Efficient Deployment of Connected Sensing Devices Using Packing Algorithms

bumpedappleΚινητά – Ασύρματες Τεχνολογίες

21 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

105 εμφανίσεις


1


Efficient Deployment

o
f

Connected Sensing

D
evices
Using

Packing Algorithms


Abstract


In this paper,
we consider the deployment
of

either homogenous or heterogeneous
sensing

devices

on one
of two fields which are static and differentiated fields.
The
static field

components

are assumed to have
same monitoring requirements while
differentiated field components

may
involve

different monitoring
requirements.
The
deployment
objective
s

are

to maximize the coverage of the monitored field and use the
best of the
sensing
devices characteristics as well as developing

a connected deployment scheme
. We
propose different

solution
algorithms to deal with different problem settings.
The
intended
a
lgorithms are
inspired
from

the packing theories
in
computational geometry
. The algorithms
efficiency
is examined using
d
ifferent sets of e
xperiments.


1.
Introduction


With the advances of sensing technolog
ies

and requirements of the
monitored
fields
, sensor
deployment
becomes

very challenging
.

Differen
t vendors have developed sensing devices
with different
capabilities.
These devices
may
differ in
their operational parameters such as detection

reliabilities, sensing
ranges, and communication ranges.

In addition
, the deployment field may
involve

different monitoring
requirements (some areas may require more coverage than the others
) which

add to the complexity of the

deployment process.

Our work
, in this paper,
focus

on
the
deployment
o
f
homogenous
and/or heterogeneous
sensing
devices

on
static

and/or differentiated monitored

field
s
. A static fie
l
d
is assumed to have t
he
same
monitor
ing requirements over all
of
its areas while
a
differentiated
field
is associated with areas that may
demand

different

monitoring requirements
.


Packing
-
based algorithm
s

are

proposed to provide

efficient
deployment schemes that maximize the coverage
, enhance the security of the monitored field by using

the

most

reliable sensors on the most important areas,
and guarantee s
ensors connectivity.

C
onsider
ing
coverage and connectivity at the same time

during the deployment process could lead
to efficient usage of the sensor networks
(
Zhang, 2004
).
The
c
overage is considered in different contexts in
the literature.

For example, Cardei et al. (Cardei, 2004)

categorized the coverage in static wireless sensor
networks into area, point, and barrier coverage. Gage in
(Gage, 1992)

classified the coverage into blanket,

2

barrier, and sweep coverage.
In addition
, Poduri et al
. in
(Poduri, 2004) measure

the sensor network quality
of service by finding the uncovered or low observed areas and highly observed areas in the monitored field.
I
n this paper,
our

definition of coverage is slightly different. Coverage refers to monitorin
g the
most

important areas in the deployment field by the
most

reliable sensors. For example, in border security
applications, covering the mountain areas might not be as important as covering the flat areas that people or
vehicles can easily pass through.

Thus,
our definition involves the usage of
the
most

reliable sensors
to

cover hot

spots in the monitored fi
e
l
d.

Dep
loyment of non
-
connected sensor networks

is studied in many areas such as geometry and
computer science. Early contribution to th
is kind of

networks
returns to Chvatal, 1975

(Chvatal, 1975)
, who
introduced the
Art Gallery
P
roblem

(AGP)
.

Each sensor is usually wired to the control cent
er
via cable.

In
this problem, the goal is to determine the minimum number of observers required to secure an art gallery
with a non
-
uniform geometry. Different versions of this problem have been studied to include mobile guard
and guards wi
th limited visibility (e.g.
(O’Rourke, 1983)
).

However,

in wireless sensor network, the
network operation is

usually

based on multi
ple
hope
s

communication; therefore, connectivity is a major
concern

for
efficient
operation of the network
. For example, sensors
need to

communicate to
report

their
data to the base
-
station/
sink node(s).

Nevertheless,
deployment of connected wireless sensor networks is

considered in the literature
. In particular, Howard et.

al.
i
n

(
Howard
, 2002)
introduced

an

incremental
deployment
algorithm
in which the new sensors placement are based on the sensed information from the
deployed ones.
However, this algorithm relies on some assumptions such as the usage of homogenous
sensors and static fields.
O
ur work differ in
considering
homogenous/
heterogen
eous sensors in terms of
sensing range, communication range, and reliability as
well as studying
the deployment on
static
/
differentiated

fields.

Similarly, u
nmanned vehicle
(Corke, 2004)

and
a flying robot
(Corke June 2004)

have
been

used to deploy nodes incrementally
.

T
he unmanned vehicle
/
robot

sensor is used to help in
collecting information about the deployed
sensors
. Although these techniques are efficient; but they are
costly compare to the cost of the tiny deployed sensors. In a
ddition, having a powerful sensor or unmanned
vehicle might not be available for each application.
Moreover,
(Kar, 2003) (Zou, 2003)
(Bai, 2006)

((XING, 2005) (Zhang, 2004)
studied the deployment of sensors
for the purpose of
coverage and
connectivity
. T
h
e

goal is different from our work in which the
authors consider
only cases where there is
a

3

binary
relationship between sensors

communication and sensing ranges. Our

algo
rithms accommodate both
range
-
constrained

and

range
-
free

sensors
.


The
rest of this paper is
organized as follows.
Assumptions and definitions used throughout this
paper
are

introduced
in section
two
.
The deployment problem and motivations are described in section
three
. Section
four

presents the packing
-
based algorithms
.
T
he

performance o
f these algorithms is
illustrated
in
s
ection
five through different sets of experiments
. Finally, the paper concludes in section
six
.


2.
Assumptions

and Definitions



In this section, the

assumptions and the
key definitions
used
throughout this paper

are described
.

Assumption 2.1.
[Disk
-
based sensors]

We assume that sensor
s

sensing and communication
ranges are represented by a disk
-
based model. A senso
r s

is associated with
a sensing range r
s
and

a

communication range r
c

. Any point in the monitored field that falls
in

a disk of radius r
s

is assumed to be
covered by the sensor s.
In addition, t
he sensing disk of sensor s centered at location u is denoted by

D
s
(u).
Similarly, the communication disk
of a sensor

s
c
e
ntered

at

location v is denoted by D
s
(v).

Assumption 2.2.
[Heterogeneous vs. homogenous sensors]
We assume that heterogeneous
sensors may differ in their sensing ranges, communication ranges, and/or reliabilit
ies
. On the other hand,
homogenous sensors have the se
nsing and communication ranges the same, as are their reliabilities.

Assumption

2.
3
.
[Static
-
field]

A static field is a
deployment
field with even monitoring
requirements. These monitoring requirements are represented by a fixed weight W.

Assumption
2.
4
.
[
Differentiated
-
field]
A

differentiated field

is a
deployment field

that
is divided
into a grid of zones
A
and each zone is assigned a weight

W
z
based on its monitoring
requirements

level
.



Definition 2.
1
.

[Sensors coverage
-
contribution]
S
ensors contribute to the coverage of the
monitoring field by the area of its sensing disk
s
.
However, if two
or more
sensors compete (overlap) fully
or partially on the same zone,

only

the

overlapped area of the most

reliable sensor will be considered a
s
a

cont
ribution to the overall coverage
.
We realize that multiple
-
coverage (k
-
coverage)

(Zhou, 2004)

could be
required in some applications. However, this
problem
is
beyond
the

scope of this paper.

Definition 2.2.

[Weighted
-
coverage]

The coverage, in this context, is measured by the sum over
the zones’ weight and the reliability of the sensor that monitors this zone. If more than
one sensor is
sharing the zones

monitoring

without overlapping
, the coverage contribution by each sensor
is computed

4

and multiplied by the zones weight as well as the reliability of each sensor.
Otherwise,
only the

contribution of the most reliable sensor is considered
.


Definition 2.
3
.

[
d
uv
]

We assume that the distance between any
two sensors s
i

and s
j

centered at u
and v

locations
, respectively, is measured by the Euclidian distance d
u
v
.


Definition 2.
4
.

[Connected sensors]
Two sensors

s
i

and s
j


with communication ranges
cj
ci
r
and
r


centered at u and v locations
, respectively
,
are
called
connected
if and only if
:
)
r
r
(
Min
)
v
,
u
(
d
cj
,
ci



Definition 2.
5
.

[Potential
-
placement

point
constraints
]

Given a current
deployment state, the
next sensor s to be deployed has to touch two other items such as two deployed sensors (circles), one of the
deployment
zones’

border and a deployed sensor, or two borders of the deployment
zone
.


This may lead to
more than one potent
ial placement point.



Figure
1(a)
shows an example on the potential placement points in a field that has three
deployed
sensors
s1, s2
, and
s3
.
The potential placement points of
s4

(
dashed line circle
s
)

are limited to two
points,

p1

and
p2
,
assuming

that it can not communicate to
s1

and
s2

while it is able to connect to
s3
.



(a)

(b)

Figure 1 : (a) Example on the potential placement points, (b) example on the potential points evaluation


Definition 2.
6
.
[Potential
-
placement

evaluation
]


To
select the final placement point for a sensor
s
,
each point is
evaluated

and

the
point

with the minimum value

is selected. First, points that are not within
the communication range of the current sensor
and the deployed ones are eliminated
.
If no point
s left, the
current sensor has
to intersect with one or more of the deployed sensors.
Conversely
, if more than one
potential point
is still

available,
the

distance d
i
j

is computed from each point

P
i

to the untouched sensors
(ci
rcles)

s
j

. After that,
the minimum distance d
min
(P
i
) is assigned to
P
i

; this value

represents the value

of
this point

(how much the point is far from the untouched circles)
.
A point with minimum
value

is selected
to be the final deployment
position
for

the sensor.


5

Figure

1
(b) shows two potential points
P
1

and
P2

for
s6
.
The distances, dished line
s
, between each
point and the untouched sensors


disk
s

are computed. The minimum distance from each point is selected
which is represented by the solid lines in the figure. The point that associated with the shortest distance is
chosen

for the final deployment

position

of the sensor which is, in this case,
P
2
.


3.
Problem Definition
and Motivations


In this section, we define the deployment problem and motivate the importance of it along with the
usage of t
he packing theory
.

A fi
e
l
d
F(A)

is given to be monitored by a set of sensors
S.

T
he field i
s divided into set of zones
A
.

A

is

assigned

to
one

if a
static field is used
; otherwise it
is

set to

greater than one
.
Each zone
z

is assigned a
weight
W
z

which

represents the
level of the
monitoring requirements
(importance)
of this zone.
The
given
set of sensors
S

could be homogenous or heterogeneous. Homogenous sensors, as
mentioned

before
, have
the same sensing and communication ranges as well as the same reliabilities.

On

the contrary,
heterogeneous sensors may vary in their sensing, communicatio
n and/or reliability capabilities.

Based on
the given sensors and the monitored field characteristics, t
he problem is to find the best deployment scheme
that maximizes the coverage of the field, minimizes the overlapp
ed

areas, and guarantees the connectiv
ity

of the deployed sensors
. The coverage is maximized
when the
highest weight zon
es
are observed
by the
most reliable sensors.
The connectivity is guaranteed by deve
loping a connected graph among

the sensors.
Therefore, t
he resulting deployment scheme ens
ures that sensors can communicate to each other via one or
multiple
hops.

Motivations:
Wireless sensor networks have been used in many applications such as border
security
and critical
infrastructure monitoring

(
Tanenbaum, 2006
)

(
Hussain, 2006) (Mehta,
2004)
.
Such
applications
require large number of sensors to satisfy the
ir

monitoring requirements. In addition, different
parts of the monitored
field

may require different types of sensors as well as different monitoring
capabilities. In reality, due to t
he budget limitations and interferences from
the deployed
sensors
and/or

from other sources, deploying a large number of sensors is impractical. Therefore, an
optimized

deployment scheme that uses the best of the sensors to secure the important spots in th
e monitored field is
required. At the same time
,
for sensors to
carry out
their sensing data to the sink and/or to each other, the
deployment scheme has to guarantee the connectivity of the sensors.


6

A naïve solution to the deployment problem
starts by
dep
loy
ing the first sensor
at the center of the
field and deploy other sensors around

it
.

As
illustrated in figure 2 (a)
, t
h
e

solution wasted a considerabl
e
area from the monitored field.
However, u
sing
a simple packing strategy

such as the one
introduced by
Stephenson

in
(
Stephenson, 2005), more sensors can be accommodated

as shown in figure
2

(b
).
This
example clearly explains the similarities between the deployment problem and the packing algorithms.
The
p
acking problem
, more
specifically

the
circle packing problem
, is

a
well defined problem in the field of
computation
geomet
ry

and well

known
as NP
-
hard problem (
(Lenstra, 1979)

(Williams, 1979”) (Collins,
2003) (Mark, 2000) (
O’Rourke
, 1997))
.

However, some polynomial time algorithms are
developed for
efficient packing
.


O
ur
solution
s

to the deployment problem are

inspired
by

the packing algorithm
introduced
by
Huang et. al. in
(Huang, 2005)
.
The authors studied the packing of unequal size circles into a rectangle
without overlapping.

Nevertheless, our
work

differ from their work in
six

folds 1)
i
n addition to the circle

radius (sensor sensing range) ,
we consider two
other

paramete
rs

which are

the communication range and
sensors’ reliability, 2) we consider the packing
i
nto

multiple
zones (rectangular or square shapes)

vers
u
s
one
zone
/rectangle, 3) we
study

the differentiated monitoring requirements which is

not a part
of their work,
4)
o
ur solution
s

ad
a
pt the packing theory for sensor deployment
; nevertheless,

their work is a gener
al
solution to a geometric problem, 5) we simplified the packing idea to fit the requirements of sensor
deployment, so we can
have

an efficient
solution in a very reasonable
time, 6) our
algorithms

minimiz
e

the
overlapp
ed areas

and

guarantee the conn
ectivi
ty
;

nonetheless
.

their work focus

on non
-
overlapped circles.
The details of the packing algorithm
s

are explained in the next section.



(a)


(b)

Figure
2
: Solutions to the deployment problem (a) a naïve algorithm (b) heuristic algorithm





7

4.
Packing
-
based

Algorithm
s


In this section, the
deployment
problem is
categorized into

four categories.

Different packing
algorithms are used to solve these problems.
In the first
category
,
homogenous sensors are given to monitor
a static field. In the second
category
, homogenous set of sensors are assumed to be used on a differentiated
field. The third
category

discusses the deployment of heterogeneous sensors on a static field. Finally,
dep
loyment of heterogeneous sensors on a differentiated field is explored.



4.1. Deployment of Homogenous Sensors on a Static Field


In this problem, a set of homogenous sensors
S

is given to
observe
a field

F(1)

with the same
monitoring requirement
W
.

Sensors
are assumed to have
the same

operational characteristics such as

sensing range
r
s
, the communication range
r
c
, and the detection reliability
R
.
T
he objective is to maximize
the coverage of the monitored field and minimiz
e

the overlapped areas.
F
or

the connectivity
purposes,

s
ensors are deployed sequentially

such that the current sensor has to be connected to
at least
one of the
previously
deployed
sensors
.

The potential sensor to be deployed could be
selected randomly or
b
ased on
the sensors

ident
ifier
.

F
igure
3

shows the details of
the
deployment algorithm in which t
he first sensor is
deployed at one of the

field’s

corners

(
lines 2 and 3
)

such

that the sensor
’s

disk touches
at least
two of the
field
’s

borders. For
a

potential

sensor to be deployed, it has to touch
at least
one
of the deployed sensors
and

one of the borders
, two of the deployed sensors, and/or two of the field’s borders
. A potential
placement points are
identified

while connectivity between the current sensor a
nd th
e deployed ones are
considered (
line 5
)
.
If

no potential point satisfies the connectivity condition, the sensor has to be deployed
at a point that gives mini
mum overlapping

(
lines 6 and 7
)
. Otherwise, p
otential

placement points are
evaluated

and the c
urrent sensor is placed on a point that has the
minimum value

(
lines
9
,
10
,
and
11
)
. This
process continues until there is no sensors available

or the fie
l
d is totally covered

(
line 10
)
. Then, the
contribu
tion of each sensor is computed (
line 1
6
)
.
Since

the
number of potential placement points is

limited
by

the borders of the deployment field and the communication range of the sensing devices, it is
obvious that the algorithm has a bounded polynomial time.

The worst case to f
ind

potential placement
point
s
is
))
m
|
S
(|
m
(
O

operations
, where m is the deployed sensors.

This occurs when

there is no
limit on the communication range of the given sensing devices
. Therefore,
each potential sensor forms a

8

potential placement point with each deployed sensor

and the field’s border
.
Computing the distances
between the untouched circles and the potential p
o
ints requires
))
m
|
S
(|
m
(
O
2


operations. Finding the
point that has the minimum distance requires
)
m
(
O

operations, as is finding the

touched circles with the
largest communicati
on range in case of overlapping

is a must
. Therefore, the estimated worst case
complexity of algorithm 1 is
)
|
S
(|
O
3
.




4.2. Deployment of Homogenous Sensors on a Differentiated Field



A set of homogenous sensors, as described in section 4.1 is given to
observe

a field with
differentiated monitoring requirements. The
field is divided
i
nto
a set of
zones

A

and each zone is assigned
a weight
W
z
.
The relationship between the monitor
ed

field

and the given sensors could be one of three cases
1) only one sensor can fit in a zone 2
) a sensor can cover
more than one

zone,
or 3
) a zone can
accommodate multiple sensors
.

In the first case,
as shown in figure 4,
the deployment method is obvious in
w
hich the
first sensor is deployed on the highest weight
ed

zone. Zones that
guarantee

the connectivity with
the deployed sensor(s) are identified; then, the current sensor is deployed on the

center of the zone

that
maximizes the coverage. If there is no zone
satisf
ying

the connectivity

constraint
, the
current
sensor may
overlap with one of the deployed s
ensors in which its coverage contribution is the
maxim
um
.
The worst
case complexity of this algorithm is
)
|
A
(|
O
2

which affected by identifying the potential zones that
guarantee the connectivity with the deployed sensors.

In the second case,

as

shown in figure 5,
where a sensor’s area can cover more than one zone, a
heuristic approach is used in w
hich sensors are always deployed at the center of the

specified

zones.


The

heuristic
starts by
deploy
ing

the first selected sensor
at the center of

the
zone that gives the highest
coverage
.

In other words, zones are scanned to find the zone that gives the maximum coverage and the
sensor is deployed at its center
. The scanning operation is used due to
the highest weight
ed

zone might be
surrounded by low weight
s’

zones

or vic
e versa
.
This

proc
ess is repeated until no more sensors or zones are
available.

The worst case complexity of this algorithm is also
)
|
A
(|
O
2

which
is
affected by the scanning
operation that looks for best zone that maximizes the coverage.
Advantages and
disad
vantages of this
heuristic

are explored in the experiment results section

(
section 5
)
.


9

The

third case

shown in figure 6 handles

the deployment of multiple sensors in a zone which is
similar to the problem described in section 4.1. However, the algorithm starts the deployment of sensors on
the highest weigh
t

zone. Once the zone is covered, no more sensors can fit in the zone
,
anoth
er zone
i
s
selected for the deployment. Since sensors have the same communication range
r
c
,
the next highest zone
that guarantee the connectivity to the deployed sensors

is selected. Within the selected zone,
the sensor is
deployed at the close
st

corner to

the deployed sensors

with applying the
Potential
-
placement point
constraints

defined in sections 2.
If more than one corner is allowed, a corner surrounded by the highest
weigh
t

zones is selected due to its importance.
As noticed, if overlapping is not re
quired
, this pattern will
be
repeated. However, crossing the border of a zone to cover another zone might require some sensors to
overlap or to be used as a bridge between the
two zones
.

The worst case complexity o
f

this algorithm
is
)
|
S
|
A
(
O
3
. This is due to
the

need
for

find
ing

the next highest zones that maximize th
e coverage and
persuade sensors

connectivity
.



Algorithm 1

1:

Select the potential sensor s
i

to be deployed

2:

IF (i = 0)
then


3:

Deploy the sensor at one of the field corners such that
it touches at least two of its items.

4:

ELSE

5:

Compute the potential placements points such that the distance between the potential point and the
deployed sensors center is equal to the communication range r
c

6:


IF

no potential point satisfies the connectivity condition

7:


Place the sensor at the point that gives minimum overlapping

8:


ELSE

9:


Compute dij between each potential point for the current sensor and the untouched deployed


sensors.

10:


Compute d
min

for each point.

11:


Place the sensor at the point that has the minimum value.

12:


End IF

13:


IF

there is no more sensors and/or or the
field

is totally covered got to 16,
ELSE

go to 1.

14:


End IF

15:

End IF

16:

Compute the coverage
contribution of each sensor

17:

Stop.

Figure 3: Deployment of homogenous sensors on a static field


4.3. Deployment of Heterogeneous Sensors on a Static Field

Given a set of heterogeneous sensors that may differ in their sensing and communication ranges
and/
or reliability; these sensors are used to monitor a static field
F
(1)
. The problem, in this case, is similar
to packing of unequal size circles in a zone. However, the connectivity is one of our major concerns. The

10

deployment algorithm for this problem is
comparable

to the algorithm mentioned in section 4.1

in terms of
its steps and complexity
. Nevertheless, sensors are sorted based on one of their characteristics.

Algorithm 2

Only one sensor can fit in a zone

1:

Select the potential sensor s
i

to be deployed (Randomly or based on sensors’ identifier)

2:

IF (i = 0)

then

3:


Deploy the sensor at one of the highest weight zone

4:

ELSE

5:


Identify the potential zones that guarantee the connectivity with the deployed sensors

6:


IF

there is no potential

zones

7:


Deploy the sensor at the zone that gives the highest coverage contribution with overlapping


with any of the deployed sensors

8:


ELSE

9:


Deploy the sensor at the center of the zone that maximizes the coverage

10:


END IF

11:


IF

no more sensors or the field is totally covered

12:


Compute the overall coverage by summing over the contribution of each sensor

13:


Stop

14:


ELSE

15:


Go to step 1

16:


END IF

17:

END IF

Figure 4:
Deployment of homogenous sensors on a differentia
ted field


only one sensor can fit a zone


Algorithm 3

The sensor covers more than one zone

1:

Select the potential sensor s
i

to be deployed

2:

IF (I = 0)

then

3:


Define MaxCoverage = 0

4:


For z = 1 to |A|

5:


Coverage [z] = the coverage contribution of the sensor s
i

when it is deployed at the center of

zone z

6:


End for

7:


Deploy the sensor virtually at zone z that has the Max (Coverage[1], Coverage[2], .. ,

Coverage[|A|] )

8:

ELSE

9:



Identify the
zones, sensor s
i

is assumed to be at the center of the zone, that satisfy the


connectivity constraints with the deployed sensors.

10:


IF

there is no zones satisfy the connectivity constraint

11:


Deploy the sensor at the zone that gives

minimum overlapping and guarantees the


connectivity with the deployed sensors


the location doe not have to be at the zone’s center.

12:


ELSE

13:


Deploy s
i
at the center of the zone that contributes the maximum.

14:


END IF

15:



IF

no more sensors or the field is totally covered

16:


Compute the overall coverage by summing over the contribution of each sensor

17:


Stop

18:


ELSE

19:


Go to step 1

20:

END IF

Figure 5:
Deployment of homogenous sensors on a differentiated fie
ld


the sensor covers more than one
zone





11

Algorithm 4: A zone accommodates multiple sensors

1:

Select the potential sensor s
i

to be deployed

2:

IF

i is the first sensor

3:


Select the highest weight zone

4:


Apply Algorithm 1 shown in figure 3.

5:

Else

6:


Identify the potential zone that guarantee the connectivity with the deployed sensors and


maximize the coverage


sensors are virtually deployed at the zones corners.

7:


IF

no zones satisfy the constraints in step 6

8:


Deploy the senso
r at a corner that guarantees connectivity, minimizes the overlapping, and


maximizes the coverage.

9:


ELSE

10:


Deploy the sensor at one of the closest corners to the deployed sensors and guarantees


connectivity while applying

the
potential
-
placement point constraints
.


11:


END IF

12:


IF

there is no more sensors got to 12,
ELSE

go to 1.

13:


Compute the overall coverage by summing over the contribution of each sensor.

14:


Stop.

15:


END IF

16:

END IF

Figure 6: Deployment of homog
enous sensors on a differentiated field


a zone accommodates multiple
sensors


Algorithm 5

1:

Sort the sensors based on one of its characteristics

2:

Select the potential sensor s
i

to be deployed

3:

Check the case: case 1 got to 4 | case 2 go to 6 | case 3 go to 8

4:

Case 1
: only one sensor can fit in a zone

5:


Execute Algorithm 2 steps 2
-
9

6:

Case 2
: Sensor i covers more than one zone

7:


Execute Algorithm 3 steps 2
-
13

8:

Case 3
: A zone
accommodates multiple sensors.

9:


Execute Algorithm 4 steps 2
-
10

10:

IF

there is no more sensors got to 12, else go to 2.

11:


Compute the coverage contribution of each sensor.

12:


Stop.

13:

END IF

Figure 7: Depl
o
yment of Heterogeneous Sensors on a Diffe
rentiated Field


4.4. Deployment of Heterogeneous Sensors on a Differentiated Field


A differentiated field
F(A)

is given to be monitored by a heterogeneous set of sensors

S
, where
A

is the number of zones; each zone is assigned a weight
W
z
. This

problem is a combination of the problems
mentioned in sections 4.2 and 4.3 in which multiple zones and heterogeneous sensors are introduced,
respectively. Therefore, sensors may have one of
three

cases which are 1) only one sensor can fit in a zone
2) a
sensor can cover more than one zone, or 3) a zone can accommodate multiple sensors.
However
,
sensors disks are unequal. The details of the
deployment
algorithm
are

illustrated in figure 7
; the algorithm

12

a combination of

algorithms 2, 3, and 4
.
The worst

c
ase

complexity

of this algorithm is still the same as the
worst case of algorithm 4 which is
)
|
S
|
A
(
O
3

operations.


5.
Experiment Results




In this section,
different case studies are conducted to show the correctness and the performance of
the packing
-
based algorithms.
In t
he first case

study,
we

investigate the correctness of the algorithms with
different problem settings. In the second case study, the effec
t of sensors characteristics on the coverage
performance

is discussed.
In the final case study, the

deployment of special networks where there is a
binary relationship between the sensors communication and sensing ranges
is explained.

All of the
experimen
ts introduced in this section are conducted on
a Dell machine with 2.2
G
HZ processor and 1 GB
memory
. T
he algorithms are simulated using our
own
simulator in a dot net environment (C#).



Case Study 1: Successful Deployment of the Packing
-
Based Algorithm
s


D
ifferent sets of experiments are conducted to show various successful deployment scenarios
with

different problem setting
s

using packing
-
based algorithms introduced in the previous section.

Deployment of Homogenous Sensors on a Static Field


Figure 8
(a)

shows 25

sensors
that
are used to monitor 600m by 600m static field

with weight value
equal to one
.
Sensors are

assumed to be homogenous in which their communication range is double the
sensing range (120m) and their reliabilities are assumed to
be
80%

(R=.8)
.
A
lgorithm 1
was able to deploy
the sensors

successfully

with maximizing the covered areas and
avoiding

overlapping.

The coverage
percentage in this case is 81% which is the maximum coverage that can be achieved without overlapping.
In the second e
xperiment

shown in figure 8(b)
, sensors with co
mmunication range (140m) less than the
sensing range (80m)
are used with the same reliability.
The algorithm deployed the sensors with minimum
overlapped areas
and

satisfied

the connectivity constraints. Adding more sensors lead to more overlapping
between the sensors areas as shown in figure 8(c).

Deployment of Homogenous Sensors on a Differentiated Field

Different sets of experiments are
conducted to

show successful deploy
ment
schemes resulted from

algorithms
2
,
3
, and
4
. In the first set of
experiments, shown in figure 9(a),
a 600m x 600m field is divided
into 144 zones.

Zones are assigned random weights based on a uniform distribution U(1,
1
00). In addition, a

13

set of homog
enous sensors with 25m sensing range, 50m communication range, and 80% reliability

(R=.8)

are deployed into the field. As shown in the figure, Algorithm 2 was successfully able to deploy the sensors
in the center of each zone to guarantee the connectivity.

In addition, when the number of sensors is reduced
to 100, the algorithm was able to distribute the sensors on the highest weight
ed

zones ta
king into
consideration the connectivity, as shown in figure 9(b).

In the second set of experiments
,
the same
deployment
field is divided into 144 zones
.
Zones
assigned weights randomly using a uniform distribution U(1,100).
The field is monitored by
30

homogenous
sensors with

the communication range
is twice
the sensing range
. U
sing algorithm 3,
the
overlapping a
mong the sensors is very high since the sensors are deployment at the center of the zone as
well as
sensors cover almost ha
lf

of its neighbors’ zones.
In addition, the highest weighted zones are
concentrated in a small area of the monitored field (the righ
t side of the field).
However
, the algorithm
produces

an efficient deployment
scheme with

less number of sensors and high communication ranges as
shown in figure
9

(
d
).
Sensors are well organized
over the highest weight
ed

zones within the limit of their
c
ommunication ranges
.


The final set of experiments consider
deployment

of homogenous sensors
on a differentiated field
such that more than one sensor can fit into a zone.
F
igure
9
(
e
)

shows

a scenario where
108

sensors are
deployed into
a differentiated
field

with 36 zones.

Algorithm
4

was able to arrange the sensors into these
zones in a way that guarantees connectivity among sensors in the same zone as well as sensors in different
zones. The communication range in this case is assumed to be double the
sensing range. Figure
9
(
f
)
illustrates

another successful scenario where
the
algorithm was able to deploy 90 sensors in a field with
four zones z1, z2, z3, and z4 assigned different weights 50, 100, 150, and 60, respectively. The algorithm
started
by
deplo
ying the sensors in z3 which
has

the highest weight

(150);

then,
it
deploy
s

only one sensor
in z4

followed by the rest of the sensors on z2

whic
h has the second highest weight
(100
).
Th
e

deployed

sensor
in z4
between the two sets of sensors


in z3 and z2.

Deployment of Heterogeneous Sensors on a Static Field

Figure 10 (a) presents a deployment scheme of 20 heterogeneous sensors in terms of the reliability,
sensing range, and communication rang
e

on a static fie
l
d (600m x 600m) with
W=1
.

Sensors are sorted
based on their

sensing range and sequentially deployed into the fie
l
d using algorithm 1.
As can be seen,

14

sensors are packed efficiently with minimum overlapping due to the shortage in the

sensors

communication
ranges.
Sensors with small

sensing range
are accommodated properly in the field.

Deployment of Heterogeneous Sensors on a Differentiated Field


40 sensors are used as a test case to show the performance of algorithm
5

described in section 4.4.
These sensors are deployed
on a field

with 144 zones
. Zones weight

is

randomly generated
using
a uniform
distribution U (1,100). In addition,
sensors ranges are uniformly generated between 1m and 120m. These
s
ensors are sorted based on their sensing ranges and deployed sequentially in the fie
ld

as shown in figure
10(b)
.
The

algorithm shows three different cases where

a zone is covered by only one sensor (e
.
g. z16,
zones are counted from left to right), a zone is covered by more than one sensor (e.g. z24), and a sensor can
cover more than a zone (e.g.
the sensor in
z3). This deployment scheme confirms the efficiency and
correctness of the algorithm.


Case St
udy 2: Effect of Sensors Characteristics on the
Coverage Performance


In this set of experiments, a field with 144 zones is monitored by a set of 40 sensors. The field and
sensor characteristics are configured as mentioned in the previous section. In addit
ion, sensors reliability is
generated randomly using a uniform distribution
R(0,1),

where 0 and 1 represent 0% and 100% reliability,
respectively. Before the deployment, sensor
could be
sorted based on the sensing range
(r
s
)
, communication
range
(r
c
)
, or t
he reliability
(R)
. Algorithm
5

is chosen to run these experiments since it is a general packing
algorithm. The average results over 20 runs are summarized in figure 11. The
results
conclude that using
heterogeneous sensors, sensors


characteristics are eq
ually important and give almost the same coverage
performance. Multi
-
criteria sorting techniques might be needed for better performance. At the same time,
the running time is slightly higher by a few seconds when the reliability is used as sorting base. Th
is
difference
comes from
the large variance
of
the sensing ranges and/or the communication ranges that the
algorithm has to take care of.


Case Study 3: Effect of the
Binary
Relationship between the Communication and Sensing Ranges on
the Coverage
Performance



A differentiated field is configured
,

as mentioned above
,

to be
monitored by diffe
rent number of
sensors

that

range from
5

to

40
.
10

curves, as depicted in figure 12, are used to show the coverage
performance
with

different communication rang
e values.

For each sensor, t
he communication range is

15

represented
as

a percentage of the sensing range. The
average
results

produced

by applying algorithm
5
,
illustrated in figure
7
, show that increasing the communication range increases the coverage performance of
the monitored field. Nevertheless, increasing the communication range to more than double the sensing
range is greatly affected the coverage performance. For example, a c
ommunication range that doubles the
sensing range gives 90,000 units of coverage while increasing the communication range to 225% adds
30000 units of coverage. This value is increased to 88,000 coverage units when sensors communication
range is increased t
o 250% of the sensing range. These results confirm that the algorithm was
able

to
collect the highest weights zones to be monitored by the most reliable sensors.


(a)


(b)


(c)

Figure 8: deployment of homogenous sensors on a static field (a)
s
c
r
2
r


(b)
s
c
r
2
r

(c) adding more
sensors increases the overlapped areas


(a)


(b)


(c)


(d)


(e)


(f)

Figure 9: Deployment of homogenous sensors on a differentiated field




16

6. Conclusion and Future Work


In this paper, we studied different deployment problems. We considered the deployment of
homogenous and/or heterogeneous sensors on a static or differentiated field. Packing
-
based algorithms are
introduced for

efficient deployment schemes. The objectives a
re to maximize the coverage of the monitored
field, us
e

the best of the sensors characteristics, and produce a connected deployment scheme. We
experimented with different problem setting
s and their

results showed that the proposed algorithms were
able to
produce efficient connected deployment schemes for the given sensors. Our future work will
consider some other important parameters such as the monitoring time, sensors energy, and mobility.


(a)


(b)

Figure 10: Deployment of heterogonous sensors on
a differentiated field



(a)


(b)

Figure 11: : Effect of Sensors Characteristics on the Algorithms Performance

0
200000
400000
600000
800000
1000000
R
rc
rs
Sensors parameters
Objective performance
10
30
50
70
90
110
130
150
R
rc
rs
Sensors parameters
Running time (s)

17


Figure 12: The relationship between the Communication and Sensing Ranges


7.
References


1.

R. Williams
,

"Circle Packing
’s

Pl
ane Tessellations, and Networks,
"
Section
2.3 in
Geometrical
Foundation of Natural Structure: A Source Book of Design.

New York: Dover, pp. 34
-
47, 1979
.

2.

C.
Collins

and

K.

Stephenson , “A Circle Packing Algorithm,” in Computational Geometry:
Theory and Applications vol. 25, pp. 233
-
256, 2003.

3.


M
.

de Berg
,
M
.

Kreveld
,
M
.

Overmars
, and
O
.

Schwarzkopf
, Computational Geometry:
Algorithms and Applications, Second Edition,
Springer
-
Verlag

, ISBN: 3
-
540
-
65620
-
0
, 2000.

4.

J
.

Goodman and J
.

O’Rourke, Handbook of Discrete and Computational Geometry, CRC Press
LLC,
ISBN: 1584883014,
1997.

5.

A.
Howard,
J. Mataric, and G. Sukhatme,


An incremental self
-
deployment algorithm for mobil
e
sensor networks
, ” IN

Autonomous Robots, Special Issue on Intelligent Embedded Systems,
vol.
13

, pp.
113


126, 2002.

6.

X. Bai, S. Kumar, Z. Yun, D. Xuan, and T. Lai.
, “

Deploying wireless sensors to achieve both
coverage and connectivity
,” In

Proceeding
s of the Seventh International Symposium on Mobile
Ad Hoc Networking and Computing (ACM MobiHoc),

pp. 131
-

142


,

2006.

7.

C. Huang and Y. Tseng., “
The coverage problem in a

wireless sensor network
,”

In ACM
International Workshop

on Wireless Sensor Networks and Applications,
pp.

115


121, 2003.

0
50000
100000
150000
200000
5
10
15
20
25
30
35
40
Number of sensors
Coverage performance
25%
50%
75%
100%
125%
150%
175%
200%
225%
250%
rc as a percentage of rs

18

8.

D. Tian and N. Georganas.
,
“A

node scheduling scheme for

energy conservation in large wireless
sensor networks
, “

In

Wireless Communication and Mobile Computing (WCNC),

pp.
271

290,
2003.

9.

H. Zhang and J. Hou
. , “

Maintaining sensing coverage and

connectivity in large sensor networks
,


In NSF International

Workshop on Theoretical and Algorithmic Aspects of Sensor

Ad Hoc
Wireless, and Peer
-
to
-
Peer Networks,

pp. 89
-
124,

2004.

10.

K. Kar and

S.
Banerjee,
“Node

placement for connected coverage

in sensor networks
,”

In
Proceedings of WiOpt, 2003.

11.

Y. Zou and K. Chakrabarty
,
“Sensor

deployment and target

localization based on virtual forces
,”

In INFOCOM,
pp.

1293

1303, 2003.

12.

J. Lenstra and A
.

Rinnoo
y,

Complexity

of packing, covering and parti
tioning problems,


Mathematical Centre Tracts,

vol.

106 pp. 275
-
290
, 1979.

13.

G. Xing, X. Wang, Y. Zhang, C. Lu, R. Pless and C. Gill,


Integrated
Coverage and Connectivity
Configuration for Energy Conservation in Sensor Networks,


in
ACM Transactions on Sensor
Networks,
vol.
1
, pp.

36
-
72
, 2005
.

14.

Y. Wang, C. Hu, and Y. Tseng, “
Efficient Deployment Algorithms for Ensuring Coverage and
Connectivity of Wireless Sensor Networks
”, Wireless Internet Conf. (WICON), 2005.

15.

M. Cardei and J. Wu, “Coverage in Wireless Sensor Networks,” in Handbook of Sensor Networks,
M
.Ilyas and I. Mahgoub (eds.), CRC Press, ISBN 0
-
8493
-
1968
-
4, 2004.

16.

C. Huang and Y. Tseng., “ The coverage problem in a wireless sensor network,” In
WSNA ’03

17.

Z. Zhou, S
.

Das,

and

H
.

Gupta, “
Connected K
-
Coverage Problem in Sensor Networks
,” Proc. of
the 13th Int. Conf. on Computer Communication and Networks (IC3N), Chicago, IL,
pp. 373
-

378,
2004.


18.

S. Poduri,,

S.
Sukhatme, “
Constrained Coverage for Mobile Sensor Networks,”
IEEE
International Conference on Robotics and Automation, New Orleans, LA, USA, pp. 165
-
172,
2004

19.

D. Gage,

Command Control for Many
-
Robot Systems,
” in

Proc. of the Nineteenth Annual AUVS
Technical Symposium pp. 22
-
24, 1992.


19

20.

V. Chvatal, "A Combinatorial Th
eorem in Plane Geometry", Journal of Computorial Theory (B),
vol. 18,

pp
.

39
-
41
, 1975
.

21.

J. O’Rourke, "Galleries Need Fewer Mobile Guards: A Variation on Chvatal’s Theorem",
Geometriae Dedicata ,vol. 14 , pp 273
-
283 ,1983.

22.

P
.

Corke and S
.

Hrabar
,
R
.

Peterson
,

D
.

Rus
,

S
.

Saripalli
,

and G
.

Sukhatme.” Autonomous
Deployment and Repair of a Sensor Network using an Unmanned Aerial Vehicle”, Proceedings of
the IEEE 2004 International Conference on Robotics and Automation, pp. 3602
-
3608, 2004.

23.

P. Corke and S. Hrabar
,

R. Peterson
,

D. Rus
,

S. Saripalli
,

and G. Sukhatme. “Deployment and
Connectivity Repair of a Sensor Net with a Flying Robot”. In Proceedings of the Ninth
International Symposium on Experimental Robotics, 2004.

24.

A.
Tanenbaum,
,

C.
Gamage,
B.
Crispo,
“Taking Sensor Networks from the Lab to the Jungle, ”
IEEE Computer Society


,
vol.
39, Issue 8,
pp.

98


100, 2006.

25.

S
.

Hussain
,
M
.

Rafiqul
,
E
.

Shakshuki
, and
M. Zaman

, “Agent
-
Based Petroleum Offshore
Monitoring Using Sensor Networks,” In
17th International Conference on Database and Expert
Systems Applications (DEXA'06)



pp. 103
-
107, 2006.

26.

V. Mehta and M. Zarki, “A Bluetooth based sensor network for civil infrastructure health
monitoring,” In Wirele
ss Networks v
ol
.
10
, issue
4, Special Issue on Ad
-
Hoc Networking
,

pp.
401
-
412, 2004.

27.

K.
Stephenson,
Introduction to Circle
Packing:

The Theory of Discrete Analytic Functions.

New
Y
ork: Cambridge University Press,

ISBN
-
10: 0521823560,
2005
.