邊界覆蓋是無線感測網路中的重要問題，應用於偵測企圖穿越保護區邊界的入
侵者。本論文考慮三個邊界覆蓋的相關問題。問題 1 是在混合的感測網路中，
尋找最佳的可移動的傳感器的位置，在達成 1-邊界覆蓋的前提下最大化網路壽
命。有文獻中曾經給出問題 1 的演算法，可惜的是，我們發現該演算法有錯。
在本論文中，我們提出問題 1 的正確演算法且只需要線性時間。問題 2 和問題
3 都是考慮一般的感測網路中，在達成 k-邊界覆蓋的前提下，最大化網路壽命。
其中，問題 2 的次要目標是最小化路徑開關數量，問題 3 的次要目標則是最小
化 k-邊界開關數量。有文獻曾經給出路徑開關數量的理論下界的證明，可惜的
是，我們發現該證明不完備。在本論文中，我們完成了開關數量的理論下界的
證明，提供了兩種達到理論下界的建構方式。 k-邊界開關是本論文提出的新概
念，我們證明了在某些前提下，問題 2 的其中一個達到理論下界的建構方式可
以同時得到問題 3 的最佳解。

Barrier coverage is an important problem in wireless sensor networks and is used to guarantee the detection of an intruder that tries to cross the barrier of a protected region. In this thesis, three problems of barrier coverage will be discussed. Problem 1 is to find an optimal arrangement of the mobile sensors in the hybrid sensor network to maximize lifetime of the network on the premise of achieving 1-barrier coverage. There is an algorithm for Problem 1 in the literature. Unfortunately, we find that the algorithm is incorrect. In this thesis, we propose a correct algorithm for Problem 1 and prove that only linear time is required. Both Problems 2 and 3 consider the problem of maximizing the lifetime of k-barrier coverage in the ordinary sensor network. Under the premise of achieving k-barrier coverage, maximize the network lifetime. Among them, the second objective of Problems 2 is to minimize the number of path switches, whereas that of Problems 3 is to minimize the number of k-barrier switches. There is a proof of the theoretical lower bound on the number of path switches in the literature. Unfortunately, we find that the proof is incomplete. In this thesis, we have completed the proof and provided two constructions to achieve the theoretical lower bound. The concept of k-barrier switches is a new concept proposed in this thesis. We have proven that under certain premises, one of the optimal constructions for achieving the theoretical lower bound in Problems 2 can simultaneously obtain an optimal solution for Problem 3.

Contents
誌謝 i
Abstract ii
中文摘要 iii
Content v
List of Figures vi

1 Introduction 1

2 Preliminaries 6

3 Problem 1 8
3.1 System Assumptions of Problem 1 8
3.2 Application Background 9
3.3 Counterexample 11
3.4 Our Solution to Problem 1 13

4 Problem 2 21
4.1 System Assumptions of Problem 2 21
4.2 Problem Definition 22
4.3 Theoretical Lower Bound 24
4.4 Two Constructions That Can Achieve The Lower Bound 35
4.4.1 The First Construction 35
4.4.2 The Second Construction 41

5 Problem 3 46
5.1 Problem Definition 47
5.2 Some Properties 48

6 Concluding Remarks 54

References 54

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