Optimal Patterns for Four-Connectivity 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 four-connectivity 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 well-known 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 four-connectivity and full coverage.We also

discover another new deployment pattern called the Double-strip 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 four-connectivity 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,four-connectivity,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 four-con-

nectivity and full coverage.Four-connected wireless sensor

networks are popular in practice.Several research projects

(e.g.,data segmentation [21],routing [12],and storage [10])

are based on four-connected wireless sensor networks.In

practice,people often deploy wireless sensors in a square

grid pattern to achieve four-connectivity [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 strip-based 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 strip-based 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 one-connectivity.If

higher degree connectivity is desired,a variant of the strip-

based pattern that achieves two-connectivity 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 43210-1277.E-mail:{baixia,xuan,lai}@cse.ohio-state.edu.

.Z.Yun is with the School of Mathematical Science,Suzhou University,

Suzhou,215006,P.R.China.E-mail: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.

E-mail: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 e-mail to:

tmc@computer.org,and reference IEEECS Log Number TMC-2008-02-0041.

Digital Object Identifier no.10.1109/TMC.2009.143.

1536-1233/10/$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 four-connectivity 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 two-connectivity (and full coverage) for which the

aforementioned strip-based 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 well-known

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 Double-strip 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 four-connectivity 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 four-connectivity 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 (four-connected sensor network).A sensor

network N is said to be four-connected 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 four-connectivity 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 well-known 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 FOUR-CONNECTIVITY 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

four-connectivity 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 four-connectivity 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 six-sided 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 six-sided 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 six-sided 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 six-sided 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 four-connectivity 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 four-connectivity and full coverage is

to use a two-connectivity,full coverage pattern and deploy

two sensors at each location.

1

We describe such an

approach in the next section.

4.2 The Double-Strip Pattern

The Double-strip pattern is constructed by deploying two

sensors at each location of the Strip-based pattern,which

was introduced and proved to be asymptotically optimal

for two-connectivity and full coverage in [4].As illustrated

in Fig.7,the Double-strip or Strip-based 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 Double-strip

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 FOUR-CONNECTIVITY AND FULL COVERAGE IN WIRELESS SENSOR NETWORKS

439

Fig.4.The square pattern to achieve four-connectivity 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 double-strip pattern.It is the same as the

“strip-based” 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 Double-strip 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 Double-strip

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 Double-strip

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 Double-strip 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 Double-strip 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 Double-strip 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 Double-strip 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 hardware-based approaches,e.g.,using

different bandwidths,and software-based approaches,e.g.,

using scheduling and synchronization protocols.We also

notice that the long-communication path problem can exist

in the Double-strip pattern.However,the long-communica-

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.,five-connectivity.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 Double-strip 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 four-connectivity

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 four-connectivity and

440 IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL.9,NO.3,MARCH 2010

Fig.8.Intuitive interpretation of why the double-strip pattern can

outperform the square pattern.

Fig.9.The local view of the double-strip 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 Double-strip 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 Double-strip pattern.The

regular triangle lattice pattern requires 86.3 percent

more sensors.However,we recall that the Double-

strip pattern has the drawback of long-communica-

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

sensor-dependent 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 FOUR-CONNECTIVITY 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 four-connectivity and full

coverage.r

c

=r

s

varies from 1.3 to 1.8.

Fig.12.Numbers of nodes needed to achieve four-connectivity and full

coverage.r

c

=r

s

varies from 0.9 to 1.43.

Fig.13.Numbers of nodes needed to achieve four-connectivity 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 disk-shaped.

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 four-connectivity 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 four-connectivity.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 one-connectivity 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 square-shaped 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 FOUR-CONNECTIVITY 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 dashed-lined

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 real-world 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 sensor-to-gateway communication

range as r

s

and the gateway-to-gateway communication

range as r

c

.Ensuring that the entire network is full covered

(with coverage range r

s

) with four-connectivity (with

communication range r

c

) means that each sensor in the

network can communicate with at least one gateway,while

each gateway has four-connectivity 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 two-connectivity 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 well-known

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 Double-strip 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 square-shaped 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 k-sided polygon in G

d

,if 2=k ,

where is the standard angle as defined in Definition 8,

we transform this polygon into a k-sided 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,four-connectivity 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 FOUR-CONNECTIVITY 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 þ1Þ-sided

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.AMSRD-ACC-R 50521-CI;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 long-lasting

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 Real-Time 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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