Mobile Networks and Applications 10, 519–528, 2005
C
2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.
The Coverage Problem in a Wireless Sensor Network
CHI-FU HUANG and YU-CHEE TSENG
∗
Department of Computer Science and Information Engineering, National Chiao-Tung University, 1001 Ta Hsueh Road, Hsin-Chu, 30050, Taiwan
Abstract. One of the fundamental issues in sensor networks is the coverage problem, which reflects how well a sensor network is monitored
or tracked by sensors. In this paper, we formulate this 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, where k is a given parameter. The sensing ranges of sensors can
be unit disks or non-unit disks. We present polynomial-time algorithms, in terms of the number of sensors, that can be easily translated to
distributed protocols. The result is a generalization of some earlier results where only k = 1isassumed. Applications of the result include
determining insufficiently covered areas in a sensor network, enhancing fault-tolerant capability in hostile regions, and conserving energies
of redundant sensors in a randomly deployed network. Our solutions can be easily translated to distributed protocols to solve the coverage
problem.
Keywords: ad hoc network, computer geometry, coverage problem, ubiquitous computing, wireless network, sensor network
1. Introduction
The rapid progress of wireless communication and embedded
micro-sensing MEMS technologies has made wireless sensor
networks possible. Such environments may have many inex-
pensive wireless nodes, each capable of collecting, storing,
and processing environmental information, and communicat-
ing with neighboring nodes. In the past, sensors are connected
by wire lines. Today, this environment is combined with the
novel ad hoc networking technology to facilitate inter-sensor
communication [13,17]. The flexibility of installing and con-
figuring a sensor network is thus greatly improved. Recently,
a lot of research activities have been dedicated to sensor net-
works, including design issues related to the physical and me-
dia access layers [15,22,24] and routing and transport pro-
tocols [3,5,7]. Localization and positioning applications of
wireless sensor networks are discussed in [2,4,11,14,19].
Since sensors may be spread in an arbitrary manner, one
of the fundamental issues in a wireless sensor network is the
coverage problem.Ingeneral, this reflects how well an area is
monitored or tracked by sensors. In the literature, this prob-
lem has been formulated in various ways. For example, the
Art Gallery Problem is to determine the number of observers
necessary to cover an art gallery (i.e., the service area of the
sensor network) such that every point in the art gallery is mon-
itored by at least one observer. This problem can be solved
optimally in a 2D plane, but is shown to be NP-hard when
extended to a 3D space [12]. Reference [8] proposes polyno-
mial time algorithms to find the maximal breach path and the
maximal support path that are least and best monitored in the
sensor network. How to find the minimal and maximal expo-
sure path that takes the duration that an object is monitored
by sensors is addressed in [9,20]. Localized exposure-based
A preliminary version of this paper has appeared in the Workshop on Wireless
Sensor Networks and Applications, 2003, San Diego, CA, USA.
∗
Corresponding author.
coverage and location discovery algorithms are proposed in
[10].
On the other hand, some works are targeted at particular
applications, but the central idea is still related to the coverage
issue. For example, sensors’ on-duty time should be properly
scheduled to conserve energy. Since sensors may be arbitrar-
ily deployed, if some nodes share the common sensing region
and task, then we can turn off some of them to conserve energy
and thus extend the lifetime of the network. This is feasible
if turning off some nodes still provide the same “coverage”
(i.e., the provided coverage is not affected). Slijepcevic and
Potkonjak [16] proposes a heuristic to select mutually exclu-
sive sets of sensor nodes such that each set of sensors can
provide a complete coverage of the monitored area. Also tar-
geted at turning off some redundant nodes, Ye et al. [23] pro-
poses a probe-based density control algorithm to put some
nodes in a sensor-dense area to a doze mode to ensure a long-
lived, robust sensing coverage. A coverage-preserving node
scheduling scheme is presented in [18] to determine when a
node can be turned off and when it should be rescheduled to
become active again.
In this work, we consider a more general sensor cover-
age problem: given a set of sensors deployed in a target area,
we want to determine if the area is sufficiently k-covered,
in the sense that every point in the target area is covered by
at least k sensors, where k is a given parameter. As a result,
the aforementioned works [18,23] can be regarded as a special
case of this problem with k = 1. Applications requiring k > 1
may occur in situations where a stronger environmental mon-
itoring capability is desired, such as military applications. It
also happens when multiple sensors are required to detect an
event. For example, the triangulation-based positioning pro-
tocols [13,14,19] require at least three sensors (i.e., k ≥ 3) at
any moment to monitor a moving object. Enforcing k ≥ 2
is also desirable for fault-tolerant purpose. The work [21]
also considers the same coverage problem combined with the
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HUANG AND TSENG
communication connectivity issue. However, it incurs higher
computational complexity to determine a network’s cover-
age level as compared to the solution proposed in this pa-
per. The arrangement issue [1,6], which is widely studied
in combinatorial and computational geometry, also consid-
ers how a finite collection of geometric objects decomposes
a space into connected elements. However, to construct ar-
rangements, only centralized algorithms are proposed in the
literature, whilst what we need for a wireless sensor network
is a distributed solution. The solutions proposed in this paper
can be easily translated to distributed protocols where each
sensor only needs to collect local information to make its
decision.
In this paper, we propose a novel solution to determine
whether a sensor network is k-covered. The sensing range
of each sensor can be a unit disk or a non-unit disk. Rather
than determining the coverage of each location, our approach
tries to look at how the perimeter of each sensor’s sensing
range is covered, thus leading to an efficient polynomial-
time algorithm. Note that this step can be executed by each
sensor based on location information of its neighbors. This
can lead to an efficient distributed solution. As long as the
perimeters of sensors are sufficiently covered, the whole area
is sufficiently covered. The k-coverage problem can be fur-
ther extended to solve several application-domain problems.
In Section 5, we discuss how to use our results for discover-
ing insufficiently covered areas, conserving energy, and sup-
porting hot spots. At the end, we also show how to extend
our results to situations where sensors’ sensing regions are
irregular.
This paper is organized as follows. Section 2 formally
defines the coverage problems. Our solutions are presented
in Section 3. Section 4 presents our simulation results and
demonstrates a tool that we implemented to solve the k-
coverage problem. Section 5 further discusses several pos-
sible extensions and applications of the proposed solutions.
Section 6 draws our conclusions.
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(a) (b)
Figure 1. Examples of the coverage problem: (a) the sensing ranges are unit disks, and (b) the sensing ranges are non-unit disks. The number in each
sub-region is its coverage.
2. Problem statement
We are given a set of sensors, S ={s
1
, s
2
,...,s
n
},inatwo-
dimensional area A. Each sensor s
i
, i = 1,...,n, is located at
coordinate (x
i
, y
i
) inside A and has a sensing range of r
i
, i.e.,
it can monitor any point that is within a distance of r
i
from s
i
.
Definition 1. A location in A is said to be covered by s
i
if
it is within s
i
’s sensing range. A location in A is said to be
j-covered if it is within at least j sensors’ sensing ranges.
Definition 2. A sub-region in A is a set of points who are
covered by the same set of sensors.
We consider two versions of the coverage problem as fol-
lows.
Definition 3. Given a natural number k, the k
-Non-unit-disk
C
overage (k-NC) Problem is a decision problem whose goal
is to determine whether all points in A are k-covered or not.
Definition 4. Given a natural number k, the k
-Unit-disk
C
overage (k-UC) Problem is a decision problem whose goal
is to determine whether all points in A are k-covered or not,
subject to the constraint that r
1
= r
2
=···=r
n
.
3. The proposed solutions
At the first glance, the coverage problem seems to be very dif-
ficult. One naive solution is to find out all sub-regions divided
by the sensing boundaries of all n sensors (i.e., n circles), and
then check if each sub-region is k-covered or not, as shown
in figure 1. Managing all sub-regions could be a difficult and
computationally expensive job in geometry. There may exit
as many as O(n
2
) sub-regions divided by the circles. Also, it
may be difficult to calculate these sub-regions.
T
HE COVERAGE PROBLEM IN A WIRELESS SENSOR NETWORK 521
Figure 2. Determining: (a) the segment of s
i
’s perimeter covered by s
j
, and (b) the perimeter-coverage of s
i
’s perimeter.
3.1. The k-UC problem
In the section, we propose a solution to the k-UC problem,
which has a cost of O(nd log d), where d is the maximum
number of sensors whose sensing ranges may intersect a sen-
sor’s sensing range. Instead of determining the coverage of
each sub-region, our approach tries to look at how the perime-
ter of each sensor’s sensing range is covered. Specifically, our
algorithm tries to determine whether the perimeter of a sensor
under consideration is sufficiently covered. By collecting this
information from all sensors, a correct answercan be obtained.
Definition 5. Consider any two sensors s
i
and s
j
.Apoint on
the perimeter of s
i
is perimeter-covered by s
j
if this point is
within the sensing range of s
j
.
Definition 6. Consider any sensor s
i
.Wesay that 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.
Similarly, 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.
Below, we propose an O(d log d) algorithm to determine
whether a sensor is k-perimeter-covered or not. Consider two
sensors s
i
and s
j
located in positions (x
i
, y
i
) and (x
j
, y
j
),
respectively. Denote by d(s
i
, s
j
) =
|x
i
− x
j
|
2
+|y
i
− y
j
|
2
the distance between s
i
and s
j
.Ifd(s
i
, s
j
) > 2r, then s
j
does
not contribute any coverage to s
i
’s perimeter. Otherwise, the
range of perimeter of s
i
covered by s
j
can be calculated as
follows (refer to the illustration in figure 2(a)). Without loss
of generality, let s
j
be resident on the west of s
i
(i.e., y
i
= y
j
and x
i
> x
j
). The angle α = arccos(
d(s
i
,s
j
)
2r
). So the arch of s
i
falling in the angle [π −α, π +α]isperimeter-covered by s
j
.
The algorithm to determine the perimeter coverage of s
i
works as follows.
1. For each sensor s
j
such that d(s
i
, s
j
) ≤ 2r, determine the
angle of s
i
s arch, denoted by [α
j,L
,α
j,R
], that is perimeter-
covered by s
j
.
522
HUANG AND TSENG
s
i
s
i
(a) (b)
Figure 3. Some examples to utilize the result in Theorem 1.
2. For each neighboring sensor s
j
of s
i
such that d(s
i
, s
j
) <
2r, place the points α
j,L
and α
j,R
on the line segment
[0, 2π ], and then sort all these points in an ascending order
into a list L. Also, properly mark each point as a left or
right boundary of a coverage range, as shown in figure
2(b).
3. (Sketched) Traverse the line segment [0, 2π]byvisiting
each element in the sorted list L from left to right and
determine the perimeter-coverage of s
i
.
The above algorithm can determine the coverage of each
sensor’s perimeter efficiently. Below, we relate the perimeter-
coverage property of sensors to the coverage property of the
network area.
Lemma 1. Suppose that no two sensors are located in the
same location. Consider any segment of a sensor s
i
that divides
two sub-regions in the network area A.Ifthis segment is
k-perimeter-covered, the sub-regionthat is outside s
i
’s sensing
range is k-covered and the sub-region that is inside s
i
’s sensing
range is (k + 1)-covered.
Proof. The proof is directly from Definition 6. Since the
segment is k-perimeter-covered, the sub-region outside s
i
’s
sensing range is also k-covered due to the continuity of
the sub-region. The sub-region inside s
i
’s sensing range is
(k + 1)-covered because it is also covered by s
i
.
An example is demonstrated in figure 2(b). The gray areas
in figure 2(b) illustrate how the above lemma works .
Theorem 1. Suppose that no two sensors are located in the
same location. The whole network area A is k-covered iff each
sensor in the network is k-perimeter-covered.
Proof.For the “if” part, observe that each sub-region inside A
is bounded by at least one segment of a sensor s
i
’s perimeter.
Since s
i
is k-perimeter-covered, by Lemma 1, this sub-region
is either k-covered or (k + 1)-covered, which proves the “if”
part.
For the “only if” part, it is clear by definition that for
any segment of a sensor s
i
’s perimeter that divides two sub-
regions, both these sub-regions are at least k-covered. Fur-
ther, observe that the sub-region that is inside s
i
’s sensing
range must be covered by one more sensor, s
i
, and is thus at
least (k + 1)-covered. So excluding s
i
itself, this segment is
perimeter-covered by at least k sensors other than s
i
itself,
which proves the “only if” part.
Note that Theorem 1 is true when all sensors are claimed
to be k-perimeter-covered. When a specific sensor s
i
is k-
perimeter-covered, it only guarantees that each point right
outside s
i
’s perimeter is k-covered, and each point right inside
s
i
’s perimeter is (k + 1)-covered. However, it does not guar-
antee that all points inside s
i
’s perimeter is (k + 1)-covered.
An example is shown in figure 3. In figure 3(a), sensor s
i
is
2-perimeter-covered since each segment of its perimeter is
covered by two sensors. This only implies the coverage lev-
els of the points nearby the perimeter of s
i
. The gray area,
which is outside the coverage of s
i
’s neighboring sensors,
is only 1-covered. In fact, the segments that bound the gray
area are only 1-perimeter-covered. If we add another sensor
to cover these segments (shown in thick dotted line) as shown
in figure 3(b), then s
i
’s sensing region will be 2-covered.
Below, we comment on several special cases which we
leave unaddressed on purpose for simplicity in the above dis-
cussion. When two sensors s
i
and s
j
fall in exactly the same
location, Lemma 1 will not work because for any segment of
s
i
and s
j
that divides two sub-regions in the network area, a
point right inside s
i
’s and s
j
’s sensing ranges and a point right
outside their sensing ranges will differ in their coverage levels
by two, making Lemma 1 incorrect (refer to the illustration in
figure 4(a)). Other than this case, all neighboring sub-regions
in the network will differ in their coverage levels by exactly
one. Since in most applications we are interested in areas that
are insufficiently covered, one simple remedy to this prob-
lem is to just ignore one of the sensors if both sensors fall in
T
HE COVERAGE PROBLEM IN A WIRELESS SENSOR NETWORK 523
(a) (b)
Figure 4. Some special cases: (a) two sensors falling in the same location (the number in each sub-region is its level of coverage), and (b) the sensing range
of a sensor exceeding the network area A.
exactly the same location. Another solution is to first run our
algorithm by ignoring one sensor, and then increase the cov-
erage levels of the sub-regions falling in the ignored sensor’s
range by one afterward. The other boundary case is that some
sensors’ sensing ranges may exceed the network area A.In
this case, we can simply assign the segments falling outside
A as ∞-perimeter-covered, as shown in figure 4(b).
3.2. The k-NC problem
For the non-unit-disk coverage problem, sensors’ sensing
ranges could be different. However, most of the results de-
rived above remain the same. Below, we summarize how the
k-NC problem is solved.
First, we need to define that how the perimeter of a sen-
sor’s sensing range is covered by other sensors. Consider two
sensors s
i
and s
j
located in positions (x
i
, y
i
) and (x
j
, y
j
) with
sensing ranges r
i
and r
j
, respectively. Again, without loss of
generality, let s
j
be resident on the west of s
i
.Weaddress
how s
i
is perimeter-covered by s
j
. There are two cases to be
considered.
Case 1. Sensor s
j
is outside the sensing range of s
i
, i.e.,
d(s
i
, s
j
) > r
i
.
α
α
α
α
(a) (b)
Figure 5. The coverage relation of two sensors with different sensing ranges: (a) s
j
not in the range of s
i
, and (b) s
j
in the range of s
i
.
(i) If r
j
< d(s
i
, s
j
)−r
i
, then s
i
is not perimeter-covered by
s
j
.
(ii) If d(s
i
, s
j
) −r
i
≤ r
j
≤ d(s
i
, s
j
) +r
i
, then the arch of s
i
falling in the angle [π −α, π +α]isperimeter-covered
by s
j
, where α can be derived from the formula:
r
2
j
= r
2
i
+ d(s
i
, s
j
)
2
− 2r
i
· d(s
i
, s
j
) · cos(α). (1)
(iii) If r
j
> d(s
i
, s
j
) +r
i
, then the whole range [0, 2π]ofs
i
is perimeter-covered by s
j
.
Case 2. Sensor s
j
is inside the sensing range of s
i
, i.e.,
d(s
i
, s
j
) ≤ r
i
.
(i) If r
j
< r
i
− d(s
i
, s
j
), then s
i
is not perimeter-covered by
s
j
.
(ii) If r
i
− d(s
i
, s
j
) ≤ r
j
≤ r
i
+ d(s
i
, s
j
), then the arch of s
i
falling in the angle [π − α, π + α]isperimeter-covered
by s
j
, where α is as defined in equation (1).
(iii) If r
j
> d(s
i
, s
j
) +r
i
, then the whole range [0, 2π ]ofs
i
is
perimeter-covered by s
j
.
The above cases are illustrated in figure 5. Based on such
classification, the same algorithm to determine the perimeter
coverage of a sensor can be used. Lemma 1 and Theorem 1
