Optimal deployment of intelligent mobile air quality systems

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

MASTER’S GRADUATION THESIS
Optimal deployment of intelligent mobile
air quality systems
NGUYEN VIET DUNG


Major: Data Science and Artificial Intelligence (Elitech)

Thesis advisor:
Institute:

Assoc.Prof. Do Phan Thuan _________________

School of Information and Communication
Technology

HA NOI, 09/2022


CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM
Độc lập – Tự do – Hạnh phúc

BẢN XÁC NHẬN CHỈNH SỬA LUẬN VĂN THẠC SĨ
Họ và tên tác giả luận văn: Nguyễn Việt Dũng
Đề tài luận văn: Triển khai tối ưu các hệ thống quan trắc khơng khí di động
thơng minh

Chun ngành: Khoa học dữ liệu và Trí tuệ nhân tạo
Mã số SV: 20202342M

Tác giả, Người hướng dẫn khoa học và Hội đồng chấm luận văn
xác nhận tác giả đã sửa chữa, bổ sung luận văn theo biên bản họp Hội đồng
ngày 29/10/2022 với các nội dung sau:
- Thêm giới thiệu chi tiết hơn về các nghiên cứu có liên quan trong
chương 2.
- Đổi tên chương 3 từ “Problem formulation & hardness” thành
“Problem formulation”.
- Thêm phát biểu về bài toán opportunistic sensing optimization trước
khi viết tắt thành OSO.
- Đổi tên phần 3.2 thành “Mathematical formulation of OSO”.
- Thêm giải thích rõ hơn về hàm mục tiêu và các điều kiện trong mục
3.2.
- Thêm lý do giải thích vì sao sử dụng thuật toán quy hoạch động:
“In this simplified scenario, our dynamic programming approach
guarantees that the set found by the submaxSet function is always
maximum. thus the number mentioned in the previous section
5.1.1.2 will be equal to 1. Later we will show that we cannot use
dynamic programming in the general scenario, and we will need
another greedy sub-process which has a lower performance ratio for
that.”
- Thêm một số giải thích chi tiết về các thuật toán meta-heuristics và lý
do lựa chọn sử dụng chúng, cụ thể như sau:
+ “They are appropriate methods to verify efficiency of the
approximation algorithm, since their tremendous performance in
practice was shown in numerous research papers, especially
researches related to air monitoring systems. If the greedy
approximation approach is decent, the experimental results produced
SĐH.QT9.BM11

Ban hành lần 1 ngày 11/11/2014

by it should be competitive to the ones produced by the chosen metaheuristics. It is indeed true, and we will show the experimental results
supporting this observation later in this thesis.”
+ “Two meta-heuristics, the genetic algorithm and the simulated
annealing algorithm, are chosen to solve the OSO problem because of
their simplicity and efficiency in practice. Related researches about air
monitoring systems also deployed these methods to solve challenging
problems, and the results usually show that they are good choices for
creating a solution.”
- Thêm giải thích cho các hình vẽ và bảng biểu.
- Thêm mơ tả input và output cho các thuật toán.
- Thêm mục 6.4. “Comparison of results between the approximation
algorithm and the meta-heuristics” và chuyển mục 6.4 cũ thành mục
6.5. “Discussion”.
Ngày
Giáo viên hướng dẫn

tháng

năm

Tác giả luận văn

CHỦ TỊCH HỘI ĐỒNG

SĐH.QT9.BM11

Ban hành lần 1 ngày 11/11/2014

Graduation Thesis Assignment
Name: Nguyen Viet Dung
Phone: +84 399629097

Email :

Student ID: 20202342M

Class: 20BKHDL-E

Thesis title: Optimal deployment of intelligent mobile air quality systems
Thesis code: 2020BKHDL-KH01
Affiliation : Hanoi University of Science and Technology
I – Nguyen Viet Dung - hereby warrants that the work and presentation in this thesis
performed by myself under the supervision of Assoc.Prof. Do Phan Thuan. All the results
presented in this thesis are truthful and are not copied from any other works. All references
in this thesis including images, tables, figures and, quotes are clearly and fully documented
in the bibliography. I will take full responsibility for even one copy that violates school
regulations.
Hanoi, 28th September, 2022
Author

Nguyen Viet Dung
Attestation of thesis advisor :
I certify that the thesis entitled “Optimal deployment of intelligent mobile air quality
systems” submitted for the degree of Master of Science (M.S.) by Mr. Nguyen Viet Dung
is the record of research work carried out by him during the period from 10/2020 to
10/2022 under my guidance and supervision, and that this work has not formed the basis
for the award of any Degree, Diploma, Associateship and Fellowship or other Titles in this

University or any other University or institution of Higher Learning.
Hanoi, 28th September, 2022
Thesis Advisor

Assoc.Prof. Do Phan Thuan
3


Acknowledgements
In order to obtain this master's thesis, apart from my own efforts, it is impossible not to
mention the help of many other people.
First, I would like to thank Associate Professor Do Phan Thuan and Dr. Nguyen Phi Le,
my direct mentors. From the time I got my thesis title to the time I finished it, there was
not a moment that they didn't encourage me to run to the finish line. I am where I am today
in large part because of their support.
Next, I have to mention the funding source of VinIF. Their financial support helped me to
pay my tuition fees and complete my studies with peace of mind.
Finally, I would like to express my sincerest thanks to my teachers, friends and family.
Without them by my side, I wouldn't have made it to the end of the road.
Two years of wonderful lectures and extremely helpful time doing research will be in my
heart forever.

4


Abstract
Monitoring air quality plays a critical role in the sustainable development of developing
regions where the air is severely polluted. Air quality monitoring systems based on static
monitors often do not provide information about the area each monitor represents or
represent only small areas. In addition, they have high deployment costs that reflect the

efforts needed to ensure sufficient quality of measurements. Meanwhile, the mobile air
quality monitoring system, such as the one in this work, shows the feasibility of solving
those challenges. The system includes environmental sensors mounted on buses that move
along their routes, broadening the monitoring areas. In such a system, we introduce a new
optimization problem named opportunistic sensing that aims to find (1) optimal buses to
place the sensors and (2) the optimal monitoring timing to maximize the number of
monitored critical regions.
We investigate the optimization problem in two scenarios: simplified and general bus routes.
Initially, we mathematically formulate 1 the −targeted1 problem and prove its NP-hardness. Then, we propose a
polynomial-time 2 -, 2 −1 - approximation algorithm for the problem with the simplified, general routes, respectively.
To show the proposed algorithms’ effectiveness, we have evaluated it on the real data of real bus routes in Hanoi,
Vietnam. The evaluation results show that the former algorithm guarantees an average performance ratio
of 75.70%, while the latter algorithm achieves the ratio of 63.96%. Notably, −1 when the sensors can be on (e.g., enough energy) during (1 the−1)whole route, the 2
-approximation

algorithm achieves the approximation ratio of . Such ratio, which is almost twice as
78.42%.

2

−1

−1

−1

, enlarges the average performance ratio to

To further test the efficiency of the greedy approximation algorithm and optimize the
results, we propose two more meta-heuristic algorithms for this problem: genetic algorithm

and simulated annealing algorithm. Experiments show that the above meta-heuristic
algorithms only increase the goodness of the results by 1% to 3% on average, but have a
much larger running time than the greedy algorithm. From there, we see that the
approximation algorithm in particular is already a feasible solution in practice without
mentioning any other complicated tools.

5


Content
Graduation Thesis Assignment

3

Acknowledgements

4

Abstract

5

Content

6

List of Figures

8

List of Tables

9

Acronyms

10

Chapter 1. Introduction

11

1.1. Mobile air quality monitoring systems

11

1.2. Opportunistic sensing optimization (OSO) problem

12

1.3. Thesis contribution

12

1.4. Structure of thesis

12

Chapter 2. Related works

13

Chapter 3. Problem formulation

17

3.1. Problem statement

17

3.2. Mathematical formulation of OSO

18

3.3. Hardness of OSO

22

Chapter 4. Theoretical background

24

4.1. Approximation algorithms

24

4.2. Meta-heuristic algorithms

24
6

4.3. Research methodology

27

Chapter 5. Proposed solution

29

5.1. Approximation algorithms

29

5.2. Meta-heuristic algorithms

38

Chapter 6. Experimental results

42

6.1. Experimental settings

42

6.2. Numerical results of approximation algorithms

45

6.3. Numerical results of meta-heuristic algorithms

51

6.4. Comparison of results between the approximation algorithm and the meta-heuristics 61
6.5. Discussion

61

Chapter 7. Conclusion

63

Published papers

64

References

65

7


List of Figures
Figure 1. A map of size 4 × 4 with 3 bus routes and 6 critical squares. When

= 2, an

Figure 2. Illustration of observable

positions,

example of the sensor’s turn-on positions on bus 1 is shown. With such selected

that sensor can observe 5 critical squares , , , and
17
boundary, observable square, and observable segment.
. ................................................

19

..............................................................................................................................................

Figure 3. Illustration of Theorem 3.1’s proof ( is an arbitrary point on a bus route segment
. is the leftmost observable bound

to

then it is also observable by
and

={ ,

}.

Figure 4. A corresponding
3

. If is a critical square observable by

........................................................................................... closest

,

).
bus map when

20
= 3,

1={

, , ,

, },

2

={ ,

,

, },
23

..................................................................................................................

Figure 5. The remaining map after removing bus 1 from the map in Fig. 1, and the greedy
process continues.. ............................................................................................................... 29

observed by

Figure 6. (a) [lAb,

Ab]

is the unique close segment that contains all sensor’s turn-on positions

observed by a sensor turned on at somewhere

on the bus route where the critical square
turning on sensor from bus route

is observed. (b) There are critical squares
(in this figure,
= 5). Each square can be

in the middle of the interval [lib,
have critical points which are the left endpoints (l , where = 1, … , ) of

ib].

We then

ib

such intervals.

33

..............................................................................................................................................

Figure 7. Efficiency heatmap.. ............................................................................................ 45
Figure 8. Performance in the simplified scenario with
= 10, = 12. ..............................
46
Figure 9. Performance in the simplified scenario with

= 25, = 30. ..............................

47

Figure 10. Performance in the simplified scenario with

= 30,

= 36. ............................

47

= 42, = 50. ............................

47

Figure 11. Performance in the simplified scenario with
Figure 12. Performance in the general and special

scenario with
Figure 14. Performance in the general and special scenario with
Figure 15. Performance in the general and special scenario with

= 10,

= 12

48

= 30,

= 36

49

= 42, = 50. ..............50

8


List of Tables
Table 1. Notation list………………………………………………………………...…… 18
Table 2. Meta-heuristics performance compared to the approximation algorithm’s results in
the simplified scenario…………………………………………………...……………….. 51
Table 3. Meta-heuristics performance compared to the approximation algorithm’s results in

the general scenario...................................................................................................................................55

9


Acronyms

Abbreviations and terms

Meaning

OSO

Opportunistic sensing optimization

GA

Genetic algorithm

SA

Simulated annealing

Fig.

Figure

10


Chapter 1. Introduction
1.1. Mobile air quality monitoring systems
The fast industrialization and urbanization, especially in developing countries, cause air
pollution in urban areas. According to WHO, the polluted air is the main reason causing 36%
of deaths due to lung cancer, 27% of heart attacks, 34% of strokes, and 35% of deaths from
respiratory [1]. In such circumstances, it is indispensable to have a comprehensive solution for
monitoring air quality on a large scale for citizens and local governments. Accordingly, there

have been many air quality monitoring systems in literature, which can be roughly classified
into two main categories: stationary and mobile. The stationary system uses fixed stations to
monitor air quality, either outdoor [2] or indoor [3]. The air quality monitoring system operates
as a wireless sensor network (WSN) [4–6]. While the sensor nodes monitor the surrounding
environment, the base stations are in charge of storing and processing the sensory data On the
one hand, the sensor nodes monitor their surrounding environments. On the other hand, the
sensory data is either stored at the sensor’s local memory or transferred to the base station.
Despite the wide adoption, the stationary systems still suffer from an inherent critical
limitation: the low-resolution sensing data. That is because the fixed monitoring station has the
sensed data for only a limited area. Besides, the stations require high deployment and
maintenance costs. It is, therefore, challenging to deploy them densely. For example, in Hanoi,
Vietnam, the local government and other organizations have less than 50 stations in the total
2

area of 3329 km [7].
Unlike the stationary system, the mobile one makes use of mobile sensors to broaden the
monitoring areas. The sensors can leverage unmanned aerial vehicles (UAVs) [8,9] or landbased ones [10]. Despite having a significant advantage in capturing spatial information, the
UAV-based approach copes with many critical issues, including high deployment cost, energy
constraint, etc. Therefore, this work focuses on the mobile air monitoring approach that
exploits land-based vehicles. We consider the public buses, on which a battery-powered sensor
senses the environmental information along the bus route. In such a system, it is essential to
determine which buses to place the sensors on and schedule the measurement, considering the
limited number of sensors and their battery capacity constraint.

11


1.2. Opportunistic sensing optimization (OSO) problem
This thesis addresses the issues of the vehicle-based mobile air quality monitoring system.
We form a novel optimization problem named opportunistic sensing optimization (OSO),

that need to be
which is as follows. Given the bus routes’ trajectories, each bus route includes two paths
available monitoring sensors, and the locations of critical regions

sharing endpoints, the
monitored. Each sensor can measure at most
due to the energy and computational resource constraints.
determine optimal buses to place the sensors and 2
to perform the air quality measurement. The

positions on each bus path
The OSO problem then asks to

positions on each sensor’s bus route
objective is to maximize the number of

observable critical regions. OSO can be seen as a hybrid problem that jointly optimizes the
trajectory (i.e., the bus route) and the schedule (the positions to perform the measurement)
of the mobile sensors.

1.3. Thesis contribution
• We provide a theoretical model and prove the NP-hardness of the OSO problem.
• We propose the polynomial-time constant approximations for the OSO problem −1 . More specifically, in general, our algorithm guarantees
the performance ratio of 2 −1. In a simplified case where the two paths of 1 each bus route are identical, the guaranteed performance ratio of
our algorithm is 2. We present theoretical proofs about these performance ratios.

• We utilize two meta-heuristics algorithms: genetic algorithm and simulated annealing, to
further verify the efficiency of the approximation algorithm. Experimental results showed
that these meta-heuristics helped increase only 1-3% the number of observable critical
squares on average, and they were slower than the simple greedy approach.

• We present extensive experiments to evaluate the proposed algorithms’ performance.

1.4. Structure of thesis
The remainder of the thesis is organized as follows. Chapter 2 presents the related works. We
formulate the OSO problem and prove its NP-hardness in Chapter 3. Chapter 4 is a brief
explanation about the techniques used to solve OSO in this thesis. Chapter 5 describes our
proposed algorithms and theoretical analysis about their effectiveness. The algorithms’
performance in practice is discussed in Chapter 6. In the end, chapter 7 concludes the thesis.

12


Chapter 2. Related works
There are many efforts in the previous works aiming to build air monitoring systems.
However, most of them use static monitoring sensors. The works in [10,11] introduced a
concept similar to our investigated system. However, they focus on realizing the sensor
device, systems rather than optimizing the deployment. The OSO problem in this work is
close to the sensor placement optimization and scheduling under the target coverage
constraint in static WSNs and mobile WSNs.
In [12], the relay node placement problem is mathematically formulated as an NP-hard
Steiner minimum tree problem with a minimum number of Steiner points and bounded
edge length. The authors then proposed two heuristic algorithms whose performance ratios
are 2.5 and 3.0, respectively. In [13], F. Senel et al. proposed to divide the target coverage
problem into sub-problems, each of which contains only three sensors. In [14], Anxing
Shan et al. considered a network comprised of omnidirectional probabilistic sensors. The
authors studied how to activate the least number of sensors to detect all targets with a
probability higher than a threshold . In [15], the authors investigated the optimal
deployment in the wireless rechargeable sensor networks. Specifically, they studied how to
deploy a minimum number of sensors to cover all the targets under the sensors’ limited
sensing angle and the mobile charger’s energy constraint. The problem of deterministic

deployment in 3D underwater WSN is addressed in [16]. The authors exploited a natureinspired evolutionary algorithm named Cuckoo search to determine the optimal position
for placing sensors. The objective is to maximize the target coverage capability with a
minimum number of sensors. The authors in [17] addressed, at the same time, the target
coverage, connectivity, and fault tolerance problems in wireless sensor networks. They
proposed a hybrid algorithm that combines the greedy approach and spanning tree
technique to determine a minimal number of sensors. Unfortunately, all of the algorithms
mentioned above consider only networks with static sensors.
Concerning the target coverage problem in mobile WSNs, there are relatively rare related
works. In this chapter we highlight four remarkable researches related to our problem,
which are [18], [19], [20] and [21].
In [18], the authors considered the target coverage problem to minimize the moving distance of
all sensors. The problem was named k-Sink Minimum Movement Target Coverage (k-

13


MMTC). m): They have k sink stations to send mobile sensors and to cover all targets on
an Euclidean space, k-MMTC is to schedule the sensor movement trajectories and
minimize the sum of moving distance. They proved that k-MMTC was NP-hard.
To solve that problem, they proposed a polynomial-time approximation scheme (PTAS),
named Energy Effective Movement Algorithm (EEMA). They divided EEMA into two
phases. In the first phase, they proposed a novel method to divide the surveillance region
into some sub-areas according to the locations of targets. The sensors in the same sub-area
can cover the same target set. In the second phase, they scheduled the mobile sensors and
move the sensors to cover all targets. They proved that ε > 0, EEMA can be an (1+ε)in time O(
approximation algorithm for k-MMTC problem that runs

∀

). For large scale

1/ 2
pa rt icular, to ke ep the

networks, they proposed a distributed version named D-EEMA. In
connectivity of the sensors, they used some mobile sensors for communication. They
called these sensors as communication sensors which do not have sensing tasks. The
communication sensors just need to move around the targets and the stations to collect
sensing data. D-EEMA was divided into two phases. In the first phase, they divided the
surveillance region into some subareas and got the positions of the targets. In the second
phase, they grouped the targets and dealt with different groups respectively. They also
provided experiments to validate the effectiveness and efficiency of EEMA and D-EEMA.
In all, EEMA was the first PTAS for sensor movement scheduling for target coverage
problem.
Nguyen et al. in [19] focused on mobile WSNs where the sensors cannot cover all the
targets. In mobile wireless sensor networks, the movement of sensors consumes much
more power than that in sensing and communication. In that research, the targets are
weighted by their importance. The more important a target is given a higher weight. These
requirements make the problem interesting, and also difficult. The aim of that study is to
study a more general and practical problem in terms of target coverage and network
connectivity, namely the Maximum Weighted Target Coverage and Sensor Connectivity
with Limited Mobile Sensors (TAR-CC) problem. Originally, the TAR-CC problem is to
schedule a limited number of mobile sensors to appropriate locations to cover targets and
form a connected network such that the total weight of the covered targets is maximized. In
addition, when the transmission range is assumed to be large enough for any
communication, a subproblem of the TAR-CC problem, termed the Reduced TAR-CC
(RTARCC) problem was also introduced.
An approximation algorithm, termed the weighted maximum-coverage-based algorithm
(WMCBA), with an approximation ratio of 1−1/e is proposed for the RTARCC problem,
where e denotes the base of the natural logarithm, was proposed. In the WMCBA, all
14

possible sets of targets that can be covered by a mobile sensor located at any point in the
sensing field are considered. Then, a greedy method is used to select suitable sets of targets to
be covered by mobile sensors. Based on the WMCBA, the Steiner-tree-based algorithm
(STBA) is proposed for the TAR-CC problem. In the STBA, the Fermat points and a nodeweighted Steiner tree algorithm are used to find a tree such that the number of mobile sensors
deployed by the tree structure to form a connected network is minimized. Simulation results
demonstrate that even if the number of mobile sensors is high enough such that a connected
network can always be formed to cover all targets, the STBA requires a significantly lower
total movement distance than the best solution proposed for the MSD problem. In addition,
when the mobile sensors may be not enough to cover all targets, the STBA works better than
the greedy method proposed in the simulation section of that paper.
In [20], Rout et al. addressed the target coverage problem with the consideration of obstacle
avoidance. In that piece of work, they proposed a localized self-deployment scheme, named as
Obstacle Avoidance Target Involved Deployment Algorithm (OATIDA), for deployment of
randomly scattered mobile sensor nodes to cover predefined targets while maintaining
connectivity with the base station in the presence of obstacles. The proposed deployment
scheme is based on the following assumptions. They were: (i) All the sensor nodes have
locomotion capability and can move independently, (ii) The base station is fixed in any place
inside the region of interest and bears all the information about the targets. (iii) Initially, all the
sensor nodes are randomly deployed within the communication range of the base station
(iv) Each sensor node has one unique ID, (v) Every sensor node has the ability to know its own
coordinates by some localization method (e.g., GPS, triangulation and multilateration),
(vi) Every sensor node is able to acquire the relative position of the other sensor nodes within
its communication range, (vii) All the sensor nodes have circular sensing and communication
areas, (viii) The sensing field contains obstacles of arbitrary shapes, and (ix) Every sensor node
is able to detect the shape and position of any obstacles in its sensing range and can calculate
the nearest distance from the obstacle by using the time-of-flight method.
They used the concept of potential field theory and relative neighborhood graph for selfdeployment of mobile sensor nodes in an unknown environment to achieve target coverage
while preserving connectivity with the base station. Their proposed approach is localized in the

sense that each decision taken by the sensor node is strictly based on information acquired
from its neighboring sensors that are part of the relative neighborhood graph. That algorithm
works well in scenarios of single target, multiple targets and moving targets in the presence of
single or multiple obstacles. The proposed algorithm preserves connectivity during the
deployment procedure and minimizes the number of sensor nodes to maintain the connectivity
so that large numbers of sensor nodes are available for monitoring targets.

15


The problem of minimizing the mobile sensors’ moving distance under the target coverage
constraint is re-visited by Choudhuri et al. in [21]. The existing sensor relocation
algorithms for target coverage and connectivity are based on the assumption of the free
mobility models. Under these models, each sensor can move any amount in any direction.
But for real life situations, the movement of the sensors may be restricted. Here they
intended to solve the problem of target coverage with a rectilinear mobility model where a
sensor can move only along two mutually perpendicular directions. Since the (x, y)
coordinates of a location can be transformed to another set of coordinates by a simple
rotation, they assumed in that paper that a sensor can move along directions parallel to the
x and y-axes only. Thus, if a sensor moves from point u to v, the distance covered by it is
the Manhattan Distance between u and v.
Their algorithm worked in two phases. In the first phase, a subset of sensors are moved to
new locations such that all targets are covered. The sensors which are essential to ensure
coverage are called assigned sensors. Only if the first phase results in some assigned
sensors not connected to BS, the second phase is initiated to move some unassigned
sensors to achieve connectivity. The proposed algorithm initiates movement of the sensors
only if it is strictly necessary. Even though that work gives a way of relocating sensors
when the initial deployment does not ensure coverage, it can be also used at a later stage
when some of the sensors die out resulting in either loss of coverage or connectivity.
All existing works above on mobile WSNs focus only on either the trajectory or the

sensors’ schedule. This thesis takes a more general approach, in which we address at the
same time both the optimal trajectories to place the mobile sensors and the optimal
positions to perform the measurement. We establish a complete process from formulating
to solving our problem and conducting experiments, which makes the results valuable in
research and applicable in real-life cases.

16


Chapter 3. Problem formulation
3.1. Problem statement
We introduce a new problem named opportunistic sensing optimization (OSO). A map of
bus routes is given on a grid of ×
squares on a 2-dimensional plane (
and are
marked either critical or non-critical. The critical
predefined integers). Each square is
squares are regions that need to be monitored. The grid has a total of critical squares ( ≤
arrival
× ). There are
bus routes on the map, each of which consists of departure,
endpoints, and two bus paths connecting two endpoints. A bus path is assumingly a fixed
polyline. A bus departs on one path and returns on another path. We assume that every bus
route has exactly one bus. There are ( ≤ ) air quality monitoring sensors that need to be
installed on buses, where each bus can have at most one sensor installed. We adopt the diskbased model to represent the sensing capability of the sensors. Each sensor can observe
bus route to measure the air quality (and thus, 2 positions in total on the two paths). Otherwise,
it has to be turned off immediately after a quick measurement. The measurement

all points in the disk of radius
resource constraint, a sensor

centered at its position. Due to the energy and computation
can only be turned on at exactly positions on each path of a

time is negligible, and the turn-on positions of sensors are free to be chosen. However, they
are fixed before installation on the buses (see Fig. 1 for an example). We assume that the air
the sensor.

quality of every point inside the same square is almost identical. A square is then said
observable by a sensor if it intersects the circle of radius centered at a turn-on position of

Figure 1. A map of size 4 × 4 with 3 bus routes and 6 critical squares. When = 2, an
example of the sensor’s turn-on positions on bus 1 is shown. With such selected positions,
that sensor can observe 5 critical squares , , , and .


17


Table 1. Notation list.
Notation

Meaning

p

Number of columns in the bus map grid

q

Number of rows in the bus map grid

c

Number of critical squares

n

Number of bus routes

m

Number of sensors

r

Sensing radius of a sensor

k

Maximum number of time a sensor can be turned on a path

O( )

Observable boundary of C
The set of the leftmost and rightmost observable bounds of a bus

C(b)

route b

The OSO problem asks to choose

bus routes to install sensors and decide the sensor’s

turn-on positions on each of these

routes to maximize the number of observable critical
scenarios: t he general scenari o and a simpl ifie d sce nari o

squares. This work considers two
where the two paths on each bus route are identical.

3.2. Mathematical formulation of OSO
In this section, we introduce the definition and mathematically formulate the targeted
problem. To facilitate readability, we summarize all the notations in Table 1.
a distance of to ’s boundary is called the
Definition 3.1 (Observable Boundary). Let be a critical square. The outer contour that has
observable boundary of , and denoted as O( ).
Fig. 2 illustrates two critical squares ( 1, 2) and their observable boundaries ( 1) and ( 2).
is the set of all observable squares of .

Definition 3.2 (Observable Square). Let

be a point on the plane. An observable square of
a sensor l oca ted at ; the observabl e square set of

is a critical square that is monitored by
X2, while
In Fig. 2,

1

is an observable square of

observable square set of X1 consists of

X1

1

and

and

2,

2

is observable by only X1. The

while that of X2 contains only 1.

18