Symposium on Foundations of Computer Science,~pp.~300--309,~1998.
-
-
-\bibitem{Gallais06}
- A.~ Gallais,~ J.~ Carle ,~D. ~ Simplot-Ryl and I.~ Stojmenovic,~emph {Localized Sensor Area Coverage with Low Communication Overhead},~Proceedings of the Fourth Annual IEEE International Conference on Pervasive Computing and Communications,pp.~328--337,~2006.
-
-\bibitem{Tian02}
- D. ~ Tian and N.~ D.~ Georganas, ~emph {A coverage-preserving node scheduling scheme for large wireless sensor networks},~Proceedings of the 1st ACM international workshop on Wireless sensor networks and applications,~pp.~ 32--41,~2002.
-
-
-\bibitem{Ye03}
-F. ~Ye,G.~ Zhong,~J.~ Cheng,~S.~ Lu and L. ~ Zhang,~emph {PEAS: A Robust Energy Conserving Protocol for Long-lived Sensor Networks},~ Proceedings of the 23rd International Conference on Distributed Computing Systems ICDCS'03,~pp.~ 28--37,~2003.
-
-
-
-\bibitem{Huang:2003:CPW:941350.941367}
- C. ~ Huang and Y. ~ Tseng,~emph {The coverage problem in a wireless sensor network},~Proceedings of the 2nd ACM international conference on Wireless sensor networks and applications WSNA '03,~San Diego, CA, USA,~pp.~115-121,~ 2003.
-
-
-
-
-\bibitem{Zhang05}
- H.~ Zhang and J.~ C.~ Hou,~emph {Maintaining Sensing Coverage and Connectivity in Large Sensor
- Networks},~Ad Hoc {\&} Sensor Wireless Networks,~ vol.~1,no.~ 1-2,~2005.
-
-
-\bibitem{HeinzelmanCB02}
- W.~ B.~ Heinzelman ,~ A.~ P.~ Chandrakasan and H.~ Balakrishnan,~emph {An application-specific protocol architecture for wireless microsensor networks},~IEEE Transactions on Wireless Communications,~ vol. ~1,~no. ~4,pp.~ 660-670,~2002.
-
-
-
-\bibitem{pedraza2006}
- F. ~Pedraza,~A.~ L.~ Medaglia and A. ~ Garcia,~emph {Efficient coverage algorithms for wireless sensor networks},~Systems and Information Engineering Design Symposium,~ pp.~ 78 --83,~2006.
-
-
-\bibitem{chin2007}
-C.~ T. ~ Vu, ~ emph {An Energy-Efficient Distributed Algorithm for k-Coverage Problem in Wireless Sensor Networks},~GeorgiaState University,~2007.
-
-
-
-
-
-
-
-
-
-\end{thebibliography}
-
-
-
-
+%\section*{Acknowledgment}
+\bibliographystyle{IEEEtran}
+\bibliography{bare_conf}
% that's all folks
\end{document}
