Optimal Patterns for FourConnectivity and
Full Coverage in Wireless Sensor Networks
Xiaole Bai,Ziqiu Yun,Dong Xuan,Ten H.Lai,and Weijia Jia
Abstract—In this paper,we study optimal deployment in terms of the number of sensors required to achieve fourconnectivity and full
coverage under different ratios of sensors’ communication range (denoted by r
c
) to their sensing range (denoted by r
s
).We propose a
new pattern,the Diamond pattern,which can be viewed as a series of evolving patterns.When r
c
=r
s
ﬃﬃﬃ
3
p
,the Diamond pattern
coincides with the wellknown triangle lattice pattern;when r
c
=r
s
ﬃﬃﬃ
2
p
,it degenerates to a Square pattern (i.e.,a square grid).We
prove that our proposed pattern is asymptotically optimal when r
c
=r
s
>
ﬃﬃﬃ
2
p
to achieve fourconnectivity and full coverage.We also
discover another new deployment pattern called the Doublestrip pattern.This pattern provides a new aspect to research on optimal
deployment patterns.Our work is the first to propose an asymptotically optimal deployment pattern to achieve fourconnectivity and full
coverage for WSNs.Our work also provides insights on how optimal patterns evolve and how to search for them.
Index Terms—Wireless sensor networks,topology,full coverage,fourconnectivity,optimal deployment pattern.
Ç
1 I
NTRODUCTION
D
EPLOYMENT
is an important issue in wireless sensor
networks (WSNs).There are two categories of deploy
ment methods.One is random deployments and the other is
planned deployments.With planned deployments,sensors are
placed at planned,predetermined locations.In planning
where to deploy sensors,it is often desirable that the pattern
requires the minimum number of sensors.In general,
finding the optimal development pattern (in terms of the
number of sensors) has practical significance.First,sensor
nodes still cost about $100 apiece.Deploying the minimum
number of sensors needed has obvious economic benefits.
Second,insights obtained from optimal deployment pat
terns can be used to guide the development of heuristic
algorithms for topology control and sensor scheduling [15],
as well as to measure the relative performance of these
heuristics as compared to optimal patterns [28].
We investigate the problem of optimal deployment
patterns in planned deployments that achieve fourcon
nectivity and full coverage.Fourconnected wireless sensor
networks are popular in practice.Several research projects
(e.g.,data segmentation [21],routing [12],and storage [10])
are based on fourconnected wireless sensor networks.In
practice,people often deploy wireless sensors in a square
grid pattern to achieve fourconnectivity [1].Then questions
often arise about the efficiency of this deployment pattern
and the most efficient way to deploy sensors.
1.1 Related Work
Finding optimal patterns for WSNs is a hard problem,and
very few results thereon are available in the literature.For
many years,the only result known to us was a theorem
proved in 1939,which states that the regular triangular
lattice pattern (triangle pattern for short) is asymptotically
optimal in terms of the number of circles required to entirely
cover a given area in the plane [16].This result,formulated
as one for sensor deployment,was proved again in [28]
using a different method.In many WSN applications,not
only must sensors cover an entire area,but they must also
form a connected communication network.When both
coverage and connectivity are required,the triangle pattern
remains optimal when r
c
=r
s
ﬃﬃﬃ
3
p
,where r
c
and r
s
are
sensors’ communication range and sensing range,respec
tively.In practice,the value of r
c
=r
s
has a wide range,not
necessarily greater than
ﬃﬃﬃ
3
p
.For example,while the reliable
communication range of the Extreme Scale Mote (XSM)
platform is 30 m,the sensing range of the acoustics sensor
for detecting an All Terrain Vehicle is 55 m[2] in which case
r
c
=r
s
¼ 30=55
ﬃﬃﬃ
3
p
.This has piqued researchers’ interests
in finding optimal deployment patterns that achieve both
connectivity and coverage for a complete range of r
c
=r
s
.In
2005,a stripbased pattern was proposed that can achieve
both connectivity and coverage,but without any study of
optimality [26].That pattern was later independently
described and proved to be near optimal when r
c
=r
s
¼ 1
[15].In 2006,the stripbased pattern was proved to be not
only near optimal but asymptotically optimal not only for
r
c
=r
s
¼ 1,but for all values of r
c
=r
s
[4].The connectivity
considered in these results is the simple oneconnectivity.If
higher degree connectivity is desired,a variant of the strip
based pattern that achieves twoconnectivity and full cover
age was proved to be asymptotically optimal,again for all
values of r
c
=r
s
[4].
In general,optimal deployment pattern in WSNs is
related to the covering problemin computational geometry.
Covering points using a minimum number of given
IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL.9,NO.3,MARCH 2010 435
.X.Bai,D.Xuan,and T.H.Lai are with the Department of Computer
Science and Engineering,Ohio State University,2015 Neil Avenue,
Columbus,OH 432101277.Email:{baixia,xuan,lai}@cse.ohiostate.edu.
.Z.Yun is with the School of Mathematical Science,Suzhou University,
Suzhou,215006,P.R.China.Email:yunziqiu@public1.sz.js.cn.
.W.Jia is with the Department of Computer Science,City University of
Hong Kong,83 Tat Chee Avenue,Kowloon,Hong Kong.
Email:wei.jia@cityu.edu.hk.
Manuscript received 6 Feb.2008;revised 30 Sept.2008;accepted 4 Aug.2009;
published online 11 Aug.2009.
For information on obtaining reprints of this article,please send email to:
tmc@computer.org,and reference IEEECS Log Number TMC2008020041.
Digital Object Identifier no.10.1109/TMC.2009.143.
15361233/10/$26.00 2010 IEEE Published by the IEEE CS,CASS,ComSoc,IES,& SPS
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geometric bodies have been extensively studied for disks on
a large area [14],[16],disks on a bounded square [18],[20],
orthogonal rectangles [11],fat convex bodies [9],[24],etc.
However,the literature in computational geometry only
considers coverage,not connectivity.This is understand
able,since connectivity is a typical networking problem
beyond pure mathematical interests.To the best of our
knowledge,the optimal planned deployment problem that
considers both coverage and connectivity simultaneously
has not been addressed before.
There are many research efforts in the design of algo
rithms that efficiently organize or schedule sensors that have
been previously deployed (especially in randomly deploy
ment) to achieve certaindegree of coverage andconnectivity,
e.g.,[3],[23],[25].These works’ problems and objectives
differ fromour foci and are beyond the scope of this paper.
1.2 Our Contribution
In this paper,for the first time,we propose and prove the
asymptotic optimality of a deployment pattern that achieves
both fourconnectivity and full coverage for various values
of r
c
=r
s
.As it turns out,there is no single pattern that is
optimal for all values of r
c
=r
s
.This is in contrast to the case
of twoconnectivity (and full coverage) for which the
aforementioned stripbased pattern is optimal for all values
of r
c
=r
s
.Our results are summarized as follows:
.WeproposeaDiamondpattern,whichcanbeviewedas
a series of evolving patterns.When r
c
=r
s
ﬃﬃﬃ
3
p
,the
Diamond pattern coincides with the wellknown
triangle lattice pattern;when r
c
=r
s
ﬃﬃﬃ
2
p
,it degener
ates toaSquare pattern.We prove the Diamondpattern
to be asymptotically optimal when r
c
=r
s
>
ﬃﬃﬃ
2
p
.
.When discussing sensor deployment,researchers
often implicitly assume that no two sensors can be
placed at the same spot.What effects does this
assumption have on optimal patterns?To answer
this question,we describe a Doublestrip pattern,
which accommodates two sensors per location,and
we show that it outperforms the Square pattern
when r
c
=r
s
< 16=17.This newly discovered pattern
provides a new aspect to research on optimal
deployment patterns.
.Our search for optimal deployment patterns is not
carried out in an ad hoc manner.Rather,it is
systematic.In doing so,we hope to give insight on
how to search for optimal deployment patterns for
WSNs.
Paper Organization.The rest of the paper is organized as
follows:In Section 2,we give the definitions and assump
tions used throughout the paper.In Section 3,we discuss
our exploration on the optimal deployment pattern to
achieve fourconnectivity and full coverage.In Section 4,we
discuss the evolution of deployment patterns.In Section 5,
we compute the number of nodes needed when different
patterns are used.We discuss practical considerations in
Section 6.Finally,Section 7 concludes the paper.
2 P
RELIMINARIES
We assume that both the sensing and the communication
scopes are binary disks with radius r
s
and r
c
,respectively.
That is,a sensor is capable to detect events that occur within
its sensing range r
s
for all directions,and the packet
reception ratio for two sensors reaches a desired level if and
only if the distance between them is not larger than r
c
.
We understand the limitations of disk models,and in
reality,the sensing and the communication ranges are likely
to be nonisotropic or even roughly conform to a normal
distribution probability model over all directions [8],[29],
[30].Disk models are adopted here because results obtained
with them are still useful in many applications.They have
been adopted in a great amount of literatures especially
theoretical ones,e.g.,in [4],[5],[6],[7],[15],[17],[28].
Furthermore,abstractions are inevitable to achieve suffi
cient generality when we are trying to establish certain
theoretical foundations.More discussion on nondisk sen
sing and communication models is presented in Section 6.
This paper studies asymptotically optimal deployment
patterns to achieve fourconnectivity and full coverage.A
deployment pattern is said to be asymptotically optimal if the
pattern is optimal when the deployment area is fixed and
the sensing range approaches zero,or equivalently,when
the sensing range is fixed and the deployment area
approaches in all directions infinity.Informally,it means
that the pattern is optimal if the dimension of deployment
area is so large compared to the sensing range that we can
ignore the boundaries of the deployment area and consider
only the interior nodes.If boundaries are not ignored,very
few can be said about optimal deployment patterns.A
pattern that is optimal for a region may not be optimal for
another region (of a different shape or different area).
Definition 1 (Voronoi polygon).Let P ¼ fa
1
;a
2
;...;a
p
g be a
set of p points on an euclidean plane S.The Voronoi polygon
V ða
i
Þ is the set of all points in S that are closer to a
i
(in terms
of euclidean distance) than to any other point in P,i.e.,
V ða
i
Þ:¼ fx 2 S:8a
j
2 P;dðx;a
i
Þ dðx;a
j
Þg:
Definition 2 (interior node).A node whose Voronoi polygon
has no edge on the boundaries of the deployment area.
Definition 3 (fourconnected sensor network).A sensor
network N is said to be fourconnected if for every two
interior nodes of N,there are at least four node disjoint paths
joining them.
Note that in a full coverage deployment,each Voronoi
polygoncorresponding to aninterior sensor node is enclosed
in a sensing disk.Thus,as illustrated in Fig.1a,each edge of
the Voronoi polygon resides on a common chord between
two sensing disks.The common chord that contains an edge
of a Voronoi polygon is said to be an edge chord.For instance,
in Fig.1b,edge a
0
b
0
of the Voronoi polygon resides on chord
ab.Thus,chord ab is the edge chord of edge a
0
b
0
.
The following terms are defined with respect to given r
c
and r
s
,which satisfy r
c
< 2r
s
:
Definition 4 (standard chord).The common chord between
two intersecting sensing disks is called a standard chord if the
distance between the two sensors is equal to the communica
tion range r
c
.
Definition 5 (long chord).If the common chord between two
intersecting sensing disks is longer than their standard chord,
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it is called a long chord.(The distance between the two sensors
is smaller than r
c
.)
Definition 6 (connection chord).A connection chord is either
a long chord or a standard chord.(The distance between the
two sensors is smaller than or equal to r
c
.)
If two sensing disks have a connection common chord,
then the two sensors can communicate directly with each
other,i.e.,they are connected by an edge in the sensors’
communication network—thus,the name connection chord.
Definition 7 (short chord).If the common chord between two
intersecting sensing disks is shorter than their standard chord,
it is called a short chord.(The distance between the two sensors
is greater than r
c
.)
If two sensing disks have a short common chord,then
the two sensors can not communicate directly with each
other;they are not connected by an edge in the sensors’
communication network.
Definition 8 (standard angle ).The angles corresponding to a
standard chord at the centers of two sensing disks are called
standard angles. ¼ 2 arccosðr
c
=2r
s
Þ.
Fig.2 illustrates the above definitions.The polygons
referred to in the following definitions are not necessarily
Voronoi polygons;and again,the definitions are made
relative to a given r
c
and r
s
.
Definition 9 (regular connection polygon).A polygon that
can be inscribed in a sensing disk,with all its edges of equal
length and no shorter than a standard chord.
Definition 10 (semiregular connection polygon).A polygon
with k 4 sides that can be inscribed in a sensing disk,with
four edges each of the length of a standard chord,and the
remaining k 4 edges being of equal length.
Definition 11 (reference polygon).It is either a regular
connection polygon or a semiregular connection polygon.
3 O
PTIMAL
P
ATTERN
E
XPLORATION
In this section,we describe our journey of exploration for an
optimal sensor deployment pattern that provides four
connectivity and full coverage.
3.1 Theoretical Foundation
In our journey,we think of a sensor deployment as a
collection of Voronoi polygons,which forms a tessellation
over a region.There are several benefits by employing
Voronoi polygons.First,as the Voronoi polygons form a
tessellation,we can regard each Voronoi polygon as the
corresponding sensor’s effective contribution to coverage.
If all Voronoi polygons are of the same size,say A,then the
number of sensors needed to cover a region of area R is
approximately R=A.We can estimate the number of
sensors needed by measuring the average size of each
Voronoi polygon.Second,polygon tessellation has been
extensively studied.Thinking in terms of Voronoi poly
gons,we are able to benefit more from the rich literature of
polygon tessellations.
Consider a rectangle of area R,over which we wish to
deploy sensors.For a sensor deployment d over R that
achieves fourconnectivity and full coverage,let G
d
denote
the set of Voronoi polygons generated by the sensors.Let G
d
be the collection of all possible G
d
s.(Each element in G
d
is a
set of Voronoi polygons.) Our goal is to find a G
d
2 G
d
with
the smallest jG
d
j,where jG
d
j denotes the cardinality of G
d
.
We denote the smallest jG
d
j by C
min
.
It is difficult to directly search G
d
for a certain element,
since we lack knowledge of this set.Therefore,we will
construct another set G
r
(to be described soon) satisfying the
following condition:for any G
d
2 G
d
,there exists a G
r
2 G
r
such that jG
d
j ¼ jG
r
j.With this set,we have
minfjG
d
j:G
d
2 G
d
g minfjG
r
j:G
r
2 G
r
g:ð1Þ
We denote the smallest jG
r
j by C
0
min
.From (1),we have
C
min
C
0
min
,which implies that the lower bound for jG
r
j
must also be the lower bound for jG
d
j.If we can find a
G
d
2 G
d
such that jG
d
j ¼ C
0
min
,then this jG
d
j must be equal to
C
min
.That is,this G
d
must have the smallest cardinality in
G
d
,and we will have found an optimal deployment pattern.
To construct the aforementioned G
r
,we first state a basic
result,which can be easily proved using a wellknown Euler
formula.In order not to interrupt the presentation of our
main ideas,we defer the lemma’s proof to the Appendix.
Lemma 1.Let denote a tessellation over a fixed region consisting
of F polygons.If each vertex of ,except those at the corner,is
on at least three edges,then the average number of sides of the
polygons of is not larger than 6 when F approaches 1.
Lemma 1 indicates that,when sensors are deployed to
achieve full coverage over a rectangular region,the average
number of edges of the Voronoi polygons generated by
them is asymptotically less than or equal to 6.Note that
Lemma 1 is not a known result.The known property that
average number of edges of a Voronoi region is less than 6
only holds for a bounded region,while Lemma 1 in this
paper presents the conclusion that holds asymptotically.
BAI ET AL.:OPTIMAL PATTERNS FOR FOURCONNECTIVITY AND FULL COVERAGE IN WIRELESS SENSOR NETWORKS
437
Fig.1.The solid and dashed circles denote sensing disks.Sensors are
represented by dark dots.The shaded area denotes the Voronoi
polygon of a sensor.(a) In a full coverage deployment,each Voronoi
polygon is constructed by the common chords of the intersecting
sensing disks.(b) Chord ab is the edge chord of edge a
0
b
0
.
Fig.2.Let Ddenote the distance between two sensors and assume that
r
c
< 2r
s
.(a) ab is a short chord when r
c
< D < 2r
s
;(b) ab is a standard
chord when D¼ r
c
, is the standard angle;(c) ab is a long chord when
D < r
c
.The common chords ab in (b) and the common chord ab in
(c) are connection chords.
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Even though the bound six is only an asymptotic one,when
constructing G
r
,we use Lemma 1 as a heuristic and confine
ourselves to those sets of polygons whose average number
of edges is no more than 6.
Based on Lemma 1 and the deployment requirement of
fourconnectivity and full coverage,we denote by G
r
any set
of polygons that satisfies the following conditions:1) the
average edge number of polygons is not larger than6,2) each
polygon is a reference polygon,3)
P
P2G
r
areaðPÞ R,
where areaðPÞ denotes the area of polygon P.(Note that
the polygons in G
r
are not necessarily the Voronoi polygons
of a sensor deployment.As a matter of fact,they don’t even
have to forma tessellation.)
Let G
r
denote the set of all such G
r
s.The following
lemma indicates that when r
c
=r
s
>
ﬃﬃﬃ
2
p
,it is possible to
“embed” G
d
in G
r
,so that (1) holds.We will present the
proof in the Appendix.(Unfortunately,we are unable to
prove the same result for the case where r
c
=r
s
ﬃﬃﬃ
2
p
.We
will discuss this case in Section 4.)
Lemma 2.If r
c
=r
s
>
ﬃﬃﬃ
2
p
,then for any G
d
2 G
d
,there exists a
G
r
2 G
r
such that jG
r
j ¼ jG
d
j.
We next establish a lower bound on jG
r
j for any G
r
2 G
r
.
This bound must also be a lower bound on jG
d
j for any G
d
in G
d
,due to Lemma 2.
Lemma 3.If r
c
=r
s
>
ﬃﬃﬃ
2
p
,then for any set G
r
2 G
r
,
jG
r
j R=ð2sin’ þsinð2’ÞÞr
2
s
;
where ’ ¼ maxð=3;Þ and is the standard angle.
Once again we defer the proof for Lemma 3 to the
Appendix.
We comment that the lower bound on jG
r
j is obtained
from R divided by the maximum average coverage
contribution of each individual sensor.This lower bound
does not tell us a specific deployment.Nevertheless,we can
use it to judge if a given deployment is optimal or not.
3.2 The Diamond Pattern
In the following,we present an optimal deployment pattern
called the Diamond pattern when r
c
=r
s
>
ﬃﬃﬃ
2
p
.
The Diamond pattern is shown in Fig.3.The Voronoi
polygon generated by each sensor,shown in Fig.3a,is a six
sided reference polygon.As r
c
=r
s
increases from
ﬃﬃﬃ
2
p
,the
length of long chords will decrease while the length of short
chords will increase.When r
c
=r
s
¼
ﬃﬃﬃ
3
p
,this polygon
becomes a regular hexagon.The shape will not change as
r
c
=r
s
further increases.Fig.3b illustrates the relative
positions of sensors in this pattern,
d
1
¼ 2r
s
cos
’
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2ð1 cos ’Þ
p
;ð2Þ
and
d
2
¼ 2r
s
cos
’
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2ð1 þcos ’Þ
p
;ð3Þ
where ’ ¼ maxð2arccosðr
c
=2r
s
Þ;=3Þ.In this pattern,the
coverage contribution of each individual sensor is d
1
d
2
=2.
Note that we use d
1
and d
2
to describe the positions of
sensors in this deployment to provide convenience in
practical.Though the Diamond pattern may look compli
cated,we can ease our real deployment by taking two steps.
We first deploysensors at the endpoints of eachgridusing d
1
and d
2
,and finally,deploy a sensor at the center of each grid.
Theorem 1.The Diamond pattern is an asymptotically optimal
deployment pattern that achieves fourconnectivity and full
coverage when r
c
=r
s
>
ﬃﬃﬃ
2
p
.
Proof.From Definition 8, ¼ 2arccosðr
c
=2r
s
Þ.When
ﬃﬃﬃ
3
p
r
c
=r
s
>
ﬃﬃﬃ
2
p
,the Voronoi polygon generated by each
sensor in the Diamond pattern is a sixsided semiregular
connection polygon.And we have =2 > =3.Form
(2) and (3),we obtain
d
1
¼ 2r
s
cosð=2Þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2ð1 cos Þ
p
and d
2
¼ 2r
s
cosð=2Þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2ð1 þcos Þ
p
.
Then the area of such a semiregular hexagon is
A
1
¼ d
1
d
2
=2 ¼ 4cos
2
ð=2Þ sinr
2
s
¼ 2ð1 þcos Þ sinr
2
s
¼ ð2sin þsinð2ÞÞr
2
s
:
When the Diamond pattern is used to cover a large
area R where the boundary condition can be ignored,the
number of such sixsided semiregular connection poly
gons needed is
N
1
¼ R=A
1
¼ R=ð2sin þsinð2ÞÞr
2
s
:ð4Þ
Similarly,when r
c
=r
s
ﬃﬃﬃ
3
p
,the Diamond pattern
becomes the regular triangle pattern,where the Voronoi
polygon generated by each sensor is a sixsided regular
connection polygon.We have 2 arccosðr
c
=2r
s
Þ =3.
Form (2) and (3),we obtain d
1
¼
ﬃﬃﬃ
3
p
r
s
and d
2
¼ 3r
s
.
In this case,the area of such a regular hexagon is
A
2
¼ d
1
d
2
=2 ¼ 3
ﬃﬃﬃ
3
p
r
2
s
=2:ð5Þ
When the Diamond pattern is used to cover a large area R
where the boundaryconditioncanbe ignored,the number
of such sixsided regular connection polygons needed is
N
2
¼ R=A
2
¼ 2
ﬃﬃﬃ
3
p
R=9r
2
s
¼ R
2sin
3
þsin
2
3
r
2
s
:ð6Þ
Equations (4) and (6) can be rewritten as
R=ð2 sin’ þsinð2’ÞÞr
2
s
;ð7Þ
where ’ ¼ maxð2arccosðr
c
=2r
s
Þ;=3Þ,which is the ex
actly the lower bound stated in Lemma 3.t
u
438 IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL.9,NO.3,MARCH 2010
Fig.3.The Diamond pattern that achieves fourconnectivity and full
coverage,where r
c
=r
s
>
ﬃﬃﬃ
2
p
.The coverage contribution of each
individual sensor can be considered as its Voronoi polygon’s area,
denoted by the shaded hexagon.
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4 P
ATTERN
E
VOLUTION
We have shown the Diamond pattern to be asymptotically
optimal when r
c
=r
s
>
ﬃﬃﬃ
2
p
.In this section,we investigate the
case where r
c
=r
s
ﬃﬃﬃ
2
p
.We introduce two other patterns
named Square and Double strip.As will become clear shortly,
the Square pattern can be viewed as a degenerated
Diamond pattern.
4.1 The Square Pattern
The Square pattern can be easily described using a diagram.
Fig.4 shows such a diagram,where d
1
¼ d
2
¼
ﬃﬃﬃ
2
p
r
c
.By
rotating the diagramin Fig.4 for 45 degrees,we obtain Fig.5
in which it is easy to see that each Voronoi polygon is a
square,with d
0
1
¼ d
0
2
¼ r
c
.The effective contribution of each
individual sensor to coverage is therefore
A
S
¼ d
0
1
d
0
2
¼ r
2
c
:ð8Þ
We display the Diamond pattern and the Square pattern
together in Fig.6,in an attempt to see their relationship.
The figure actually shows the Voronoi polygons corre
sponding to various patterns rather than the patterns
themselves.To see how these Voronoi polygons evolve as
r
c
=r
s
changes value,let us assume that the sensing range r
s
is fixed and the communication range r
c
goes from large to
small.When r
c
is sufficiently large so that r
c
=r
s
ﬃﬃﬃ
3
p
,the
regular Diamond pattern,whose Voronoi polygons are
regular connection hexagons (as defined in Definition 9),is
optimal.As r
c
gets smaller,in order to maintain four
connectivity of the network,some sensors need to get closer
to each other.As we have proved in Theorem 1 and
depicted in Fig.6,the semiregular Diamond pattern,whose
Voronoi polygons are semiregular connection hexagons (as
defined in Definition 10),is still optimal for the case where
ﬃﬃﬃ
3
p
> r
c
=r
s
>
ﬃﬃﬃ
2
p
.
If we let the semiregular connection hexagon continue to
shrink,it becomes a square when r
c
=r
s
¼
ﬃﬃﬃ
2
p
.The square
gets smaller and smaller as r
c
=r
s
continues to decrease.An
interesting question arises:is the Square pattern optimal for
r
c
=r
s
ﬃﬃﬃ
2
p
?
We speculate that the Square pattern is optimal for
r
c
=r
s
¼
ﬃﬃﬃ
2
p
.What about the case r
c
=r
s
<
ﬃﬃﬃ
2
p
?Is there any
evidence to suggest the optimality of the Square pattern one
way or the other?We will look at this question from a
different,interesting perspective.
When discussing sensor deployment,researchers often
implicitly assume that no two sensors can be placed at the
same spot.If this implicit restriction is lifted (i.e.,if multiple
sensors can be placed at each location),then a straightfor
ward way to achieve fourconnectivity and full coverage is
to use a twoconnectivity,full coverage pattern and deploy
two sensors at each location.
1
We describe such an
approach in the next section.
4.2 The DoubleStrip Pattern
The Doublestrip pattern is constructed by deploying two
sensors at each location of the Stripbased pattern,which
was introduced and proved to be asymptotically optimal
for twoconnectivity and full coverage in [4].As illustrated
in Fig.7,the Doublestrip or Stripbased pattern has as
many horizontal strips of sensors as needed for full
coverage,with each strip connected.In addition,two
vertical strips are placed at the east and west boundaries,
respectively,to connect the horizontal strips.
The data in Section 5 will show that the Doublestrip
pattern with two sensors at each location outperforms the
Square pattern in terms of saving the number of sensors
when r
c
=r
s
is smaller than a certain “turning point” value.
In the following,we first explain the intuitive reason (not
proof) for this phenomena and then derive the turning point
value of r
c
=r
s
.
BAI ET AL.:OPTIMAL PATTERNS FOR FOURCONNECTIVITY AND FULL COVERAGE IN WIRELESS SENSOR NETWORKS
439
Fig.4.The square pattern to achieve fourconnectivity and full coverage
when r
c
=r
s
ﬃﬃﬃ
2
p
.The coverage contribution of each individual sensor is
denoted by a shaded square.
Fig.5.Another view of the square pattern.
Fig.6.The Voronoi polygons generated in different deployment patterns
are denoted by the shaded areas.They are also the amount of coverage
contribution from each individual sensor.
1.If such deployments are legitimate,one may wonder if the Diamond
pattern’s optimality established in Theorem 1 will still hold.The answer is
positive since in our optimality proof,we made no assumption about sensor
locations.
Fig.7.A global view of the doublestrip pattern.It is the same as the
“stripbased” pattern [4] except that there are now two sensors at each
location.
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If an area is covered by a set of sensors,then each sensor
has a certain amount of “effective” contribution to the
coverage.The contribution is not simply r
2
s
,instead,it is
r
2
s
minus some overlap.More “effective” contribution to
the coverage fromeach sensor implies fewer sensors needed
to cover an area.Thus,if each sensor in pattern A has more
effective contribution than each sensor does in another
pattern B,then pattern Ais more efficient than pattern B.To
intuitively illustrate the reason why the Doublestrip pattern
can be more efficient than the Square pattern,we roughly
calculate the amounts of effective contribution from each
sensor in these two patterns,respectively,and then compare
them.As illustrated in Fig.8a,when there are no multiple
sensors deployed at the same location,the sensing disk of
each sensor is overlapped by four neighboring sensors from
four different directions.Denote the overlapped area
created by one neighboring sensor by O
a
,which is shown
shaded in Fig.8.The effective contribution fromeach sensor
in the Square pattern then can be expressed roughly as
r
2
s
4O
a
.When there are two sensors deployed at the same
location as illustrated in Fig.8b,the sensing disk of each
sensor is overlapped by four neighboring sensors from two
different directions.The effective contribution from each
sensor in this case can be expressed roughly as
ðr
2
s
2O
a
Þ=2 ¼ r
2
s
=2 O
a
.When comparing the above
two expressions,we can see that the coefficients of negative
items will dominate the results when O
a
is relatively large.
That is,when O
a
is large enough,the result of r
2
s
4O
a
will
be smaller than that of r
2
s
=2 O
a
,which explains why the
Square pattern can be not as efficient as the Doublestrip
pattern.We note that O
a
will increase as r
c
decreases when
r
s
is given.Hence,when r
c
=r
s
decreases below a certain
value,O
a
will be large enough to dominate the effective
contribution from each sensor,and then the Doublestrip
pattern will outperform the Square pattern in terms of
saving the number of sensors.With these informal argu
ments,we are ready to present a rigorous analysis below.
Directly calculating overlapped area based on O
a
is
complicated.Instead,the area of Voronoi polygon generated
by each sensor can be derived to represent the effective
contribution to the coverage accurately.Note here that we
consider the asymptotical optimality.The two vertical strips
then can be ignoredsince they are negligible comparedto the
large number of horizontal strips.The Voronoi polygons of
the Doublestrip pattern are as depicted in Fig.9.Since there
are two sensors at each location,each Voronoi polygon must
be shared by two sensors.The amount of coverage contribu
tionof eachindividual sensor isthereforeA
D
¼ d
1
d
2
=4,where
d
1
and d
2
are as indicated in Fig.9b and can be computed as
d
1
¼ r
c
;d
2
¼ 2r
s
þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
4r
2
s
r
2
c
q
:
There is a formula in [4] for computing A
D
,which is
A
D
¼
1
2
r
2
s
sin þ2r
2
s
sin
2 2
4
;ð9Þ
where ¼ 2arccosðr
c
=ð2r
s
ÞÞ.
Comparing A
D
with A
S
(8),we have A
D
¼ A
S
when
r
c
=r
s
¼ 16=17;A
D
< A
S
when r
c
=r
s
> 16=17;and A
D
> A
S
when r
c
=r
s
< 16=17.That is,the Square pattern is more
efficient than the Doublestrip pattern when r
c
=r
s
> 16=17;
and the opposite is true when r
c
=r
s
< 16=17.The two of
them are equally good when r
c
=r
s
¼ 16=17.
The Doublestrip pattern can save a great amount of
nodes when r
c
=r
s
is small.As shown in Section 5,when
r
c
=r
s
¼ 0:5,the Doublestrip pattern requires 70 percent less
nodes than the Square pattern does.Deploying multiple
sensors at the same location provides a new lead to obtain
multiple connectivity in optimal patterns.It seems contro
versial to deploy multiple sensors at the same location due to
the possible interference in communications.This problem
can be handled by hardwarebased approaches,e.g.,using
different bandwidths,and softwarebased approaches,e.g.,
using scheduling and synchronization protocols.We also
notice that the longcommunication path problem can exist
in the Doublestrip pattern.However,the longcommunica
tionpathproblemis not necessarily anoutcome of deploying
multiple sensors at the same location.We do not expect that
this problem will be inherited in optimal patterns with
higher connectivity,e.g.,fiveconnectivity.Deploying multi
ple sensors at the same location,therefore,offers a promising
newoptioninour further explorationof optimal deployment
patterns in WSNs.
Byconcatenating the Diamondpattern,the Square pattern
and the Doublestrip pattern together along the value of
r
c
=r
s
,the whole evolution picture is shown in Fig.10.
The Diamond pattern,later evolving to the Square,is
better until when r
c
=r
s
¼ 16=17.Afterward,if multiple
sensors are allowed to put at the same location,the optimal
pattern possibly takes a turn and follows a different path.
We note that optimal patterns that achieve fourconnectivity
and full coverage when r
c
=r
s
ﬃﬃﬃ
2
p
remain unknown.
5 N
UMERICAL
R
ESULTS
In this section,we compare the numbers of nodes needed
for various patterns to provide both fourconnectivity and
440 IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL.9,NO.3,MARCH 2010
Fig.8.Intuitive interpretation of why the doublestrip pattern can
outperform the square pattern.
Fig.9.The local view of the doublestrip pattern.Each Voronoi polygon
is shared by two sensors.
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full coverage over a deployment region of size 1;000 m
1;000 m with r
s
¼ 30 m,and 9 m r
c
54 m,i.e.,r
c
=r
s
varies from 0.3 to 1.8.
In Fig.11,we compare the performances of three
patterns—the Diamond pattern,the regular triangle lattice
pattern,and the Square pattern—with r
c
=r
s
varying from
1.3 to 1.8 (r
c
changing from 39 to 54 m).We make the
following observations:
1.It is convenient to view the Diamond pattern as a
series of transiting patterns as shown in Fig.6.As
such,the Diamond pattern coincides with the Square
pattern when r
c
=r
s
is small.At r
c
=r
s
¼
ﬃﬃﬃ
2
p
,i.e.,
1.4142,the two patterns diverge.Afterward,the
Diamond pattern stands alone until r
c
=r
s
¼
ﬃﬃﬃ
3
p
,i.e.,
1.7321 when it converges with the regular triangle
lattice pattern.
2.The Diamond pattern outperforms the regular
triangle lattice pattern when they are separate (i.e.,
when r
c
=r
s
<
ﬃﬃﬃ
3
p
).For instance,compared with the
regular triangle lattice pattern,the Diamond pattern
can save about 13.7 percent of nodes when
r
c
=r
s
¼ 1:5,and 14.9 percent when r
c
=r
s
¼
ﬃﬃﬃ
2
p
.
3.Also,the Diamond pattern outperforms the Square
patternwhenthe twoare separate.The Square pattern
costs 11.6 percent more sensors when r
c
=r
s
¼ 1:5,and
costs 25 percent more when r
c
=r
s
¼
ﬃﬃﬃ
3
p
,compared
with the Diamond pattern.This difference keeps
unchanged as r
c
=r
s
further increases.
In Fig.12,r
c
=r
s
varies from 0.9 to 1.43 (16=17
r
c
=r
s
ﬃﬃﬃ
2
p
).In this range,the Diamond and the Square
patterns are identical,and they outperform the regular
triangle lattice pattern constantly by 14.9 percent.
In Fig.13,r
c
=r
s
varies from 0.3 to 1.We observe the
following:
1.The 14.9 percent difference in performance between
the Square pattern (or the Diamond pattern) and the
regular triangle lattice pattern keeps unchanged
within this range.
2.The Doublestrip pattern outperforms the Square
pattern and the regular triangle pattern when
r
c
=r
s
< 16=17.Their difference in performance gets
larger as r
c
=r
s
turns smaller.For instance,when
r
c
=r
s
¼ 0:5,the Square pattern requires 70 percent
more sensors than the Doublestrip pattern.The
regular triangle lattice pattern requires 86.3 percent
more sensors.However,we recall that the Double
strip pattern has the drawback of longcommunica
tion paths,and it mounts two sensors at each
location,which may be undesirable.
6 P
RACTICAL
C
ONSIDERATIONS
Although abstraction is inevitable in the aim to establish
general theoretical foundations,it does not always hold in
practice.Disk sensing and communication models do not
always coincide with empirical observations,and deploy
ment errors,geographical constraints,and heterogeneous
sensor nodes should be considered.In the following,we
discuss these practical considerations.
6.1 Impacts of Nondisk Sensing Models
There exist several practical sensing models that stem from
real device experiments.Megerian et al.[19] propose that the
sensing quality can be expressed as =v
,where and are
sensordependent parameters and v is the distance between
the sensor and the detection target.In this model,the quality
of sensing gradually attenuates with increasing distance.In
[31],Zhou and Chakrabarty propose a probabilistic sensing
BAI ET AL.:OPTIMAL PATTERNS FOR FOURCONNECTIVITY AND FULL COVERAGE IN WIRELESS SENSOR NETWORKS
441
Fig.10.The whole evolution picture of patterns.The Voronoi polygons
generated in different deployment patterns are denoted by the shaded
areas.They are also the amount of coverage contribution from each
individual sensor.
Fig.11.Numbers of nodes needed to achieve fourconnectivity and full
coverage.r
c
=r
s
varies from 1.3 to 1.8.
Fig.12.Numbers of nodes needed to achieve fourconnectivity and full
coverage.r
c
=r
s
varies from 0.9 to 1.43.
Fig.13.Numbers of nodes needed to achieve fourconnectivity and full
coverage.r
c
=r
s
varies from 0.3 to 1.
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model in which two values,R
1
andR
2
(R
1
R
2
),are defined
from empirical observations.When the distance from the
target to the sensor is less than R
1
,it will be detected with
probability 1;when the distance is larger than R
2
,the
detection probability is 0;when the distance is between R
1
and R
2
,the detection probability will exponentially decrease
with the increasing distance similar to that in [19].We note
that,the optimal deployment patterns canstill be constructed
fromthe disk model when the above models are considered.
The disk model can be obtained by setting a desirable
threshold for sensing quality or probability,and exploiting
this threshold to determine the largest possible distance
between the sensor and the target.This distance is then
considered as the sensing range r
s
for the disk model.
For some types of sensors,sensing capability varies in
different directions.In such cases,the sensing area is
nondisk even after a threshold has been set.One typical
model obtained from real device experiments is proposed
by Cao et al.[8].Capturing the essential irregularity of
sensing capability in different directions,it suggests that the
sensing capability roughly follows a Gaussian distribution
over all directions.Denote the average sensing range over
all directions by and the standard deviation by
2
.Let the
sensing range define a random variable X.In this nondisk
model,the probability that the sensing range is x in a
particular direction is given by
PfX ¼ xg ¼
1
ﬃﬃﬃﬃﬃﬃ
2
p
e
ðxÞ
2
2
2
:ð10Þ
As increases,the sensing range in a particular direction is
more likely to be the average value,i.e.,it is more likely that
the overall sensing range is diskshaped.
In the following,we study,by simulation,the impact
from such sensing irregularities on coverage in our optimal
deployment patterns.In our simulation,sensors are
deployed over a 1;000 m1;000 m deployment region.
Each sensor has an average sensing range of 30 m in each
direction,i.e., ¼ 30 m.The communication range r
c
varies
from 10 to 54 m.The corresponding optimal deployment
patterns to achieve fourconnectivity are determined by the
values of r
c
and the average sensing range.We study four
cases where the sensing irregularity is defined by
2
¼ 2;5;10,and 20,respectively.For each case in a
particular deployment,we randomly generate 10,000 points
in the deployment region and then check how many of
themare within the sensing range of at least one sensor.The
percentage of these detected points reflects the percentage
of the area that is covered.The results are shown in Fig.14.
Fromthe results in Fig.14,we notice that higher sensing
irregularity results in lower overall coverage when other
parameters are given.We also observe that smaller r
c
values
can help overcome the shrinking coverage to which sensing
irregularity contributes.These observations can be ex
plained from the view of overlapped areas.Deployment
patterns that generate more overlapped areas are more
tolerant to sensing irregularity.Thus,when sensing
irregularity must be considered,we can deploy optimal
patterns by conservatively adopting a smaller communica
tion range to increase sensor density.
6.2 Impacts of Nondisk Communication Models
In reality,wireless communication signals are attenuated
and disrupted by various physical phenomena.In the
following,we discuss the impact from a practical commu
nication model on optimal deployment pattern connectivity.
We consider a widely used model suggested by Zuniga
and Krishnamachari [32],[33].This model establishes the
communication link quality,measured by packet reception
rate (PRR),as a function of the distance v between the
transmitter and receiver.From experiments,the PRR at
distance v can be expressed as
PRRðvÞ ¼ 1
1
2
e
P
t
PLðvÞP
n
2
8‘
;ð11Þ
where P
t
is the transmitter’s output power,PLðvÞ is the
path loss at distance v [22],P
n
is the noise floor,and ‘ is the
frame length.Refer to [33] for a detailed derivation.
When the above practical communication model is
considered,we can consider a connection established
betweentwo nodes onlyif their PRRfromeachother is above
a certain desirable threshold.By simulation,we investigate
the effect of the above model onthe probabilityfor one sensor
in our deployment to connect to its four neighbors.
We consider a connection established when PRR 0:95.
We let r
c
=r
s
vary from 0.3 to 1.8 such that all patterns are
covered.We pick the transmission power P
t
in the above
model to show its impact on the PRR as it is an adjustable
property of a physical device and does not depend on the
external environment.For each combination of P
t
and an
optimal deployment pattern,we run our simulation
10,000 times.The probability is then the ratio of number of
times when a link with PRR 0:95 can be established over
10,000.Other parameters are fromempirical data in [33].
From Fig.15,we observe that the probability transition
from 1 to 0 is sharp.This implies that connectivity will
quickly deteriorate when the transmission power decreases
in our deployment patterns.
6.3 Impacts of Deployment Errors
In practice,sensors may not be deployed exactly at planned
locations.Coverage andconnectivity are thus affected.There
are two kinds of deployment errors in practice:misalign
ment and random errors [27].In the following,we discuss
how these errors will affect our optimal pattern design.
442 IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL.9,NO.3,MARCH 2010
Fig.14.The impact in terms of coverage from sensing irregularity on
optimal deployment patterns to achieve fourconnectivity.The region
size is 1;000 m1;000 m.The average r
s
over all directions is 30 mand
the standard deviation is 2,5,10,and 20,respectively.r
c
varies from10
to 54 m.
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Misalignment occurs when a machine drops sensors row
by row or column by column but with some measurement
error from the first sensor.This misalignment error can be
horizontal,as illustrated in Figs.16a and 16b,or vertical,as
illustrated in Fig.16c.Denote the horizontal misalignment
distance by"
h
and the vertical misalignment distance by"
v
.
Let d
1
and d
2
be decided in (2) and (3),respectively.We note
that a left misalignment"
h
is equivalent to a right
misalignment d
1
"
h
in the sense that they lead to Voronoi
polygons of the same shape.Consider two left misalign
ments"
h
> 0 and"
0
h
> 0.They lead to the same Voronoi
polygons as long as ð"
h
mod d
1
Þ ¼ ð"
0
h
mod d
1
Þ.Without loss
of generality,we consider 0 <"
h
< d
1
in our following
analysis.From Definition 1,a necessary and sufficient
condition to achieve full coverage is that each vertex of the
Voronoi polygon should be covered.Fig.17 shows the
corresponding Voronoi polygons for misaligned sensors
shown in Fig.16.In Figs.17a and 17b,the distance from
each Voronoi polygon vertex to a
0
is the same.In Fig.17c,
the vertex with the maximumdistance to a
0
is e.In general,
the distance of a
0
e,denoted by
a
0
e,can be obtained by
solving the following (12):
arccos
a
0
b
2
a
0
e
þarccos
a
0
c
2
a
0
e
¼ ﬀba
0
c:ð12Þ
In this equation,ﬀba
0
c can be obtained by
cos ﬀba
0
c ¼
a
0
b
2
þ
a
0
c
2
d
2
1
=
2
a
0
b
a
0
c
:
For horizontal misalignment,
a
0
b ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðd
1
=2 "
h
Þ
2
þðd
2
=2Þ
2
q
and
a
0
c ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðd
1
=2 þ"
h
Þ
2
þðd
2
=2Þ
2
q
;for vertical misalignment,
a
0
b ¼
a
0
c ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðd
1
=2Þ
2
þðd
2
=2 þ"
v
Þ
2
q
:
We obtain
a
0
e as follows:
a
0
e ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð
a
0
b=2Þ
2
þtan
2
ð
a
0
c=2cos
a
0
b=2Þ
2
q
:ð13Þ
To ensure coverage,d
1
and d
2
can be adjusted such that
a
0
e r
s
is satisfied.We can use the same approach to
overcome compound errors from horizontal and vertical
misalignments.That is,we first construct Voronoi polygons,
and then use the maximum distance between their vertices
and the misaligned sensors to adjust the deployment
pattern.The calculation is straightforward,and thus,
omitted in this paper.The criteria to adjust the deployment
pattern to ensure connectivity are relatively simple.We can
calculate distances from the sensor at a
0
to its six nearest
neighbors,and then set a threshold for adjusting d
1
and d
2
such that at least four of them are within the range of r
c
.
While misalignment leads to correlated deployment
errors,randomerror refers to independent errors associated
with individual sensors.Denote the maximumrandomerror
by"
r
.To ensure the coverage and connectivity,we can use
r
0
s
¼ r
s
"
r
and r
0
c
¼ r
c
"
r
to decide the values of d
1
and d
2
.
6.4 Impacts of Geographical Constraints
In practice,the sensor deployment field is always bounded
and it may also have various corners and obstacles.Thus,
exploring optimal deployment patterns in a field with
particular geographical constraints is important.Wang
et al.[26] propose several deployment algorithms to ensure
coverage and oneconnectivity in fields with boundaries,
corners,and obstacles.Zhou and Chakrabarty [31] propose a
deployment strategy based on virtual force in a field with
obstacles.Their work is valuable.However,the optimality of
the deployments generated by these algorithms is unknown.
Apparently,optimality is heavily affected by specific
geographical constraints.Numerous scenarios can occur if
different boundary shapes,corners,and obstacles are
considered.Each scenario has a specific form of optimal
deployment pattern.One pattern that is optimal for a certain
scenario cannot be applied easily to another scenario.
Yet determining how much boundaries affect optimal
deployment in terms of the required number of nodes is of
interest.In the following,we investigate their impact.
We consider a bounded squareshaped deployment
region.We first consider how to efficiently cover the
upper boundary.To achieve this,we take the distance
BAI ET AL.:OPTIMAL PATTERNS FOR FOURCONNECTIVITY AND FULL COVERAGE IN WIRELESS SENSOR NETWORKS
443
Fig.15.The probability for a sensor to establish links with PRR 0:95 to
four specific neighbors for different transmission power P
t
and distance
v.r
s
¼ 30 m.
Fig.16.Denote the distance of horizontal misalignment by"
h
and the
distance of vertical misalignment by"
v
.The “should be” locations of
sensors are illustrated by stars,denoted by a.The locations of the
corresponding misaligned sensors are denoted by a
0
.The dashedlined
polygons are the Voronoi polygons for the misaligned sensors.Let d
1
and d
2
be decided in (2) and (3),respectively.(a) 0 <"
h
d
1
=2.
(b) d
1
=2 <"
h
< d
1
.(c) 0 <"
h
< d
2
=2.
Fig.17.The corresponding Voronoi polygons for the misaligned sensors
shown in Fig.16.In (a) and (b),the distance fromeach Voronoi polygon
vertex to a
0
is the same.In (c),the vertex with the maximumdistance to
a
0
is e.
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between the nodes to the boundary d
v
as d
v
¼ d
2
=2 r
c
¼
r
s
cosð’=2Þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2ð1 þcos ’Þ
p
r
c
such that the upper boundary
exactly passes through the sensing disks’ upper intersec
tion points,as shown in Fig.18a.Next we determine the
relative position between corner A and the leftmost sensor
at the top row.To cover A most efficiently,the sensed area
outside the boundaries should be minimal.Hence,A can
be determined as the left intersection point of the upper
boundary and the leftmost sensor’s sensing disk.Thus,the
distance between the sensor to the left boundary d
h
is d
h
¼
d
1
=2 ¼ r
s
cosð’=2Þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2ð1 cos ’Þ
p
.We then cover the re
maining area by using the proper optimal pattern decided
by the relationship between r
c
and r
s
.For the right and
bottom boundaries,coverage may be more inefficient.
Fig.19 shows the number of sensors needed when
boundaries are considered and not considered,respectively.
The side length of the square region changes from 200 to
2,000 m.We assume for each sensor that the communication
range r
c
is 1.5 times the sensing range r
s
,which is 30 m.In
Fig.19,if boundaries are not considered,the number of
sensors is obtained by dividing the area of deployment
region by d
1
d
2
=2,i.e.,by the coverage contribution of each
individual sensor in the Diamond pattern.When bound
aries are considered,we first carefully cover the upper left
corner of the region as we described before,and then cover
the remaining area using the Diamond pattern.
We observe from Fig.19 that more sensors are needed
when boundaries are considered because each sensor
deployed at the boundary contributes coverage less than
d
1
d
2
=2 within the region.This decreases the overall average
coverage contribution of each sensor,and hence,more
sensors are needed.The difference increases with area since
more sensors must be deployed at boundaries.However,as
the figure shows,this difference increases much more
slowly than the total number of sensors does.The
percentage difference becomes negligible as the target
deployment area increases in size.
We adopt a simplified model in this paper to achieve
sufficient generality.Our results can act as theoretical
references that guide realworld deployments in order to
avoid ad hoc deployments,especially when the deployment
area is large.
6.5 Impacts of Sensor Nodes Heterogeneity
Sensor nodes may not be homogenous.It may also happen
that there are certain gateways (a multitiered sensor
network structure),where the gateway routes data between
sensors and the base station.In such cases,optimality must
be provided to both sensors and gateways.Our proposed
optimal patterns are still valuable.A simple example is as
follows:Consider the sensortogateway communication
range as r
s
and the gatewaytogateway communication
range as r
c
.Ensuring that the entire network is full covered
(with coverage range r
s
) with fourconnectivity (with
communication range r
c
) means that each sensor in the
network can communicate with at least one gateway,while
each gateway has fourconnectivity to other gateways.
There may be many more specific scenarios of hetero
geneous sensor nodes.Exploring optimal patterns for all
these scenarios is very hard,if not impossible.
7 C
ONCLUSIONS
This work is the first to study optimal deployment patterns
for more than twoconnectivity and full coverage simulta
neously.We proposed a Diamond pattern,which could be
viewed as a series of evolving patterns.When r
c
=r
s
ﬃﬃﬃ
3
p
,
the Diamond pattern coincides with the wellknown
triangle lattice pattern;when r
c
=r
s
ﬃﬃﬃ
2
p
,it degenerates to
the Square pattern.We proved the Diamond pattern to be
asymptotically optimal when r
c
=r
s
>
ﬃﬃﬃ
2
p
.
Revealing optimal patterns in planned deployment of
wireless sensor networks is a hardproblem.The answer may
be different depending on whether or not we allowmultiple
sensors to be mounted at one location.To demonstrate this,
we described a Doublestrip pattern and showed it to need
fewer sensors than other patterns when r
c
=r
s
< 16=17.
We wish that our discoveries and proof techniques can
shedlight onfurther exploringoptimal deployment patterns.
A
PPENDIX
Proof of Lemma 1
Proof.The proof technique here is inspired by [16].Let V
denote the number of vertices and E denote the number
of edges in the tessellation .Let C denote the number of
corners,which is a constant for a fixed region.Since each
vertex except for those at corners is on at least three
edges and each edge is on two vertices,we have
3V C 2E:ð14Þ
444 IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL.9,NO.3,MARCH 2010
Fig.18.We consider a bounded squareshaped deployment region as a
typical example.Corner A is the upper left corner of the region and
corner B is the lower right corner.
Fig.19.Sensor nodes required as the side length of the square region
changes from 200 to 2,000 m when boundaries are considered and not
considered,respectively.For each sensor,r
c
¼ 1:5r
s
and r
s
¼ 30 m.
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We also have the Euler relation for :
V E þF ¼ 1:
Substituting (14) into the Euler relation for ,we obtain
E 3F 3 þC:
Nowlet e
i
ði ¼ 1;2;...;FÞ denote the number of edges on
the ith polygon.Let B denote the number of boundary
edges,which are on only one polygon.Since all other
edges are each on two polygons,we have
X
F
i¼1
e
i
X
F
i¼1
e
i
þB ¼ 2E 6F 6 þ2C:ð15Þ
Dividing F into (15) yields
1
F
X
F
i¼1
e
i
6
6
F
þ
2C
F
;
the right side of which equals 6 as F approaches 1.
Note that when F is finite,ð1=FÞ
P
F
i¼1
e
i
6 may
not hold;it depends on C.t
u
Proof of Lemma 2
Proof.We prove this Lemma by carrying out transforma
tions from any given G
d
to a G
r
2 G
r
.
Since each polygon in G
d
is a Voronoi polygon
generated by a sensor,each edge of them must reside
on one common chord between two sensing disks as
shown in Fig.1.This chord is called an edge chord.The
transformation is carried out on these edge chords such
that the polygons are changed to desired shapes.
Since we are looking for G
r
2 G
r
,transformation is
only allowed if the following three constraints are all
satisfied.First,the total area of the polygons in G
r
will be
larger than or equal to the total area of G
d
;second,the
average number of edges of polygons in G
r
is not larger
than 6;third,each polygon in G
r
has at least four edge
chords that are connection chords.The above three
constraints together guarantee that G
r
obtained after
transformation be in the set G
r
.
Considering one ksided polygon in G
d
,if 2=k ,
where is the standard angle as defined in Definition 8,
we transform this polygon into a ksided regular
connection polygon as defined in Definition 9 by letting
each edge be overlapped totally with its edge chord and
of the same length.Fig.20 illustrates an example.
After transformation,the area will not decrease since
the regular polygon has the maximum area when k is
given,which can be proved using Lagrangian multipliers
[13].At the same time,fourconnectivity will not be
violated.Since the number of edges is not changed,the
average edge number of polygons will not change.
Now we consider the case where 2=k < .First,if
among k edge chords there are more than four connec
tion chords,we randomly delete some of themto let only
four be left,and add one short chord when necessary.
Fig.21 illustrates this transformation.Then,if among the
four connection chords there are some long chords,we
change theminto standard chords.The transformation is
made by fixing one end point of these long chords and
rotating them toward outside until they become stan
dard chords.Fig.22 illustrates this step.Next,we shift
standard chords along the circumference until not any
two standard chords intersect each other within the
sensing disk.Then use k 4 short chords together with
four standard chords to construct a polygon with vertices
all on the circumference.The purpose of this step is to
make the overlapped area among any three sensing disks
to be as small as possible.This transformation is always
feasible since r
c
=r
s
>
ﬃﬃﬃ
2
p
.Fig.23 illustrates this transfor
mation.Finally,we transformpolygons into semiregular
connection polygon as defined in Definition 10 by
shifting standard chords along the circumference and
letting short chords equally share the remaining arc.
BAI ET AL.:OPTIMAL PATTERNS FOR FOURCONNECTIVITY AND FULL COVERAGE IN WIRELESS SENSOR NETWORKS
445
Fig.20.The large circle denotes the sensing disk.The solid lines denote
the connection chords.The dashed lines denote the short chords.
(a) The polygon before transformation.(b) Change the original polygon
to a regular polygon with the same number of edges.
Fig.21.The large circle denotes the sensing disk.The solid lines denote
the connection chords.The dashed lines denote the short chords.
(a) The polygon in G
d
before transformation.(b) If we remove the
connection chord cd,no short chord needed to add.(c) If we remove the
connection chord ab,a short chord a
0
b
0
is needed to connect the “open
side” that results from the removing in the circumference.
Fig.22.The large circle denotes the sensing disk.The solid lines denote
the connection chords.The dashed lines denote the short chords.
(a) The polygon before transformation.(b) Rotate long chord ab toward
outside until it becomes a standard chord ab
0
.We let short chord meet ab
at the circumference when necessary.
Fig.23.The large circle denotes the sensing disk.The solid lines denote
the connection chords.The dashed lines denote the short chords.
(a) The polygon before transformation.(b) Shift standard chords such
that there are no standard chords intersecting each other within the
sensing disk.Connect their ends with three short chords.
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Since r
c
=r
s
>
ﬃﬃﬃ
2
p
,the angle made at the center of the
sensing disk by the remaining arc is always larger than 0.
Fig.24 illustrates this final transformation.
Now we denote the set of polygons after transforma
tion by G
r
.G
r
consists of only reference polygons and
G
r
¼ G
d
.
As we can see from the transformation procedure,
transformations do not decrease the total area of
polygons.Also they do not increase the edge number
since only deleting chords are allowed.They further do
not violate the condition that at least four edge chords of
each polygon are connection chords.Hence,G
r
2 G
r
.
Note that the Voronoi polygon generated by a
noninterior node has at least one edge that is con
tributed by boundaries.The above transformation
cannot always success since the position and size of
such an edge are fixed.t
u
Proof of Lemma 3
Let M
k
(k 5) denote the area of a polygon in G
r
,which is
the set of reference polygons after the transformations in
Lemma 2.We have
M
k
¼ 2r
2
s
sin’ þ
ðk 4Þr
2
s
2
sin
2 4’
k 4
;ð16Þ
where ’ ¼ maxð2=k;Þ.Recall that is the standard angle
as defined in Definition 8, ¼ 2arccosðr
c
=2r
s
Þ.
To prove Lemma 3,we need to first prove the
following lemma:
Lemma 4.For k 5,
0 < M
kþ1
M
k
< M
k
M
k1
:ð17Þ
Proof.It can be proved by extensively using Taylor’s
expansion for sinx.
Let fðkÞ is defined as fðkÞ ¼ M
kþ1
M
k
.The proof
can be divided into the following four cases:
Case 1.When ’ 2=ðk þ1Þ,M
kþ1
,M
k
,and M
k1
are
the areas of the regular connection polygons.Therefore,
at this time,the claim that “fðkÞ > 0 and fðkÞ is
decreasing as k increases” follows directly from [16,
Lemma 2] (Lemma 2 in [16] states that the area of regular
polygons will increase as the number of edges increases.
But the amount of area increment will decrease as the
number of edges increases).
Case 2.When ’ 2=ðk 1Þ,M
kþ1
,M
k
,and M
k1
are
the areas of the semiregular connection polygons.This
case essentially has no difference from case 1,since the
polygons are of the same type.Hence,exactly the same
technique used in proof of [16,Lemma 2] can show that
the claim holds at this time.
The first inequality in (17) can be easily proved using
Taylor’s expansion for sinx.We now prove the second
inequality.we have
fðkÞ ¼
ðk 3Þr
2
s
2
sin
2 4’
k 3
ðk 4Þr
2
s
2
sin
2 4’
k 4
:
Taking derivatives of both sides of the above equation,
we get
dfðkÞ
dk
¼
r
2
s
2
sin
2 4’
k 3
2 4’
k 3
cos
2 4’
k 3
r
2
s
2
sin
2 4’
k 4
2 4’
k 4
cos
2 4’
k 4
:
Since sinx xcos x is an increasing function of x in
ð0;Þ and ð2 4’Þ=ðk 3Þ;ð2 4’Þ=ðk 4Þ 2 ð0;Þ
for k 5,dfðkÞ=dk < 0 for k 5.Hence,fðkÞ is a
decreasing function,and thus,(17) holds when
2=ðk 1Þ ’.
Case 3.When 2=ðk þ1Þ < ’ 2=k,for k 4,let
M
r
k
¼
r
2
s
2
ksin
2
k
and
M
s
k
¼ 2r
2
s
sin’ þ
r
2
s
ðk 4Þ
2
sin
2 4’
k 4
:
Then,M
kþ1
¼ M
s
kþ1
,M
k
¼ M
r
k
,and M
k1
¼ M
r
k1
.To
show at this time fðkÞ > 0,we have
M
s
kþ1
M
r
k
¼
ðk 3Þr
2
s
2
sin
2 4’
k 3
þr
2
s
sin’
kr
2
s
2
sin
2
k
;
which is greater than 0 for 2=ðk þ1Þ < ’ 2=k.
For the claim that fðkÞ is decreasing as k increases in
this case,we need to prove M
s
kþ1
M
r
k
< M
r
k
M
r
k1
,
which follows fromthe conclusion for the case 2 and the
fact that M
s
kþ1
< M
r
kþ1
.The latter inequality holds since
M
s
kþ1
is the maximum area possible of any ðk þ1Þsided
polygon inscribed in a sensing disk.
Case 4.When 2=k < ’ < 2=ðk 1Þ,for the claim
that fðkÞ > 0,we notice that 0 < M
s
kþ1
M
s
k
is obvious.
For the claimthat fðkÞ is decreasing as k increases in this
case,we need to prove M
s
kþ1
M
s
k
< M
s
k
M
r
k1
,which
follows by noting that
M
s
k
M
r
k1
M
s
kþ1
M
s
k
¼ ðk 4Þr
2
s
sin
2 4’
k 4
r
2
s
2
ðk 5Þ sin
2
k 5
r
2
s
2
ðk 3Þ sin
2 4’
k 3
is greater than 0 when 2=k < ’ < 2=ðk 1Þ by using
Taylor’s expansion for sinx.
The four cases together prove that fðkÞ > 0 and fðkÞ is
decreasing as k increases for all values of ’,where fðkÞ is
defined as fðkÞ ¼ M
k
M
k1
,where M
k
is expressed in
(16) and k 5.t
u
446 IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL.9,NO.3,MARCH 2010
Fig.24.The large circle denotes the sensing disk.The solid lines denote
the connection chords.The dashed lines denote the short chords.
(a) The polygon before transformation.(b) Shift standard chords such
that all standard chords are together.Let three short chords equally
share the left arc.
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Now we are ready to prove Lemma 3.
Proof.Let n
i
ði ¼ 3;4;...;mÞ denote the number of polygons
in G
r
with i edges,so we have
P
m
i¼3
¼ jG
r
j.
Since G
r
2 G
r
,the average number of sides of the
polygons of G
r
is smaller than 6,i.e.,
P
m
i¼3
in
i
6
P
m
i¼3
n
i
,which can be rewritten as
P
m
i¼7
ði 6Þn
i
P
5
i¼3
ð6 iÞn
i
.
By Lemma 4 that states fðkÞ is decreasing as k
increases,we can strengthen the above as
X
m
i¼7
ði 6ÞðM
7
M
6
Þn
i
X
5
i¼3
ð6 iÞðM
6
M
5
Þn
i
:ð18Þ
The fact that fðkÞ is decreasing as k increases implies
that the interval from M
q
to M
p
consists of ðp qÞ
subintervals among which the shortest is M
p
M
p1
and
the longest is M
qþ1
M
q
.
We then have for p > q 5,
ðp qÞðM
p
M
p1
Þ ðM
p
M
q
Þ;ð19Þ
and
ðM
p
M
q
Þ ðp qÞðM
qþ1
M
q
Þ:ð20Þ
Hence,by (19),we have
ð6 iÞðM
6
M
5
Þ ðM
6
M
i
Þ;i < 6;ð21Þ
and by (20),
M
i
M
6
ði 6ÞðM
7
M
6
Þ;i > 6:ð22Þ
Then from (18),(21),and (22),
X
m
i¼3
M
i
n
i
M
6
X
m
i¼3
n
i
¼ jG
r
jM
6
;
where
P
m
i¼3
M
i
n
i
R.
We then have R jG
r
jM
6
.This concludes our
proof.t
u
A
CKNOWLEDGMENTS
An earlier version of this work was published in the 27th
IEEE Conference on Computer Communications (INFO
COM) 2008.This work was supported in part by the US
National Science Foundation (NSF) CAREER Award CCF
0546668;the Army Research Office (ARO) under grant
No.AMSRDACCR 50521CI;NSFC grant No.10971185;
and grants from the Research Grants Council of the Hong
Kong SAR,China,Nos.CityU114908 and CityU114609 and
CityU Applied R&D Funding (ARD(Ctr)) No.9681001.
Any opinions,findings,conclusions,and recommendations
in this paper are those of the authors and do not necessarily
reflect the views of the funding agencies.Ziqiu Yun was the
corresponding author for this paper.
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Xiaole Bai received the BS degree from South
east University,China,in 1999,and the MS
degree from the Helsinki University of Technol
ogy,Finland,in 2003.He is currently working
toward the PhD degree in the CSE Department
at the Ohio State University.His research
interests include networking,security,distribu
ted computing,and algorithms.
Ziqiu Yun received the PhD degree from the
Helsinki University,Finland,in 1990.He is a
professor in the Department of Mathematics at
Suzhou University,China.He was the vice chief
director of the Department of Mathematics at
Suzhou University from 1995 to 1998.From
2002 to 2006,he was invited by the Helsinki
University and the Finish National Academy of
Science.He is well recognized in the field of
topology for having solved several longlasting
open problems.His work on the metrizability on Nagata space is well
accepted as one of the most notable result in the field of topology in the
recent years.
Dong Xuan received the BS and MS degrees in
electronic engineering from Shanghai Jiao Tong
University (SJTU),China,in 1990 and 1993,
respectively,and the PhD degree in computer
engineering fromTexas A&MUniversity in 2001.
He is an associate professor in the Department
of Computer Science and Engineering at the
Ohio State University.He was the faculty of
Electronic Engineering at SJTU from 1993 to
1997.In 1997,he worked as a visiting research
scholar in the Department of Computer Science at the City University of
Hong Kong.From 1998 to 2001,he was a research assistant/associate
in RealTime Systems Group at the Department of Computer Science,
Texas A&M University.He is a recipient of the US National Science
Foundation (NSF) CAREER award.His research interests include real
time computing and communications,network security,sensor net
works,and distributed systems.
Ten H.Lai is a professor of computer science
and engineering at the Ohio State University.A
pioneer of Zen Networking,he is interested in
the art of applying Zen to teaching and research
of protocol design.He served as a programchair
of ICPP 1998,general chair of ICPP 2000,
program cochair of ICDCS 2004,and recently,
general chair of ICDCS 2005.He is/was an
editor of the IEEE Transactions on Parallel and
Distributed Systems,ACM/Springer Wireless
Networks,Academia Sinica’s Journal of Information Science and
Engineering,International Journal of Sensor Networks,and International
Journal of Ad Hoc and Ubiquitous Computing.
Weijia Jia received the BSc and MSc degrees
from the Central South University,China,in
1982 and 1984,respectively,and the master of
applied science and PhD degrees from the
Polytechnic Faculty of Mons,Belgium,in 1992
and 1993,respectively,all in computer science.
He is a professor in the Department of Computer
Science and the director of Future Networking
Center,ShenZhen Research Institute at the City
University of Hong Kong (CityU).He joined the
German National Research Center for Information Science (GMD) in
Bonn (St.Augustine) from 1993 to 1995 as a research fellow.In 1995,
he joined the Department of Computer Science,CityU,as an assistant
professor.His research interests include next generation wireless
communication,protocols and heterogeneous networks,distributed
systems,and multicast and anycast QoS routing protocols.
.For more information on this or any other computing topic,
please visit our Digital Library at www.computer.org/publications/dlib.
448 IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL.9,NO.3,MARCH 2010
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