Sensor Coverage in Wireless Sensor Networks

swarmtellingMobile - Wireless

Nov 21, 2013 (3 years and 6 months ago)

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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 sensors 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
Wus 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 nodes 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 querys 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
querys 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
13

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 sensors 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 its 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 us 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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