Optimal deployment of intelligent mobile air quality systems

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 radi

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

MASTER’S GRADUATION THESIS

Optimal deployment of intelligent mobile

air quality systems

NGUYEN VIET DUNG

Dung.NV202342M@sis.hust.edu.vn

Major: Data Science and Artificial Intelligence (Elitech)

Technology

HA NOI, 09/2022

SĐH.QT9.BM11 Ban hành lần 1 ngày 11/11/2014

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

- Thêm gi ới 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 bi ểu về bài toán opportunistic sensing optimization trước khi vi ết tắt thành OSO

- Đổi tên phần 3.2 thành “Mathematical formulation of OSO”

- Thêm gi ải 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 𝛼𝛼 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 m ột 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

by it should be competitive to the ones produced by the chosen heuristics It is indeed true, and we will show the experimental results supporting this observation later in this thesis.”

meta-+ “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 gi ải thích cho các hình vẽ và bảng biểu

- Thêm mô 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”

Name: Nguyen Viet Dung

Phone: +84 399629097 Email : Dung.NV202342M@sis.hust.edu.vn 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

Graduation Thesis Assignment

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

Acknowledgements

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 the targeted problem and prove its NP-hardness Then, we propose a polynomial-time 1

2 -,

𝑒𝑒−1 2𝑒𝑒−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, when the sensors can be on (e.g., enough energy) during the whole route, the 𝑒𝑒−1

2𝑒𝑒−1 -approximation

algorithm achieves the approximation ratio of (1 −1𝑒𝑒) Such ratio, which is almost twice as

𝑒𝑒−1

2𝑒𝑒−1, enlarges the average performance ratio to 78.42%

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

Abstract

Graduation Thesis Assignment 3

1.2 Opportunistic sensing optimization (OSO) problem 12

Figure 1 A map of size 4 × 4 with 3 bus routes and 6 critical squares When 𝑘𝑘 = 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 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷 and 𝐸𝐸 17

Figure 2 Illustration of observable boundary, observable square, and observable segment. 19

Figure 3 Illustration of Theorem 3.1’s proof (𝑋𝑋 is an arbitrary point on a bus route segment 𝑃𝑃 𝑌𝑌 is the leftmost observable bound closest to 𝑋𝑋 If 𝐶𝐶 is a critical square observable by 𝑋𝑋, then it is also observable by 𝑌𝑌) 20

Figure 4 A corresponding bus map when 𝛽𝛽 = 3, 𝑉𝑉1 = {𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷, 𝐹𝐹}, 𝑉𝑉2 = {𝐴𝐴, 𝐶𝐶, 𝐷𝐷, 𝐸𝐸}, and 𝑉𝑉3 = {𝐵𝐵, 𝐹𝐹} 23

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

Figure 6 (a) [l Ab, 𝑟𝑟Ab] is the unique close segment that contains all sensor’s turn-on positions on the bus route 𝑏𝑏 where the critical square 𝐴𝐴 is observed (b) There are 𝑑𝑑 critical squares observed by turning on sensor from bus route 𝑏𝑏 (in this figure, 𝑑𝑑 = 5) Each square 𝑖𝑖 can be observed by a sensor turned on at somewhere in the middle of the interval [l ib, 𝑟𝑟ib] We then have 𝑑𝑑 critical points which are the left endpoints (l ib, where 𝑖𝑖 = 1, … , 𝑑𝑑) of such intervals 33

Figure 7 Efficiency heatmap 45

Figure 8 Performance in the simplified scenario with 𝑝𝑝 = 10, 𝑞𝑞 = 12 46

Figure 9 Performance in the simplified scenario with 𝑝𝑝 = 25, 𝑞𝑞 = 30 47

Figure 10 Performance in the simplified scenario with 𝑝𝑝 = 30, 𝑞𝑞 = 36 47

Figure 11 Performance in the simplified scenario with 𝑝𝑝 = 42, 𝑞𝑞 = 50 47

Figure 12 Performance in the general and special scenario with 𝑝𝑝 = 10, 𝑞𝑞 = 12 48

Figure 14 Performance in the general and special scenario with 𝑝𝑝 = 30, 𝑞𝑞 = 36 49

Figure 15 Performance in the general and special scenario with 𝑝𝑝 = 42, 𝑞𝑞 = 50 50

List of Figures

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

List of Tables

Abbreviations and terms Meaning

Acronyms

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 area of 3329 km2 [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 land-based 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

Chapter 1 Introduction

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), which is as follows Given the 𝑛𝑛 bus routes’ trajectories, each bus route includes two paths sharing endpoints, the 𝑚𝑚 available monitoring sensors, and the locations of critical regions that need to be monitored Each sensor can measure at most 𝑘𝑘 positions on each bus path due to the energy and computational resource constraints The OSO problem then asks to determine 𝑚𝑚 optimal buses to place the sensors and 2𝑘𝑘 positions on each sensor’s bus route

to perform the air quality measurement The 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 More specifically, in general, our algorithm guarantees the performance ratio of 𝑒𝑒−1

• 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

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 𝜖𝜖 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 nature-inspired 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-Chapter 2 Related works

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)-approximation algorithm for k-MMTC problem that runs in time O(𝑛𝑛1/𝜀𝜀 2

) For large scale networks, they proposed a distributed version named D-EEMA In particular, to keep the 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,

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 node-weighted 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 deployment 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

self-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

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 𝑝𝑝 × 𝑞𝑞 squares on a 2-dimensional plane (𝑝𝑝 and 𝑞𝑞 are

predefined integers) Each square is marked either critical or non-critical The critical

squares are regions that need to be monitored The grid has a total of 𝑐𝑐 critical squares (𝑐𝑐 ≤

𝑝𝑝 × 𝑞𝑞) There are 𝑛𝑛 bus routes on the map, each of which consists of departure, arrival 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 𝑚𝑚 (𝑚𝑚 ≤ 𝑛𝑛) air quality monitoring sensors that need to

be installed on buses, where each bus can have at most one sensor installed We adopt the disk-based model to represent the sensing capability of the sensors Each sensor can observe all points in the disk of radius 𝑟𝑟 centered at its position Due to the energy and computation resource constraint, a sensor can only be turned on at exactly 𝑘𝑘 positions on each path of a bus route to measure the air quality (and thus, 2𝑘𝑘 positions in total on the two paths) Otherwise, it has to be turned off immediately after a quick measurement The measurement 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 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 𝑟𝑟 centered at a turn-on position of the sensor

Figure 1 A map of size 4 × 4 with 3 bus routes and 6 critical squares When 𝑘𝑘 = 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 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷 and 𝐸𝐸

Chapter 3 Problem formulation

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

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

Definition 3.1 (Observable Boundary) Let 𝐶𝐶 be a critical square The outer contour that has

a distance of 𝑟𝑟 to 𝐶𝐶’s boundary is called the observable boundary of 𝐶𝐶, and denoted as O(𝐶𝐶)

Fig 2 illustrates two critical squares (𝐶𝐶1, 𝐶𝐶2) and their observable boundaries (𝐶𝐶1) and (𝐶𝐶2)

Definition 3.2 (Observable Square) Let 𝑋𝑋 be a point on the plane An observable square of

𝑋𝑋 is a critical square that is monitored by a sensor located at 𝑋𝑋; the observable square set of

𝑋𝑋 is the set of all observable squares of 𝑋𝑋

In Fig 2, 𝐶𝐶1 is an observable square of X 1 and X 2, while 𝐶𝐶2 is observable by only X 1 The

observable square set of X 1 consists of 𝐶𝐶1 and 𝐶𝐶2, while that of X 2 contains only 𝐶𝐶1

Figure 2 Illustration of observable boundary, observable square, and observable segment

Proposition 1 A critical square 𝐶𝐶 is an observable square of 𝑋𝑋 if and only if 𝑋𝑋 stays on the boundary or inside of O( 𝐶𝐶)

Proof 𝐶𝐶 is observable by 𝑋𝑋 if and only if the distance from 𝑋𝑋 to 𝐶𝐶’s boundary does not exceed 𝑟𝑟 This condition is equivalent to that 𝑋𝑋 stays on the boundary or inside O(𝐶𝐶)

Definition 3.3 (Observable Segment) Suppose 𝐶𝐶 is a critical square and 𝑃𝑃 is a straight bus

route segment The observable segment of 𝑃𝑃 concerning 𝐶𝐶 is defined by the portion of 𝑃𝑃

staying on the boundary or inside O(𝐶𝐶)

Moreover, we define the leftmost and rightmost observable bounds of 𝑃𝑃 concerning O(𝐶𝐶) as

the first and last point of the observable segment when we move from one end of the other end of the bus route

Fig 2 represents a bus route 𝑏𝑏 and its observable segments Segment P 1 P 2 stays inside

O(𝐶𝐶1 ), thus its observable segment concerning 𝐶𝐶1 is itself Segment P 5 P 6 has the left end

point, i.e., P 5 , stays inside O(𝐶𝐶2 ), thus the observable segment of P 5 P 6 concerning P 5 P 6

comprises of two end points, one is P 5 and the other is the intersection of P 5 P 6 and O(𝐶𝐶2 ), i.e., I 6 As the two end points of segment P 8 P 9 stays outside of O(𝐶𝐶2 ), P 8 P 9’s observable segment concerning 𝐶𝐶2 comprises of the two intersection points of P 8 P 9 and O(𝐶𝐶2 ), i.e., I 8,

I 9 In a special case, the observable segment of P 7 P 8 concerning 𝐶𝐶2 deduces to a point P 7

Proposition 2 Let 𝑃𝑃, 𝑋𝑋 be a straight bus route segment and a point on 𝑃𝑃 ; 𝐶𝐶 is a critical square Suppose I 1 I 2 is the observable segment of 𝑃𝑃 concerning 𝐶𝐶 𝐶𝐶 is an observable square

of 𝑋𝑋 if and only if 𝑋𝑋 stays between I 1 and I 2

Proof According to Proposition 1, 𝐶𝐶 is an observable square of 𝑋𝑋 if and only if 𝑋𝑋 stays on

the boundary or inside of O( 𝐶𝐶) On the other hand, 𝑋𝑋 stays on 𝑃𝑃 , thus, 𝐶𝐶 is an observable

square of 𝑋𝑋 if and only if 𝑋𝑋 belongs to the intersection of 𝑃𝑃 and O(𝐶𝐶) i.e., I 1 I 2

Theorem 3.1 Suppose 𝑏𝑏 is a bus route and C(𝑏𝑏) is the set of 𝑏𝑏’s leftmost and rightmost observable bounds concerning all the critical squares Let 𝑋𝑋 be an arbitrary point of 𝑏𝑏 such that 𝑋𝑋’s observable square set is not null Suppose 𝑌𝑌 is the leftmost bound of 𝑋𝑋’s observable segment, closest to 𝑋𝑋 Then the observable square set of 𝑋𝑋 is the subset of 𝑌𝑌 ’s

Proof Let us denote by 𝑃𝑃 the straight bus route segment containing 𝑋𝑋, then obviously 𝑃𝑃 must also contain 𝑌𝑌 Suppose 𝐶𝐶 is an observable square of 𝑋𝑋, we will prove that 𝐶𝐶 is also an observable square of 𝑌𝑌 We denote by P u P v the observable segment of 𝑃𝑃 with respect to 𝐶𝐶,

where P u and P v are the leftmost and rightmost observable bounds As 𝐶𝐶 is an observable square of 𝑋𝑋, according to Proposition 2, 𝑋𝑋 must stay between P u and P v Specifically, 𝑋𝑋 must

stay on the right side of P u and the left side of P v As 𝑌𝑌 is the leftmost observable bound closest to 𝑋𝑋, 𝑌𝑌 must either overlap or stay on the right side of P u On the other hand, as P v

stays on the right side of 𝑋𝑋, it also stays on the right side of 𝑌𝑌 Consequently, 𝑌𝑌 stays between

P u and P v Thus, 𝐶𝐶 must be an observable square of 𝑌𝑌 (see Fig 3)

Figure 3 Illustration of Theorem 3.1’s proof (𝑋𝑋 is an arbitrary point on a bus route segment

𝑃𝑃 𝑌𝑌 is the leftmost observable bound closest to 𝑋𝑋 If 𝐶𝐶 is a critical square observable by 𝑋𝑋, then it is also observable by 𝑌𝑌)

From Theorem 3.1, we deduce that to decide the optimal turn on locations of sensors on buses, we just need to determine the optimal turn on points among the leftmost observable

bounds We call these leftmost observable points critical points

In the following, we present the mathematical formulation of the considered problem We define binary variables 𝑠𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖1, 𝑠𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖2, 𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖1, 𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖2, 𝑏𝑏𝑗𝑗𝑖𝑖1, 𝑏𝑏𝑗𝑗𝑖𝑖2, 𝑎𝑎𝑖𝑖 as follows

• 𝑠𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖1 equals 1 if the critical square 𝑡𝑡 can be observed from the 𝑗𝑗th critical point on the first path of bus route 𝑖𝑖, is 0 otherwise

• 𝑠𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖2 equals 1 if the critical square 𝑡𝑡 can be observed from the 𝑗𝑗th critical point on the second path of bus route 𝑖𝑖, is 0 otherwise

• 𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖1 equals 1 if we choose to observe the critical square 𝑡𝑡 from the 𝑗𝑗th critical point on the first path of bus route 𝑖𝑖, is 0 otherwise

• 𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖2 equals 1 if we choose to observe the critical square 𝑡𝑡 from the 𝑗𝑗th critical point on the second path of bus route 𝑖𝑖, is 0 otherwise

• 𝑏𝑏𝑗𝑗𝑖𝑖1 equals 1 if the 𝑗𝑗th critical point on the first path of bus route 𝑖𝑖 is chosen to be a turn-on position, is 0 otherwise

• 𝑏𝑏𝑗𝑗𝑖𝑖2 equals 1 if the 𝑗𝑗th critical point on the second path of bus route 𝑖𝑖 is chosen to be a

turn-on posititurn-on, is 0 otherwise

• 𝑎𝑎𝑖𝑖 equals is 1 if a sensor is installed on bus route 𝑖𝑖, is 0 otherwise

Moreover, denoting 𝑚𝑚i1, 𝑚𝑚i2 as the number of critical points on the first and second path of bus route 𝑖𝑖, our problem is mathematically formulated as follows

�

𝑚𝑚𝑖𝑖2𝑗𝑗2=1

The objective function that has to be maximized is simply the number of critical squares

covered A square t is covered when four conditions below are satisfied:

- A bus route i is chosen to install sensor (i.e 𝑎𝑎𝑖𝑖 = 1)

- The square t can be observe from the j th critical point on the first path of i, i.e 𝑠𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖1 = 1 (or

the second path of i, i.e 𝑠𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖2 = 1)

- The j th critical point on the first path is chosen to observe t, i.e 𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖1 = 1 (or on the second

path is chosen to observe t, i.e 𝑠𝑠𝑗𝑗𝑗𝑗𝑖𝑖2 = 1)

- The j th critical point on the first path is chosen to be a sensor’s turn-on position, i.e 𝑏𝑏𝑗𝑗𝑖𝑖1 =

1 (or on the second path, i.e 𝑏𝑏𝑗𝑗𝑖𝑖2 = 1)

That’s why the objective function is written as above To make sure that an observed critical

square is not count twice or more, and to guarantee that we don’t use more than m sensors

or turn on them more than k times on some bus path, we need three additional constraints

Constraint (1) ensures each observable critical square is counted at most once in the objective function Constraint (2) guarantees that there are at most 𝑘𝑘 turn-on positions on any path The last constraint depicts that there are 𝑚𝑚 bus routes picked to install a sensor

Theorem 3.2 OSO belongs to the NP-hard class

Proof We prove this by reducing from MCP to OSO First, we set a bus map on a grid of

size |𝑈𝑈| × 3 (columns × rows), where 𝑈𝑈 = ∪𝛽𝛽𝑖𝑖=1 𝑉𝑉𝑖𝑖 Let the critical regions be at all squares

of the second row, hence each element in 𝑈𝑈 represents a square We then need to build a bus route including the depart and return parts for each set 𝑉𝑉i, ∀𝑖𝑖 = 1 … 𝛽𝛽, such that one of its paths does not touch any square on the map’s second row; and the other path goes through only the squares represented in 𝑉𝑉i Such a route always exists by passing alternatively these three rows at each column containing a square represented in 𝑉𝑉i (see Fig 4 for an example)

Figure 4 A corresponding bus map when 𝛽𝛽 = 3, 𝑉𝑉1 = {𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷, 𝐹𝐹}, 𝑉𝑉2 = {𝐴𝐴, 𝐶𝐶, 𝐷𝐷, 𝐸𝐸}, and 𝑉𝑉3 = {𝐵𝐵, 𝐹𝐹}

It is clear that if there is an optimal solution for OSO, the subset 𝑉𝑉 ′ containing 𝑉𝑉 ’s elements, which correspond to the routes in 𝑉𝑉 , is also an optimal solution for MCP The optimal solution 𝑉𝑉 ′ for MCP can also be converted to an optimal set of bus routes and turn-on positions for such instance of OSO in polynomial time Therefore, MCP is proven to be not harder than OSO That means OSO is NP-hard

In chapter 5, we introduce a greedy approximation algorithm for OSO We prove that in a general case, the algorithm gives an 𝑒𝑒−1

2𝑒𝑒−1 Moreover, in a simplified case where the

approximation ratio of two paths on each bus route are identical, it gives an approximation ratio of 1

2

4.1 Approximation algorithms

In computer science and operations science, approximation algorithms produce near-optimal results, which the furthest distance from the optimal result is always provable Because of the belief that P is different from NP, some problems are said to be unsolvable in polynomial time, hence approximation algorithms are proposed as an alternative to finding the optimal algorithm

Any approximation algorithm must be accompanied by a theoretical proof that certifies the efficiency of the result produced in the worst case Algorithms of this class differ from meta-heuristics, such as genetic or simulated annealing algorithms, in that meta-heuristic algorithms do not guarantee the quality of results by theoretical proof, although in practice they are often superior to approximation algorithms

In this thesis, the greedy approximation algorithms provide results with provable efficiency, thereby serving as the basis for other algorithms to compare

In this thesis, two meta-heuristic algorithms, genetic algorithm and simulated annealing algorithm, are used as independent models to solve the OSO problem, which is an NP-hard problem They are appropriate methods to verify efficiency of the approximation algorithm,

Chapter 4 Theoretical Background

approach is decent, the experimental results produced by it should be competitive to the ones produced by the chosen meta-heuristics It is indeed true, and we will show the experimental results supporting this observation later in this thesis

4.2.1 Genetic algorithm

Genetic algorithm (GA) is a technique for solving combinatorial optimization problems Among the meta-heuristic algorithms, GA is a well-known algorithm, which is inspired from the biological evolution process [28] GA mimics the Darwinian theory of survival of the fittest in nature GA was proposed by J.H Holland in 1992 The basic elements of GA are chromosome representation, fitness selection, and biological-inspired operators Holland also introduced a novel element namely, Inversion that is generally used in implementations

of GA [29]

GA simulates the evolution of an organism Its basic idea is to treat each solution (each individual) as a chromosome, which contains many genes Initially, a population of many individuals is created Over generations, depending on a given fitness function, individuals will interbreed to create new individuals, or they will be mutated, or they will be eliminated

by natural selection, and so on The process will stop when certain conditions are satisfied, for example the quality of the population does not increase after a few consecutive generations During the whole process, the best instance that ever existed will be selected as the result of the algorithm

The skeleton of a simple genetic algorithm will have the following structure:

Algorithm 1: Genetic algorithm

Input: The definition of individual (solution) and gene, a fitness function

Output: The best solution found after the algorithm halted

Initialize a population P of POP_SIZE solutions

Initialize best solution s* = the best solution in P

while not meet STOP CONDITION do

Crossover (by probability pcross) pairs of solutions from P -> new solutions

Mutate (by probability pmut) solutions in P -> new solutions

Assign s* = the best solution found in P and newly created solutions

Build a new population P’ from P and newly created solutions based on f

Assign P = P’

return s*

There are lots of other variations for genetic algorithms, and the above procedure is one of the most general instances Note that:

- The initialized population should contain randomized solutions Sometimes, the solutions may be chosen from a specific area of the search space where optimal solutions are likely to

be found

- The crossover operator takes two or more solutions (parents) and merges them in a way that creates new feasible solutions (children) The children are supposed to inherit good characteristics of their parents, hence likely being a better solution than their ancestors In each generation, many crossover operations can be done, depending on a given probability

pcross

- The mutation operator takes one solution and modifies it to make a new solution, by a given probability pmut It enhances the diversity of the population, and avoids converging to a local optimum

- The new population P’ of the next generation can be greedily built from P and newly created solutions Occasionally, dummy or bad solutions are kept to ensure the diversity of the population

- There are many kinds of stopping conditions, for example: the best solution has not changed for several generations, or the average fitness of the population has not increased for a long time, etc

4.2.2 Simulated annealing

Simulated annealing (SA for short) was first applied to optimization problems by S Kirkpatrick et al [30] and V.Cerny [31] In the book "Metaheuristics: From design to implementation" of El-Ghazali Talbi [32], the author described almost every aspect of SA

in detail It is a meta-heuristic to approximate an optimal solution in a large search space for

an optimization problem The idea of the SA algorithm is derived from physical metallurgy The metal is heated to high temperatures and cooled slowly so that it crystallizes in a low energy configuration

SA is chosen as a method to solve this OSO problem because of its simplicity and efficiency

It allows for a more extensive search for the global optimal solution, and can even find a global optimal solution if it runs for enough time

The skeleton of a simulated annealing algorithm will have the following structure:

Algorithm 2: Simulated annealing algorithm

Input: A way to compare two solutions, the function p(T, s, s’), the cooling function

Initialize current solution s = any solution s0

Initialize best solution s* = s

Initialize T = Tmax

while T > 0 do

for i from 1 to L do

Generate a random neighbor s’ of s

if s’ is better than s then

- In theory, the initial solution s0 can be any valid solution and it does not affect the quality

of SA However, when the solution searching space is too large, a good initial solution can

be a suitable approximation for the global optimum in a short amount of time

- A neighbor s’ of s is a solution which is not much differ from s

- The probability p(T, s, s’), also called move acceptance probability, must satisfy that better solutions have more chance to be chosen; and the higher the temperature T, the more likely that the move is going to be done

- The cooling function cooling(T) returns a number slightly smaller than T, which demonstrates a cooling process in metallurgy In theory, the higher Tmax and L are the higher chance for the optimal solution to be discoverable However, to save computation energy, these three hyperparameters should be carefully tuned

4.3 Research methodology

The research method to solve the OSO problem includes the following steps:

- Step 1: Prove that the OSO problem is an NP-hard problem (done in the previous chapter) Once proven, the optimal algorithm will no longer be searched, but instead we will look for good approximation algorithms and meta-heuristic algorithms

- Step 2: Build an approximation algorithm The approximation algorithms are prioritized to

be searched first because the goodness of the solution they provide compared to the optimal solution can be guaranteed From there, we can use it as a comparison base for other algorithms

- Step 3: Construct meta-heuristic algorithms In experiment, meta-heuristic algorithms are really effective problem solving tools In problems with large search spaces such as OSO problems, meta-heuristics are often one of the first options to solve the problem that experienced researchers will mention

- Step 4: Create tests and run tests of the built algorithms This is an indispensable step to confirm whether the algorithms are effective in practice The dataset is built directly from the bus map of the city (in particular, Hanoi), thereby ensuring the objectivity of the test

- Step 5: Based on the results, draw conclusions about the feasibility of the proposed algorithms

5.1 Approximation algorithms

5.1.1 Greedy algorithm and proof of effectiveness

5.1.1.1 Greedy approximation algorithm

We develop a greedy algorithm named grOSO to solve the problem The algorithm will find

approximation ratios for both the general and the simplified scenarios The main idea is as

follows For each bus route, we desire to determine the maximum observable critical square set, which is the biggest set of critical squares observable by a sensor mounted on that bus,

when the sensor is turned on at most 𝑘𝑘 times on each path of the bus route We will use a heuristic to find an observable critical square set whose size is as large as possible The set

is called sub-maximum observable critical square set The following process is repeated 𝑚𝑚 times:

1 For each bus route, we heuristically determine a sub-maximum observable critical square set

2 The bus route with the largest sub-maximum observable critical square set is selected to install a sensor

3 After that, we remove the chosen bus route from the map and change all items of its maximum observable critical square set into non-critical squares

sub-Figure 5 The remaining map after removing bus 1 from the map in Fig 1, and the greedy

process continues

Chapter 5 Proposed solution

Finally, we have 𝑚𝑚 selected bus routes with 2𝑘𝑘 sensor’s turn-on positions on each route, which are our greedy algorithm’s results Fig 5 shows the remaining map after removing bus 1 from the map in Fig 1 The critical squares 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷 and 𝐸𝐸 are changed to non-critical squares

The grOSO pseudo-code is presented in Algorithm 3 The input includes the bus map, the

sensor radius (i.e., 𝑟𝑟), the number of sensors (i.e., 𝑚𝑚), and the number of turning on points per sensor (i.e., 𝑘𝑘) The map 𝑏𝑏𝑏𝑏𝑠𝑠𝑏𝑏𝑎𝑎𝑝𝑝 consists of the bus routes on a 𝑝𝑝 × 𝑞𝑞 grid and positions

of critical squares The output is a set 𝑆𝑆 of observable critical squares For the sake of brevity,

we do not output the selected bus routes and the sensors’ turn-on positions The function

submaxSet(busMap, r, k, j) returns a sub-maximum observable critical square set of the bus

route 𝑗𝑗 given the information of the bus map 𝑏𝑏𝑏𝑏𝑠𝑠𝑏𝑏𝑎𝑎𝑝𝑝, the sensor radius 𝑟𝑟 and the sensor’s turn-on limit 𝑘𝑘 This submaxSet function is calculated in polynomial time by a greedy

process discussed in Section 5.1.2

Algorithm 3: Greedy approximation algorithm

Input: Bus map busmap, sensor radius r, number of sensors m, no turn-on time k

Output: The constructed solution S

𝑌𝑌 ←∅

if |𝑌𝑌 | ≥ |𝑋𝑋| then

𝑋𝑋 ← 𝑌𝑌, 𝑏𝑏 ← 𝑗𝑗

𝑆𝑆 ←𝑆𝑆 ∪𝑋𝑋 Bus route 𝑏𝑏 is marked as chosen Squares in 𝑋𝑋 is changed to be non-critical return 𝑆𝑆

5.1.1.2 The lower bound of performance ratio

The efficiency of our greedy algorithm mainly depends on the function submaxSet The

obtained approximation ratio follows the approximation method of C Chekuri and A Kumar [23] for MCP with group budget constraints We prove that for any 𝛼𝛼 ≥ 1, if the algorithm submaxSet always returns a sub-maximum observable critical square set which can cover at

1

then our greedy approach is a 1

𝛼𝛼+1-approximation algorithm By using this claim, in Section

5.1.2, when a greedy method is used for the function submaxSet with 𝛼𝛼 = 1 − 1𝑒𝑒, our algorithm for the OSO problem is proved to be 1

𝛼𝛼+1= 2𝑒𝑒−1𝑒𝑒−1-approximation After that, by applying a dynamic programming method to a simplified scenario of OSO, we extend this result to show another approximation algorithm of ratio 1

2 At the end of this section, we present a simpler version of C Chekuri and A Kumar’s proof that includes modifications fitting our problem formulation

We assume that the greedy algorithm chooses the bus route 𝑖𝑖 in the 𝑖𝑖th iteration of the outer for-loop by renumbering the bus routes Let 𝑂𝑂𝑃𝑃𝑂𝑂 be a fixed optimal solution, and 𝑗𝑗1, 𝑗𝑗2, …,

𝑗𝑗m be the indices of the bus routes that 𝑂𝑂𝑃𝑃𝑂𝑂 installs sensors We additionally assume that if both the solutions of the greedy algorithm and 𝑂𝑂𝑃𝑃𝑂𝑂 choose the bus route 𝑖𝑖, then 𝑗𝑗i = 𝑖𝑖, by reordering the chosen bus routes in 𝑂𝑂𝑃𝑃𝑂𝑂 Let 𝑆𝑆i be the set of new observable critical squares determined by the greedy algorithm when the bus route 𝑖𝑖 is chosen Let 𝑂𝑂i be the covered set of new observable critical squares determined by 𝑂𝑂𝑃𝑃𝑂𝑂 when the bus route 𝑗𝑗i is chosen Moreover, 𝑆𝑆 = ∪𝑚𝑚

𝑖𝑖=1 𝑆𝑆𝑖𝑖 and 𝑂𝑂 =∪𝑚𝑚

𝑖𝑖=1 𝑂𝑂𝑖𝑖 are denoted as the numbers of observable critical squares that are covered by the greedy algorithm and 𝑂𝑂𝑃𝑃𝑂𝑂 We need to prove that for any 𝛼𝛼

≥ 1, if the algorithm submaxSet always returns a sub-maximum observable critical square

set which can cover at least 1

𝛼𝛼 as many critical squares as a maximum observable critical

square set of a bus route, then |𝑆𝑆| ≥ 𝛼𝛼+11 |𝑂𝑂| To fulfill this task, the following critical observation (Lemma 5.1) is indispensable

Lemma 5.1 For any 𝛼𝛼 ≥ 1, if the algorithm submaxSet always returns a sub-maximum observable critical square set that can cover at least 1

𝛼𝛼 as many critical squares as a

maximum observable critical square set of a bus route, then ∀𝑖𝑖, 1≤ 𝑖𝑖≤ 𝑚𝑚: |𝑆𝑆 i | ≥ 𝛼𝛼1|𝑂𝑂i − 𝑆𝑆|

Proof If 𝑆𝑆i = Oi , Oi ⊆ 𝑆𝑆 So |𝑆𝑆i| ≥ 𝛼𝛼1 |Oi − 𝑆𝑆| = 0 We assume that 𝑆𝑆i ≠ Oi, which means Oi

is not chosen to be covered in the 𝑖𝑖th operation of the outer for-loop in the greedy algorithm

Since submaxSet always returns a sub-maximum observable critical square set with the

covering condition, 𝑆𝑆i will be at least 1

𝛼𝛼 as large as the maximum size of all maximum

observable critical square sets of all bus routes Consequently, we have |𝑆𝑆i| ≥ 𝛼𝛼1 |Oi -

∪𝑖𝑖−1

ℎ=1 𝑆𝑆ℎ| In addition, ∪𝑖𝑖−1

ℎ=1 𝑆𝑆ℎ⊆ 𝑆𝑆, therefore |𝑆𝑆i| ≥ 𝛼𝛼1|𝑂𝑂i − 𝑆𝑆|

Lemma 5.1 shows that the number of newly covered squares 𝑆𝑆i is always not less than 1

𝛼𝛼 as many critical squares as a

maximum observable critical square set of a bus route, then the greedy algorithm grOSO is

a 1

𝛼𝛼+1-approximation algorithm for the OSO problem

Proof We have to prove that |𝑆𝑆| ≥ 𝛼𝛼+11 |𝑂𝑂| Since

|𝑆𝑆| = ∑𝑚𝑚𝑖𝑖=1 |𝑆𝑆𝑖𝑖| ≥ ∑𝑚𝑚𝑖𝑖=1 1𝛼𝛼|𝑂𝑂𝑖𝑖− 𝑆𝑆| ≥1𝛼𝛼(| ∪𝑚𝑚𝑖𝑖=1 𝑂𝑂𝑖𝑖| − |𝑆𝑆|) =𝛼𝛼1|𝑂𝑂| −1𝛼𝛼|𝑆𝑆| ,

therefore 𝛼𝛼+1

𝛼𝛼 |𝑆𝑆| ≥1𝛼𝛼|𝑂𝑂| ⇔ |𝑆𝑆| ≥𝛼𝛼+11 |𝑂𝑂|

5.1.2 Sub-maximum observable set calculation

In grOSO, the function submaxSet(busMap, r, k, j) returns a sub-maximum set of observable

critical squares on the bus route 𝑗𝑗 when the bus map 𝑏𝑏𝑏𝑏𝑠𝑠𝑏𝑏𝑎𝑎𝑝𝑝, the sensor radius 𝑟𝑟, and the sensor’s turn-on limit 𝑘𝑘 are given To guarantee the polynomial running time of grOSO, submaxSet ’s time complexity has to be polynomial In the following, we introduce our

approaches for the simplified and general scenarios

5.1.2.1 Simplified scenario

5.1.2.1.1 Dynamic programming-based approximation algorithm

The simplified scenario indicates the two paths of every bus route are identical Specifically, each bus route’s two paths are the same polyline; and the total number of allowed turn-on positions to be determined on that polyline is 2𝑘𝑘

The NP-hard proof in this scenario is straightforward since the same method in Section 3.3 can be used We first make an assumption which is reasonable since it usually appears in practical as follow For any pair of (𝐴𝐴, 𝑏𝑏), where 𝐴𝐴 is a critical square and 𝑏𝑏 is a bus route,

if a sensor can observe 𝐴𝐴 from some positions on 𝑏𝑏, all sensor’s turn-on positions on 𝑏𝑏 where

𝐴𝐴 is observed are lying on exactly one closed segment [l Ab, 𝑟𝑟Ab], see Fig 6a for an illustration This interval can be found in polynomial time by using computational geometry

techniques We then present a dynamic programming approach for the function submaxSet

to construct the maximum observable critical square set for any bus route In this simplified

submaxSet function is always maximum thus the number 𝛼𝛼 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

Suppose that there are 𝑑𝑑 critical squares which can be observed from at least one sensor’s turn-on position on a bus route, as the above assumption, there are 𝑑𝑑 critical points which

are the left endpoints (l Ab ) of the observable intervals ([l Ab, 𝑟𝑟Ab]) (an illustration in Fig 6b)

It is obvious that a sensor should only be turned on at critical points for the highest efficiency

Figure 6 (a) [l Ab, 𝑟𝑟Ab] is the unique close segment that contains all sensor’s turn-on positions

on the bus route 𝑏𝑏 where the critical square 𝐴𝐴 is observed (b) There are 𝑑𝑑 critical squares observed by turning on sensor from bus route 𝑏𝑏 (in this figure, 𝑑𝑑 = 5) Each square 𝑖𝑖 can be

observed by a sensor turned on at somewhere in the middle of the interval [l ib, 𝑟𝑟ib] We then have 𝑑𝑑 critical points which are the left endpoints (l ib, where 𝑖𝑖 = 1, … , 𝑑𝑑) of such intervals The below steps describe our method

1 The 𝑑𝑑 points are sorted in an increasing order of their distances to the bus route’s starting point

2 We compute 𝑔𝑔(𝑖𝑖, 𝑗𝑗) for all pair (𝑖𝑖, 𝑗𝑗), 0 ≤ 𝑖𝑖 < 𝑗𝑗 ≤ 𝑑𝑑, which is the number of critical squares observable from the 𝑗𝑗th critical point but not observable from the 𝑖𝑖th critical point (the 0th

critical point is set as −∞)

3 Let 𝑓𝑓 (𝑖𝑖, 𝑗𝑗) be the maximum number of observable critical squares if we are allowed to turn on the sensor 𝑗𝑗 times at critical points from the (𝑖𝑖+1)th point to the 𝑑𝑑th point (a sensor

can be turned on more than one time at the same point) The function submaxSet should

return 𝑓𝑓(0, 2𝑘𝑘) The dynamic programming formula for 𝑓𝑓 is following

• Base case: 𝑓𝑓(𝑖𝑖, 0) = 0, where 0 ≤ 𝑖𝑖 ≤ 𝑑𝑑;

• General case: 𝑓𝑓(𝑖𝑖, 𝑗𝑗) = maxu (𝑓𝑓(𝑏𝑏, 𝑗𝑗 − 1) + 𝑔𝑔(𝑖𝑖, 𝑏𝑏)), where 0 ≤ 𝑖𝑖 < 𝑏𝑏 ≤ 𝑑𝑑 and 𝑗𝑗 > 0

The pseudo code of submaxSet is represented in Algorithm 4

Algorithm 4: Dynamic programming for submaxSet

Input: Bus map busmap, sensor radius r, no turn-on time k, the current bus b

Output: The maximum number of critical squares a sensor on b can observe, and a

maximum set of critical squares that sensor can observe

procedure submaxSet(𝑏𝑏𝑏𝑏𝑠𝑠𝑏𝑏𝑎𝑎𝑝𝑝, 𝑟𝑟, 𝑘𝑘, 𝑏𝑏)

∀ critical square 𝐴𝐴, calculate the interval [l Ab, 𝑟𝑟Ab]

Sort 𝑑𝑑 critical points in increasing order

Compute 𝑔𝑔(𝑖𝑖, 𝑗𝑗) for all (𝑖𝑖, 𝑗𝑗), 0 ≤ 𝑖𝑖 < 𝑗𝑗 ≤ 𝑑𝑑

Since Algorithm 4 produces the maximum observable critical square set for any bus route 𝑏𝑏,

from Theorem 5.1, the algorithm grOSO is 1

𝛼𝛼+1= 1+11 =12 -approximation for this simplified version of OSO

5.1.2.2 General scenario

5.1.2.2.1 Greedy approximation algorithm

The general scenario indicates there is no constraint on the bus paths Following Proposition

1 in Section 3.2, a sensor should only be turned on at critical points for the highest efficiency