在圖論中偵測關節點是一個很基本的問題，此問題可以應用於
網路流通穩定性的分析。在此篇論文中，我們主要解決於圖形之邊
容錯模型中維持關節點的偵測問題。在此邊容錯模型的設定中，
一個圖形中的邊允許被刪除(我們解釋為連結失敗)，抑或已經被
刪除的邊可以再被加回到圖形中(我們解釋為連結恢復)。此論文
達到使用O(mlog n) 的前置處理時間，經過此時間後，我們可
以花費~O(n + k^2)的時間回報在圖形中的所有關節點，其中m 與n
分別表示一個圖中的邊數與點數，而k 則表示在一個圖中經過
多少次的邊刪除或邊恢復的動作. 此篇論文研究結果相較於近期
Baswana et al. [SODA 2016:730{739] 的~O (nk)更新時間分析結果，其
於更具有一般性的完全動態模型中解決偵測關節點問題(此模型亦允
許圖形中邊或點的加入)，而我們在動態模型的特殊情況中，也就是
容錯模型，達到比Baswana et al. [SODA 2016:730{739]更有效率的更
新時間分析結果。
我們的研究結果主要發源於Baswana et al. [SODA 2016:730{739]
以及Nakamura and Sadakane [WALCOM 2017:295{307] 在近期研究
中的技巧，再巧妙地結合一種新的資料結構，此結構我們稱為「無後
邊樹」，並在此篇論文中給予此資料結構定義及使用. 更直覺來說，
前者的的技巧者主要貢獻於搜尋的加速；而後者(無後邊樹) 則使我
們相較於目前的方法，在找尋關節點上可以更有效率的處理資訊。

Identication of articulation points is a fundamental problem in
graph theory, which has applications in network stability analysis. In
this thesis, we focus on maintaining articulation points in a graph G in
the fault tolerant model, where edges in the graph may be deleted (representing
link failure) or deleted edges may be added back (representing
link recovery). We show that by using O(mlog n) preprocessing time,
we can report all articulation points in G in ~O(n + k^2) time, where m
and n denote the number of edges and vertices in G, respectively, while
k denotes the number of edge deletions or recoveries. This result has
improved a special case in the recent work by Baswana et al. [SODA
2016:730{739], which maintains articulation points in ~O(nk) time in the
more general fully dynamic model, where new vertices and new edges
may also be inserted to G.
Our result stems from a non-trivial combination of the recent techniques
by Baswana et al. [SODA 2016:730{739] and by Nakamura and
Sadakane [WALCOM 2017:295{307], with the use of a novel data structure
called no-back-edge tree dened in this thesis. Intuitively speaking,
the former techniques are for query speed-up, while the latter data
structure allows us to maintain and process less information than the
existing approach.

Contents
1 Introduction 6
1.1 Motivation and background . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Problem denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Preliminaries 8
3 DFS-based Scan-First Trees 12
4 Fault Tolerant Scan-First Tree Algorithm 16
4.1 Overview of our technique . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Procedure Traversal-within-Subtree . . . . . . . . . . . . . . . . . . . . . 17
4.3 Procedure Traversal-across-subtrees . . . . . . . . . . . . . . . . . . . . . 20
5 Time complexity analysis 24
6 Discussion 25
7 Conclusion 26

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