Research on Optimal Coverage Prob
lem of Wireless Sensor Networks
Xueqing Wang
1
, Fayi Sun
2
and Xiangsong Kong
1
1
School of Mechanics & Civil Engineering, Chin
a University of Mining & Technology Beijing
100083 Beijing, China
wxqease@163.com
2
Weifang University of Science and Technology, 262700 Shouguang, Shandong, China
Abstract: 
In a wireless sensor network, the size of radio nodes has direct relation to the cost of total wireless
sensor networks, and at the same time, the problem is closely connected to wireless sensor networks’
performance, such as robust, faulttolerance, and further more, it is considered at first as wireless sensor
networks are designed. Therefore, the research on the size of radio nodes has significant meanings of theory
and practice to design of wireless sensor networks. The paper simplifies complex coverage problem step by
step. By means of math modeling, theoretical analysis and formula deducting, classical geometric theories and
the method of mathematics induction are adopted, at last, the analysis formula of minimum number of nodes is
theoretically educed under circumstances of entire and seamless coverage in WSN. In addition, it is worth
mentioning that the conclusion is the full same with
computing the number of communication nodes of wireless
sensor networks.
KeyWords: 
Wireless sensor networks; coverage problem; sensor field; sensor nodes
1.
INTRODUCTION
Since 1990s, more and more researchers paid great
attention to micro sensors capable of sensing, computing
and wireless communicating, and Wireless Sensor
Networks (WSN) made up of them with rapid
development of the technologies of wireless
communicating, embedded computing, sensing and
MicroElectroMechanical Systems (MEMS)
[1, 2]
. Micro
nodes are selforganized, distributed at random, integrated
with sensor unit, data processing unit and communication
unit. Sensors cooperatively and realtime sense, monitor
and collect physical phenomena people are interested in.
And then, the information are processed, thereby,
knowledge is acquired in details and with accuracy. Huge
application merit and development promise attracts
attention of them who are people from industry, academic
researchers, and military department
[3]
.
In wireless sensor networks, nodes are usually
deployed at random in sensor field, therefore, coverage
problem is one of basic problem, and the problem has
some influence on monitoring and tracking object.
Reference [4] proposed polynomial time algorithms to find
the maximal breach path and the maximal support path
that are least and best monitored in the sensor networks. A
coveragepreserving node scheduling scheme is presented
in [5] to determine when a node can be turned off and
when it should be rescheduled to turn on again. How to
find the minimal and maximal exposure path that takes the
duration that an object is monitored by sensors is
addressed in [6, 7, 8]. Localized exposurebased coverage
and location discovery algorithms are proposed in [9].
Reference [10] presents that different kinds of holes can
form in such networks creating geographically correlated
problem areas such as coverage holes, routing holes,
jamming holes, sink/black holes and worm holes, etc,
detail different types of holes, discuss their characteristics
and study their effects on successful working of a sensor
network. Reference [11] proposes a probebased density
control algorithm to put some nodes in a sensordense area
to a doze mode to ensure a longlived, robust sensing
coverage. The effectiveness of clusterbased distributed
sensor networks depends to a large extent on the coverage
provided by the sensor deployment in [12] and the paper
proposes a virtual force algorithm (VFA). Document [13]
proves the set of active nodes selected by a connected
2009 International Conference on Communications and Mobile Computing
9780769535012/09 $25.00 © 2009 IEEE
DOI 10.1109/CMC.2009.231
548
2009 International Conference on Communications and Mobile Computing
9780769535012/09 $25.00 © 2009 IEEE
DOI 10.1109/CMC.2009.231
548
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dominating set (CDS) provides full coverage and
connectivity. Document [14] addresses how to
dynamically maintain two important measures on the
quality of the coverage of a sensor network: the bestcase
coverage and worstcase coverage distances, and gives
relevant algorithm.
In this paper, the problem is researched: minimum
number of radio nodes is demanded in a sensor field if the
field is covered entirely and seamlessly.
2. PROBLEM DESCRIPTION
To illustrate in figure 1, theoretical hypotheses are
shown below:
Hyp.1 a sensor’s detecting ability is omnidirectional, that
is , its coverage range is a disk whose radius is
r
and
whose area is
D
(
D
=
π
r
2
).
Hyp.2 in a sensor field, all sensors’ radio power is uniform,
that is, the radio radius r of all sensors is equal.
Hyp.3 in a sensor field, all sensors are in the same plane.
nodes
cover age r ange
r
r
r
r
r
r
r
r
r
r
Fig.1 Sensors’ sensing range and WSN’ S sensor field
According to above hypotheses, the minimum
number of nodes, that is demanded in order to entirely and
seamlessly cover the sensor field to illustrate in figure 1 is
the number if coverage area of every node in it is maximal.
Expression in math is:
For , , , and
is maximal, or max
x
F i x D D
i j
j N
D
j
j N
∀ ∈ ∃ ∈
∈
∈
∪
∪
so as t o
Where
x
is any point in the sensor field,
F
is the sensor
field,
D
is disk of every node,
N
is number of nodes and
∪
is union.
3. RELATED THEORIES
Theorem 1.
Area of inscribed equilateral triangle is
maximal in all inscribed triangle of a circle.
Proof: To illustrate in figure 2, in circle
⊙
A
,
△
C1C2C3
and
△
C1’C2C3
are inscribed triangle of it.
C1P1
⊥
C2C3
and
C1P1
is through the center of
⊙
A
.
C1’P2
⊥
C2C3
and
C1’
is any point except for point
C1
.
Obviously, for
1'1'1,
C A C C
∀ ∈ ≠
⊙ ，
i f
then 
C1P1

＞

C1’P1
 (
C1P1
denotes the length of segment
C1P1
).
Moreover
∵

C2C3
=
C2C3

∴
area of
△
C1C2C3
＞
△
C1’C2C3
∵
C1’
is discretional
∴
area
S
of
△
C1C2C3
is maximal
Here, 
C1C2
=
C1C3

Moreover,
∵
symmetry
∴
in a similar way, 
C2C1
=
C2C3
 and 
C3C1
 = 
C3C2

∴

C1C2
=
C2C3
=
C3C1

∴
△
C1C2C3
is equilateral triangle and its area is
maximal.
∟
∟
A
C1
C1'
C2
C3
P1
P2
Fig.2 Graphic illustration of theorem 1
Theorem 2.
To illustrate in figure 3, if seamless topology
area of 3 seamless topology disks:
D1
,
D2
and
D3
get
maximum, then 3 circles:
C1
,
C2
and
C3
correspondingly
encircling 3 disks must intersect at only point
A
. That is,
D1
∩
D2
∩
D3
≠
Φ
, if max(
D1
∪
D2
∪
D3
), then
C1
∩
C2
∩
C3
={
A
}.
Proof: Max(
D1
∪
D2
∪
D3
)
→
min(
D1
∩
D2
∩
D3
)
∵
D1
∩
D2
∩
D3
≠
Φ
According to the definition of intersection
∴
C1
∩
C2
∩
C3
={
A
}
( a) (b)
D1
A
D2
D3
B
C1
C2
C3
D1
A
D3
D2
B
C1
C3
C2
Fig.3 Graphic illustration of theorem 2
Theorem 3.
Seamless topology area of 3 seamless
topology disks:
D1
,
D2
and
D3
is maximal and its value is
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2
4 3 3
2
r
π
＋
if 3 circles:
C1
,
C2
and
C3
correspondingly
encircling
D1
,
D2
and
D3
intersect at point
A
and
△
C1C2C3
is equilateral triangle.
Proof: Let max(
D1
∪
D2
∪
D3
) be true, according to
theorem 2, then
⊙
C1
,
⊙
C2
and
⊙
C3
must intersect at
point
A
.
Furthermore,
∵

AC1
=
AC2
=
AC3
=
r
∴
point
C1
,
C2
and
C3
are concyclic
∴△
C1C2C3
is inscribed triangle of
⊙
A
let area of
△
C1C2C3
be maximal, according to theorem 1,
then
△
C1C2C3
must be equilateral triangle.
Since area of
△
C1C2C3
get maximum, here, area of gray
field in figure 4(a) is minimal to a certainty. That is, area
of gray field in figure 4(b) is minimal, i.e. max(
D1
∪
D2
∪
D3
).
(b)
(
a
)
Fig.4 Graphic illustration of theorem 3
Area
S
1 of gray field in figure 5(b):
1 2'2 3
S AC A AC P
＝
ar ea of sect or
－
ar ea of
△
2
30 1
2 3 * 3
360 2
r
C P AP
π
D
D
＝ －
( )
( )
2
2
1
2
* *
2 2
12 2
r
r r
r
π
＝ － －
2 3 3
2
24
r
π
－
＝
(1)
∵
symmetry
∴
area
S
2 of gray field in figure 5(c)
2 3 3
2
2 4* 1
6
S S r
π
－
＝ ＝
(2)
∵
symmetry
∴
area
S
3 of gray field in figure 5(d)
2
2 3 3
3 2* 2
3
S S r
π
－
＝ ＝
(3)
∴
Area
S
4 of gray field in figure 5(e)
2 2
2 3 3
4 3
3
S D S r r
π
π
−
－
＝ ＝ －
2
3 3
3
r
π
+
＝
(4)
∴
Area
S
5 of gray field in figure 5(f)
4 2
5 3 3
S S S
= ∗ + ∗
2
3 3
3
2 3 3
2
6
3 3
r
r
π
π
+
= ∗ + ∗
－
4 3 3
2
2
r
π
+
=
(5)
∴
the problem proves to be true.
(b)
(c)
(d)
(
e
)
C1
C2
C3
A
A'
P3
(
f
)
r/2
60°
3
0
°
r
3
r
(a)
C1
C2
C3
B
Fig.5 Graphic illustration of computing area
From the process of proving, we notice that
∠
AC2B
=60
°
,
∴
360
°∕∠
AC2B
= 360
°∕
60
°＝
6,
i.e. A circle
⊙
C1
is exactly covered by 6 circles
⊙
C
2
,
⊙
C
3
,
⊙
C
4
,
⊙
C
5
,
⊙
C
6
,
and
⊙
C7. The case is
illustration in figure 6.
C1
C3
C4
C5
C6 C7
C2
Fig.6 Graphic illustration that one circle is seamlessly
covered by 6 circles
550
550
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4. MINIMUM NUMBER OF NODES IN WSN THAT
IS COVERED ENTIRELY AND SEAMLESSLY
According to the above theorem 3, in a given sensor
field
F
, the illustration in figure 7 shows the topology
graph. From the figure , it is known that a node is added
every time, then the increment
δ
of coverage area is
2
2 3 3
2 2
6
3 3
r r
D S
π
δ π
= − ∗ = − ∗
－
3 3
2
2
r
=
(6)
∴
the number N of nodes in the sensor field (Some of
boundary nodes are ignored) is
2
3 3
2
2
2
3 3
r
r
F
F F
N
δ
= = =
(7)
Fig.7 Graphic illustration of counting the number of nodes
5. CONCLUSION
In wireless sensor networks, the paper simplifies
complex coverage problem step by step. By means of math
modeling, theoretical analysis and formula deducting, the
analysis formula of minimum number of nodes is
theoretically educed under circumstances of entire and
seamless coverage in WSN. It resolves a math problem of
WSN in theory. The research on the number of nodes has
significant meanings of theory research and network
designing of WSN. On its basis, the theory has some
influence on algorithm research and protocol designing.
This is the content that I am going to work later.
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