Efficient Deployment of Connected Sensing Devices 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 objectives 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 algorithms are
inspired from the packing theories in computational geometry. The algorithms efficiency is examined using
different sets of experiments.
1. Introduction
With the advances of sensing technologies and requirements of the monitored fields, sensor
deployment becomes very challenging. Different 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 of homogenous and/or heterogeneous sensing
devices on static and/or differentiated monitored fields. A static field is assumed to have the same
monitoring requirements over all of its areas while a differentiated field is associated with areas that may
demand different monitoring requirements. Packing-based algorithms 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 sensors connectivity.
Considering coverage and connectivity at the same time during the deployment process could lead
to efficient usage of the sensor networks (Zhang, 2004). The coverage 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,
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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.
In this paper, our definition of coverage is slightly different. Coverage refers to monitoring 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 field.
Deployment of non-connected sensor networks is studied in many areas such as geometry and
computer science. Early contribution to this kind of networks returns to Chvatal, 1975 (Chvatal, 1975), who
introduced the Art Gallery Problem (AGP). Each sensor is usually wired to the control center 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 with limited visibility (e.g. (O’Rourke, 1983)). However, in wireless sensor network, the
network operation is usually based on multiple hopes 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. in (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. Our work differ in considering homogenous/heterogeneous sensors in terms of
sensing range, communication range, and reliability as well as studying the deployment on static
/differentiated fields. Similarly, unmanned vehicle (Corke, 2004) and a flying robot (Corke June 2004)
have been used to deploy nodes incrementally. The 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 addition, 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. The goal is different from our work in which the authors consider only cases where there is a
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binary relationship between sensors communication and sensing ranges. Our algorithms 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. The performance of these algorithms is
illustrated in section 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 sensors sensing and communication
ranges are represented by a disk-based model. A sensor s is associated with a sensing range rs and a
communication range rc . Any point in the monitored field that falls in a disk of radius rs is assumed to be
covered by the sensor s. In addition, the sensing disk of sensor s centered at location u is denoted by Ds(u).
Similarly, the communication disk of a sensor s centered at location v is denoted by Ds(v).
Assumption 2.2. [Heterogeneous vs. homogenous sensors] We assume that heterogeneous
sensors may differ in their sensing ranges, communication ranges, and/or reliabilities. On the other hand,
homogenous sensors have the sensing 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 Wz based on its monitoring requirements level.
Definition 2.1. [Sensors coverage-contribution] Sensors contribute to the coverage of the
monitoring field by the area of its sensing disks. 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 as a
contribution 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
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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. [duv] We assume that the distance between any two sensors si and sj centered at u
and v locations, respectively, is measured by the Euclidian distance d uv.
Definition 2.4. [Connected sensors] Two sensors si and sj with communication ranges rci and rcj
centered at u and v locations, respectively, are called connected if and only if: d(u, v)  Min (rci, rcj )
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 potential 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 circles) 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 points 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 dij is computed from each point Pi to the untouched sensors
(circles) sj . After that, the minimum distance dmin(Pi) is assigned to Pi ; 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.
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Figure 1(b) shows two potential points P1 and P2 for s6. The distances, dished lines, between each
point and the untouched sensors’ disks 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, P2.
3. Problem Definition and Motivations
In this section, we define the deployment problem and motivate the importance of it along with the
usage of the packing theory.
A field F(A) is given to be monitored by a set of sensors S. The field is 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 Wz 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, communication and/or reliability capabilities. Based on
the given sensors and the monitored field characteristics, the problem is to find the best deployment scheme
that maximizes the coverage of the field, minimizes the overlapped areas, and guarantees the connectivity
of the deployed sensors. The coverage is maximized when the highest weight zones are observed by the
most reliable sensors. The connectivity is guaranteed by developing a connected graph among the sensors.
Therefore, the resulting deployment scheme ensures 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 their 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 the 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 the 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.
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A naïve solution to the deployment problem starts by deploying the first sensor at the center of the
field and deploy other sensors around it. As illustrated in figure 2 (a), the solution wasted a considerable
area from the monitored field. However, using 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
packing problem, more specifically the circle packing problem, is a well defined problem in the field of
computation geometry 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.
Our solutions 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) in addition to the circle
radius (sensor sensing range) , we consider two other parameters which are the communication range and
sensors’ reliability, 2) we consider the packing into multiple zones (rectangular or square shapes) versus
one zone/rectangle, 3) we study the differentiated monitoring requirements which is not a part of their work,
4) our solutions adapt the packing theory for sensor deployment; nevertheless, their work is a general
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 minimize the
overlapped areas and guarantee the connectivity; nonetheless. their work focus on non-overlapped circles.
The details of the packing algorithms are explained in the next section.
(b)
(a)
Figure 2: Solutions to the deployment problem (a) a naïve algorithm (b) heuristic algorithm
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4. Packing-based Algorithms
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,
deployment 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 rs, the communication range rc, and the detection reliability R. The objective is to maximize
the coverage of the monitored field and minimize the overlapped areas. For the connectivity purposes,
sensors 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 based on
the sensors identifier. Figure 3 shows the details of the deployment algorithm in which the 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 and the 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 minimum overlapping (lines 6 and 7). Otherwise, potential placement points are
evaluated and the current 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 field is totally covered (line 10). Then, the
contribution of each sensor is computed (line 16). 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 find potential placement
points is
O( m (| S | m )) 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
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potential placement point with each deployed sensor and the field’s border. Computing the distances
between the untouched circles and the potential points requires
O( m 2 (| S | m )) operations. Finding the
point that has the minimum distance requires O( m ) operations, as is finding the touched circles with the
largest communication range in case of overlapping is a must. Therefore, the estimated worst case
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complexity of algorithm 1 is O(| S |
).
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 into a set of zones A and each zone is assigned
a weight Wz. The relationship between the monitored 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
which the first sensor is deployed on the highest weighted 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 satisfying the connectivity constraint, the current sensor may
overlap with one of the deployed sensors in which its coverage contribution is the maximum. The worst
case complexity of this algorithm is
O(| A | 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 which sensors are always deployed at the center of the specified zones. The
heuristic starts by deploying 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 weighted zone might be
surrounded by low weights’ zones or vice versa. This process is repeated until no more sensors or zones are
available. The worst case complexity of this algorithm is also
O(| A | 2 ) which is affected by the scanning
operation that looks for best zone that maximizes the coverage. Advantages and disadvantages of this
heuristic are explored in the experiment results section (section 5).
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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 weight zone. Once the zone is covered, no more sensors can fit in the zone, another zone is
selected for the deployment. Since sensors have the same communication range rc, the next highest zone
that guarantee the connectivity to the deployed sensors is selected. Within the selected zone, the sensor is
deployed at the closest 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
weight zones is selected due to its importance. As noticed, if overlapping is not required, 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 of this algorithm is
O(A | S |3 ) . This is due to the need for finding the next highest zones that maximize the coverage and
persuade sensors connectivity.
Algorithm 1
1: Select the potential sensor si 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 dmin 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
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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 si 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 differentiated field – only one sensor can fit a zone
Algorithm 3 –The sensor covers more than one zone
1: Select the potential sensor si 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 si 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 si 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 si 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 field – the sensor covers more than one
zone
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Algorithm 4: A zone accommodates multiple sensors
1: Select the potential sensor si 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 sensor 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 homogenous 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 si 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: Deployment of Heterogeneous Sensors on a Differentiated 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 Wz. 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
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a combination of algorithms 2, 3, and 4. The worst case complexity of this algorithm is still the same as the
worst case of algorithm 4 which is
O(A | S |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 the first case study, we investigate the correctness of the algorithms with
different problem settings. In the second case study, the effect 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
experiments introduced in this section are conducted on a Dell machine with 2.2 GHZ processor and 1 GB
memory. The algorithms are simulated using our own simulator in a dot net environment (C#).
Case Study 1: Successful Deployment of the Packing-Based Algorithms
Different sets of experiments are conducted to show various successful deployment scenarios with
different problem settings 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). Algorithm 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 experiment shown in figure 8(b), sensors with communication 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 deployment 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(1,100). In addition, a
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set of homogenous 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 weighted zones taking 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. Using algorithm 3, the
overlapping among the sensors is very high since the sensors are deployment at the center of the zone as
well as sensors cover almost half of its neighbors’ zones. In addition, the highest weighted zones are
concentrated in a small area of the monitored field (the right 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 weighted zones within the limit of their
communication 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. Figure 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 deploying the sensors in z3 which has the highest weight (150); then, it deploys only one sensor
in z4 followed by the rest of the sensors on z2 which has the second highest weight (100). The 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 range on a static field (600m x 600m) with W=1. Sensors are sorted
based on their sensing range and sequentially deployed into the field using algorithm 1. As can be seen,
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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
sensors are sorted based on their sensing ranges and deployed sequentially in the field 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 Study 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 addition, 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 (rs), communication
range (rc), or the 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 equally 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. This
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 different 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 range values. For each sensor, the communication range is
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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 communication 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 to 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)
Figure 8: deployment of homogenous sensors on a static field (a)
(c)
rc  2rs (b) rc  2rs (c) adding more
sensors increases the overlapped areas
(a)
(d)
(b)
(c)
(e)
(f)
Figure 9: Deployment of homogenous sensors on a differentiated field
15
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 are to maximize the coverage of the monitored
field, use the best of the sensors characteristics, and produce a connected deployment scheme.
We
experimented with different problem settings 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
800000
Running time (s)
Objective performance
1000000
600000
400000
200000
0
R
rc
rs
150
130
110
90
70
50
30
10
R
Sensors parameters
rc
rs
Sensors parameters
(b)
(a)
Figure 11: : Effect of Sensors Characteristics on the Algorithms Performance
16
rc as a percentage of rs
Coverage performance
25%
150%
50%
175%
75%
200%
100%
225%
125%
250%
200000
150000
100000
50000
0
5
10
15
20
25
30
35
40
Number of sensors
Figure 12: The relationship between the Communication and Sensing Ranges
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