Coverage problems in sensor networks

32

Coverage Problems in Sensor Networks: A Survey

BANG WANG, Huazhong University of Science and Technology

Sensor networks, which consist of sensor nodes each capable of sensing environment and transmitting data,have lots of applications in battlefield surveillance, environmental monitoring, industrial diagnostics, etc.Coverage which is one of the most important performance metrics for sensor networks reflects how wella sensor field is monitored. Individual sensor coverage models are dependent on the sensing functions ofdifferent types of sensors, while network-wide sensing coverage is a collective performance measure forgeographically distributed sensor nodes. This article surveys research progress made to address variouscoverage problems in sensor networks. We first provide discussions on sensor coverage models and designissues. The coverage problems in sensor networks can be classified into three categories according to thesubject to be covered. We state the basic coverage problems in each category, and review representativesolution approaches in the literature. We also provide comments and discussions on some extensions andvariants of these basic coverage problems.

Categories and Subject Descriptors: A.1 [General Literature]: Introductory and Survey—Survey; C.2.2[Computer-Communication Networks]: Network Protocols—Coverage problems; H.1.0 [InformationSystems]: General—Sensor networks

General Terms: Algorithms, Performance, Theory, Verification

Additional Key Words and Phrases: Coverage, sensor networks, sensor coverage model, network coveragecontrol, network protocol design

ACM Reference Format:Wang, B. 2011. Coverage problems in sensor networks: A Survey. ACM Comput. Surv. 43, 4, Article 32(October 2011), 53 pages.DOI = 10.1145/1978802.1978811 http://doi.acm.org/10.1145/1978802.1978811

1. INTRODUCTION

Generally speaking, a sensor is a device which responds to physical stimulus (such asheat, light, sound, pressure, magnetism, etc.), and converts the quantity or parameterof a physical stimulus into recordable signals (such as an electrical signals, mechanicalsignals, etc.) [Wilson 2005]. These signals are then digitalized to produce sensing data.A sensor node normally encapsulates one or more sensor units, a power supply unit,a data processing unit, data storage, and a data transmission unit. A sensor networkconsists of sensor nodes that are deployed in different geographical locations within asensor field to collectively monitor physical phenomena. A sensor network also includesone or more sinks which collect data from sensor nodes. A sink can be regarded as aninterface between a sensor network and the people operating the sensor network.Applications of sensor networks are in a wide range, including battlefield surveillance,

Author’s address: B. Wang, The Department of Electronics and Information Engineering, Huazhong Uni-versity of Science and Technology Hongshan Luoyu Lu 1037, Wuhan, Hubei, 430074, China; email: [email protected] to make digital or hard copies of part or all of this work for personal or classroom use is grantedwithout fee provided that copies are not made or distributed for profit or commercial advantage and thatcopies show this notice on the first page or initial screen of a display along with the full citation. Copyrights forcomponents of this work owned by others than ACM must be honored. Abstracting with credit is permitted.To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of thiswork in other works requires prior specific permission and/or a fee. Permissions may be requested fromPublications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212)869-0481, or [email protected]© 2011 ACM 0360-0300/2011/10-ART32 $10.00

DOI 10.1145/1978802.1978811 http://doi.acm.org/10.1145/1978802.1978811

ACM Computing Surveys, Vol. 43, No. 4, Article 32, Publication date: October 2011.

32:2 B. Wang

environmental monitoring, biological detection, smart spaces, industrial diagnostics,etc. (see, e.g., Akyildiz et al. [2002] and Yick et al. [2008] surveys of sensor networks).

In sensor networks, an interesting and important question is: how well does a sensornetwork works? Coverage, which reflects how well a sensor field is monitored, is one ofthe most important performance metrics to measure sensor networks. Sensor coveragemodel, like many other sensor characteristics such as transfer function, sensitivity,dynamic range, accuracy, etc., can be used to measure the sensing capability and qualityof a single sensor. Simply said, sensor coverage models are abstraction models trying toquantify how well sensors can sense physical phenomena at some location, or in otherwords, how well sensors can cover such locations. Network sensing coverage, on theother hand, can be considered as a collective measure of the quality of service providedby sensor nodes1 at different geographical locations. In many cases, we may interpretthe coverage concept as a nonnegative mapping between the space points of a sensorfield and the sensor nodes of a sensor network.

In sensor networks, coverage problems can arise in all network stages and can beformulated in various ways with different scenarios, assumptions, and objectives: Inthe design stage, one may want to know at least how many sensor nodes are neededsuch that every space point in a sensor field can be covered. In the deployment stage,sensor nodes may be deterministically placed at the desired locations, or simply scat-tered within the sensor field from an aircraft. In the operation stage, one may wantto schedule different sensor nodes to work alternatively in order to prolong networkoperational time while still preserving network coverage. Some other network perfor-mance metrics, such as energy consumption and network connectivity, may also needto be considered together with coverage problems.

Coverage problems have also been researched in other fields, for example, the ArtGallery problem [O’Rourke 1987] and the Circle Covering problem [Williams 1979],and computational geometry can be applied to solve such coverage problems. However,they may not be directly applied to some coverage problems in sensor networks due tothe specific properties of sensor networks, such as limited energy and ad hoc topology.Depending on different network scenarios and application objectives, various coverageproblems have bee proposed and studied in sensor networks. Some surveys on coverageproblems in sensor networks can be found in Cardei and Wu [2004], Huang and Tseng[2005], Wang and Xiao [2006], and Ghosh and Das [2008]. In this article, we try topresent a tutorial-like and up-to-date survey on almost all types of coverage problemsin sensor networks. In particular, we first elaborate various sensor coverage models anddesign issues of coverage problems. Based on the coverage subject, three categories ofcoverage problems are identified: They are point (or target), area, and barrier coverageproblems. In each category, we group coverage problems of similar objectives, state thebasic coverage problem in each group, and review representative solution approachesin the literature. We also provide comments and discussions on some extensions andvariants of these basic coverage problems.

2. SENSOR COVERAGE MODELS

Sensor coverage models measure the sensing capability and quality by capturing thegeometric relation between a space point and sensors. In almost all cases, a sensorcoverage model can be formulated as a function of the Euclidean distances (and theangles) between a space point and sensors. The inputs of such a coverage function arethe distances (and angles) between a particular space point and sensors’ locations,

1In this article, a single sensor node is assumed equipped with only one sensor. Sometimes, we use the term“sensor” and “sensor node” interchangeably in this article.


Coverage Problems in Sensor Networks: A Survey 32:3

Fig. 1. Illustrations of (a) a directional Boolean sector coverage model; (b) a space point being 3-covered bythree sectors; (c) an omnidirectional Boolean disk coverage model; (d) a space point being 3-covered by threedisks.

and the output is called coverage measure of this space point, which is a nonnegativereal-valued number.

We introduce the concept of coverage function in the context of a two-dimensionalplane. Let us consider a space point z, and a set of sensors S = {s1, s2, . . . , sn}. We used(s, z) (d(s, z) ≥ 0) to denote the Euclidean distance between a sensor s and a spacepoint, and in the two-dimensional space

d(s, z) .=√

(sx − zx)2 + (sy − zy)2, (1)

where (sx, sy) and (zx, zy) are the Cartesian coordinates of the sensor s and the spacepoint z, respectively. We use φ(s, z) (0 ≤ φ(s, z) < 2π ) to denote the angle betweenthem. For a sensor s, we draw a horizontal line starting from the sensor and pointingto right. We connect the sensor s and the space point u with another line sz. Thenφ(s, z) is the anticlockwise angle between the two lines, starting from the horizontalline and ending at the line sz. We use dn = (d(s1, z), d(s2, z), . . . d(sn, z)) to denote thevector of such distances, and φn = (φ(s1, z), φ(s2, z), . . . , φ(sn, z)) to denote the vector ofsuch angles between the set of sensors and the space point. A sensor coverage modelcan be formulated as a coverage function f mapping from (dn,φn) to a nonnegativereal-valued number, that is,

f : (dn,φn) → R+, (2)

where R+ stands for the set of nonnegative real numbers. We call f (dn,φn) the coveragemeasure of a space point with respect to the sensors s1, s2, . . . , sn. Similar definition canalso be applied in three-dimensional space yet with some simple modifications.

Many sensor coverage models have been proposed in the literature. In some models,the inputs of a coverage function are only the distance and angle between a spacepoint and one sensor. In some other models, the inputs of a coverage function are thedistances and angles between a space point and more than one sensor. In our view, twotypes of coverage functions can be classified: One type is a kind of Boolean coveragemodels, where the coverage measure is either 0 or 1 for one space point; and the othercan be called general coverage models, where the coverage measure can take variousnonnegative values. If the angle argument is not included in the coverage function,then such coverage models are called omnidirectional coverage models. On the otherhand, if it is included, such coverage models are called directional coverage models. Inwhat follows, we elaborate some commonly used coverage models in details.

2.1. Boolean Sector Coverage Models

The Boolean sector coverage model (sometimes called as sector model), which might bemotivated from a directional camera, is a Boolean directional coverage model [Ma andLiu 2007]. Figure 1(a) illustrates such a sector model, where φs is called orientationalangle and ω is called visual angle of the sector model, and Rs is called sensing range.


32:4 B. Wang

The coverage function of the sector model is given by

f (d(s, z), φ(s, z)) ={

1, if d(s, z) ≤ Rs, and φs ≤ φ(s, z) ≤ φs + ω

0, otherwise,, (3)

where d(s, z) is the Euclidean distance between a sensor s and a space point z, andφ(s, z) is their angle. This coverage function defines a sector: All space points withinsuch a sector have a coverage measure of 1, and are said covered by this sensor. Allspace points outside such a sector have a coverage measure of 0, and are said notcovered by this sensor. In Figure 1(a), the space point marked by a star has a coveragemeasure of 1, and is covered by the sensor.

The orientational angle of a directional sensor might be adjustable after a sensorhas been deployed [Ai and Abouzeid 2006; Cai et al. 2007; Liu et al. 2008b; Fusco andGupta 2009]. Obviously, the area that can be covered by such a sensor will be differentwhen it takes different orientational angle. A space point might be covered by morethan one sector. With the Boolean sector coverage model, the coverage measure of aspace point relative to a set of sensors can be the addition of the coverage measureof the point relative to each individual sensor. Formally, the coverage function can bedefined as

f (dn,φn) =n∑

i=1

fi(d(si, z), φ(si, z)), (4)

where fi is the coverage function of a sensor si and is given by Eq. (3). If f (dn,φn) = k(k ≥ 1), then we say that the point is k-covered. Obviously, if a point is k-covered, it isalso (k − 1)-covered. Figure 1(b) illustrates an example of space point being 3-covered,where the space point marked by the star is within the sensing sectors of sensors s2,s4, and s5.

2.2. Boolean Disk Coverage Models

The Boolean disk coverage model (often simplified as disk model) might be the mostwidely used sensor coverage model in the literature. The coverage function of the diskmodel is given by

f (d(s, z)) ={

1, if d(s, z) ≤ Rs,

0, otherwise,(5)

where d(s, z) is the Euclidean distance between a sensor s and a space point z, and theconstant Rs > 0 is called sensing range. Indeed, this function defines a disk (often calleda sensing disk) centered at the sensor with the radius of the sensing range. Figure 1(c)illustrates a disk coverage model. The disk coverage model is an omnidirectional cov-erage model. All space points within such a disk have a coverage measure of 1, andare said covered by this sensor. All space points outside such a disk have a coveragemeasure of 0, and are said not covered by this sensor.

The sensing range Rs is used to characterize the sensing capability of a sensor.Normally, different sensor types are assumed to have different sensing ranges. Someresearchers even argue that a single sensor unit may have different sensing ranges andcan choose one sensing range as its working sensing range [Cardei et al. 2005b; Wangand Medidi 2007; Zhou et al. 2009]. It is generally assumed that a sensor consumesmore energy when it uses a larger sensing range. A space point may be located withinmore than one sensing disks. Under the disk coverage model, the coverage measureof a space point relative to a set of sensors can be the addition of the coverage mea-sure of the point relative to each individual sensor. Formally, the coverage function is



Fig. 2. Illustration of (a) an attenuated disk coverage model; (b) a truncated attenuated disk coverage model;(c) a truncated multilevel attenuated disk coverage model.

defined as

f (dn) =n∑

i=1

fi(d(si, z)), (6)

where fi(·) is the coverage function of a sensor si, and is given by Eq. (5). If f (dn) = k,then we say that the point is k-covered. Obviously, if a point is k-covered, it is also (k−1)-covered. Figure 1(d) illustrates an example of space point being 3-covered, where thespace point marked by the star is within the sensing disks of sensors s2, s4, and s5.

2.3. Attenuated Disk Coverage Models

Some researchers argue that the sensing quality of a sensor reduces with the increaseof the distance away from the sensor [Megerian and Koushanfar 2002; Veltri et al.2003]. An attenuated disk coverage model is used to capture such attenuated sensingqualities. An example attenuated disk coverage model is given by

f (d(s, z)) = Cdα(s, z)

, (7)

where α is the path attenuation exponent, and C a constant. Since it is a nonnegativefunction, a single sensor enforces its coverage measure to any point in the space.Figure 2(a) illustrates such an attenuated disk coverage model. The coverage measureof z1 is larger than that of z2, as it is closer to the sensor.

There may be more than one sensor in a sensor field. Under the attenuated diskcoverage model, the coverage measure of a space point relative to a set of sensors isthe addition of the coverage measure of the point relative to each individual sensor.Formally, the coverage function is modified as

f (dn) =n∑

i=1

Cdα(si, z)

. (8)

In some cases, only the sensors close to a space point are included in the computationof the preceding equation for simplification.

2.4. Truncated Attenuated Disk Models

In the attenuated disk coverage model, the coverage measure becomes very small whenthe distance between a space point and a sensor becomes very large. In such cases,the coverage measure might be neglected, and some approximations can be made bytruncating the coverage measure for larger values of distance. For example, Zou andChakrabarty [2005] propose the following truncated attenuated coverage function

f (d(s, z)) ={

Ce−αd(s,z) if d(s, z) ≤ Rs,

0, otherwise.,(9)


32:6 B. Wang

where α is a parameter representing the physical characteristics of the sensor unit,and Rs the sensing range. Figure 2(b) illustrates such a coverage model.

Another truncated attenuated disk model [Zou and Chakrabarty 2004a] is definedas

f (d(s, z)) =⎧⎨⎩

1 if d(s, z) ≤ Rs − Ru

e−α(d(s,z)−(Rs−Ru))β if Rs − Ru < d(s, z) ≤ Rs,0, if Rs < d(s, z),

(10)

where Rs is the sensing range, Ru is called the uncertain range, and α and β areconstants. The use of Ru is to capture the reducing but not yet vanishing of the sensingquality when the distance between a sensor and a space point increases. Figure 2(c)illustrates such a coverage model.

2.5. Detection Coverage Models

An important application of sensor networks is to detect some event occurred at somelocation. In the context of detection application, the sensing quality of a sensor canbe represented by its detection probability. The detection probability of a space pointby a single sensor is also related to, among other factors, the distance between them.However, the detection probability of a space point relative to a set of sensors is nolonger simply computed as the addition of the detection probability of the point relativeto each individual sensor (otherwise, it might be larger than one). Instead, value fusionor decision fusion can be used to derive the detection probability. Based on differentevent scenarios and detection techniques, many detection coverage models have beenproposed in the literature [Clouquer et al. 2003; Onur et al. 2004b; Xing et al. 2004;Ahmed et al. 2005e; Wang et al. 2007; Hefeeda and Ahmadi 2007; Tian et al. 2008;Yang et al. 2008; Tsai 2008; Yang and Qiao 2009].

Let us consider a general signal propagation model where the signal parameter θ(e.g., the sound pressure of a sound source) attenuates along with the signal propaga-tion. Depending on the hypothesis of whether the target is present (H1) or not (H0), thereadings at the sensor sk are given by

H0 : xk = nk, (11)

H1 : xk = θ

dαk

+ nk, (12)

where α is the attenuation exponent, dαk = dα(sk, z) the Euclidean distance between the

sensor sk and the space point z, and nk is the measurement noise (e.g., circuity thermalnoise). It is often assumed that the noise follows a Gaussian distribution with zeromean and variance σ 2

k .Given a threshold A, a sensor makes its detection decision of whether a target is

present by

xkH1

≷H0

A. (13)

That is, if the measurement is larger than A, it decides that a target is present; and ifthe measurement is less than A, it decides that a target is not present. When a targetis present at the space point z, the detection probability Pd

k of the sensor sk is given by

Pdk = Pr

[θ

dαk

+ nk ≥ A]

= Q

(A− θ

dαk

σk

), (14)



where Q(·) is the Q-function defined by

Q(x) =∫ ∞

x

1√2π

e− t22 dt. (15)

Since Q-function is a monotonous decreasing function, the detection probability Pdk

decreases when the distance dk increases. Indeed, Eq. (14) defines an attenuated diskcoverage model. Furthermore, if we define a threshold for detection probability, Pd

th, andonly those points with detection probability equal to or larger than such a threshold,that is, Pd

k ≥ Pdth, are considered as covered by this sensor, then we actually define a

truncated attenuated disk coverage model. If we do not discriminate the points withdetection probability not less than the detection threshold and simply call these pointsbeing covered by the sensor, then finally we get a Boolean disk coverage model. Insuch a case, the points with the detection probability equal to the threshold consistof a circle, and their distances to the sensor are also equal and often regarded as thesensing range, Rs. That is,

Q

(A− θ

Rαs

σk

)= Pd

th =⇒ Rs =(

θ

A− σkQ−1(Pk

th

)) 1

α

, (16)

where Q−1(·) denotes the inverse function of Q(·).When K sensors are used to cooperatively detect an event, the value fusion technique

can be used to compute the detection probability of a space point by these sensors. Letxk, k = 1, 2, . . . , K denote the readings of the kth sensor. With the value fusion, wecompare the sum of xk and a threshold to make a decision whether or not a target ispresent. We assume that all the noises nk (k = 1, 2, . . . , K) are independent Gaussiannoises with zero mean and variance σ 2. When a target is present at the point z, thedetection probability by these sensors is given by

PdK = Pr

[K∑

k=1

(θ

dαk

+ nk

)≥

√KA

]= Q

⎛⎝

√KA−∑K

k=1θdα

k√Kσ

⎞⎠ , (17)

where√

KA is the value fusion threshold. Again, we can use the threshold of detectionprobability Pd

th, and the points with detection probability not less than the detectionthreshold are called covered by these sensors. In such a case, the covered points by Ksensors satisfy the following distance inequality

K∑k=1

1dα

k≥

√K

Rαs

, (18)

where dk is the distance between a point and a sensor sk, and Rs given by Eq. (16).Indeed, Eq. (18) defines a Boolean detection model for K sensors, that is,

f (dK) ={

1, if∑K

k=11dα

k≥

√K

Rαs.

0, otherwise.(19)

Figure 3 marks out the space points that are considered as being covered when usingEq. (19) (α = 1.0) as the coverage model. The points within a disk are consideredas being covered when only one sensor is used. They are also considered as beingcovered when more than one sensor is used. Furthermore, those points colored by yellow(and outside the disks) are not covered by only a single sensor, but are considered asbeing covered by more than one sensor. These additionally covered space points can


32:8 B. Wang

Fig. 3. Illustration of the space points covered by using the detection coverage model of: (a) 2 sensors;(b) 3 sensors; (c) 4 sensors.

be regarded as a kind of cooperation gain by using more than one sensor for a samesensing task.

There are also many decision fusion techniques that can be used to derive the detec-tion probability by a set of sensors. For example, the following decision fusion computesthe overall detection probability by a set of sensors

PdK = 1 −

K∏k=1

(1 − Pdk ), (20)

where Pdk is given by Eq. (14). Note that Pd

k is dependent on the distance between asensor and the space point. Eq. (20) can also be used to define a coverage model (it hasbeen called as probabilistic coverage in some papers [Ahmed et al. 2005; Hefeeda andAhmadi 2007; Tian et al. 2008). Again, we can set a threshold and define that a pointis covered by K sensors (s1, . . . , sK) if its overall detection probability is not less thansuch a threshold.

2.6. Estimation Coverage Models

Another important application of sensor networks is to estimate signal parameters. Inthe context of estimation application, the sensing quality of a sensor can be representedby its estimation error. The estimation error of a space point by a single sensor is alsorelated to, among other factors, the distance between them. However, when multiplesensors are used in estimation, the estimation error of a parameter of some signal ata space point is no longer simply computed as the addition of the estimation error ofthe point relative to each individual sensor. Instead, different estimation techniquescan be used, and their estimation errors are also different. Based on different eventscenarios and estimation techniques, some estimation coverage models have been pro-posed [Wang et al. 2005b; Venkataraman et al. 2006, 2007a; Wang and Cao 2007; Liuet.al 2008c].

We use a simple signal estimation scenario to illustrate an estimation coveragemodel. We assume that a signal occurs at some space point z, and its signal parameterθ attenuates along with the signal propagation. For example, θ can be the acoustic am-plitude due to a motor engine or due to a leakage of gas barrel. For magnetic wave suchas acoustic wave, its amplitude is attenuated when propagating. The measurement ofthe signal parameter by a sensor sk is given by

xk = θ

dαk

+ nk, (21)



where α is the attenuation exponent, dαk = dα(sk, z) the Euclidean distance between the

sensor sk and the space point z, and nk is the measurement noise (e.g., circuity thermalnoise). It is often assumed that the noise follows a Gaussian distribution with zeromean and variance σ 2

k , denoted by N (0, σ 2k ). We note that this measurement model is

the same as the one in Eq. (12) when the target is present.A parameter estimator can be used to estimate θ based on the measurements xk,

k = 1, 2, . . . , K. Let θ and θ = θ − θ denote the estimate and the estimation error,respectively. If an estimation error is small, the estimate of the signal parameter isobtained with high confidence level. We can use the probability that the absolute valueof the estimation is less than or equal to a predefined constant A, that is, Pr[|θK| ≤ A],to measure how well a point is monitored (Pr[|θK| ≤ A] is called information exposurein Wang et al. [2005a]). Some standard estimators can be used to perform the estima-tion. For example, if the Best Linear Unbiased Estimator (BLUE) estimator [Mendel1995] is used, and all noises are assumed to have the same variances, that is, σ 2

k = σ 2

for all k = 1, 2, . . . , K, then we have

Pr[|θK| ≤ A] = 1 − 2Q

⎛⎜⎝ A

σ

(K∑

k=1

d−2αk

) 12

⎞⎟⎠ , (22)

where Q-function is defined in Eq. (15).We can see that Eq. (22) is dependent on the distances between the K sensors and the

space point, and can be used as a coverage function to define an estimation coveragemodel. Now let us consider that only one sensor is used in estimation. In such a case,Eq. (22) is given by

Pr[|θk| ≤ A] = 1 − 2Q(

Aσ

d−αk

). (23)

Since Q(·) is a monotonous decreasing function, Pr[|θk| ≤ A] decreases when the dis-tance dk increases. Indeed, Eq. (23) defines an attenuated disk coverage model. Fur-thermore, if we define a threshold ε (0 ≤ ε ≤ 1), and only those points with Pr[|θ | ≤ A]equal to or larger than such a threshold, that is, Pr[|θk| ≤ A] ≥ ε, are considered ascovered by this sensor, then we actually define a truncated attenuated disk coveragemodel. If we do not discriminate the points within such a disk, then finally we get aBoolean disk coverage model. In such a case, the points with Pr[|θk| ≤ A] = ε consistof a circle, and their distances to the sensor are also equal and can be regarded as thesensing range, Rs. That is,

1 − 2Q(

Aσ Rα

s

)= ε =⇒ Rs =

(A

σ Q−1( 1−ε

2

)) 1

α

, (24)

where Q−1(·) denotes the inverse function of Q(·).We can also define a Boolean estimation coverage model of K sensors by comparing

Pr[|θK| ≤ A] with the threshold ε. In such a case, the covered points by K sensors satisfythe following distance inequality

K∑k=1

1d2α

k

≥ 1R2α

s, (25)

where dk is the distance between a point and a sensor sk, and Rs given by Eq. (24).Indeed, Eq. (25) defines a Boolean estimation coverage model for K sensors (which is


32:10 B. Wang

Fig. 4. Illustration of the space points covered by using the estimation coverage model of: (a) 2 sensors;(b) 3 sensors; (c) 4 sensors.

Fig. 5. Examples of: (a) point (target) coverage; (b) area coverage; (c) barrier coverage.

called information coverage model in Wang et al. [2005b]), that is,

f (dK) ={

1, if∑K

k=11

d2αk

≥ 1R2α

s

0, otherwise.(26)

Figure 4 marks out the space points that are considered as being covered when usingEq. (26) (α = 1.0) as the coverage model. It is also seen that when using more than onesensor for the same sensing task (estimation in this case), the covered space points aremore than those by only using one single sensor. Again, the increased coverage areacan be seen as a kind of cooperation gain.

3. DESIGN ISSUES AND COVERAGE PROBLEMS

3.1. Design Issues

In this article, we classify the following design issues for coverage problems: namely,coverage type, deployment method, coverage degree, coverage ratio, activity scheduling,and network connectivity.

Coverage Type. Coverage type refers to the subject to be covered by a sensor network.Cardei and Wu [2004] argue that, according to the subject to be covered, coverage insensor networks can be classified into three types, namely, point (target) coverage, areacoverage, and barrier coverage, with examples shown in Figure 5. In point (target)coverage problems, targets are often modeled as a set of discrete space points withinthe sensor field. Area coverage problem, on the other hand, equally treats every pointin the sensor field. Barrier coverage concerns with constructing a barrier for intrusiondetection, or finding a penetration path across the sensor field with some desiredcoverage characteristics.

Deployment Method. Deployment method concerns with how a sensor network isconstructed. In general, a sensor network can be constructed by deterministically plac-ing sensor nodes at desired locations, or by randomly scattering sensor nodes into the



sensor field. Deterministic sensor placement can be applied to a small to medium sen-sor network in a friend environment. When the network size is large or the sensor fieldis remote and hostile, random sensor deployment might be the only choice. The mostcommonly used random deployment model is the uniform deployment where each nodehas equal likelihood of falling at any location in the sensor field, independently of theother nodes.

Coverage Degree. Coverage degree describes how a point is covered. For example, inthe sensing disk coverage model, coverage degree refers to how many sensors cover apoint. Using more than one sensor to cover a point can improve coverage robustness.If a point is covered by k sensors, then it can tolerate up to k− 1 failed sensors. Similardefinition can also be applied to other coverage models. For the sake of simplicity,throughout this article, what we call a covered point means that this point is coveredby at least one sensor. We will explicitly state higher coverage degree when necessary.Coverage degree is considered as one of the application coverage requirements.

Coverage Ratio. Coverage ratio measures how much area of a sensor field or howmany targets satisfy the application requirement of coverage degree. For example, if8 out of 10 targets are covered, then the coverage ratio is 80%. We sometimes usecomplete coverage to refer to 100% coverage ratio, that is, every point within sensorfield (or every target in the target set) achieves the required coverage degree. Similarly,we use partial coverage to state the situation that not all points in the sensor field (ornot all targets in the target set) can be covered with the required coverage degree.Coverage ratio is often regarded as one of the application coverage requirements.

Activity Scheduling. Activity scheduling is to schedule the activation and deactiva-tion of nodes’ sensor units. If the area covered by one sensor can also be covered byother sensors, such a sensor can be regarded as redundant and can be temporarilytransited into the energy-saving sleep state. The basic objective of activity schedulingis to decide which sensors are in which states and for how long a time, such that theapplication coverage requirement can be guaranteed and network operational timecan be prolonged. Many distributed algorithms and centralized algorithms for activityscheduling have been proposed in the literature, yet based on different assumptionsand objectives. In the distributed algorithms, the decision process is localized in eachindividual sensor node, and only information from neighboring nodes are used for theactivity decision. In centralized algorithms, a central controller makes all decisionsand distributes the results to sensor nodes.

Network Connectivity. Network connectivity concerns with how to guarantee thateach sensor node can find a route to the sink. Although normally this is the task of thenetwork layer, it may also be incorporated into the design of coverage control algorithmsas a cross-layer approach. Two nodes are directly connected if they can transmit andreceive the data to and from each other directly. Two nodes can also be connected bymultihop transmissions with some other nodes serving as relays. A connected networkensures that the sensing data of any sensor node can be transmitted to other nodes aswell as to the sink, possibly via multihop transmissions. A commonly used transmissionmodel is the disk model where a node can communicate with other nodes within a diskcentered at itself with the radius of its communication range.

3.2. Coverage Problems

In this article, we classify coverage problems into three categories based on the coveragetypes. In each category, we first state the basic coverage problem, and then presentrepresentative solution approaches to these problems.

Point Coverage Problems. Section 4 is devoted to the point coverage problems, wherethe subject to be covered is a set of discrete points. These points can be used to repre-sent some physical targets in the sensor field. Section 4.1 studies the node placement


32:12 B. Wang

optimization problem for coverage configuration before network deployment, wherethe objective is to find the optimal locations to place sensor nodes to minimize networkcost. Section 4.2 investigates the coverage lifetime maximization problem by control-ling coverage characteristics in a randomly deployed network, where the objective is tooptimally schedule sensors’ activities in order to extend network lifetime.

Area Coverage Problems. Section 5 is dedicated to the area coverage problems, wherethe subject to be covered is the whole sensor field. Section 5.1 discusses the CriticalSensor Density (CSD) problem for coverage configuration before network deployment,where the objective is to find the least number of sensor nodes per unit area to providecomplete coverage for the whole sensor field. Section 5.2 looks into the sensor activityscheduling problem of controlling network coverage characteristics in a randomly de-ployed network, where the objectives are to identify coverage redundant sensors andschedule sensors’ activity in order to prolong the network lifetime.

Barrier Coverage Problems. Section 6 discusses the barrier coverage problems. In thebarrier coverage problem, the objective is to identify the desired coverage characteris-tics, if it exists, for a sensor network. Section 6.1 examines the coverage problems ofbuilding intrusion barriers for detecting intrusions of a mobile object when it traversesfrom one side to the other side of the sensor field. The trajectory of an intrusion mobileobject is called its traverse path. The objective is to enable the covered points to forman intrusion barrier, stretching across the sensor field and intersecting with every po-tential traverse path. Section 6.2 reviews the coverage problems of finding penetrationpaths. A penetration path is a continuous curve with arbitrary shape, spanning fromone side to the other side of a sensor field. We assign a coverage measure (a real value)to represent the coverage characteristics of a single space point, for example, the Eu-clidean distance between a point and its closest sensor. The objective is to identify sucha penetration path on which every single point satisfies the required coverage measure.

4. POINT COVERAGE PROBLEMS

In point coverage problems, the subject to be covered is a set of discrete space points.These points can be some particular space points to represent the sensor field (e.g., thevertices of a grid) or are used to model some physical targets in the sensor field (e.g.,the missile launchers in a battlefield). In order to cover these points, sensor nodes canbe deterministically placed or randomly deployed in the sensor field.

4.1. Node Placement Optimization

Deterministic node placement is to place the nodes only at the desired locations suchthat each sensor node can cover at least one target. Deterministic node placement isapplicable if the sensor field is friendly reachable and the network size is not too large.The objective of a deterministic node placement can be summarized as to answer thefollowing question:

Where are the optimal locations (among the available locations) to place sensor nodessuch that the number of nodes (or the network cost) can be minimized and the pointcoverage requirements can be satisfied?

4.1.1. Modeling Node Placement. The locations of the targets (i.e., the space points) tobe covered are assumed known before placement; and the available locations to placenodes (called sites) are limited. Suppose there are I sites and J targets. We use subscripti to index a site and j to index a target. Let xi denote an indicator function of whethera sensor is placed at the site i, that is,

xi ={

1, if a sensor is placed at site i,0, otherwise.



Let δi j denote an indicator function of whether a target j can be covered by a sensorlocated at the site i, that is,

δi j ={

1, if target j is covered by a sensor located at site i,0, otherwise.

The problem of placing the least number of sensors to cover all targets can be formulatedas the following Integer Linear Programming (ILP) problem.

MinimizeI∑

i=1

xi (27)

Subject toI∑

i=1

δi j > 0, j = 1, . . . , J (28)

xi ∈ {0, 1}, i = 1, . . . , I (29)

The constraint Eq. (28) requires that all targets are covered by at least one sensor.A generalization of the previous ILP problem is to minimize the network cost, if

different sensor nodes have different costs and coverage capabilities [Chakrabartyet al. 2002; Patel et al. 2005; Wang and Zhong 2006; Xu and Sahni 2007; Altinel et al.2008]. Sometimes, we may have other constraints such as the distance between anypair of sensor nodes should not be too close [Sen et al. 2007]. For example, in thescenario of placing different types of sensors, suppose that we have B types of sensors,each type with cost cb and sensing range rb, b = 1, . . . , B. Normally, a larger coveragerange corresponds to a larger cost. Let Db(i) denote the set of targets that can be coveredby a type b sensor placed at the site i, that is,

Db(i) = { j|d(i, j) ≤ rb}, i = 1, . . . , I,

where d(i, j) is the Euclidean distance between a site i and a target j. Again, weuse xb

i = 1 to denote a type b sensor being placed at site i; and otherwise, xbi = 0.

Furthermore, the coverage requirement is to cover each target with at least k sensors.The objective is to minimize the total sensor placement cost

MinimizeI∑

i=1

B∑b=1

cbxbi ,

Subject toB∑

b=1

∑j∈Db(i)

xbi ≥ k, i = 1, . . . , I

B∑b=1

xbi ≤ 1, i = 1, . . . , I.

The first constraint ensures that each target is covered by at least k sensors, and thesecond constraint ensures that each site can be occupied by at most one sensor.

It is also possible to allow some points not covered by any sensor. This might be thecase when we use the grid approach to approximate area coverage. Accordingly, someother variants of sensor placement problems include: (a) maximizing the number ofcovered targets subject to a given number of sensors; or (b) minimizing the networkcost subject to a minimum target coverage ratio. For the directional coverage model,another objective is to optimize sensor orientational angles to maximize target cov-erage [Horster and Lienhart 2006]. All these types of sensor placement problems are


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Table I. GREEDY-SET-COVER(X,F)

1. U ← X.2. C ← ∅3. while U �= ∅4. do select an S ∈ F that maximizes |S ∩ U |.5. U ← U − S6. C ← C ∪ {S}7. return C

basically optimization problems, and can be formulated as mathematical programmingproblems.

4.1.2. Greedy Placement Algorithms. For small problem instances, exhaustive search canbe used to find the global optimum by trying every possible placement. If we have Iavailable sites, we can choose i sites each time for placing sensors, and examine thecost of the placement. The number of all possible placements that need to be exam-ined are

∑Ii=1( I

i ) = 2I − 1. That is, the computation complexity for exhaustive searchincreases exponentially with the number of available sites. For the mathematical pro-gramming problems discussed in the previous subsection, some standard optimizersuch as CPLEX [Inc. 2009] can be used to solve these problems with faster computationspeed than the exhaustive search. However, it is worth of noting that the computationis still a very time-consuming process, especially when the problem instance is large.

The problem of placing the least number of sensors to cover all discrete targets can beequated to the canonic set-covering problem [Cormen et al. 2001]. In the node placementproblem, a sensor node can only be placed at one of the available sites, and can coverat least one target if it is placed at this site. Furthermore, all targets can be covered ifall of these available sites are occupied by sensors. In the set-covering problem, let Xdenote the finite set of targets with known locations. Let F denote a family of subsetsof X, and let S ∈ F denote its element. The cardinality of F is the number of availablesites to place sensors. Every element of F corresponds to the set of targets that canbe covered if a sensor is placed at one site. We say that S covers some targets. Theset-covering problem is to find a minimum-size subset C ⊆ F whose members cover alltargets of X,

X =⋃S∈C

S. (30)

We say that any C satisfying Eq. (30) covers X, or C is a set-cover of X. The size ofC is defined as the number of sets it contains, rather than the number of individualelements in these sets.

The classical greedy algorithm [Cormen et al. 2001], as listed in Table I, can be usedto solve the set-covering problem in polynomial time. The greedy algorithm works byselecting, at each step, the set S that covers the largest number of remaining elementsthat are uncovered. The set U contains, at each stage, the set of remaining uncoveredtargets. The set C contains the cover being constructed. Line 4 is the greedy decision-making step. A subset S is chosen if it covers as many uncovered targets as possible(with ties broken arbitrarily). After S is selected, its elements (the already coveredtargets) are removed from U , and S is placed in C. When the algorithm terminates, theset C contains a subfamily of F that covers X. The number of iterations of the loop onsteps 3–6 is bounded from above by min(|X|, |F |), and the loop body can be implementedto run in time O(|X||F |).

In general, an algorithm is called greedy if it divides the problem into subproblemsto be solved in consecutive stages, and always takes the best local solution at eachstage. The greedy decision made in each stage may depend on the decisions made so



far but not on future decisions. With such greedy decisions, the original problems areiteratively reduced to many smaller subproblems. A greedy algorithm terminates if apredefined optimal threshold has been achieved or the maximal allowable stages havebeen performed. Many variants of the simple GREEDY-SET-COVER algorithm havebeen proposed to solve various node placement problems [Dhillon et al. 2002; Dhillonand Chakrabarty 2003; Zou and Chakrabarty 2004b; Wang and Zhong 2006; Xu andSahni 2007; Fang and Wang 2008; Wang 2008]. For example, we can normalize the costof placing a type b sensor at the site i by cb/nj . nj is the number of uncovered targetsthat can be covered if a type b sensor is placed at site i. In the decision stage, the sitewhich requires the least normalized cost is selected, and allocated to the sensor typewith the least normalized cost. The newly covered targets are then removed from theset of uncovered targets, and the process is terminated when all targets are covered.

4.1.3. Notes and Comments. The point coverage problem can be linked to the area cov-erage problem via a grid approach. If all the vertices (or called grid points) of a well-defined grid embedded within the sensor field are covered, then the whole sensor fieldis said to be completely covered. A grid point is covered by the sensor node located atthis grid point. The placement problem is trivial if a sensor at a grid point can onlycover this grid point. On the other hand, for a given grid where a sensor node locatedat one grid point can also cover a few of its neighboring grid points, the problem ofplacing the least number of sensors only at the grid points such that all the grid pointsare covered is a NP-complete problem [Ke et al. 2007].

In deterministic node placements, some other sensor coverage models have beenused as coverage measure, such as the attenuated disk model and detection coveragemodels [Dhillon et al. 2002; Dhillon and Chakrabarty 2003; Zou and Chakrabarty2004b; Zhang et al. 2006; Stolkin et al. 2007; Stolkin and Florescu 2009]. For example,with the detection coverage model, the overall detection probability should be largerthan a predefined threshold for all targets, and the objective is to place the least numberof sensors to achieve such detection requirements. Similar to the set-covering problem,this placement problem can be solved by using a modified greedy algorithm.

Besides the greedy algorithm, some other well-known approximation algorithmshave been applied to find approximate solutions, such as simulated annealing [Linand Chiu 2005b, 2005c, 2005a] and genetic algorithms [Xu and Yao 2006; Wu et al.2007; Zhao et al. 2007; Seo et al. 2008]. Simulated annealing is a generic probabilisticheuristic for locating a good approximation to the global extremum of a given function.Genetic algorithms, which are inspired by biological evolutionary processes, model andapply biological inheritance, mutation, selection, and crossover in the search of globaloptimal solution.

4.2. Coverage Lifetime Maximization

In random node deployment, it is common to scatter many sensor nodes around eachtarget. In such a randomly deployed sensor network, a target may be covered by morethan one sensor, and a sensor may also cover more than one target. We can partitionsensors into different covers (each a subset of sensors that can satisfy the coveragerequirement), and activate these covers alternatively. The target coverage lifetime canbe computed as the sum of these sensor covers’ runtime. The objective of the coveragelifetime maximization is to answer the following question:

How to partition sensors into different sensor covers and schedule their operatingintervals, such that the coverage requirement can be satisfied by these covers and thetarget coverage lifetime can be maximized?

4.2.1. Maximize Target Coverage Lifetime. Let us first consider a simple example of targetcoverage. As illustrated in Figure 6(a), there are 6 sensors and 4 targets in a randomly


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Fig. 6. Illustration of (a) a randomly deployed sensor network for covering targets; and (b) the correspondingsensor-target bipartite graph.

deployed network. If we consider a disk coverage model, then the targets z2 and z4are each covered by two sensors; and the targets z1 and z3 are each covered by threesensors. Figure 6(b) uses a sensor-target bipartite graph to represent the coveragerelations among all sensors and all targets.

In the context of target coverage, it is a common prerequisite that all targets canbe covered if all sensors are activated for sensing. However, activating all the sensorsat the same time is not energy efficient. If every sensor can only operate for one timeunit in a continuously active state, then activating all sensors all the time results ina total network lifetime of also one time unit. Instead, we can alternatively activatesensors. For example, in the network shown in Figure 6, we can activate C1 = {s1, s3, s6}for one time unit, and C2 = {s2, s4, s5} for another time unit. Since all targets are stillcovered by either C1 or C2, the coverage requirements are not sacrificed. Furthermore,the target coverage lifetime can be extended to two time units. The objective of thetarget coverage problem is to find the optimal subsets and their active intervals suchthat the coverage requirements can be satisfied and the total target coverage lifetimecan be maximized.

We consider a sensor network of N sensors and M targets, and let S = {s1, . . . , sN}and Z = {z1, . . . , zM} denote the sensor set and target set, respectively. For the sensingdisk coverage model, it is commonly assumed that if a sensor is active, it can cover alltargets within its sensing disk. A sensor may cover several targets, and a target mightbe covered by multiple sensors. We use Z(si) to denote the set of targets covered by thesensor si, and its cardinality |Z(si)| is the number of covered targets. We use S(zj) todenote the set of sensors covering the target zj , and its cardinality |S(zj)| is the numberof sensors covering the target zj . For example, in Figure 6, Z(s3) = {z1, z2}, |Z(s3)| = 2and S(z3) = {s4, s5, s6}, |S(z3)| = 3.

We use C to denote a collection of the subsets of Z. The ith element in C is the targetscovered by the sensor si, that is, C = {Z(s1), . . . , Z(sN)}. For example, in Figure 6,C = {{z1}, {z1}, {z1, z2}, {z2, z3}, {z4, z5}, {z4, z5}}. We call a subset of sensors a (sensor) setcover, if it can cover at least one target. We use the subscript k to denote the kth setcover Ck. The cardinality of the set cover Ck, |Ck|, is the number of sensors in Ck. Weuse an inclusion indicator function δ(si, Ck) to indicate whether the sensor si is includedin the set cover Ck. That is, δ(si, Ck) = 1, if si ∈ Ck; and otherwise, δ(si, Ck) = 0. With alittle abuse of symbols, we use {Ck} to denote the set of the targets covered by a set coverCk, and use |{Ck}| to denote the number of covered targets. For example, in Figure 6,the set cover C1 consists of sensors s1, s3 and s6, and its cardinality |C1| = 3. The set oftargets covered by C1 is {C1} = {z1, z2, z3, z4} and |{C1}| = 4.

4.2.1.1 Maximum Set Cover (MSC) for Complete Target Coverage. The completetarget coverage requires that all targets should be covered all the time. This indicatesthat {Ck} = Z for all set cover Ck. We use ek(si, Ck), ek(si, Ck) ≥ 0, to denote the energyconsumption of sensor si when the set cover Ck operates for tk time units. Note thatek(si, Ck) = 0 implies si �∈ Ck. The energy constraint indicates that

∑Kk=1 ek(si, Ck) ≤ Ei,



where Ei is the initial energy of sensor si. The maximal set cover problem is defined asfollows.

Definition 4.1 (Maximum Set Cover (MSC) Problem). Given a collection C of sub-sets of a finite set Z, find a schedule (Ck, tk) of the set covers C1, C2, . . . , CK with timeintervals t1, t2, . . . , tK to maximize

∑Kk=1 tk, such that each set cover can cover all targets

({Ck} = Z) for all Ck), and each sensor does not consume more than its initial energy(∑K

k=1 ek(si, Ck) ≤ Ei for all si).

The MSC problem is a NP-complete problem [Cardei et al. 2005a]. Let us first considerthe simplest MSC problem where all sensors have the same lifetime of one time unit,and all targets need to be covered by at least one target at all the time. The optimizationversion of this MSC problem can be formulated as follows.

Maximize: T ≡K∑

k=1

tk

Subject to:K∑

k=1

δ(si, Ck)tk ≤ 1, for all si (energy constraint)

{Ck} = Z, for all Ck (coverage constraint)δ(si, Ck) ∈ {0, 1}, for all si, Ck (inclusion constraint)

The energy constraint states that each sensor cannot be activated for more than onetime unit. The coverage constraint requires that each target should be covered by atleast one sensor in each set cover.

In the preceding optimization problem, the term δ(si, Ck)tk is not linear. We can con-vert it to a linear programming problem by setting xik = δ(si, Ck)tk, where xik = 0or xik = tk ≤ 1. Furthermore, the complete coverage constraint is rephrased as∑

i∈S(zj ) xik ≥ tk for all zj ∈ Z. The Linear Programming (LP) formulation of the MSCproblem is as follows.

Maximize: T ≡K∑

k=1

tk (31)

Subject to:K∑

k=1

xik ≤ 1, for all si (32)

∑i∈S(zj )

xik ≥ tk, for all zj ∈ Z and k = 1, 2, . . . , K

0 ≤ xik ≤ tk ≤ 1

The number of the set covers K, however, is unknown, and needs to be preset in orderto apply some LP solvers.

One approach to decide K is to apply an exhaustively search for all irreducible setcovers [Berman et al. 2004, 2006]. A set cover is called irreducible if the coveragerequirement cannot be satisfied by removing any sensor from the set cover. The worstcase of the search space can be as large as 2N, where N = |S| is the total numberof sensor nodes. For the complete coverage requirement, the worst case of the searchspace is upper bounded by 2Nmax

z , where Nmaxz = maxzj∈Z |S(zj)| is the maximum number

of sensors covering a target. We use Call to denote the set of irreducible set covers in


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which each set cover can satisfy the coverage requirement, and set Kall = |Call|. For thepreceding LP problem, we can cancel the coverage constraint, and rewrite it as

Maximize: T ≡Kall∑k=1

tk

Subject to:Kall∑k=1

xik ≤ 1, for all si

xik ≥ 0.

This is a standard form LP problem, and the solution can be obtained via some com-mercial LP solver.

The exhaustive search for all irreducible set covers is computation intensive, andthe value of Kall might be very large, which again increases the computation of the LPproblem. Another approach is to guess a small value of K to solve the small size LP. Forexample, K can be set to the number of sensors [Cardei et al. 2005a]. However, withsuch a predefined K, the solution of the LP (denote as x∗

ik and t∗k ) may be a suboptimal

result, and the network lifetime can be further improved. Cardei et al. [2005a] proposea heuristic which iteratively solves the LP problem to improve the network lifetime.

We can also use a modified greedy algorithm to solve the MSC problem. In the greedyalgorithm [Cardei et al. 2005a], a partition of set covers is constructed in a greedy way,where a most beneficial sensor covering a critical target is selected in each stage. Acritical target is the most sparsely covered, both in terms of the number of sensorscovering it as well as with regard to the residual energy of these sensors. A sensorhas larger benefit if it covers a larger number of uncovered targets and if it has moreresidual energy. After the set cover construction, all sensors in the set cover are setto be active for τ time units, and their lifetime is adjusted accordingly. The networklifetime is computed as τ × Kgreedy, where Kgreedy is the number of set covers returnedby the greedy algorithm.

—Set K-Cover for Minimum Coverage Breach. Covering all the targets all the timeis a strict coverage requirement. Sometimes, we can relax this exacting requirement ofcomplete coverage by allowing some coverage breach [Cheng et al. 2007]. If a target isnot covered by any active sensor, then it is said breached. In other words, the coveragerequirement becomes a partial target coverage requirement, and a set cover is allowedto cover only a fraction of targets. A set cover can only be activated for a short timeduration, and all set covers are alternatively activated. With such rotative activationsof set covers, a target that is not covered in this round may be covered in the nextround.

Instead of finding the maximum number of set cover for complete target coverage, theset K-cover problem with allowable coverage breach is to construct K set covers (i.e., Kis predefined) to minimize the coverage breach while maximizing the network lifetime.However, they are two conflicting objectives in most cases. We need to balance the twoobjectives. For example, we can set a threshold as the acceptable minimum networklifetime, and then minimize the coverage breach. On the other hand, we can set athreshold as the acceptable maximum allowable coverage breach, and then maximizethe network lifetime.

The solution to the set K-cover problem is a schedule (Ck, tk), k = 1, 2, . . . , K. Thenetwork lifetime is given by T ≡ ∑K

k=1 tk. The total coverage breach is defined as∑Kk=1 tk × (|Z| − |{Ck}|), and the average coverage breach rate is defined as

∑Kk=1 tk ×

(|Z| − |{Ck}|)/∑K

k=1 tk. The set K-cover for minimum breach is defined as follows.



Fig. 7. Illustration of the connected target coverage problem: (a) one potential solution; (b) another potentialsolution.

Definition 4.2 (Set K-Cover for Minimum Breach). Given a collection C of subsets ofa finite set Z, a positive integer K, and the network lifetime threshold Tth > 0, constructa schedule (Ck, tk) of K set covers C1, C2, . . . , CK with time intervals t1, t2, . . . , tK, suchthat the network lifetime is not less than Tth, that is,

∑Kk=1 tk ≥ Tth, and the coverage

breach∑K

k=1 tk × (|Z|− |{Ck}|) is minimized. The schedule should also satisfy the energyconstraint,

∑Kk=1 tkδ(si, Ck) ≤ 1 for all si and Ck, where δ(si, Ck) = 1 if sensor si in the set

cover Ck; and otherwise, δ(si, Ck) = 0.

This problem is also NP-complete. Again, we can formulate it as a linear programmingproblem, and solve it with modified greedy algorithms [Wang et al. 2007c, 2009b].

4.2.2. Maximizing Connected Target Coverage Lifetime. In the previous sections, we haveintroduced the target coverage lifetime maximization problems without consideringthe network connectivity. Such problems might be applicable to a kind of one-hopnetwork where all sensor nodes send their sensing data directly to the sink. In someother scenarios, the deployed sensor nodes together with the sink(s) are often modeledas a multiplehop network where a sensor may need to send its data to the sink via amultihop path consisting of other sensor nodes. In such a multihop path, a node maynot be necessarily sensing active for monitoring any target, but only serves as a relayto receive data and retransmit the data.

Let us use an example to illustrate the target coverage problem with the connectivityrequirement. As shown in Figure 7, there are 1 sink, 4 targets, and 13 randomlydeployed sensor nodes. The sensors that can cover one or more targets are indicated bytheir circles: solid circles for active sensors and dashed circles for other sensors. Someof nodes may not be able to cover any target, say for example, the sensor nodes s2, s8, s13in the figure. However, such sensor nodes may still be useful as they can serve as relaysto relay other sensors’ data. Arrow lines are used to denote the routes used to relaydata from sources to the sink. In Figure 7(a), only three sensors (s7, s9, s10) are neededto cover all targets, and another two sensor nodes (s8, s11) are used to relay sensingdata. A different selection is shown in Figure 7(b), where another subset of sensors areused for covering targets (s3, s4, s5, s6) and relaying sensing data (s1, s2, s4).

We use connected target coverage problem to refer to the network lifetime maxi-mization problem for target coverage with the connectivity requirement. It is normallyassumed that in the network initialization: (1) all the sensor nodes can reach the sinkvia either single-hop or multihop communications, and (2) each target is covered by


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at least one sensor. A sensor is called a source if it is selected to be active to performthe monitoring task, and it generates sensing data at a certain rate. A sensor node iscalled a relay if it is selected to be active to relay data only. A relay does not performthe sensing task and does not generate sensing data. An active sensor node can serveas either a source or a relay or both. A sensor node that is not active enters into anenergy-saving sleep state.

The network lifetime for connected target coverage is defined as the time period fromthe time when the network is set up to the time that: (1) the coverage requirementcannot be satisfied; or (2) the connectivity requirement cannot be satisfied (e.g., a routecannot be found to send the sensing data to the sink), whichever is earlier. Again,we assume that all sensor nodes have limited lifetime. And hence the network haslimited lifetime. A schedule is a series of subsets of active sensors (along with therole allocation for each active sensor) and their active intervals. Furthermore, eachsubset of active sensors should satisfy the coverage and connectivity requirements.The connected target coverage problem is to find the optimal schedule to maximize thenetwork lifetime.

We next elaborate a connected target coverage problem with the complete 1-coverageand 1-connectivity requirements. With such requirements, each target should be cov-ered by at least one active source; and each source can find one route to the sink. Inother words, each subset not only serves as a set cover, but also forms a routing tree. Inour context, the routing tree is rooted at the sink, and all sources can find a path to thesink. We call such a subset a cover tree. We define the following maximum cover treeproblem with the objective of maximizing network lifetime and with the constraints oftarget coverage and network connectivity.

—Maximum Cover Tree (MCT) Problem for Connected Complete Target CoverageAgain, we use S = {s1, . . . , sN} (|S| = N) and Z = {z1, . . . , zM} (|Z| = M) to denote theset of deployed sensor nodes and the set of targets, respectively. We use R to denotethe sink. We assume that all sources have the same data generation rate, and eachsource generates a fixed amount of bits, denoted by B(t), in a time interval t. In eachinterval, the state (active/sleep) and the role (source/relay/source+relay) of a sensorremain unchanged within the interval.

The energy consumption model considers three types of energy dissipation, namely,energy consumed for producing, receiving, and sending one data bit. Signaling over-heads are assumed very small when compared with the sensing data, and are notincluded in the energy consumption model. An energy unit es is consumed by a sourcefor producing a bit data. An energy unit energy er is consumed by a relay for receivinga bit data. And an energy unit et(d) is consumed by a sender (a source or a relay)for sending a bit to a receiver separated with d meters. Here, we consider a distance-dependent transmission energy dissipation et(d) = a+b×dc, where a is the transmitterelectronics energy, b is the transmitter amplifier energy, and c is the radio attenuationfactor [Heinzelman et al. 2002]. For simplicity, we use et to represent et(d) for a givensender and receiver pair. A source needs to consume es × B(t) energy for its sensingtask and at least et × B(t) energy for sending out its sensing data in an active intervalt. The energy consumed by a relay is dependent on the number of bits it will transmitor receive, where the latter is further dependent on how we construct a connected treefrom all the sources to the sink.

We use Csk and Cr

k to denote the set of sources and the set of relays in an interval tk,respectively. The set of active nodes is Ck = Cs

k ∪ Crk, and the set of nodes serving as

both source and relay is Csk ∩Cr

k. We use Tk = (Csk ∪Cr

k ∪R, Ek) to denote the constructedrouting tree in the interval tk, where Ek is the set of edges connecting the active sensorsand the sink. The tree Tk has the following properties: (1) The root of the tree is the



sink; (2) each leaf of the tree is a source sensor; (3) each target can directly connect toat least one source in the tree (that is, each target is covered by at least one source).We call Tk a cover tree since it not only covers all targets but also connects every activenode to the sink.

In a cover tree Tk, we call a sensor si a descendant of another sensor si′ if sensor sineeds si′ to relay its data to the sink; and si′ is called the ancestor of si. Let D(si, Tk)denote the number of sources among the descendants of sensor si in a given cover treeTk. If a sensor si′ is a leaf, that is, si′ has no descendants, then D(si′ , Tk) = 0. Obviously,D(R, Tk) = |Cs

k|. From the previous discussion, for a given cover tree Tk operating in aninterval tk, the energy consumption for a sensor si is given by

ek(si, Tk) =

⎧⎪⎨⎪⎩

(es + et)B(tk), si ∈ Csk and si �∈ Cr

k,(er + et)B(tk)D(si, Tk), si ∈ Cr

k and si �∈ Csk,

(es + et)B(tk) + (er + et)B(tk)D(si, Tk), si ∈ Csk ∩ Cr

k,0, si �∈ Cs

k ∪ Crk.

(33)

A source consumes energy for sensing and sending, and has no descendants. Henceits energy consumption is es B(tk) + et B(tk). A relay consumes energy for receivingand sending, and it has D(si, Tk) descendants. Hence its energy consumption is(er + et)B(tk)D(si, T ). A source plus relay node consumes energy for sensing, receiving,and sending, and hence its energy consumption is (es + et)B(tk) + (er + et)B(tk)D(si, Tk).A sleep node performs no tasks, hence its energy consumption is zero.

The preceding energy consumption model exploits the state and role allocation as wellas the tree structure. Although the details of a tree are to be decided, this energy modelimplicitly guarantees the network connectivity requirement. Furthermore, it has alsoimplicitly included the flow conservation rule via the aid of number of descendants.Again, we call (Tk, tk) a schedule of cover trees Tk with their operating intervals tk,k = 1, 2, . . .. The maximum cover tree problem is defined as follows.

Definition 4.3 (Maximum Cover Tree (MCT) Problem). Given a set of sensors S, aset of targets Z, and a sink R satisfying the initial network configuration assumptions,find a schedule (Tk, tk) of the cover trees T1, . . . , TK with operating intervals t1, . . . , tK tomaximize

∑Kk=1 tk. Each cover tree Tk = (Cs

k ∪ Crk ∪R, Ek) consists of vertices and edges.

The vertices of each cover tree consist of the set of sources Csk, the set of relays Cr

k, andthe sink R; and the edges Ek specify the routes from sources to the sink. Furthermore,each cover tree can cover all targets ({Cs

k} = Z for all Csk), and each sensor does not

consume more than its initial energy (∑K

k=1 ek(si, Tk) ≤ Ei for all si. Ei is the initialenergy of si).

In this definition, the number of intervals is denoted by K. Given a finite initial energyand a minimum time interval, the value of K is finite but unknown. The MCT prob-lem is NP-complete [Zhao and Gurusamy 2008b]. Again, we can use modified greedyalgorithms to solve the MCT problem. Zhao and Gurusamy [2006, 2008b] propose aCommunication Weighted Greedy Cover (CWGC) heuristic to assign nodes with differ-ent roles. The basic idea of CWGC is first to construct an energy-aware routing treefrom each node to the sink and then apply a greedy approach to select a node as asource. With the energy-aware routing tree, a node can determine the total energy con-sumption for its produced data if it is selected as a source. The greedy source selectionneeds to take into account the number of uncovered target and total energy consump-tion for its data transmission. Furthermore, the routing tree needs to be updated aftereach source selection step. The selection process terminates if all targets are covered.

4.2.3. Notes and Comments. In coverage lifetime maximization problems, some otherconstraints have been considered in the literature. For example, an interesting


32:22 B. Wang

constraint is to allow only disjoint set covers to be constructed. Two sets are calleddisjoint if their intersection is an empty set, that is, Ck ∩ Ck′ = ∅ for Ck �= Ck′ . Supposethat all sensors have the same amount of initial energy and have the same energy con-sumption rate in the active state. If a set cover is scheduled to be continuously active,then all sensors in the set cover will die at the same time. Each disjoint set cover isactivated until its death, and all the disjoint set covers are activated one by one. Withsuch an arrangement, the target coverage lifetime equals to the number of these setcovers times the runtime of a single set cover.

The problem of maximum disjoint set cover for complete target coverage [Cardeiand Du 2005; Slijepcevic and Potkonjak 2001] is similar to the MSC problem, yetwith the disjoint constraint. It has been noted in Cardei et al. [2005a] (with a goodexample) that without such a disjoint constraint, the network lifetime can be furthermaximized. Likewise, the problem of disjoint set K-cover problem for minimum coveragebreach [Abrams et al. 2004; Cheng et al. 2007; Deshpande et al. 2008, 2009] is similarto the set K-cover problem, yet with the disjoint constraint. A very simple randomizedalgorithm can be used for the disjoint set K-cover problem: A sensor si randomly choosesan integer k uniformly distributed from 1 to K, and assigns itself to the cover Ck. Therandomized algorithm requires no processing, and is very easy to implement. We useM(ζ ) =∑K

k=1 |{Ck}| to denote the total number of covered targets by the constructed setcovers. It has been shown in Abrams et al. [2004] that the expected value of M(ζrand)is a 1 − 1

e approximation to OPT = M(ζ ∗), where ζrand is the partition of the set coversconstructed by the randomized algorithm, and ζ ∗ is the partition used by the optimumsolution.

There are also some other problem variants based on different assumptions, objec-tives, or other coverage models. In the sensing disk model, it is often assumed thatan active sensor can cover all the targets within its sensing disk. In some cases, anactive sensor may choose to monitor only one target or a subset of the targets withinits sensing disk [Aly et al. 2005; Liu et al. 2005b, 2006a; Zhao and Gurusamy 2005].Also in the sensing disk coverage model, it is usually assumed that all active sensornodes have the same energy consumption rate, regardless how many targets they arecovering. Sometimes, the energy consumption rate may be dependent on the number ofcovered targets [Gu et al. 2009]. The reliable coverage normally requires that a targetshould be covered by more than one active sensor [Liu et al. 2006b; Pyun and Cho2009]; while the differentiated coverage may ask that different targets should be cov-ered by different numbers of active sensors [Pyun and Cho 2009]. The target coveragelifetime maximization problem has also been studied under other coverage models,such as the disk coverage model with adjustable sensing ranges [Cardei et al. 2005b],the disk coverage model with multiple sensing modalities [Shih et al. 2009; Ahujaet al. 2009], the directional coverage model [Ai and Abouzeid 2006; Cai et al. 2007;Wang et al. 2007d], and the estimation coverage model [Wang et al. 2006; Vashisthaet al. 2007a].

In connected target lifetime maximization problems, a generalization is to enforcek1-coverage and k2-connectivity requirements such that every target is covered by k1different active sensors, and such active sensors can form a k2-connected network[Li et al. 2007; Yang et al. 2005, 2006]. There are also some other variants for con-nected target coverage. For example, only disjoint cover trees are allowed to be con-structed [Jaggi and Abouzeid 2006]. Some consider the scenario that each active sensorcan only cover one or a subset of targets within its sensing range [Liu et al. 2006c, 2007;Zhao and Gurusamy 2007, 2008a]. In the sensing disk model, each sensor may havedifferent sensing ranges, and the sensing range also needs to be scheduled [Lu et al.2009]. Some other sensing models such as estimation coverage model [Wang et al.2005b] have also been considered for connected coverage [Wang et al. 2007b].



Fig. 8. Tessellation of using: (a) regular triangles; (b) regular squares; (c) regular hexagons.

5. AREA COVERAGE PROBLEMS

In the area coverage problem, the subject to be covered is the whole sensor field. Thatis, all points within the sensor field are treated equally. This section discusses areacoverage problems in both the network deployment stage and network operation stage.

5.1. Critical Sensor Density

Before network deployment, one may want to know how many sensor nodes are needed,such that every point of the sensor field can be covered by at least one sensor. Sensordensity is defined as the number of nodes per unit area. In a homogeneous sensornetwork, the Critical Sensor Density (CSD) provides an insight on the minimal numberof nodes required for complete area coverage. In deterministic sensor placements, abasic placement pattern can be repeated to tile-up the whole sensor field. In randomsensor deployments, the mathematical analysis provides a lower bound of CSD for asensor field with finite area. In particular, the following questions are addressed in thissubsection:

What is the optimal placement pattern in deterministic node deployments and how toderive the critical sensor density for random node deployments?

5.1.1. Deterministic Node Placement. In deterministic node placements, a basic place-ment pattern consisting of only a few nodes can be repeated to tile-up the whole sensorfield. A regular tiling of polygons in a two-dimensional place is also called a tessellation.A placement pattern can be a polygon, and sensor nodes are put at the polygon’s ver-tices. If a polygon is completely covered by the sensors at its vertices, then the wholeplane can be completely covered by the tessellation of such polygons. For example,Figure 8 illustrates three canonical tessellations by regular triangles, regular squares,and regular hexagons.

The critical sensor density λcsd for a tessellation can be computed as follows. LetAp denote the area of a placement pattern (a polygon), Np the number of nodes thatcompose a pattern, and Nn the number of pattern blocks that share a node. Then λcsdcan be computed as

λcsd = Np

ApNn. (34)

Let Rs denote the radius of the sensing disk. Then we can compute the CSD for regulartriangular tessellation λt

csd, regular square tessellation λscsd, and regular hexagonal

tessellation λhcsd as follows.

λtcsd = 3

3√

34 R2

s × 6= 2

√3

9× 1

R2s

(35)

λscsd = 4

2R2s × 4

= 12

× 1R2

s(36)


32:24 B. Wang

Fig. 9. A strip-based placement pattern for complete coverage and 1-connectivity, a = min{Rc,√

3Rs},b = Rs +

√R2

s − a2/4. When Rc <√

3Rs, one vertical strip is added to ensure 1-connectivity.

λhcsd = 6

3√

32 R2

s × 3= 4

√3

9× 1

R2s

(37)

Obviously, λtcsd < λs

csd < λhcsd and the regular triangular tessellation has the minimum

density requirement among the three tessellations. Actually, regular triangular tessel-lation is the optimal tessellation in terms of the minimum number of nodes requiredfor complete area coverage. This can be deduced from the following theorem proved byR. Kershner [Kershner 1939] in 1939.

THEOREM 5.1 (THEOREM IN KERSHNER [1939]). Let A denote the volume of a boundedplane point set A and let N(r) be the minimum number of disks of radius r which cancompletely cover A. Then

limr→0

πr2N(r)A

= 2√

3π

9. (38)

For a polygon A, Aactually is its area. The left side of (38) is the ratio between the totalarea of the disks covering A and the area of A. The constant 2

√3π

9 of the right side of(38) may be thought of as measuring the proportion of unavoidable overlapping. Nowlet us examine the regular triangular placement pattern. Suppose that the triangulartessellation is used in a very large field. Then the ratio between the total area of alldisks and the area of the whole field can be computed π2

√3R2

s9R2

s= 2

√3π

9 . This indicatesthat the regular triangular tessellation achieves the minimum overlapping. Hence itrequires the minimum number of sensing disks and is an optimal tessellation.

In order to form a connected network, the transmission range Rc should satisfyRc ≥ √

3Rs for the regular triangle tessellation, Rc ≥ √2Rs for the regular square

tessellation, and Rc ≥ Rs for the regular hexagon tessellation. Obviously, network con-nectivity poses another restriction on sensor density. The regular triangle tessellationis also optimal for network connectivity if and only if Rc ≥ √

3Rs. However, it is not un-common that Rc may be smaller than

√3Rs. Some researchers [Kar and Banerjee 2003;

Lyengar et al. 2005; Wang et al. 2005c; Bai et al. 2006; Han et al. 2008] haveproposed and analyzed a kind of strip-based node placement pattern to addressthe network connectivity for Rc <

√3Rs. As shown in Figure 9, a horizonal strip of nodes

is formed by putting together nodes at regular separation of a = min{Rc,√

3Rs}. Thesestrips are deployed horizontally with alternate rows shifted to the right by a distanceof a/2. The vertical separation between the neighboring strips is b = Rs +√R2

s − a2/4.



When Rc >√

3Rs, then a = √3Rs and b = 3

2 Rs, and the deployment pattern is ac-tually a regular triangular tessellation. When Rc <

√3Rs, the nodes within a strip

are connected, but these horizonal strips are not connected. To form a connected net-work for all nodes, some additional nodes are placed as a vertical strip to connectthese horizonal strips. As shown in Figure 9, one vertical strip is added to ensure 1-connectivity. In fact, k-connectivity can be achieved by adding k such vertical strips.When the number of horizonal strips are much larger than the number of verticalstrips, the vertical nodes contribute little to the critical sensor density. Bai et al. [2006]prove that the strip-based placement is asymptotically optimal for achieving completecoverage and 1-connectivity even when Rc <

√3Rs. Bai et al. [2008a, 2008b] have also

studied the optimal node placement patterns for higher connectivity requirement (upto 6-connectivity).

5.1.2. Random Node Deployment. When modeling a random deployment, a sensor nodeis assumed as a point in a two-dimensional plane, and a random deployment is oftenmodeled as either a uniform point process or a Poisson point process with intensityλ. Let A and A .= ||A|| denote a sensor field and its area, respectively. In the randomnode deployments of λA (or N) nodes, a sensor field might be completely covered in onerandom deployment, but might not be completely covered in another random deploy-ment. The Critical Sensor Density (CSD), λcsd, helps to decide at least how many nodesare needed to completely cover the sensor field in every random deployment. Given asensor field with area A, if λcsd A nodes or more are randomly deployed, then the fieldis completely covered almost surely in each deployment. For the sensing disk coveragemodel, complete area coverage indicates that every point in the sensor field is withinthe sensing disk of at least one sensor.

The analysis of CSD normally starts by providing bounds for the complete coverageprobability of a square field A with finite area, and then use asymptotical analysis toprovide the relation between the CSD and the area (or the side length) of the square.For a field A with area much larger than the area of of a sensing disk (i.e., A � π R2

s ),the boundary effect can be neglected by using a torus convention (refer to [Hall 1988,page 23]). We use Cλ to denote the event that a square field A is completely covered byλA sensing disks each with radius Rs. In the random deployments, the CSD is definedas the smallest possible λ such that for every λ ≥ λcsd, the probability Pr[Cλ] = 1 almostsurely. However, the expression of Pr[Cλ] is not easy to obtain. Instead of seeking forexplicit expression of Pr[Cλ], its bound can be obtained by relating the probability ofcomplete coverage to the probability of the field vacancy defined as follows. For a pointz within the square field A, let χ (z) denote the indicator function of whether the pointz is covered, that is,

χ (z) ={

1, if z is not within at least one sensing disk,

0, otherwise.(39)

The vacancy V within A is defined as the area that is not covered:

V = V (A) ≡∫A

χ (z) dz. (40)

As shown by Hall (see Theorem 3.3 and its remarks in Hall [1988]), no vacancy withinA (that is, V (A) = 0) implies the complete coverage of A. Therefore, it is desirable toprovide bounds for probability of no vacancy, that is, to bound Pr[V = 0]. To do so, thefirst step is to establish the probability of an arbitrary point is not covered, that is,Pr[χ (z) = 1], and then to bound the probability of no vacancy based on Pr[χ (z) = 1]from some special points in the field. In the literature, two approaches have been used


32:26 B. Wang

to bound the probability of no vacancy Pr[V = 0] (or equivalently Pr[V > 0]). One is toexploit the property of the sensing disk model [Hall 1988; Zhang and Hou 2004, 2005b];The other is based on a grid approach [Kumar et al. 2004, 2008].

In the first approach, the special crossing points are used to bound the probabilityof no vacancy Pr[V = 0]. A crossing is defined as the intersection point of the outmostcircles of two sensing disks (called sensing perimeter) or an intersection point of asensing perimeter and the boundary of the sensor field A. A crossing is said to becovered if it is an interior point of at least one disk. Note that a crossing is not consideredto be covered by its driving sensing perimeters. Hall provides both the lower and upperbounds for Pr[V > 0] as follows.

THEOREM 5.2 (THEOREM 3.11 IN HALL 1988). Suppose that A is a unit square, and sup-pose that the random deployment is a Poisson point process with intensity λ. Let a = π R2

s .For each λ ≥ 1 and 0 < Rs ≤ 1

2 ,

0.05 min{1, (1 + aλ2)e−aλ} < Pr[V > 0] < 3 min{1, (1 + aλ2)e−aλ}. (41)

In proving the theorem, Hall claims that the probability of vacancy can be divided intothree parts (Theorem 3.11 in Hall [1988]):

Pr[V > 0] = p1 + p2 + p3,

where

p1 ≡ Pr[no disks centered within A] = e−λ = e−λ(1−a)−aλ ≤ e−aλ,p2 ≡ Pr[at least one disk centered within A,

but none of these disks intersects any other disk]≤ Pr[≥ 1disk centered within A] × Pr[a given disk intersects no other disks]= (1 − e−λ)e−λπ(2Rs)2 ≤ e−aλ

and

p3 ≡ Pr [Anot covered, at least one disk centered within A, and at least one ofthese disks intersects with another disk].

It is easy to give upper bounds for p1 and p2. To upper bound p3, the Markov inequality(see, e.g., Yates and Goodman [1999, page 263]) can be applied. Note that p3 is basedon the property of disk coverage. By such decomposition, complete area coverage isconnected with only a few discrete crossings.

Hall’s method based on crossing points has been extended to complete k-coverage [Zhang and Hou 2005b; Wan and Yi 2006]. A field is said completely k-coveredif every point is within the sensing disks of at least k sensors. Let χk(z) denote theindication function of whether a point z is covered by at most k − 1 sensors, that is,

χk(z) ={

1, if at most k − 1 nodes cover the point z,0, otherwise.

Similarly, let the k-vacancy Vk denote the area that is covered by at most k − 1 nodes(i.e., Vk are the area not k-covered). Zhang and Hou [2005b] argue that the probabilityPr[Vk > 0] can also be decomposed into three parts:

Pr[Vk > 0] = p′1 + p′

2 + p′3,



wherep′

1 ≡ Pr{no disks centered within A},p′

2 ≡ Pr{at least one disk centered within A, but none of thedisks intersects any other disk, and none of the disksintersects the boundary of A},

p′3 ≡ Pr{A is not completely k − covered, at least one disk is

centered within A, and at least two disks intersect eachother, or at least one disk intersects the boundary of A}.

After providing bounds for Pr[Vk > 0], asymptotical analysis can be applied to providethe relationship between the sensor density λ and a scaling factor of the sensor field(e.g., the square side length l). Without considering the boundary effect, Zhang and Hou[2005b] show that the asymptotic k-coverage requires the sensor density λ growing withthe side length according to

λ = log(l2) + (k + 1) log log(l2) + c(l), (42)

with liml→∞ c(l) = ∞. Wan and Yi [2006] include the boundary effect into the asymptoticanalysis of critical sensor density and derive the asymptotic k-coverage requirement

λ = log(l2) + 2k log log(l2) + c(l), (43)

with liml→∞ c(l) = ∞.

5.1.3. Notes and Comments. In deterministic node placements, when complete k-coverage is required, a simple approach is to place k sensors at the same location.Another also simple approach is to put k layers of tessellations, where each layer oftessellation provides complete 1-coverage. The latter placement method might be moredesirable as sensors at different locations are not likely fail (e.g., due to environmentalchanges) at the same time. As such, it is desirable to place sensors not too close toeach other (minimum separation requirement) for higher degree of coverage [Kim et al.2008, 2009].

In random node deployments, one would also like to know the average vacancy if agiven number of nodes are to be randomly deployed in a field with finite area [Lazos andPoovendran 2006; Liu and Towsley 2004; Yen et al. 2006; Paillard and Ravelomananana2008]. To find the expected value of vacancy, we can use Fubinis theorem [Samko et al.1993] and exchange the order of integral and expectation

E[Vk] =∫A

E[χk(z)] dz =∫A

Pr[χk(z) = 1] dz = ||A||Pr[χk(z) = 1]. (44)

Compared with the probability of no vacancy Pr[V = 0], the average vacancy E[V ] iseasy to obtain. For example, in a random Poisson deployment an arbitrary point is notcovered if there is no sensor in the disk area centered at this point with radius Rs, thatis,

Pr[χ (z) = 1] = e−λπ R2s. (45)

Similarly, for k-coverage an arbitrary point is not k-covered if there are less than ksensors in the disk area centered at this point with radius Rs, that is,

Pr[χk(z) = 1] = e−aλ

(k−1∑i=0

(aλ)i

i!

)and a = π R2

s . (46)

There are also some CSD analysis based on other sensing models [Adlakha andSrivastava 2003; Liu and Towsley 2004; Mo et al. 2005, 2006; Wang et al. 2007a]. For


32:28 B. Wang

example, Mo et al. [2005, 2006] consider a variant of the sensing disk model. Instead ofusing a fixed sensing radius Rs, they model Rs as a random variable with mean Rs andvariance R2

s σ 2s , and the average sensing area of a sensor is a ≡ E[π R2

s ] = π R2s (1 + σ 2

s ).The derivation of the probability of a point not k-covered and the average k-vacancysimilar to those based on the sensing disk model, only with the coverage area a ≡ π R2

sof a single sensor being replaced by a. Based on the estimation coverage model, Wanget al. [2007a] provide an upper bound of a point not (k, ε)-covered, and use it to boundthe average coverage ratio.

5.2. Sensor Activity Scheduling

In one random node deployment, there may have some redundant sensors whose cov-ered area can also be covered by other sensors. Sensor activity scheduling for areacoverage maintenance is used to schedule nodes to be activated alternatively, such thatthe network operation time may be prolonged and certain area coverage requirementcan still be met. Generally speaking, the design of a sensor activity scheduling schemeshould answer the following question:

How to determine which sensors are active at which time and for how long?

5.2.1. Assumptions and Objectives. Many activity scheduling schemes proposed in theliterature have different assumptions and objectives. The basic assumption is thatthe coverage model of individual sensors or the covered area of individual sensors areknown a priori. Another assumption is on the availability of the nodes’ location ordistance information: whether each node knows its own Cartesian coordinates or twoneighboring nodes know the Euclidean distance between them. Sometimes, all sensorsare assumed to have the same coverage model, and the design of a sensor activityscheduling scheme can be based on the absence of both the nodes’ location and distanceinformation.

The basic objective is to guarantee the area coverage ratio, which is defined as thefraction between covered area and uncovered area of a sensor field. We use A to denotea sensor field, and A(A) to denote its area. Let S = {s1, s2, . . . , } and Sa ⊆ S denotethe set of all the deployed sensors and the set of selected active sensors, respectively.Let A(s) denote the area covered by a sensor s. We use A(S) = A(A)

⋂(⋃

s∈S A(s)) todenote the area of the sensor field covered by all the deployed sensors. Similarly, weuse A(Sa) = A(A)

⋂(⋃

s∈SaA(s)) to denote the area of the sensor field covered by the

selected active sensors. A sensor field is completely covered if A(S) = A(Sa), and ispartially covered if A(Sa) < A(S). The area coverage ratio is defined as A(Sa)

A(S) .Another important objective is to select active sensors as least as possible. An active

sensor consumes energy to sense physical phenomena, and to produce sensing data.Furthermore, its sensing data needs to be sent back to the sink or exchanged withother sensor nodes, which increases energy consumption for this active sensor as wellas others. This objective, however, is often in conflict with the coverage ratio objective.In general, the more the active sensors, the higher the coverage ratio. We should makea good balance between coverage ratio and the number of active sensors.

The network area coverage lifetime is also an important objective. Similar to thedefinition of target coverage lifetime, the area coverage lifetime is defined as the du-ration from the time that the network starts operation until the time that the areacoverage requirement cannot be satisfied, even if all the alive sensor nodes are active.Generally speaking, selecting the least number of active sensors helps to prolong thenetwork lifetime. But care must be taken, since data processing and disseminationalso consume energy and impact on the network lifetime. Sometimes, sensor activityscheduling may not lead to any extension of the network lifetime when preserving



Fig. 10. Illustration of using grid approach for redundancy check. Grid points represent sensor coveredarea.

Fig. 11. Illustration of a coverage hole caused by the inactivation of two redundancy-dependent sensors.

complete area coverage is required. For example, consider the case that a small area ina completely covered sensor field is only covered by one sensor. To preserve completearea coverage, this sensor has to be active all the time, and the network lifetime in thisexample cannot be prolonged via activity scheduling.

5.2.2. Preserving Complete Area Coverage. Complete coverage requires that the coverageratio equals to one. In the algorithm design, the first challenge is to determine whetherthe area covered by one sensor can also be completely covered by its active neighbors.If so, then this node is a redundant one in terms of coverage. A redundant sensor nodeis eligible to shut off its sensor unit, and enters into the energy-saving sleep state. Thesecond challenge is to determine the order of sensor nodes’ activation or deactivation.The redundancy of a node is dependent on its neighbors’ states. A node may be nolonger redundant if one of its active neighbor becomes inactive.

5.2.2.1 Redundancy Check Methods. A straightforward method for redundancycheck is to use a grid approach, where each sensor maintains a list of grid pointswithin its covered area, as shown in Figure 10. If such grid points are covered by itsactive neighbors, then it is a redundant one. For example in Figure 10, the grid pointsmarked by small blue circles are covered by the sensor s1, and all these grid points arealso covered by its neighbors. Hence, s1 is redundant if its neighbors s2, s3, s4, and s5 areactive. Using the grid approach is a simple method, however, it may be computationcomplicated, time consuming, and storage expensive. For the sensing disk coveragemodel, some of its geometric properties can be exploited for redundancy check, and webriefly introduce some representative ones.Sponsored sector: Tian and Georganas [2003] propose a concept of sponsored sector

for checking redundancy. A sensor s2 is called a sponsor to sensor s1 if d(s1, s2) ≤ Rs.Since d(s1, s2) ≤ Rs, the two sensing disks intersects. As shown in Figure 12, thecrescent-shaped area bounded by the bold arcs is the intersection of the two sensingdisks. The sponsored sector by sensor s2 to sensor s1 is the sector area of s1 withinthe intersection crescent of the two sensing disks, as shown by the shaded sector inFigure 12. The reason to use the sponsored sectors other than the intersection crescentsfor redundancy check is to reduce calculation complexity. The area of a sector can be


32:30 B. Wang

Fig. 12. Sponsored coverage for redundancy check.

Fig. 13. Perimeter coverage for redundancy check.

Fig. 14. Crossing coverage for redundancy check.

represented by its central angle (e.g, ∠p1s1 p2 in the figure) accurately and uniting theareas of two sectors is equivalent to merging two central angles. The redundancy ruleby sponsored sector is as follows: If the sensing disk of a sensor s can also be covered bythe sponsored sectors from its sponsors, then the sensor s is redundant and eligible to beinactive. In Figure 12, the sensing disk of sensor s1 is covered by the three sponsoredsectors contributed from the sensors s2, s3, and s4, and hence sensor s1 is redundantand eligible to go to the sleep state.Perimeter coverage: Huang et al. [2007] propose to use perimeter coverage to check

sensor redundancy. Two sensors’ sensing disks intersect each other if d(s1, s2) < 2Rsand each sensor is called a direct neighbor to the other. As illustrated in Figure 13,sensor s1 has 3 direct neighbors. The arc p1 p2 is a segment of s2’s sensing perimeterwithin s1’s sensing disk and is covered by sensors s3 and s4. In this regard, s1 is called acandidate for its director neighbor s2. The redundancy rule by perimeter coverage is asfollows: If a sensor s is a candidate for each of its direct neighbors, then it is redundantand eligible to be inactive. In Figure 13, sensor s1 is a candidate for its direct neighbors3 (the arc p5 p6 covered by s2 and s4) and also a candidate for its direct neighbor s4, andhence it is redundant and eligible to go to the sleep state.Crossing coverage: Xing et al. [2005] apply crossing coverage to determine redun-

dant sensors. If two sensing disks intersect each other, then they create crossings thatare the intersection points on the two disks’ perimeters. As shown in Figure 14, thecrossing p1 within s1’s sensing disk is created by s2’s sensing perimeter and s3’s sensingperimeter. Crossing points can also be caused due to the intersection between a sensor’s



Fig. 15. Voronoi vertices and intersections coverage for redundancy check.

sensing perimeter and the sensor field boundary. The points on the sensing perimeterof a sensor is not considered as covered by this sensor. Therefore, in Figure 14, thecrossing p1 is not covered by the sensor s2 and s3. But p1 is covered by the sensor s4.In this regard, the crossing p1 is considered as a covered crossing within sensor s1’ssensing disk. The redundancy rule by crossing coverage is as follows: If all crossingswithin a sensor’s sensing disk are covered, then s is redundant and eligible to be inactive.In Figure 14, there are 3 crossings (p1, p2, and p3) within sensor s1’s sensing disk andthey are all covered (p2 covered by s3 and p3 covered by s2), and hence ss is redundantand eligible to be inactive.Voronoi diagram vertices and intersections: Carbunar et al. [2006] propose to

use Voronoi diagram vertices and intersections to check redundancy. A Voronoi diagram(see, e.g., Aurenhammer [1991]) for N sensors s1, s2, . . . , sN in a plane is defined as thesubdivision of the plane into N cells each for one sensor, such that the distance betweenany point in a cell and the sensor of the cell is closer than that between this point andother sensors. Two Voronoi cells meet along a Voronoi edge and a sensor is a Voronoineighbor of another sensor if they share an Voronoi edge. When checking sensor sredundancy, a 2-Voronoi diagram is first constructed, which the Voronoi diagram of theVoronoi neighbors of s when s is excluded. In Figure 15, the bold lines form a 2-Voronoidiagram for sensor s1. The 2-Voronoi Vertices (2-VV) of a sensor s are the Voronoivertices of the 2-Voronoi diagram of s. A 2-Voronoi Intersection Point (2-VIP) of s is theintersection between an edge of the 2-Voronoi diagram and the sensing perimeter of s.In Figure 15, there are one 2-VV (2-VV1) and three 2-VIPs (2-VIP1, 2-VIP2, 2-VIP3).The redundancy rule by Voronoi diagram vertices and intersections is as follows: If allthe 2-VVs and 2-VIPs of a sensor s are covered by the Voronoi neighbors of s, then s isredundant and eligible to be inactive. In Figure 15, sensor s1 is eligible since 2-VV1 and2-VIP1,2,3 are all covered by s1’s Voronoi neighbors.

—Activity Scheduling Procedures. A sensor activity scheduling algorithm can be ei-ther centralized or distributed. A distributed scheduling algorithm is more desirable,as it can be easily scaled to large-scale sensor networks. Most existing protocols aredistributed, and only local message exchanges are incurred in the sensors’ decision pro-cess. Although these protocols differ in their redundancy check methods, their schedul-ing procedures are similar. It is often assumed in these protocols that the time-line isdivided into consecutive rounds. At the beginning of each round, there is a decisionstage where all sensor nodes should make their activity decisions in the current round.Normally, the length of the decision stage is much less than the length of a round. Atthe end of a round, all sensors are required to be active, and they enter the decisionstage again in the next round.

We identify two approaches for scheduling procedures, namely, the self-inactivationapproach and sequential activation approach. The two approaches differ in the messageexchange and process method. In the self-inactivation approach, a sensor which hasdecided to be inactive broadcasts a SLEEP message that is used to alert its neighborsto recheck their redundant eligibility. In the sequential activation approach, a sensor


32:32 B. Wang

Table II. Steps of the Self-Inactivation Approach

Step 0: Set self state as active.Step 1: Collect all neighbors’ information via local message exchanges

and build a list of active neighbors.Step 2: Perform redundancy check. If redundant, set a timer with a

random backoff time and goto Step 3.Step 3: Wait for the timer expiration. If receive a SLEEP message,

rebuild the list of active neighbors and goto Step 2.If timer expires, goto Step 4.

Step 4: Broadcast a SLEEP message and set self state as sleep.

which has decided to be active broadcasts an ACTIVE message that is used to set orreset its neighbors’ activation timers. Many protocols have included some more messagetypes and have introduced transient states as well as a state transition machines toavoid simultaneous inactivations and to combat transmission collisions and losses. Inwhat follows, we simply use two states, that is, active and sleep and two message types,that is, SLEEP and ACTIVE to describe the basic operation steps of the two approaches.Self-Inactivation. In the self-inactivation approach, each sensor node maintains

a list of its active neighbors. At the beginning of each decision stage, the list actuallyincludes all of its neighbors. A sensor node performs redundancy check based on theassumption that all neighbors in the list are active. After a sensor node has decidedits own redundancy, it can go into a sleep state. When this is done in a distributedmanner, care must be taken to avoid creating coverage holes. Since a sensor redundanteligibility is dependent on its neighbors’ active state, a coverage hole may appear iftwo redundancy-dependent sensor nodes go to sleep at the same time. For example, inFigure 11, s1 is completely covered by s2, s3, and s4 and s2 is also completely coveredby s1, s5, and s6. However, s1 and s2 are redundancy-dependent sensors and cannot besimultaneously inactive. Otherwise, some points only covered by s1 or s2 will not becovered, and a coverage hole is created. This problem can be mitigated or avoided byusing a random backoff mechanism and carefully designed message exchange processfor asynchronous activity decision-making. We sketch the basic steps required in thedecision stage of the self-inactivation approach in Table II.Sequential Activation: In the sequential activation approach, each sensor node

also maintains a list of its active neighbors. Compared with the self-inactivation ap-proach, the list is empty at the beginning of each decision stage. Each sensor sets anactivation timer. Upon the expiration of the timer, it sets itself as active and broad-casts an ACTIVE message. A sensor node which has received a new ACTIVE messageresets its timer, builds its active neighbor list, and performs redundancy check. Theredundancy check is normally based on the grid approach. A redundant sensor setsitself as sleep and ignores the following messages. As the name suggests, it is desirablethat sensor nodes are activated sequentially. At first, a sensor volunteers to be activewith a small probability, and a nonvolunteer sets a long expiration time. This firstactive sensor node then activates its neighbors by broadcasting an ACTIVE messageto reset its neighbors’ timers and such process continues until all sensors have decidedtheir states. How to adjust the activation timer determines the selection of desiredactive sensors as well as the number of active sensors. We sketch the basic steps of thisapproach in Table III.

Two representative sensor activity scheduling protocols are the Coverage Configura-tion Protocol (CCP) [Xing et al. 2005], which applies crossing-based redundancy checkand follows the self-inactivation scheduling procedure, and the Optimal Geographi-cal Density Control (OGDC) [Zhang and Hou 2005a], which applies grid-points-basedredundancy check and follows the sequential activation scheduling procedure.



Table III. Steps of the Sequential Activation Approach

Step 0: Set self state as active and set an empty list of active neighbors.Step 1: Volunteer to be active with a small initial probability. If

volunteer, goto Step 4. If not, set a timer and adjustthe volunteer probability, and goto Step 3.

Step 2: Perform redundancy check. If redundant, set self state assleep. If not, goto Step 3.

Step 3: Wait for the timer expiration. If receive an ACTIVE message,rebuild the list of active neighbors, adjust the expirationtimer, and goto Step 2.

Step 4: Broadcast an ACTIVE message.

5.2.3. Preserving Partial Area Coverage. Preserving complete area coverage is a desirableyet demanding objective. Sometimes, an activity scheduling protocol that can providehigh average coverage ratio may be of more practical interest. Analysis and simulations(see, e.g., Zhang and Hou [2004] and Wang and Kulkarni [2008]) have shown that thenetwork coverage lifetime can be greatly prolonged if only preserving partial coverageother than preserving complete coverage is required.

Normally, a sensor node does not need to perform a redundancy check, when onlypreserving partial coverage is required. Hence computation complexity and storagerequirement can be greatly reduced. Many activity scheduling protocols have beenproposed for preserving partial area coverage. We classify them into two main groups.In the first group, a sensor makes its activity decision independent of others. In thesecond group, a sensor exploits its neighbors’ information to make the activity decision.

—Random Independent Sleeping. Random Independent Sleeping (RIS) might be thesimplest sensor activity scheduling scheme where a sensor node decides its activitystates independent of other sensor nodes (see, e.g., Gui and Mohapatra [2004], Tianand Georganas [2004], Choi and Das [2006], and Abrams et al. [2004]). RIS has twomain advantages: (1) No location or distance information is required; and (2) no controlmessage is required in RIS. A RIS scheme can be implemented in either asynchronousor synchronous approaches.

An asynchronous approach can be as follows. The timeline is divided into consecutiverounds with equal length T for each sensor node but the beginning time of the veryfirst round is different across nodes, that is, the rounds are not synchronized acrosssensors. At the beginning of a round, a sensor decides its active state with the durationgiven by p × T and the remaining part of the round is the sleep state.

Another approach to implement a RIS scheme is to use synchronous decisions asfollows. The timeline is divided into rounds of equal length and the starting time ofevery round is considered to be synchronized across sensor nodes. Two versions can beimplemented following this synchronized decision approach. One is to, at the beginningof each round, let each node decide its active state for this round with probability p.This method as well as the aforementioned asynchronous RIS produce nondisjoint setsof active sensors in different rounds. That is, the intersection between the active sensorset produced in round i and that produced in round i + 1 may not be an empty set.Another version is to divide the deployed nodes into K disjoint subsets to be activatedin a round-robin manner. A simple randomized algorithm is to let each sensor randomlygenerate an integer between 1 and K to decide which set it belongs to. Obviously, thismethod produces disjoint subsets of active sensors in different rounds.

Suppose that N sensor nodes are uniformly deployed in the sensor field. The expectednumber of active sensors at any time in RIS is p× N or N/K, and they are distributeduniformly. The parameter p or K in the RIS is assumed as globally known by all sensorsand its value is dependent on the requirements of coverage ratio. Although RIS is very


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easy to implement, it may lead to low coverage ratio (when p is small or K is large) ormore selected active sensors (when p is large or K is small). Analysis or simulation canbe used to determine the relation between the value of p or K and the coverage ratio.

The relation between the number of sensors N and the area coverage ratio ρc canbe derived by using asymptotic analysis as follows [Hall 1988; Liu and Towsley 2004].Let λ denote the sensor density, that is, the number of sensor nodes per unit area. Fora given large region A with area A, the number of nodes in this area is n = λA. Theprobability of a point within A which is not covered by an arbitrary sensor (i.e., thepoint does not lie within the sensor’s sensing disk) is given by 1 − a

A, where a = π R2s .

Since sensors are uniformly located in A, then the probability that the point does notwithin any sensors equals (1 − a

A)n. In the limit when A → ∞, we have

Pr[a point not covered] = E

[(1 − a

A

)n]= e−(a/A)E(n) = e−aλ = e−λπ R2

s . (47)

The fraction of the area being covered is hence given by

ρc = 1 − e−λπ R2s. (48)

This formula can be used in the network planning to determine the required sensordensity. Given the sensor density and the area of the sensor field, it can be used todetermine the active probability p or the number of covers K for a specified coverageratio.

In RIS, a point that is within K sensors’ sensing disks may be k-covered (k ≤ K) by atleast k active sensors in some time interval (t1, t2), and not-k-covered at all in anothertime interval (t2, t3). A point is not k-covered if it is covered by less than k active sensors.The coverage of a point hence can be modeled as an alternative renewal process withtwo states: k-covered and not-k-covered. The probability or the distribution of such apoint being k-covered is important to detection and tracking applications. Let Y (k) andZ(k) be two random variables denoting the length of k-covered period and not-k-coveredperiod, respectively. Given that a point is within n sensors’ sensing disk and the sensorsare active with probability p in each round, the expectations for Y (k) and Z(k) can becomputed as follows [Hua and Yum 2007].

E[Y (k)] =

1 −∑k−1i=0

(Ki

)pi(1 − p)K−i

K(

Kk−1

)pk(1 − p)K−k+1

, k = 1, . . . , K

and

E[Z(k)] =

∑k−1i=0

(Ki

)pi(1 − p)K−i

K(

Kk−1

)pk(1 − p)K−k+1

, k = 1, . . . , K

fan Hsin and Liu [2006] also analyze the tail distribution of the probability that a pointwithin K sensors’ sensing disks cannot be covered by any active sensor.

—Neighbor-Based Scheduling. Unlike the totally independent RIS scheduling, a sen-sor node can schedule its activity based on its neighbors’ information. Such informationcan include the distances between itself and its neighbors, and the number of its activeneighbors, etc. For example, a sensor node is eligible to enter the sleep state if has oneor more active neighbors (see, e.g., Gao et al. [2003], Tian and Georganas [2004], Wuet al. [2005], [Ye et al. 2006], Bai et al. [2007], and [Wang et al. 2009a]).

The (communication) neighbors of a node are those nodes that can communicatedirectly with this node. A commonly used communication model is the communicationdisk model, which is a disk centered at a sensor with radius the transmission range Rc.



Fig. 16. Illustration of neighbor distance-based scheduling: (a) An active sensor within the probing area;(b) the intersection between two nodes’ sensing disks.

A direct communication link exists between two nodes if their Euclidean distance is notlarger than the transmission range. Transmission range is dependent on transmissionpower. If the transmission power is adjustable, so the transmission range; and thelarger the transmission power, the larger the transmission range.Neighbor Distance-Based Scheduling. A simple approach is that a sensor is eli-

gible to enter the sleep state if it finds an active neighbor not far away from itself.For example, in the Probing Environment and Adaptive Sleeping (PEAS) schedulingprotocol [Ye et al. 2006], a sensor decides to sleep if it finds at least one active neighborwithin its probing area. The probing area of a sensor is a disk centered at the sensorwith the radius the probing range, as illustrated by the dashed disk in Figure 16(a).In PEAS, each sensor sleeps for an exponentially distributed duration. When a sensorwakes up, it probes whether there exists any other active sensor within its probingarea by sending a probe message. Any active sensor that hears this probe messageshould reply. And if at least one reply is received by this probing sensor, then it entersthe sleep state again for another random interval. Otherwise, it enters the active stateuntil its death. Evidently, the probing range controls the density of the active nodes,and the larger the probing range, the smaller the coverage ratio.

Tian and Georganas [2004] propose a closest-neighbor-based scheduling algorithm,where a sensor node is eligible to sleep if the distance to its closest active neighbor isless than a predefined threshold. Indeed, if the distance between two nodes is not morethan 2Rs, then there are some intersection areas covered by both sensors. As shown inFigure 16(b), the intersection lens is the area covered by both sensor nodes, and can becomputed as follows.

A(si)⋂

A(sj) ={

R2s cos−1

(d

2Rs

)− d

2

√4R2

s − d2, d ≤ 2Rs,

0, d > 2Rs

Neighbor-Number-Based Scheduling. Wu et al. [2005] and Gao et al. [2003] analyzethe relation between the coverage of a sensor’s sensing disk and the number of itssensing neighbors. A sensor node is a sensing neighbor of another if their Euclideandistance is not larger than the sensing range Rs. Let Ni and n = |Ni| denote the setand the number of sensing neighbors of a sensor si, respectively. They show that if asensor’s sensing neighbors cover its sensing disk, that is,

⋃s∈Ni

A(s) ⊇ A(si), then wecan find a subset of Ni, denoted by N ′

i , such that⋃

sj∈N ′i

A(sj) ⊇ A(si) and 3 ≤ ∣∣N ′i

∣∣ ≤ 5.This suggests that if a sensor’s sensing disk is covered by its sensing neighbors, thenwe need to choose at least 3 sensing neighbors and at most 5 sensing neighbors tocover the sensor’s sensing disk. As shown in Figure 17(a), the node s1 has four sensingneighbors, namely, s2, s3, s4, and s5. The four sensing neighbors can completely coverthe s1’s sensing disk, and actually only three of them, namely, s2, s3, and s4, need to beactive to completely cover the s1’s sensing disk.


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Fig. 17. Illustration of the coverage by sensing neighbors: (a) The sensing disk is completely covered;(b) some areas of the sensing disk are not covered.

This result, however, is not very useful in practice, since it is usually not easy tocheck whether a sensor is completely covered without accurate geography information.Furthermore, it is also possible that the sensing neighbors cannot completely coverthe sensing disk, as illustrated by Figure 17(b). On the other hand, it may be moreuseful to estimate the probability that a sensor’s sensing disk is completely coveredand its average covered area of the sensing disk. Suppose there are n sensing neighborsrandomly and uniformly distributed within the sensing disk of a sensor node s. Let Andenote the area within this sensor’s sensing disk that are covered by these n sensingneighbors, that is, An = (

⋃nj=1 A(sj))

⋂A(s). Wu et al. [2005] provide an upper and a

lower bound of the probability that the sensor s’s sensing disk is fully covered by itssensing neighbors, that is, Pr{A(s) = An}, as follows.

1 − n0.609n−1 ≤ Pr{A(s) = An} ≤ 1 − n0.609n−1 + n(n − 1)2

(0.276)n−1

They also provide a lower bound for the average covered area E[An] as follows.

E[An] ≥ π R2s ×[1 − 0.609n −

(n6

− 0.109)

0.109n−1]

Given the coverage requirement E[An], a sensor can determine the least number of itsactive sensing neighbors from the preceding formula.

5.2.4. Preserving Area Coverage and Network Connectivity. A sensor activity schedulingalgorithm that is designed to schedule the state (sleep or active) of nodes’ sensorunits does not necessarily imply that an inactive sensor node shall also shut off itscommunication unit and cannot serve as an intermediate relay for forwarding data.In some cases, it might be desirable for a node with inactive sensor unit to turn offits communication unit as well. However, network connectivity may not be guaranteedif only those active sensor nodes that are selected by a sensor activity schedulingalgorithm are used for data transport. In these cases, sensor activity scheduling shouldensure both area coverage and network connectivity.

5.2.4.1 Relation between Area Coverage and Network Connectivity. Let us assumethat the sensor coverage model is a disk with radius Rs (sensing range), and thenode communication model is also a disk with radius Rc (communication range). Thefollowing theorem states the relation between complete area coverage and the networkconnectivity with respect to the relation between Rs and Rc.

THEOREM 5.3. Theorem 1 in Xing et al. [2005], Lemma 1 in Zhang and Hou [2005a].Suppose that a set of sensors S provides complete 1-coverage for a convex field. Thenthese sensors form a 1-connected network if Rc ≥ 2Rs.

This theorem indicates that in case of Rc ≥ 2Rs, a sensor activity scheduling algo-rithm that provides complete area coverage can also guarantee network connectivity.



Tian and Georganas [2005] extend this result to the scenario where the complete areacoverage is relaxed to the union of covered area by a set of sensors.

THEOREM 5.4. [TIAN AND GEORGANAS 2005]. Suppose that a set of sensors S can forma connected network and can cover an area A .= ⋃s∈S A(s). If a subset of sensors Sa ⊆ Scan also preserve the same coverage, Aa

.= ⋃s∈SaA(s) = A, and Rc ≥ 2Rs, then sensors

in Sa also form a connected network.

The difference between the two theorems is that although all sensor nodes in S form aconnected network, they may not be able to provide complete area coverage.

—Connected Coverage Scheduling. The objectives of a connected coverage schedulingalgorithm are to select active sensor nodes, such that the area coverage requirementcan be satisfied by these active sensors and these selected nodes (together with thesink) also form a connected network. As just discussed, if Rc ≥ 2Rs, a sensor activityscheduling algorithm preserving complete area coverage can also guarantee networkconnectivity. On the other hand, in the case of Rc < 2Rs or partial area coverage, someextra nodes may need to be activated (or some nodes cannot be deactivated) or onlyconnected nodes are selected for activation, in order to ensure complete area coverageand maintain network connectivity.Activate Extra Nodes. In this approach, a sensor activity scheduling algorithm is

first executed to select sensing active nodes. Then the network connectivity of theseactive nodes is examined. If these nodes cannot form a connected network, some extranodes are activated [Liu et al. 2006a; Tian et al. 2008].

For example, Liu et al. [2006a] propose an extra-on rule to select more active nodesto guarantee that each active node can find a path with the minimum hops to thesink. At first, a synchronous decision approach of RIS is applied to generate k disjointsensor set covers. That is, each sensor node randomly selects a number i drawn from auniform distribution from 1 to k, and activates itself in the corresponding round. Thena controlled flooding is enforced to setup the hop count relative to the sink for everynode. If a node s1 is active in a round i but none of its upstream nodes is active in thesame round, then s1 should activate at least one of its upstream node in the round i.Select Connected Nodes. In this approach, the network connectivity constraint is

embedded in the sequential selection of active nodes. At each selection step, only thosenodes that are connected to at least one of the already selected nodes are eligible to beactivated [Gupta et al. 2003, 2006; Liu and Liang 2005; Zou and Chakrabarty 2005;Funke et al. 2007].

Let Sa be the set of selected sensor nodes, and let N (Sa) denote the set of sensors thatare neighbors of at least one sensor node in Sa but not in Sa. A greedy algorithm [Guptaet al. 2003, 2006] to select connected active nodes can be as follows: At each selectionstage, only those nodes in N (Sa) are considered, and the node in N (Sa) that can max-imize a profit function is selected. Since only nodes in N (Sa) are considered at eachstage, network connectivity can be guaranteed. By using different profit functions, dif-ferent objectives such as minimal energy consumption and optimal balance betweenenergy consumption and coverage requirement can also be achieved.

5.2.5. Notes and Comments. In the redundancy check methods, the grid approach canalso be used in sensor activity scheduling to achieve differentiated coverage, whereeach grid point may be required to be covered by different numbers of sensors [Yanet al. 2008; Du and Lin 2005]. There are also some extensions or variants of the fourredundancy check approaches discussed in this section, such as the extended sponsoredarea approach [Jiang and Dou 2004; Boukerche et al. 2007; Noh et al. 2008], theextended perimeter coverage approach [Quang and Miyoshi 2008, 2009], the extended


32:38 B. Wang

crossing approach [Huang et al. 2005a; Lu et al. 2007; Gallais et al. 2008], and theextended Voronoi approach [Boukerche and Fei 2007]. These schemes have extendedthe basic redundancy check approaches in order to schedule fewer active sensors,to reduce computation complexities, or to cope with sensors with different sensingranges. Some researchers propose to use estimated distances between neighboringnodes, instead of their coordinates, to schedule sensor activity [Wu and Ssu 2005;Younis et al. 2008; Zhang et al. 2007]. For example, the basic idea behind Wu and Ssu[2005] is to guarantee that all points within a triangle are covered by the three activesensors on the triangle vertices.

In the design of activity scheduling protocols, many other variants have been pro-posed based on different assumptions, objectives, or other coverage models, such asdirectional coverage models [Tezcan and Wang 2008], disk coverage models with ad-justable sensing ranges [Wu and Yang 2005], detection coverage models [Ahmed et al.2005; Ren et al. 2007; Fusco and Gupta 2009; Hefeeda and Ahmadi 2007], and estima-tion coverage models [Vashistha et al. 2007b; Wang et al. 2008].

In the connected coverage scheduling, the greedy algorithm can also be adaptedto other coverage models, such as the disk coverage model with adjustable sensingranges [Zhou et al. 2004, 2005, 2009], and the profit function of a sensor node canalso take into consideration of the node’s residual energy and transmission energyconsumption [Wang et al. 2007b; Zhao and Zhao 2009].

6. BARRIER COVERAGE PROBLEMS

In barrier coverage problems, the objective is to identify the desired coverage charac-teristics, if it exists, for a sensor network.

6.1. Build Intrusion Barrier

Intrusion detection is a typical application of sensor networks. When mobile objectsare entering into the boundary of a sensor field or are moving across the sensor field,they should be detected by the deployed sensors. It might not be necessary to detect amoving object at every point on its trajectory. Instead, it might be enough if the objectcan be detected at least by k distinct sensors before it penetrates through the sensorfield. This requires a sensor network to provide k-barrier coverage over a sensor field.The following questions are addressed in barrier coverage problems:

What is k-barrier coverage and how to provide k-barrier coverage?

6.1.1. Sensor Barrier for Intrusion Detection. Kumar et al [2005, 2007a] introduce the fol-lowing definition for k-barrier coverage. A belt region is defined by two parallel curvesseparated by a distance w. Let d(x, l) denote the Euclidean distance between a pointx and a curve l and d(x, l) = min{d(x, y) : y ∈ l}. Two curves l1 and l2 are said to beparalleled with separation w if d(x, l2) = d(y, l1) = w for all x ∈ l1 and y ∈ l2. Thefollowing definitions are used for k-barrier coverage.

Definition 6.1 (Belt of Width w). If l1 and l2 are two parallel curves with separationw, the region between l1 and l2 is referred to as a belt region of width w. The two curvesl1 and l2 are the belt’s parallel boundaries.

Definition 6.2 (Crossing Path). A path is said to be a crossing path if it crosses fromone parallel boundary of a belt region to the other boundary of the belt region.

Definition 6.3 (k-Barrier Coverage). A sensor network deployed over a belt regionis said to provide k-barrier coverage if and only if all crossing paths through the beltintersect the sensing region of at least k distinct sensors.



Fig. 18. Illustration of k-barrier coverage: (a) The network cannot provide barrier coverage; (b) the networkcan provide 2-barrier coverage.

For ease of presentation, a belt region is assumed from left to right and two parallelboundaries are referred to as the top and the bottom boundary. Furthermore, intrusionmovement is assumed to occur from top to bottom of the belt. Figure 18 illustrates thesedefinitions in a rectangular belt. The cross paths in Figure 18(a) do not interact withany sensor’s sensing region, and hence the network cannot provide barrier coverage.The network in Figure 18(b) can provide 2-barrier coverage. It is seen that providingbarrier coverage does not imply complete area coverage.

An interesting question is, given a sensor network randomly deployed in a belt, howto determine whether the belt provides k-barrier coverage or not? Kumar et al. [2005]show that this question cannot be locally answered by individual sensors. Kumar et al.[2005] propose to use a global coverage graph to check k-barrier coverage for an openbelt. A coverage graph CG = (V, E) of a sensor network is constructed as follows. Thevertex set V corresponds to all sensor nodes. In addition, it has two virtual nodes, s andt to correspond to the left and right boundaries. An edge exists between two nodes iftheir sensing regions overlap in the belt region. An edge exists between a node and s (ort) if the sensing region of the node overlaps with the left boundary (or right boundary)of the belt. The following theorem provides an existence check method for k-barriercoverage.

THEOREM 6.4. [THEOREM 4.1 IN KUMAR ET AL 2007a]. A sensor network N is deployedover an open and ordinary belt region B. The belt region B is k-barrier covered by thesensor network N iff there exist k node-disjoint paths between the two virtual nodes sand t in the coverage graph CG.

For example, in Figure 18(b) a CG is constructed for the deployed sensor nodes, and2 node-disjoint paths (the bold lines) between s and t exist. Hence according to thepreceding theorem, the belt if 2-barrier covered.

Another interesting question is: If sensors are randomly deployed within an openbelt, what is the minimum number of sensors required to provide k-barrier coverage?This problem is the critical sensor density problem for k-barrier coverage, and hasbeen studied in Kumar et al. [2007a], Balister et al. [2007], Liu et al. [2008a], andSaipulla et al. [2009]. For example, Kumar et al. [2007a] study this problem for a kindof weak barrier coverage where only those orthogonal crossing paths are considered.That is, what is the critical density to ensure that every orthogonal crossing path canbe k-barrier covered? Their results, however, may not provide practical insights, as astealth penetration often follows a nonorthogonal crossing path. Liu et al. [2008] studythe critical conditions for the strong barrier coverage: A sensor network is said to bestrongly k-barrier covered if the probability of any crossing path is k-barrier covered


32:40 B. Wang

equals to 1 with high probability. They apply the results from the percolation theory(see, e.g., Grimmett [1999]) to derive the critical conditions for strong barrier coverage.

THEOREM 6.5. [THEOREM 1 IN LIU ET AL. 2008a]. Consider a sensor network deployed ona two-dimensional rectangular area B = [0, l] × [0, w(l)], where sensors are distributedaccording to a Poisson point process with density λ.

—If w(l) = �(log l), the network is strongly barrier covered w.h.p. (with high probability),where the sensor density reaches a certain value. There exists a positive constant βsuch that w.h.p. there exists βw(l) disjoint horizontal sensor barriers crossing the strip.

—If w(l) = o(log l), the network has no strong barrier coverage w.h.p., regardless whatthe sensor density is in the underlying sensor network.

This theorem states that the existence of the strong barrier coverage in a rectangle beltdepends on the width-to-length ratio of the rectangle region. If the width of the beltis asymptotically smaller than the logarithm of the length, then the network has nostrong barrier coverage, regardless what sensor density is in the sensor network. Onthe other hand, if the width is asymptotically larger than the logarithm of the length,the network has strong barrier coverage for a certain sensor density. Their analysisapproach has been extended to a more practical scenario where all sensor nodes arerandomly deployed in a thin belt region (the width of the belt is not too much largerthan the sensing range) [Saipulla et al. 2009]. As nodes are confined within such athin belt, their offsets to a single barrier are much smaller than that generated by arandom deployment over a rectangle field, and the critical density for establishing abarrier can be significantly reduced.

6.1.2. Sensor Scheduling for Barrier Construction. In random sensor deployments, sensoractivity scheduling can be used to activate sensors alternatively to form barriers, whichhelps to prolong the network lifetime. This is similar to the problem of scheduling sensoractivity for area coverage. However, as the objective is different, providing barriercoverage other than providing area coverage, the solution approaches are different.Several sensor activity scheduling algorithms for barrier coverage have been proposedin Kumar et al. [2007a, 2007b], Chen et al. [2007, 2008]; Liu et al. [2008a], and Yangand Qiao [2009].

The simplest scheduling algorithm again is the Random Independent Sleeping (RIS)scheme where each sensor self-activates itself for some time with some probability pand independent other sensors. In the RIS approach, the barrier coverage can only beguaranteed probabilistically. In order to obtain high probability of achieving barriercoverage, more sensors or small activation probability p are used. Kumar et al. [2007b]propose two algorithms, namely, the stint algorithm for homogeneous networks and thePrahari algorithm for heterogeneous networks, to optimally schedule sensors’ activity.The optimality means that the two algorithms can achieve the upper bound of thenetwork lifetime.

In Chen et al. [2007], Chen et al. define L-local k-barrier coverage and propose aLocalized Barrier Coverage Protocol (LBCP) to let each sensor locally schedule itsown activity state only based on the information of its neighbors. Chen et al. refer tothe barrier coverage defined in Kumar et al. [2007a] as global barrier coverage as itguarantees intrusion detection for any crossing path within the belt. As illustratedin Figure 19, the belt is not globally barrier covered as there exists a long uncoveredcrossing path. Chen et al. [2007] argue that movements are likely to follow a shorterpath when crossing a belt region and hence the trajectory is likely to be bounded withina slice of length L, called L-zone. They define L-local k-barrier coverage as follows: ForL > 0 and k = 1, 2, . . . , a belt region is said to be L-local k-barrier covered if everyL-zone in the region is k-barrier covered.



Fig. 19. Illustration of global barrier coverage and L-local barrier coverage.

6.2. Find Penetration Paths

The coverage problem of finding penetration paths is rooted in the intrusion detectionand tracking applications. A penetration path is a crossing path (a continuous curvewith arbitrary shape) which enters the sensor field from one side and leaves the sensorfield from the other side. The objective is to identify one crossing path with every pointon it whose coverage measure satisfies a predefined coverage requirement. This isdifferent from the problem of building intrusion barriers, which is mainly to guaranteethat some points of every crossing path should meet certain coverage requirements.The following questions are addressed in this subsection.

What are penetration paths and how to find a penetration path?

6.2.1. Maximal Breach Path. Megerian et al. [2005] and Meguerdichian et al. [2001a]might be the first to study the penetration path problem. They apply the Euclideandistance between a point and its closest sensor node as the coverage measure for apoint and introduce the following maximum breach path problem. Let S denote the setof sensor nodes. Given an initial location I and a final location F, let P denote a pathconnecting I and F.

Definition 6.6 (Path Breach). The breach of a path P connecting I and F is definedas the minimum Euclidean distance from P to any sensor in S.

For every point on P, we measure the Euclidean distance between the point and itsnearest sensor, and then take the minimum among these distances.

Definition 6.7 (Maximal Breach Path). Among all the paths connecting I and F, theone with the maximum breach value is called a maximal breach path, PB.

From the viewpoint of an intruder, a PB is the safest path, since the distance fromthe nearest sensor along PB is maximized. But for the defender, this is a worst pathsince the chance of detecting an intruder is minimized. This is called worst coveragein Megerian et al. [2005]. There are infinitely many paths connecting I and F, andexhaustive search for the maximal breach path is not possible. Megerian et al. [2005]apply the Voronoi diagram to provide a geometric division of the sensor field and arguethat at least one maximal breach path lies on the Voronoi edges. They also provide acentralized algorithm to find a PB. The algorithm performs binary search and breadth-first search to find PB on the graph induced from the Voronoi diagram.

After Megerian et al. [2005], many other researchers have also studied such breachpath problems and proposed improved algorithms to find a maximal breach path or tominimize path breach [Adriaens et al. 2006; Duttagupta et al. 2006, 2008; Fang andLow 2007; Huang et al. 2006, 2005b; Mehta et al. 2003; Hou et al. 2009]. For example,Mehta et al. [2003] argue that a PB can also be found on the maximum spanning tree


32:42 B. Wang

Fig. 20. Examples of: (a) the Voronoi diagram and the maximal breach path; (b) the Delaunay triangulationand the maximal support path. S and D are the source and destination points.

of the induced Voronoi graph. Mehta et al. [2003] also consider a redundant breachproblem where the coverage measure of a point is defined as the Euclidean distancebetween this point and its kth closest sensor. This corresponds to the scenario that anintruder is detected only if it is simultaneously near k sensors. To find such a redundantmaximal breach path, the kth nearest sensor Voronoi diagram is used instead of theordinary Voronoi diagram (Figure 20).

6.2.2. Maximal Support Path. Megerian et al. [2005] propose the problem of finding amaximal support path, which also uses the Euclidean distance between a point and itsclosest sensor as the coverage measure. The support of a path is defined for a giveninitial location I and a given final location F and the sensor set S, and is as follows.

Definition 6.8 (Path Support). The support of a path P connecting I and F is definedas the maximum Euclidean distance from P to the closest sensor in S.

For every point on P, we measure the Euclidean distance between the point and itsnearest sensor and then take the maximum among these distances.

Definition 6.9 (Maximal Support Path). Among all the paths connecting I and F,the one with the lowest support value is called a maximal support path, PS.

From the viewpoint of an intruder, this is the worst possible path to take as thechances of being detected are maximized. But for the defender, this is best path alongwhich the surveillance quality is the highest. This is called best coverage in Megerianet al. [2005]. To reduce search space, Megerian et al. [2005] apply the Delaunay trian-gulation to provide a geometric division of the sensor field and argue that at least onemaximal support path lies on the edges of the Delaunay triangulation. The algorithmused for searching PB can also be used to find PS with some changes.

6.2.3. Exposure Path. Exposure path is another type of penetration path which mea-sures how well a sensing field is covered in terms of the expected ability to detect amoving target. The higher the exposure, the better the coverage that the network canprovide. Similar to the breach path and support path, a distance-dependent sensingfunction is used as the coverage measure to define the exposure of a point [Megerianand Koushanfar 2002; Meguerdichian et al. 2001b; Veltri et al. 2003]. In the maximalbreach path or the maximal support path problem, the breach or the support of a pathis defined as the extremum value of the coverage measure of all points on the path.But in the exposure path problem, the exposure of a path is defined as the average ofthe coverage measure of all points on the path.

Megerian and Koushanfar [2002] define the sensibility of a point p by a sensor s asinversely proportional to their distance S(s, p) = A

[d(s,p)]α , where the constants A and



α are sensor technology-dependent parameters. Given this sensor coverage function,two models can be used to measure the sensing field intensity exerted by the sensorfield F to a point p. The closest sensor field intensity is defined IC(F, p) = S(smin, p),where smin is the sensor closest to point p. The all sensor field intensity is defined asIA(F, p) = ∑n

i=1 S(si, p), where every active sensor si contributes a certain amount ofsensitivity to the point p depending on its distance to the point.

Suppose an intruder is moving in the field F from point p(t1) to point p(t2) along thepath p(t). The exposure of this path is defined as follows.

Definition 6.10 (Path Exposure). The exposure for an intruder in the sensor field Fduring the interval [t1, t2] along the path p(t) is defined as

E(p(t), t1, t2) =∫ t2

t1I(F, p(t))

∣∣∣∣dp(t)dt

∣∣∣∣dt,

where the sensor field intensity I(F, p(t)) can be either IA(F, p(t)) or IC(F, p(t)) and|dp(t)/dt| is the element of arc length. For example, if p(t) = (x(t), y(t)), then |dp(t)

dt | =√( dx(t)

dt )2 + ( dy(t)dt )2.

The minimal exposure path problem is to find a path with the minimum exposureconnecting two known locations I and F.

Definition 6.11 (Minimal Exposure Path). Among all the paths connecting I and F,the one with the minimal exposure value is called a minimal exposure path, denotedby PE.

To an intruder, a minimum exposure path is the best stealthy path with the leastexpected detection probability. But for the defender, this is the worst coverage, and itneeds to take some measure to increase the path exposure, such as adding more nodes.Finding a PE in a sensor network with arbitrary deployment is an extremely difficultoptimization task. Megerian and Koushanfar [2002] propose to use a grid to transformthe problem from the continuous domain to a tractable discrete domain. The weight ofa line segment is computed as the exposure of this line segment by assuming a constantmoving speed. The search for PE is restricted to only the line segments of the grid andis found as the shortest path connecting I and F on this grid graph.

6.2.4. Detection Path. The coverage measure used for computing path breach and pathsupport is defined as the Euclidean distance of a point to its closest sensor, whichimplicitly assumes that a point is sensed or covered by only one sensor. The cover-age measure used for calculating path exposure allows that a point can be sensed bymore than one sensor, which considers a simple form of collaborative sensing and pro-cessing capability among sensors. For some applications with known signal processingparadigm, one may define the coverage measure as the outcome of collaborative signalprocessing among sensors.

In detection path problems, the detection probability, such as the detection proba-bility Pd

K defined in Eqs. (17) and (20), is used to define the coverage measure of apoint [Clouqueur et al. 2003a, 2003b]. Similar to other types of penetration path, thedetectability of a penetration path P connecting an initial location I and a final locationF can be defined as the net probability of detecting an intruder that moves along thepath P. An intruder using a path P is not detected if and only if it is not detected atany time while it is on that path. Therefore, the net probability (G(P)) of not detectinga target moving along the path P is the product of the probabilities of no detection ateach point p ∈ P. The path detectability hence is computed by 1 − G(P). Likewise, theproblem of finding a minimal detection path, denoted by PD, is to find a path which


32:44 B. Wang

Fig. 21. Illustration of using a grid approach and applying shortest path algorithm to find a minimaldetection path (the bold line).

minimizes (1 − G(P)). In general, the path PD can be with arbitrary shape. Similar tothe approach to find a minimal exposure path, Clouqueur et al. [2003a] apply a grid torestrict the search space, and a minimal detection path is only composed of the line seg-ments of the grid. The weight of a line segment L is assigned as wL

.= |∑p∈L log(1− PdK)|

and some classical shortest path algorithm is then applied to find a PD connecting Iand F. Figure 21, redrawn from Clouqueur et al. [2003a], illustrates a grid and a PDon the grid.

Some researchers have proposed to use other collaborative signal processingparadigms to define coverage measure [Onur et al. 2004a, 2006; Wang et al. 2005a;Chin et al. 2005; Deng and Liu 2007]. For example, Onur et al. [2006] consider usingNeyman-Pearson detector (NP-detector) for collaborative detection. The NP-detectormaximizes the detection probability while guaranteeing the maximal false alarm rateless than a required threshold. Wang et al. [2005a] define a coverage measure basedon the estimation coverage model, and use Eq. (22) to define the coverage measure fora point. Chin et al. [2005] propose a simple model to take into account the impact ofobstacles.

7. CONCLUDING REMARKS

In this article, we have presented a survey on various coverage problems in sensornetworks. Sensor coverage models measure the sensing capability and quality by cap-turing the geometric relation between a space point and sensors, while network-widesensing coverage is a collective performance measure for geographically distributedsensor nodes. Based on the subject to be covered, coverage problems can be classifiedinto point coverage, area coverage, and barrier coverage problems. We state the ba-sic coverage problem in each category, and review representative solution approachesin the literature. We also provide comments and discussions on some extensions andvariants of these basic coverage problems.

Many coverage problems are NP-complete, and different optimization techniqueshave been applied to solve these problems. Although these optimization solutions canbe used to provide insight on the achievable performance, the centralized solutionapproaches are computation-intensive and not applicable for resource-limited sensornetworks. Also centralized algorithms are not scalable and not robust enough to adaptto network dynamics. Instead, localized and distributed algorithms or protocols aremore preferred. Furthermore, the message transmission error and message exchangeoverhead should also be included into the protocol design.

Current coverage problems mainly focus on the two-dimensional plane. Althoughsome researchers claim that their solutions for the two-dimensional plane also apply tothe three-dimensional space, the extension may not be as that straightforward. Since



many practical applications of sensor networks actually involve a three-dimensionalspace, the gap between the research in two-dimensional plane and the practice inthree-dimensional space needs to be carefully addressed in future investigations.

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Received February 2008; revised December 2009; accepted January 2010


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