1

Sensor Coverage in Wireless Sensor Networks

Deying Li

1

, Hai Liu

2

1

School of Information, Renmin University of China

2

Department of Computer Science, Hong Kong Baptist University, Hong Kong

Abstract -- Coverage problem is an important and fundamental issue in sensor networks, which

reflects how well a sensor network is monitored or tracked by sensor. In this chapter, we survey

the current works on coverage problem in sensor networks. Two types of sensor coverage are

investigated: area coverage and target coverage. Combining with sensor development mechanism

(deterministic, statistical) and wireless sensor network properties (e.g. network connectivity,

energy efficient and fault tolerant for connectivity and sensing etc), various coverage problems

have been introduced and discussed in details. We focus on the most representative problems in

each domain and present a comprehensive review and analysis of various existed algorithms,

techniques.

1. Introduction

Wireless sensor networks (WSNs) have received signi ficant attention of researchers in

recent years due to its wide range of applications such as military surveillance,

environmental monitoring, forest fire detection, he althcare and other areas

[Akyildiz02, Chong2003]. A wireless sensor network composes of a large scale of

sensor devices (called sensor nodes) equipped with sensor unit, a wireless

communication unit, a battery power unit and a prog rammable embedded processor.

The sensor nodes are capable of sensing, data proce ssing, and communicating with

each other via radio transceivers. They coordinate with each other to establish a

network to remotely interact with the physical worl d, such as to monitor a

geographical region or a set of targets spread acro ss a geographical region, and to

report sensed data to the monitoring center, which is connected to the base station.

Wireless networks can be random or deterministic de ployed in physical environments

to collect information from an area of interest in a robust and autonomous manner.

An important research issue in wireless sensor net work is the coverage problem

which reflects how well the deployed sensor nodes c an monitor a set of targets.

Sensor activation scheduling under constraint on co vering of targets is called the

coverage problem in the literature. There are two types of targets: area [Cardei02,

Carle04, SP01, Tian02, Wang03, Zhang05] and point [ CardeiDu05, Kar03, CTW05,

Cheng05, Li07a, Li07b]. In area coverage problem, a set of sensors is given and

distributed over a geographical region to monitor a given area. But in point coverage

problem, a set of sensors is given and distributed over a geographical region to

monitor a set of points (or targets). The sensing r ange of a sensor is typically model as

disk in the 2D space or as a sphere in the 3D space, with the sensor located in the

center. The communication range of a sensor is mode led in the same way. A sensor

can monitor all targets that fall in its sensing ra nge. The data sent by the sensor can be

received by all sensors that fall in its communicat ion range. A sensors transmission

range is typically larger than its sensing range.

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Technology limitations of sensors could seriously a ffect the quality of service.

Stringent power supply of wireless sensor nodes is the most critical limitation,

because those nodes are usually powered by batterie s that may not be possible to be

recharged or replaced after they are deployed in ho stile or hazardous environments

[Yan 03]. Recently researchers have found that the significant energy saving can be

achieved by elaborate managing the duty cycle of no des in WSN with high node

density and it can prolong the network lifetime. In this approach, some nodes are

scheduled to sleep (or enter a power saving mode) w hile the remaining active nodes

keep working. However, the excessive-number of slee p nodes will lead to a WSN to

be disconnected, i.e. the set of working nodes will be isolated. In addition, the

over-loading of working nodes will cause these node s to be easily exhausted and

failed, which also cause a WSN to be disconnected a nd, consequently, invalidate the

data collection and transmission. It is therefore c rucial to determine a small number of

sensors that still cover the given area (or a set o f targets) or divide all sensors into

maximum number of subsets such that each subset sti ll cover the given area or a set of

targets. This is a lifetime requirement. These sele cted active sensors are connected so

that the sensor can report the detected data to the monitoring center. This is a

connectivity requirement.

. Sensors are prone to be failure. The over-loadi ng of working sensor nodes will

cause easily exhausted and failed. In addition to p ossible hardware or software

malfunctions, sensors may fail because of severe we ather conditions or other hash

physical environment in the sensor filed. It is the refore crucial to construct a

fault-tolerant WSN that will continuously provide n eeded services despite sensor

failures. This is the fault-tolerance requirement.

The fault-tolerance requirement includes two types: coverage fault-tolerance and

connectivity fault-tolerance. The sensor coverage problem can be further divided into

single coverage and multiple coverage. In single co verage, each target or point in the

area must be monitored by at least one working sens or. In multiple coverage, each

target or point in the area needs to be monitored b y at least k different working

sensors, which is called as the flat k-area-coverage problem for area coverage

[Gallais07]. There is another definition on k-area-coverage. An area is k-covered if

there exist k distinct sets of sensors so that each one can prov ide fully coverage of the

sensing area, which is called layered k-area-coverage problem [Gallais07]. For

connectivity fault-tolerance requirement, the cover age problem can be further divided

into 1-connectivity and k-connectivity coverage problem.

In this chapter, we survey recent contributions which address the coverage

problem. In section 2, we survey area coverage prob lem including multiple area

coverage problem, and connected coverage problem. S ection 3 investigates point

coverage problem, which includes connected point co verage problem, multiple

connected multiple point coverage problem, and brea ch coverage problem.

2. Area Coverage

The area coverage problem has been well studied and a review of existing

solutions for area coverage problem is described in [SSW05, Cardei06, ChenK07].

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Most of works in the survey [Cardei06] and [ChenK07 ] were published before 2003

and 2005, respectively. The survey work in [SSW05] focuses on only constructing

area-dominating sets for sensor area coverage. We p resent in the chapter a

comprehensive and updated survey on sensor area cov erage.

There are various coverage problems including area coverage, k-coverage,

m-connected k-coverage problems. The area coverage problem is de fined as follows:

Definition 1(Area Coverage problem) A set of sensors are given and distributed

over a geographical region to monitor a given area, an area coverage problem is to

find a minimum number of sensors to work such that each physical point in the area is

monitored by at least a working sensor.

Definition 2(k-coverage) An area is k-coverage if each physical point in the area

is covered by at least k (

1

k

) working (or active) sensors.

Definition 3 (m-connected) The communication graph of a given set of sensors

M is m-connected if for any two vertices in M, there are m vertex-disjoint paths

between the two vertices. A equivalent definition is, after the removal of any k-1

vertices in M, the resulted graph is still connected.

Definition 4 (m-connected k-coverage problem) A set of sensors are given and

distributed over a geographical region to monitor a given area, an m-connected

k-coverage problem is to find a minimum number of sensors to work such that each

physical point in the area is monitored by at least k active sensors and the active

sensors form a m-connected graph.

Most of algorithms or protocols for coverage problem guarantee full coverage,

that is, each physical point must be covered. There are some algorithms or protocols

for coverage problem which does not guarantee 100% coverage, such as PEAS

[YZLZ03] and the approach in [Cardei02].

There is another objective except selecting a minimal set of working nodes in the

area coverage: to divide all sensors into a maximum number of disjoint sets of sensors

(or non-disjoint sets) such that each set fully covers the area. Selecting a minimal a set

of working nodes reduce power consumption and prolongs network lifetime. In the

same way, dividing all sensors into a maximum number of disjoint(or non-disjoint)

sets which activate successively prolongs network lifetime. We now give a

comprehensive literature review of existing solutions and their contributions which

address various area coverage problems. In the following subsections, we introduce

detail algorithms and solutions.

2.1 Area coverage without connectivity guarantee

2.1.1 Maximize the number of disjoint sets

For energy efficient area coverage, the works in [SP01, Cardei02] consider a

large population of sensors, deployed randomly for area monitoring.

Slijepcevic and Potkonjak [SP01] proposed an energy conservation technique for

area coverage in wireless sensor networks. It selects and successively activates

mutually exclusive sets of sensor nodes, such that each set completely monitors the

entire monitored area. The authors propose a heuristic to this problem. It first divides

the monitored area into fields, which is a set of points. Two points belong to same

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filed if and only if they are covered by the same set of sensors. After the fields are

established, for each sensor, a list of all fields covered by that sensor is created. The

set of all fields in the area is denoted as A, and set of sensors as C. The authors

transform the coverage problem as the set k-cover problem: Does C contain k disjoint

covers for A. Then present a heuristic solution for the set k-cover to get a heuristic

solution for the coverage problem. Their method achieves energy saving by increasing

the number of disjoin covers.

The results on the set k-cover problem [SP01] solve a fair version where the

objective is to maximize k such that every cover contains all the physical points. In

many environments, requiring that a cover contain all the physical points may be too

strict. For instance, there is a single area that is monitored by only one sensor but all

other areas are monitored by hundreds of sensors. Except for that single area, all other

areas could be covered for many times by dividing the sensors into covers. But in the

fair version, the sensors can not be partitioned at all because only all sensors to

monitor all areas. Figure 1 shows this case. Fig. 1 (b) can cover the whole monitoring

area. But in Figure 1(c), (d), the small area A is not be covered. All areas except area

A can be covered by at three different sensors while small area A is only covered by

one sensor.

Figure 1. The single area A is covered by only one sensor.

Abrams et al [Abrams04] study a variation of the set k-cover problem: to find a

partition of the subsets into k covers so that the number of times that areas(fields) are

covered by the partition, is maximized. Three approximation algorithms are presented:

randomized, distributed greedy, and centralized greedy. In the randomized algorithm,

each sensor simply assigns itself to a cover chosen uniformly at random from the set

of all possible covers. In the distributed greedy algorithm, each sensor assigns itself,

in turn, to the cover with the minimum intersection between the areas the sensor

monitors and the areas monitored by the cover thus far. The centralized greedy

algorithm is similar to the distributed greedy except that an area in the intersection is

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weighted.

Cardei et al [Cardei02] propose another efficient method to achieve energy

saving by organizing the sensors nodes into a maximum number of disjoint

dominating sets which are activated successively. Only the sensors from the active set

are responsible for monitoring the monitored area and all other nodes are in a sleep

mode. The authors prove that the maximum disjoint dominating sets problem is

NP-complete, and any polynomial-time approximation algorithm has a lower bound

of 1.5. Based on the sequential coloring algorithm, the authors propose a heuristic to

compute maximum number of disjoint dominating sets in an undirected graph.

Compared to the work in [SP01], the maximum number of disjoint dominating sets is

greater or equal than the maximum number of covers. This is valid because the

sensors in one cover also form a dominating set. Therefore, approach in [Cardei02]

potentially achieves better energy saving than approach in [SP01]. However, the

approach [SP01] can achieve the full area coverage constraint, but there are small

coverage lapses in the monitored area for approach in [Cardei02]. For example, as in

Figure 2, there is only one set cover {S

1

, S

2

}, but there are two disjoint dominating

sets {S

1

} and {S

2

}. Considering disjoint dominating sets compared [Cardei02] with

disjoint covers method [SP01], in this example the longevity of the network is double

from the point of view of energy resources. However, there are some uncovered parts

of the target area in [Cardei02].

Figure 2. two sensors are deployed on monotoried area.

2.1.2 Minimize the number of active nodes

Approaches in [SP01] and [Cardei2002] are to divide sensors to maximum

number of disjoint sets (or dominating sets), each set can monitor the sensed area.

These sets are activated successively in order to prolong lifetime. Tian and Georganas

[Tian03, Tian02] propose another energy-efficient node scheduling scheme for area

coverage in synchronous networks where sensing range is equal to the transmission

range. The main objective of the algorithm is to minimize number of working nodes,

as well as maintain the original sensing coverage. It requires every node to be aware

of its own and its neighbor location information. At the beginning of each round,

each node selects a time-out interval. At the end of the interval, if a node sees that

neighbors together cover its monitoring area, the node transmits a retreat message to

all its neighbors and goes into the sleep mode. Otherwise, the node remains active, but

does not transmit any message. The process repeats periodically to allow for changes

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in monitoring status. In this scheme, each node must know its neighbor location

information and has to do accurate geometrical calculation to determine whether or

take an off-duty status. Tian and Georgannas [Tian04] propose three different

alternative node scheduling schemes for area coverage, which are location and

calculation-free. In the Nearest-neighbor based scheme, after each node collects

distance information to its all neighboring nodes, it determine its working status by

examining if its distance to the nearest neighbor is not more than the threshold. If

affirmative, the node can take off-duty status. In the neighbor number-based scheme,

each node collects its all neighbors information, and determines its working status by

examining if its neighbors number exceeds a given threshold. If affirmative, the node

will take off-duty status. In probability-based scheme, each node generates a random

number from [0,1) and checks if the number is less than the off-duty probability, if it

is, the node takes off-duty status, otherwise, it sets its status as on-duty.

2.2 Connected Area Coverage

In all schemes introduced in above sections, the working sensors may not be

connected, and thus reporting to a monitoring center can not be proceeded. In order to

collect information from the sensor nodes to monitoring center, the active sensors are

desired be connected. A frequently addressed objective is to determine a minimal

number of active sensors to maintain monitoring the given area as well as connectivity.

Next we will introduce several connected coverage mechanisms.

2.2.1 Transmission range equals to the sensing range

Ye et al [YZCLZ02, YZCLZ03] present PEAS, a distributed, probing-based density

control algorithm for robust sensing coverage. In this work, a subset of sensor nodes

operative mode maintains coverage while others are put into sleep. Each sensor node

has the same probing range R

p

and may vary its transmission power and choose a

power level to cover a circular area given a radius. In PEAS, each node has three

operation modes: sleeping, probing and working. Initially all sensor nodes are in the

sleeping mode. Each node sleeps for an exponentially distributed duration generated

according to a probability density function (PDF). When sleeping time expires, the

sensor enters the probing mode. The probing node uses an appropriate transmission

power to broadcast a PROBE message within its local probing range R

p.

Any working

node(s) within that range should respond with a REPLY message, also sent within the

range of R

p

. If the probing node hears a REPLY, it goes back to the sleeping mode for

another random period of time. If the probing node does not hear any REPLY, it enters

the working mode and starts monitoring until it fails or consumes all its energy. The

probing range can be adjusted to achieve different levels of coverage redundancy. The

choice of probing range also affects network connectivity. The authors also study the

asymptotic connectivity of PEAS. With this protocol, the probability having full

coverage of a monitored area is close to 1 if the sensing range

(1 5)

t p

R R

+. There

is a problem that PEAS does not ensure that the coverage area of a sleeping node is

completely covered by other active nodes, i.e. it does not guarantee complete

coverage. Figure 3 illustrates PEAS, with the black nodes being active and the white

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nodes being in sleep mode, because each white node is contained within R

p

to one of

the active nodes. But there is a coverage area of the white node which is not

completely covered by the active nodes. This protocol has limited usefulness because

it is probabilistic and does not ensure full area coverage.

Figure 3. PEAS for area coverage.

Carle and Simplot [Carle04] propose another mechanism for energy-efficient

connected area coverage for the case when all sensor nodes have the same range and

the communication range equals the sensing range. The goal of the algorithm is to

select the minimum number of active nodes to cover the given area. The authors

modify one of existing protocol for connected dominating-set protocol (e.g. Dai and

Wus algorithm in [Dai 03] to find area coverage rather than node coverage. In the

modification protocol, each node computes its timeout function based on its priority

and listens to messages from other nodes before deciding its dominating status at the

end of a timeout interval. A node choosing gateway status always transmits a message

(positive advertising) to all its neighbors. A node choosing not to monitor its area has

the option of transmitting this information to its neighbors (negative advertising) or

not. The protocol runs: using a simple perimeter coverage scheme [Tian02], a node

computes the area covered by each node that transmits either positive or negative

advertising and includes the transmitting node in a subset; at the end of its timeout

interval, the node computes a subgraph of its one-hop neighbors that sent

advertisements(these are its neighbors with higher priority); If this subgraph is

connected and the nodes in subgraph fully cover the nodes area, the node opts for

sleeping status; Otherwise, the node chooses active status. The distributed

dominant-pruning algorithm can prove that the set of active nodes is connected.

Figure 4 shows an example that how a node does decision for its status. (a) Node A

decides to be active because its active neighbors do not fully cover its monitoring area.

(b) Node A decides to be inactive because its monitoring area is covered by its active

neighbors that are connected. (c) Node A decides to be active because its active

neighbors are not connected.

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Figure 4. Example of configurations for area-coverage decision.

2.2.2 Transmission range is at least twice of the sensing range

Wang et al [Wang03] and Zhang et al [Zhang 03] first discuss how to combine

consideration of coverage and connectivity maintenance in a single activity

scheduling.

An important, but intuitive result for maintaining sensing coverage and

connectivity by keeping a minimal number of sensor nodes in the active mode has

been proved by Zhang and Hou [Zhang03]. The authors first investigate the

relationship between coverage and connectivity, and prove that if the transmission

range is at least twice of the sensing range, a complete coverage of a convex area

implies connectivity among the working nodes in the active mode. Second, the

authors derive, under the ideal case in which node density is sufficiently high, a set of

optimality conditions under which a subset of working sensor nodes can be chosen for

full coverage. Based on the optimality conditions, the authors propose a decentralized

and localized density control algorithm, called optimal geographical density

control(OGDC). OGDC is under assumptions: the transmission range is at least twice

of the sensing range, each node is aware of its own position, and all nodes are time

synchronized. At any time, a node is in one of the three states: UNFECIDED, ON,

OFF. Time is divided into rounds. At the beginning of each round, all the nodes

wake up, set their states to UNDECIDED, and carry out the operation of selecting

working nodes. By the end of the execution, all the nodes change their states to either

ON or OFF and remain in that state until the be ginning of the next round. This

decision is based on the power-on messages. Every node keeps a list with neighbor

information. When a node receives a power-on message, it checks whether its

neighbors cover its sensing area, and if so, it will change to OFF state. A node

decides to change into the ON state if it is the closest node to the optimal location of

an ideal working node. The process of selecting working nodes (in a decentralized

manner) in each round commences by randomly selecting a sensor node A to be the

starting node (Figure 5). Then one of its neighbors with an approximate distance of

3

r

, B, is selected to be a working node. To cover the crossing point of disk A and B,

the node, Q, whose position is closest to the optimal position C is then selected to

become a working node. The process continues until all the nodes change their states

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to either ON or OFF, and the set of modes with ON states forms the working

set.

Figure 5.

The process of selecting working nodes.

Wang et al [Wang03] also prove that the transmission range is at least twice of the

sensing range, and the area to be covered is convex, then the area coverage also

implies connectivity among the covering sensors.

Wang [Wang 03] and Zhang et al [Zhang 03, 05] provide a sufficient condition

for safe scheduling integration in those fully covered networks. However, random

node deployment often makes initial sensing holes inside the deployed area inevitable

even in an extremely high-density network. Tian and Georgnnas [Tian05] enhance

their work to support general wireless sensor networks by proving another conclusion:

the communication range is twice of the sensing ra nge is the sufficient condition

and the tight lower bound to ensure that complete coverage preservation implies

connectivity among active nodes if the original network topology (consisting of all the

deployed nodes) is connected. That is, the authors prove that if active nodes form a

completely coverage, and the original topology is connected, when the transmission

range is twice of the sensing range, then the induced subgraph by active nodes is

connected. When the transmission range is less than twice of the sensing range, then

the induced subgraph by active nodes may be disconnected.

Wu and Yang[Wu04] extend a result from [Zhang 03] where only uniform

sensing range among all sensors is used. Wu and Yang consider cases where each

sensor is able to select one of two or three adjustable ranges and the transmission

range is at least twice of the sensing range, with the goal of minimizing the

overlapped sensing area. They present two new energy-efficient models of different

sensing ranges.

Jiang and Dou[Jiang04] describe several improvements to algorithm in [Tian02].

The authors present a distributed and localized density control algorithm for wireless

sensor networks, which all nodes have the same sensing range and the transmission

range is at least twice of the sensing range. The authors apply the perimeter criterion

that a circle is covered completely if perimeters of other circle covering it are fully

covered by other covering circles. In the algorithm, a sensor is in one of the two states:

ACTIVE and NON-ACTIVE. At the beginning, all no des are in ACTIVE state.

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Network lifetime is divided into rounds, and each round has a scheduling phase

followed by a sensing. The scheduling phase is further divided into two sub-phases:

neighbor discovery phase and evaluating phase. At the beginning of the neighbor

discovery phase, node broadcast a hello message to its one-hop neighbors and sets a

timer to wait for neighbors hello message. Upon this timer expires, node has obtained

knowledge about one-hop neighbors and construct its neighbor set and effective

neighbor set. Then entering the evaluating phase, sensor begins to evaluate the density

control algorithm to decide which state it should go. In each time round, the ACTIVE

nodes work for the sensing task and the NON-ACTIVE nodes will turn off their

sensing and communication units to save energy.

2.2.3 Arbitrary Ratio of transmission range to sensing range

Gallais et al [Gallais08] generalize the approach in [Carle04] for an arbitrary ratio

of sensing range and transmission range. The approach are based on a time-out

scheme, in addition to being fully localized, has a very small communication

overhead. When a round starts, each node selects a time out and listens to messages

sent by other nodes before the time-out expires. Sensor nodes whose sensing area is

not fully covered when the deadline expires decide to remain active for the considered

round and transmit an activity message announcing it. There are four variants in the

approach, depending on whether or not withdrawal and retreat messages are

transmitted. Covered nodes decide to sleep, with or without transmitting a withdrawal

message to inform neighbors about the status. After hearing from more neighbors,

active sensors may observe that they became covered and may decide to alter their

original decision and transmit a retreat message.

In this approach, the covering criterion which has been already applied in

[Jiang04], [Xing05] and [Zhang05] is applied on the borders of the sensing area of

each sensor[Gallais 06], the node using it verifies whether or not its sensing area is

fully covered. The details of the protocol include how the time-out is decided, and

how the area coverage and connectivity tests are performed. The test for connectivity

of covering circles must be performed when the transmission range is less than twice

of the sensing range, that is, when the transmission range is less than twice of the

sensing range, a node can decide to turn off if and only if its neighbors fully cover and

are also connected.

Sheu et al [Sheu 07] study query execution over a specific geographical region.

And propose an efficient distributed protocol to find minimum number of connected

active sensor nodes to cover the queried region. Assumptions: Transmission ranges

and sensing ranges differ between sensors, and the sensing range of a sensor node

may differ from its transmission. The proposed protocol consists of two

phases-self-pruning phase and sensing nodes discovery phase. In the beginning of the

protocol, each sensor node is assumed to have the information of its 1-hop-cover

neighbors. In the self-pruning phase, each node checks whether or not its sensing area

is completely covered by its higher priority neighbors by using the perimeter covering

criterion in [Huang 03]. If no, it becomes a sensing node. The authors prove that the

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sensing nodes selected by self-pruning can fully cover the queried region when the

deployed sensor nodes cover the queried region. In the sensing nodes discovery phase,

each of the considered perimeters is subdivided into sub-perimeters, based on the

intersections with other considered circles. For each such sub-perimeter, the sensor

with the highest priority, among nodes covering this sub-perimeter, is active. After the

two phases, the selected sensing nodes are connected and can cover the queried

region.

Gupta et al [Gupta 03] study the connected sensor coverage problem: Given a

query over a sensor network, select a minimum set of sensors, called connected sensor

cover, such that a) the sensing regions of the selected sensors cover the entire

geographical region of the query, and b) the selected sensors form a connected

communication graph. The authors first prove that the connected sensor coverage

problem is NP-complete and then propose a centralized greedy algorithm. The

proposed algorithm is as follows: Let M be the set of sensors already selected for

inclusion in the connected sensor cover by the greedy algorithm at any stage.

Initially, M is an empty set. The algorithm starts with including in M an arbitrary

sensor that lies within the querys region. At each stage, the greedy algorithm selects a

sensor C along with a path of sensors P that forms a communication path between C

and some sensor in M with maximum benefit of P, add selected path P to M, till

querys region is covered by sensors in M. In the algorithm, the benefit of P is

defined as the number of uncovered valid subelements covered by P per sensor. At

any stage of the algorithm, the communication subgraph induced by M is connected.

A straightforward distributed version of the same algorithm is also given.

Zhou et al [Zhou04a] address Variable Radii Connected Sensor Cover problem

which generate the problem in [Gupta03]: Given a query region in the network, each

node has vary its sensing range and transmission range where they can not exceed the

maximum sensing range and the maximum transmission, selecting a subset of sensors

which forms connected sensor cover such that the total energy cost (including sensing

cost and transmission cost) is minimized. The authors design various centralized and

distributed algorithms-Voronoi based algorithm, Greedy algorithm and Steiner tree

based algorithm. One of the designed centralized algorithms (called CGA) is shown

as O(logn)-approximation. CGA works as follows. CGA maintain a set of selected

sensors M along with their assigned transmission and sensing range, and increases the

covered region while keeping connectivity of M. At each stage, either adds to M a

path of sensors or increases the sensing range of a sensor in M, whichever gives the

maximum benefit. CGA terminates when the given qu ery region is completely

covered by the assigned sensing regions of the sensors in M.

2. 3 k-area coverage

Sensor nodes usually are deployed into remote and inhospitable area to monitor

targets. Because severe weather conditions or other hash physical environment in the

sensor filed or the over-loading of working sensor nodes, sensors are prone to fail. It

is therefore crucial to construct a k-coverage problem (

1

k

), in which each physical

point is covered at least k different sensor nodes. There are many existing works to

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address k-coverage problem. Next, we give a survey on k-coverage problem.

Sensor networks are often desired to prolong the lifetime of operation. This is

usually achieved by putting sensors to sleep for most of their lifetime. On the other

hand, the intrusion detection applications require guaranteed k-coverage off protected

region at all times. To determine the appropriate number of sensors to deploy that

achieves both goals simultaneously becomes a challenging problem. Kumar et al

[Kumar 04] study this problem: Given an area to be protected, how many sensors

should be deployed so that every point in the region is covered by at least k sensors,

and given that the network must last for a specified length of times? The authors

consider three kinds of deployments for a sensor network on a unit square-

a

nn grid, random uniform (for all n points), and Poisson (with density n). In

all three deployments, each sensor is active with probability p. A critical condition for

three deployments is derived. And the authors show that the conditions for

deterministic deployments are similar to the conditions for random deployments.

2.3.1 k-area coverage without connectivity guarantee

The k-area coverage problem addressed in [Gallais06a] consists in building k

distinct subsets of active nodes (layers) so that each layer covers the area. The authors

propose a decentralized protocol. Sensors are randomly deployed over a square area

and activity is imagined in a rounded fashion. At each round, every node decides its

status between either monitoring for the entire round or getting passive until the next

decision phase. Every sensor is aware of required coverage degree, denoted as k. A

node A can find smallest i so that ith layer of the area covered by that node is not fully

covered by its neighbors. Then, if

i k

, A decides to be active at layer i and sends a

positive acknowledgement announcing its activity layer i and its geographical position.

Otherwise, it decides to be passive and no message is sent. Figure 6 shows that that

sensor A first evaluates the coverage provided by neighbors of layer 1(black nodes on

Figure 6(b) before deciding to evaluate the coverage at layer 2(Figure 6(c)). Finally,

Figure 6(d) shows that A is covered at all 2 layers. A takes its activity decision

depending on its required coverage degree k. If k>2, then A gets active at layer 3 and

sends a position acknowledgment. If k=2, then A gets passive without sending any

message.

Figure 6. Evaluation of coverage.

Cai et al [Cai07] propose a precise and energy-aware coverage control protocol,

named Area-based Collaborative Sleeping (ACOS). Based on the net sensing area of

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a sensor, which is covered only by the sensor and not covered by other active sensors,

the ACOS controls the mode of sensors to maximize the coverage degree, minimizing

the energy consumption. Each sensor node has four states: Sleep, PreWakeUP, Awake,

and Overdue. Initially, each sensor is Sleep with timer, when node s wake up, its

state changes from Sleep to PreWakeUP, node u sends a broadcast message, to its

neighbors within radius 2r and waits for T seconds. When any neighboring sensor v

with Active receives this message, node v sends back reply message including its

location. After u receives reply messages, u computes the net area ratio, if the net area

ratio is less than the threshold, u return back to Sleep state. If the net area ration is

more than the threshold, u changes to Awake state, and initialize its wake timer and

broadcast a Wake-Notification message. When node u is still in the Awake state and

its wake time expires, it changes from Awake to Overdue state. When a node which is

in Awake or Overdue state hears a Wake_Notification message, it re-calculates the net

area ratio to repeat the process. The state transition diagram is in Figure 7.

Figure 7. State transition diagram of ACOS.

Hefeeda and Baghen [Heffeda07] study coverage prob lem: Given n

already-deployed sensors in a target area, and a desired coverage degree

1

k

, select

a minimal subset of sensors to cover all sensor locations such that every location is

within the sensing range of at least k different sensors. The authors model the

k-coverage problem as a set system (X, R) where X is the set of sensor locations and

RC is a the collection of subsets of X created by intersecting disks of radius r with

points of X, for which an optimal hitting set corresponds to an optimal solution for

k-coverage. And propose an approximation algorithm with a logarithmic ratio for

computing near-optimal hitting sets [BG95]. A fully distributed version of the

proposed algorithm is designed and implemented.

There are various theoretical works on area coverage problem in wireless sensor

networks. Xing et al [Xing2004] presents a theoretical analysis of greedy geographic

routing protocols on wireless sensor networks that must provide sensing coverage

over a geographic area. The authors prove that the Greedy Geographic

Forwarding[Karp00, Stoj01] and their new greedy protocol always succeed in any

sensing covered network when the communication range is at least twice the sensing

range. Liu and Towsley [LiuB04] approach the coverage problem from a theoretical

perspective and explored the fundamental limits of the coverage of a large-scale

sensor network. The authors study three fundamental coverage measures of

14

large-scale sensor networks: Area coverage, node coverage, and detectability. These

measures are determined by basic network parameters and have important

implications on network planning and protocol performance of sensor networks. Ke et

al [Ke07] proves that deploying sensors on grid points to construct a wireless sensor

network that fully covers critical grids using minimum sensors (Critical-Grid

Coverage problem) and that fully covers a maximum total weight of grids using a

given number of sensors(Weighted-Grid Coverage problem) are each NP-Complete.

2.3.2 k-area coverage with the transmission range being at least twice sensing

range

The network connectivity is rarely treated in existing works on k-area coverage.

Wang et al [Wang03] prove that when the transmission range is at least twice the

sensing range, a set of working nodes that forms k-coverage a convex region forms a

k-connected communication graph. Tian et al [Tian05] enhance the result in [wang03]

for general random deployment network to prove that when the transmission range is

at least twice of the sensing range, and the system sensing coverage is completely

k-degree preserved after node scheduling, if a network graph is originally k-connected,

the induced subgraph by the active nodes must be k-connected. Most of existing

results on k-area coverage rely on this theorem to focus on area coverage only without

addressing the problem of the connectivity preservation.

Wang et al [Wang 03] generate the result in [Zhang 03]. And propose the

coverage configuration protocol (CCP) that is a decentralized protocol that only

depends on local states of sensing neighbors and can provide different degrees of

coverage requested by applications. In CCP, each node determines its eligibility using

the k-coverage eligibility algorithm based on the information about its sensing

neighbors, and may switch state dynamically when its eligibility. Given a requested

coverage degree k, a node is ineligible if every location within its coverage is already

k-covered by other active nodes in its neighborhood. The authors prove that a convex

region is k-cover if it contains intersection points between sensors or between sensors

and region boundary and all these intersection points are k-covered. Based on this, a

sensor is ineligible to turn active if all the intersection points inside its sensing circle

are at least k-covered. Every node maintains a table of known sensing neighbors based

on the beacons (hello messages) that it receives from its communication neighbors. A

node can be in one of three states: SLEEP, ACTIVE and LISTEN. All nodes start in

the SLEEP state for a random time. When the sleep timer expires, a node in the sleep

state enters LISTEN state. When a beacon (HELLO, WITHDRAW or JOIN message)

is received, a node in the listen state evaluate its eligibility. If it is eligible, it starts a

join timer, otherwise it returns to the SLEEP state. If it becomes ineligible after the

join timer is stated, it cancels the join timer. If the join timer expires, the node

broadcast a JOIN beacon and enters the active state. If the listen timer expires, it starts

a sleep timer and returns to the SLEEP state. Once a node is in the active state, it

re-evaluate the coverage eligibility every time it receives HELLO message and decide

whether to go into the SLEEP state or remain in the ACTIVE state.

If the ratio of the communication range to the sensing range is more than 2, CPP

15

can guarantee connectivity. But CPP does not guarantee connectivity when the ratio of

the communication range to the sensing range is less than 2. The authors also present

a simple approach for integrating CCP with an existing connectivity maintenance

protocol, SPAN [ChenJ01] to provide sensing coverage and communication

connectivity.

The proposed protocol in [Sheu 07] can be extended to solve k-coverage problem,

which can find a set of sensing nodes satisfy the k-coverage request. The protocol is

as follows: Assume that a set SN1 of sensing nodes is got in the self-pruning phase. If

a non-sensing node is aware of its neighboring nodes in SN1, it can delete these

sensing nodes from its 1-hop-cover neighboring set and execute the self-pruning again

to determine whether it can be a sensing node. After the second iteration, all the

non-sensing nodes can determine their roles-sensing nodes or non-sensing nodes, then

get the second coverage set SN2 to fully cover the queried region if the remaining

sensor nodes can fully cover the queried region. SN1 and SN2 form a 2-coverage.

Applying the above procedures, k-coverage can be got.

Lu et al [Lu06] address the k-coverage Maintenance Problem: Given a sensor

group S deployed in region R and a natural number k, find subset 'S with the

minimum number of sensors such that 'S is able to maintain k-coverage. That is,

for any position v in R, if v can be k-covered by S, it must be k-covered by'S;

Otherwise, the coverage degree of v in 'S is same as in S. It assumes that the

transmission range is at least twice the sensing range. The authors propose a scalable

coverage maintenance scheme (called as SCOM). SCOM assume that each node

knows its location and can acquire the location of neighbors through one-hop

communication. Time is slotted into rounds. At the beginning of each round, SCOM

runs in two phases: Decision phase and optimization phase. In the decision phase,

each sensor is initially in BOOTSTRAP state and has an empty active neighbor list.

Before making the decision of turning on or off, each sensor sets a back-off timer

depending on its residual energy. When a sensors t imer expires, the sensor checks

whether its sensing region is k-covered by the sensors in the active neighbor list using

the redundancy eligibility rule for homogenous or heterogeneous, and switches to

ACTIVE or INACTIVE state accordingly. If a sensor decides to turn into ACTIVE

state, it broadcast a TURNON beacon with its coordinates to its the neighbors. Upon

receiving the TURNON beacon, a neighbor adds the sender into the active neighbor

list. In the optimization phase, sensors optimize the coverage by turning off redundant

active sensors while still guaranteeing the required coverage.

The Sensor Scheduling for k-Coverage(SSC) problem is investigated in [Gao06].

Which requires to efficiently schedule the sensors, such that the monitored region can

be k-covered throughout the whole network lifetime with maximizing network

lifetime. All the sensors have uniform transmission range and sensing range. And the

transmission range is at least twice the sensing range. The authors model the SSC

problem to find maximum number of disjoint k-cover sets. In [Huang03], the authors

prove that the entire monitored region is k-covered if and only if each sensor in the

monitored region is k-perimeter-covered. Consider any two sensors s

i

and s

j

. A point

on the perimeter of s

i

is perimeter-covered by s

j

if this point is within the sensing

16

range of s

j

. s

i

is k-perimeter-covered if all points on the perimeter of s

i

are

perimeter-covered by at least k sensors other than s

i

itself. A segment of s

i

s perimeter

is k-perimeter-covered if all points on the segment are perimeter-covered by at least k

sensors other than s

i

itself. Figure 8 shows an example: the perimeter of s

i

between

two arrows is covered by sensor s

j

. Based on this result, Gao et al propose a greedy

algorithm, PCL-Greedy-Selection(GS). The main idea of GS is to iteratively construct

subset by choosing sensors from the area with the lowest sensor density. When

construct an individual subset, the sensor with a small PCL value is added to the

subset. In addition, the authors develop a guideline for designing a sensor deployment

by employing density control.

Figure 8. An example of perimeter-coverage.

2.3.3 Connected k-area coverage

Area coverage protocols aim at turning off redundant sensor nodes while ensuring

full coverage of the area by the remaining active nodes. Providing k-area coverage

means that every physical point of the monitored area is sensed by at least k sensors.

Connectivity of the active nodes subset must also be provided so that monitoring

reports can reach the sink stations. Existing solution hardly address these two issues

as a unified one. The works in [Zhou04b, Zhou05, Gallias07] address coverage and

connectivity as a unified one. Next, we review them.

Zhou et al [Zhou04b, Zhou05] study the k-area coverage problem and the

connectivity preservation problem. Zhou et al consider the problem of selecting a

minimum size connected K-cover, which is defined as a set of sensors M such that

each point in the sensor network is covered by at least K different sensor in M, and

the communication graph induced by M is connected. The authors design a

centralized O(logn)-approximation algorithm. The greedy algorithm is a

generalization of the centralized approximation algorithm in [Gupta 03] for the

connected 1-coverage problem. The Greedy Algorithm maintains a set of M of

selected sensors and at each stage, select a candidate sensor without belong to M and a

candidate path of sensor with maximum K-Benefit with respect M, add the selected

path to M. This is repeated until the query region is k-covered by M. The distributed

version of the Greedy algorithm is also given.

Zhou et al [Zhou 05] address a more general, variable radii sensor model,

choosing a subset of sensors such that they maintain a k

1

-connectivity and k

2

-cover,

wherein every sensor can adjust both its sensing and transmission ranges, and the

overall energy consumption is minimized. The energy consumption includes sensing

energy consumption and transmission energy consumption. The authors propose a

distributed and localized Voronoi-based algorithm. The Voronoi-based algorithm

S

i

S

j

17

works as follows. Initially, each sensor node in the sensor network is active, and

gathers locations of all the nodes in the l-hop active neighborhood. Each active sensor

node computes its k

2

th order local Voronoi cell, and the neighbors in the k

1

-RNG

over active nodes. It uses the V-R assignment method to assign itself sensing and

transmission radius. Each node computes its sleep benefit, based on the sleep benefit,

choose a sensor with the most sleeping benefit among all its local voronoi neighbors

to become inactive. A sensor node is chosen to become inactive only if the remaining

active sensors are capable of k

1

-covering the query region and maintaining

k

1

-connectivity of their communication graph. Repeat above processes. The algorithm

terminates when no more sensors can be made inactive.

Gallais and Carle [Gallais07] consider connected k-coverage problem. And

consider two definitions for the k-area coverage problem: the flat k-area coverage

problem and the layered k-area coverage problem. The authors propose a localized

algorithm that can be applied to time-synchronized networks. Each node selects a

time-out, which depends on the remaining energy, and has some random number,

while listening to messages from neighboring nodes. Once the timeout ends, u takes

its activity decision based on known neighboring nodes. It so evaluates its coverage

according to the appropriate coverage evaluation scheme.

If completely k-covered according to the flat k-area coverage issue, if u decides to

be passive and turns into sleep mode. Otherwise, u remains active and sends a positive

acknowledgment message which contains the values of its communicating and

sensing range with its position. Any node with a longer timeout that receives this

message adds u to its neighbor table.

For the layered k-area coverage issue, Nodes still listen for messages during a

given timeout before making their activity decision and choosing an activity layer

whose number is included in the messages. A node u sorts its neighbors according to

a number of layers. Then, u evaluates if at least k virtual activity layers fully cover its

area S(u). If no, u remains active and chooses the uncovered activity layer which has

the lowest number, and sends an activity message to announce its status. About

connectivity, when

2

CR SR

, connectivity is ensured. When CR<2SR, a simple

connectivity test is added in activity decision process since the knowledge of positions

and transmission ranges of active neighbors.

There is another theoretical work on k-area coverage problem in wireless sensor

networks.

Huang and Tseng [Huang03] formulate the k-coverage problem as a decision

problem, whose goal is to determine whether every point in the service area of the

sensor network is covered by at least k sensors. The authors prove that the whole

network area A is k-covered if and only if each sensor in the network is

k-perimeter-covered. Consider any two sensors s

i

and s

j

. A point on the perimeter of s

i

is perimeter-covered by s

j

if this point is within the sensing range of s

j

. s

i

is

k-perimeter-covered if all points on the perimeter of s

i

are perimeter-covered by at

least k sensors other than s

i

itself. A segment of s

i

s perimeter is k-perimeter-covered if

all points on the segment are perimeter-covered by at least k sensors other than s

i

itself.

The authors propose a polynomial time algorithm to decide if a sensor is

18

k-perimeter-covered. The algorithm to determine the perimeter coverage of s

i

work as

follows: First, for each sensor s

j

with rssd

ji

2),( , determine the angle of s

i

s arc,

denoted by ],[

,,RjLj

, that is perimeter-covered by s

j

. Secondly, place the points

RjLj,,

,

of all neighboring sensors s

j

of s

i

on the line segment [0, 2

] and sort all

these points in an ascending order into a list L. Thirdly, traverse the line segment [0,

2

] by visiting each element in the sorted list L from the left to right and determine

the perimeter-covered of s

i

.

3. Target Coverage

The target coverage problem is to cover a set of given targets. The objectives are

normally to minimize sensing cost and achieve maximum lifetime. The target

coverage problem has been studied extensively, and many solutions have been

proposed. Current work on target coverage can be divided into three categories. Work

in the first category is to place a set of sensor nodes to cover the given targets. Work

in the second category is to divide given sensor nodes into several groups and

schedule each group of sensors to cover the given target. Work in the third category

not only considers the coverage of targets, but also requires connectivity of sensor

nodes. They are introduced next.

3.1 Deployment of Sensor Networks

There are some works for the deployment of sensor network ensuring point

coverage [Chakrabary02, Kar03, Xu06, Wang06].

Chakrabary et al [Chakrabary02] address the sensor placement problems: Given a

surveillance region (grid points) and sensors of different types (with different ranges

and costs), (1) determine the placement and type of sensors in the sensor field such

that the desired coverage is achieved and cost is minimized. (2) How should the

sensors be placed at grid points such that every grid point is covered by a unique

subset of these sensors. The authors first formulate the sensor placement problem in

terms of cost minimization under coverage constraints as an integer linear

programming (ILP):

∑

∑

∑

+

i j k

kjiBkjiA

yCxC )(min

,,,,

s.t. 222 ))2,1()2,1((

1 1 1

,,,,

, k,jimybxa

i j k

kjiBkjiA

+

∑

∑

∑

Where, C

A

and C

B

denote as costs of two types of sensors respectively. x

i,j,k

and

y

i,j,k

represent if type A and B place on the grid point (i, j, k) respectively. Then use a

divide-and-conquer approach to solve it. However the divide-and-conquer approach

just solves small size the ILP problem.

Xu et al [Xu06] address the sensor network deployment problem of placing

sensors at a subset of pre-selected sites so as to minimize sensor cost while providing

19

a specified degree of coverage of the target sites, which is general of problem in

[Chakrabary02].

The authors develop an integer linear programming formulation to

find a minimum cost deployment of sensors that provides the desired coverage of a

target point set:

∑

∑

z l

zi

xitMin ))((cos

,

(1) ,),,(cov

),,(

,

ljljerx

i ljilocationsz

zi

∑

∑

(2) ,),,(0

,

lilicapacityx

zi

(3) ),(

,

iitotaltypex

z

zi

∑

(4) z),(

,

∑

ziontotalLocatx

i

zi

Where variable x

i,z

is the number of sensors of type i to be placed at each location z.

Capacity(i, l) is the number of sensors of types i that may feasibly be placed at

location l. Cost(i) is the cost of one sensor of type i. Cover(j, l) is the degree of

monitoring the coverage required at location l for modality j. A greedy algorithm to

solve the proposed general ILP is developed. Main idea is that: For (j, l), in which the

coverage required at location l for modality j is not satisfied, select an optimal

sensor-location pair (i, z) which does not violate (3) and (4) such that the incremental

coverage cost is minimized, place sensor of type i at location z. Additionally, for the

case of grid coverage[Chakrabary02], ε-approximation algorithms and a polynomial

time approximation scheme are proposed. The proposed algorithms are centralized.

Wang et al [Wang06] study minimum-cost sensor placement on a bound 3D

sensing a number of discrete target what may or not be a grid points. There are l types

of sensors available with different sensing range and different costs. The

minimum-cost sensor placement is to find a selection of sensors and a subset of points

to place these sensors such that every target is covered at least k sensors (given k) and

the total cost is minimized. The problem is formulated as an integer linear

programming:

∑∑

= =

n

i

l

v

v

iv

xCMin

1 1

∑

∑ ∑

=

=

l

v

v

i

l

v iEj

v

j

x

x

v

1

1 ][

1

Where

v

i

x represents if type t

v

sensor is placed at grid point i. Based on the optimal

of relax linear programming, propose an approximation. The authors claim their

algorithm takes O(nlogn) time. However it is not correct since the lowest time

complexity of LP is O(n

3.5

) [Schrijver1986].

20

3.2 Pure Coverage Problem

3.2.1 Fixed sensing range

We first investigate the work which assumes sensor nodes have a common fixed

sensing range. The case of adjustable sensing range will be introduced in next section.

To prolong network lifetime, one naive is to divide sensors into mutually exclusive

subsets, while every subset can cover the set of targets given. Each subset is switch

to active mode and sleep mode alternatively, so that at any time there is only one set

of sensors active. When sensors are divided into disjoint sets, maximizing the number

of subsets can extend the sensor network lifetime significantly. Then the target

coverage problem is formulated as Target Coverage Problem (disjoint-set model):

Given a set of sensors and a set of targets, and a coverage mapping from sensors to

targets, find the maximum number of disjoint subsets such that each subset can cover

all targets. This problem is NP-hard. Various approximation algorithms have been

proposed in [Cardei02, CardeiDu05, CTW05].

Cardei and Du (CardeiDu05) first prove that target coverage problem (disjoint-set

model), called as DSC, is NP-hard. And prove that DCS has no polynomial-time

approximation algorithm with performance p for any p<2, if NP≠P. In order to

compute the maximum number of disjoint covers, the authors transform DSC into a

maximum-flow problem (MIP), which is then formulated as a mixed integer

programming (MIP). Based on the solution of the MIP, design a heuristic to compute

the number of covers.

In [CTW05], each sensor is allowed to activate and sleep at any time and the

active sensor sets are organized not-necessarily disjoint. A sensor set which can cover

all targets is called sensor cover set. Cardei et al [CTW05] study non-disjoint cover

set problem: Given a set of m sensors

1 2

{,,...,}

m

s s s

each monitoring a subset of n

targets

1 2

{,,...,}

n

r r r

, find a family of sensor cover sets

1 2

{,,...,}

p

S S S

with time

weights

1 2

,,...,

p

t t t

in [0,1], respectively, to maximized

1 2

...

p

t t t

+ + +

subject to

every sensor appears in

1 2

,,...,

p

S S S

with a total weight at most 1. This problem is

proved still NP-hard.

The authors first represent this problem as an 0-1 integer programming. Then

transform the problem as 0-1 integer linear programming. Two heuristics are

proposed: LP-MSC and Greedy-MSC. LP-MSC is based on the optimal solution of

the relaxation linear programming. The LP-MSC includes two steps: First, computes

the optimal solution of relax linear programming. Second, employs a rounding

technique to obtain an approximation solution for the original target coverage

problem. Greedy-MST uses a primal-dual approach. In the primal-dual approach, a

dual feasible solution and a primal near-feasible solution satisfying a part of the

complementary-slackness condition are computed at the same time. At each iteration,

to improve the feasibility of primal solution and the dual feasible solutions until the

21

primal one becomes feasible so that it can be an approximation solution for the

non-disjoint cover set problem.

Liu et at [Liu06, Liu07] study the maximal lifetime scheduling for sensor

surveillance system in wireless sensor networks. It assumes each sensor can watch

only target at a time and each target should be watched by K sensors (K>1 [Liu06],

K=1 [Liu07]) at any time. The problem is to schedule sensors to watch the target, such

that the lifetime of the sensor surveillance system is maximized. The lifetime is

defined as the duration up to the time until there is a target can not be watched by K

sensors or sensed data can not be forwarded to the sink due to energy depletion of the

sensor nodes. The connectivity of sensor nodes is further required in [Liu07] to

forward the sensed data to the remote sink.

The problem can be solved in polynomial time. The optimal solutions [Liu06,

Liu07] consist of three steps. In the first step, the maximum lifetime scheduling

problem is formulated as a Linear Programming problem. Upper bound on the

lifetime and the workload matrix are computed. Each element of the workload matrix

denotes the amount of duration time a sensor watching a target. In the second step, a

perfect matching technique is employed and sensors and targets are represented as

two sides of the bipartite graph based on the workload matrix. It continually computes

a perfect matching (represented as a schedule matrix) on the bipartite graph until the

workload matrix is completely decomposed into a sequence of schedule matrices.

Finally, a sensor surveillance tree and table is built based on the resulting schedule

matrices. Details of the algorithm can be found in [Liu06, Liu07].

3.2.2 Adjustable Sensing Range

The most of works address the coverage problem with fix sensing range. Cardei

et al [CardeiW05] address the target coverage problem in wireless sensor networks

with adjustable sensing range. Given a set of targets and a set of sensors with

adjustable sensing ranges, the adjustable Range Set Covers(AR-SC) problem is

finding a maximum number of set covers and the ranges associated with each sensor,

such that each sensor set covers all the targets. In AR-SC problem, a sensor can

participate in multiple sensor sets, sum of the energy spent in each set is constrained

by the initial energy. Figure 9 shows an example with three sensors s

1

, s

2

, s

3

and three

targets t

1

, t

2

, t

3

. Each sensor has two sensing range r

1

, r

2

. {(s

1

, r

2

)}, {(s

1

, r

1

), (s

3

, r

1

)},

and {(s

1

, r

1

), (s

2

, r

2

)} et al forms a target cover respectively.

22

23

for point coverage and connectivity. The authors model the problem: divide sensors

into the maximum number of disjoint subsets such that each subset can ensure both

coverage and connectivity in the network. This problem is NP-hard. The authors

propose a Greedy Iterative Energy-Efficient Connected Coverage (GIECC) algorithm.

The GIECC algorithm operates in iterations. During each iteration, the algorithm

finds an active set from among the available set of sensors. After the end of each

iteration, the available set is modified by removing the sensors which belong to the

active set found in the current iteration. The algorithm halts when it is unable to find

an active set of sensors from among the available set of sensors. Each iteration

includes three phases: coverage phase and connectivity phase, redundancy phase. . In

the coverage phase: start with an empty set A, choose a target point t with minimum

coverage, which is not covered by any of sensors in the set A, and choose a senor s

which covers t, with maximum utility, add sensor s to the set A, repeat till A is a cover

set. In the connectivity phase, add new sensors to A to get B such that B is connected.

Three methods are proposed to get B from A: Shortest Path Tree, Greedy

Incremental Tree and Implicit Connectivity Tree. Then does redundancy reduction

phase to remove redundancy sensors from B.

Li et al [Li07a] address the k-connected coverage problem. The k-connected

coverage problem is : Given a set of sensors and a set of targets, and a coverage

mapping from sensors to targets, and constants k, 1

k, find a minimum number of

sensors such that each target is covered at least a sensor and the selected sensors is

k-connected. The authors first address k-connected augmentation problem， that is,

for a given graph G=(V, E) and a subset of V, add the minimum number of nodes such

that the resulting subgraph is k-connected. The k-connected augmentation problem is

NP-hard and heuristic algorithms are proposed. Based on the investigation of

k-connected augmentation problem, two heuristic algorithms (TS algorithm and

Reverse algorithm) are proposed for k-connected coverage. The main idea of TS

algorithm is that the algorithm includes two steps; the first step is to construct a

coverage of the targets using set cover algorithm; and the second step is to increase

some nodes to this coverage such that the subgraph composed by both these increased

nodes and the nodes already existing in coverage is k-connected. The main idea of the

reverse algorithm is that, initially, each sensor node in the sensor network is active,

and then change one active node to inactive node each time if it satisfies two

conditions: (1) after deleting this node, the remain nodes also form a coverage, and (2)

any two neighbors of the node has k vertex-disjoint paths in remaining graph after

deleting this node.

3.3.2. Connected multiple coverage

Yang et al [Yang06] address another type coverage problem: select a subset of

sensors to cover a rest of sensors. The authors study k-coverage set problem (called

k-CS) and connected k-coverage set problem (called k-CCS). The k-CS problem is:

Given a constant k>0, and an undirected graph G=(V, E) find a subset of nodes

C V such that each node in V is dominated by at least k different nodes in C, and

the number of nodes in C is minimized. The k-CCS is to add another constraint that

24

subgraph induced by C is connected. The k-CS is formulated as an integer linear

programming and then a centralized LP-based algorithm for k-CS is proposed.

LP-based algorithm includes two steps: the first step is to compute the optimal

solution of relax LP problem, the second step is to round the optimal solution to

solution of ILP: if the solution of optimal solution of LP is greater than some value,

set this variable to 1, otherwise, zero. Non-global solutions for k-CS/k-CSS are

proposed: cluster based algorithm and pruning-based algorithm. Cluster base

algorithm runs as follows: sequentially apply a traditional clustering algorithm k time

to get k sets of clusterheads, find gateways to connect the first set, then add other

nodes to all clusterheads and gateways to form the k-CS/k-CSS. In the pruning-based

algorithm, all nodes are initially assumed as active. Each node using 2-hop

neighborhood to determines its status. Initially, all nodes are marked. Each node u is

given a unique priority, L(u). Each node broadcast its neighbor set N(u), and build a

subset )(uC which is formed by us all neighbors with higher priorities than u.

Node u is umnmarked if C(u) is connected(this constraint is removed for k-CS) and

for any neighbor w of u, there are k distinct nodes in C(u), such that w is a neighbor of

all the k nodes. All marked nodes form k-CS/k-CSS.

Li et al [Li07b] address k-connected m-coverage problem which is different from

the coverage problem in [Yang06]. The coverage problem is : Given a set of sensors

and a set of targets, and a coverage mapping from sensors to targets, and constants k

and m, 1,1

mk, find a minimum number of sensors such that each target is

covered at least m sensors and the selected sensors is k-connected. The k-connected

coverage problem and k-connected m-coverage problem are NP-hard.

In [Li07b], the authors first study m-coverage problem, which is formulated as ILP,

then propose an approximation algorithm based on LP. Based on solution of

m-coverage problem and algorithms for k-connected augmentation [Li07a], two

heuristics (kmTS algorithm and kmReverse algorithm) are proposed for k-connected

m-coverage problem. Two algorithms include two steps: the first step is to construct a

m-coverage of targets; The second step is to increase small size nodes to this

m-coverage such that the subgraph by these increased nodes and nodes of m-coverage

is k-connected using the algorithms [Li07a].

3.3.3. Breach Coverage

Network lifetime has been recognized as an important factor in sensor network

design. To extend sensor network lifetime, one potential approach is to divide sensors

into disjoint subsets, each of which can cover all targets. Each subset is switched to

active mode and sleep mode alternatively, so that at any time there is only one set of

sensors active to prolong network lifetime. The size of sensor cover sets is not put any

constraint. However, the number of deployed sensors is usually very large and the

base station may not provide a bandwidth large enough for receiving data from all

sensors in the cover sets. In this situation, a complete coverage is sometime not

25

available. Maybe there exists some targets can not be monitored by any sensor. A

target is in breach if it is not monitored by any sensor. There are some coverage

breach problems studied in the literature [Slijepcevi01, Chengxi05, 07, WangC07,

Thai05].

Cheng et al [Chengx05, Chengx07] study three coverage breach problems:

Minimum Breach problem, Minimum Individual Breach Time problem and Minimum

Maximal Breach problem. The Minimum Breach problem is : Given a set A of fixed

points and a set S of sensors, organize sensors into disjoin subsets C

i

, i=1,2, K,

where each subset | |

i

C W

and the overall breach is minimized. The authors prove

the three problems are NP-hard. The three coverage breach problem are formulated

as 0-1 linear integer programming problems. The minimum breach problem is

formulated as a 0-1 integer programming problem as following:

1 1

min{ (1 )}

K M

kj

k j

y

= =

∑∑

,,

1

1,...,,1,...;

N

ij k i k j

i

a x y j M k K

=

= =

∑

,

1

1, 1,...;

K

k i

k

x i N

=

= =

∑

,

1

, 1...,;

N

k i

i

x W k K

=

= =

∑

,

,

{0,1} 1...,1,...;

{0,1} 1...,1....

k j

k i

y k K j M

x k K i N

= =

= =

A Greedy approximation algorithm and a heuristic based on the LP-relaxation are

proposed. In a greedy strategy, iteratively pick the most coverage-effective sensor and

put it in its fit position until all sensors are put into subsets. Each subset can have at

most W sensors. LP-based heuristic(called Relaxation) includes three steps; In the

first step, the integer programming(IP) problem is relaxed to a linear programming

(LP) problem, and compute an optimal solution for LP. In the second step, using

greedy strategy to find an integer solution based on the optimal solution of LP. In the

third step, the solution from (IP) problem is used to construct the subsets.

In [Thai05], Thai et al present two new linear programming based models,

Minimum Coverage Breach under Bandwidth constraints (MCBB) and Maximum

Network Lifetime under bandwidth constraints (MNLB) to solve the joint

optimization on energy and bandwidth utilization. MCBB problem is: Given a

collection C of subsets of a finite set R, find a family of p order pairs (S

j

, t

j

) such that

the total coverage breach is minimized. Where S

j

is a set cover and t

j

is the time

duration between 0 and 1 for S

j

to be active. MNLB problem is to find a family of p

26

order pairs (S

j

, t

j

) such that

∑

=

p

j

j

t

1

is maximized. In the two models, sensors are

organized into non-disjoint set cover. The MCBB problem and the MNLB are

NP-hard, and can be formulated as mix integer programming. The authors propose

two approximation algorithms based on the optimal solution of relax linear

programming to solve them.

4. Conclusion

In the chapter, we investigate the current works on coverage problem in sensor

networks, and classify them into two categories: sensor area coverage and target

coverage. We focus on the most representative problems in each domain and present a

comprehensive review and analysis of various existed algorithms and techniques.

Acknowledgement

This research is partially supported by the National Natural Science Foundation of

China under grant 10671208, and Key Laboratory of Data Engineering and

Knowledge Engineering (Renmin University of China), MOE.

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