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Author: 郭啟容
Chi-Jung Kuo
Thesis Title: 凱利圖上回饋問題及獨立擴展樹之研究
On the Study of Feedback Problems and Independent Spanning Trees in Cayley Graphs
Advisor: 徐俊傑
Chiun-Chieh Hsu
Committee: 王有禮
Yue-Li Wang
賴源正
Yuan-Cheng Lai
蕭培墉
Pei-Yung Hsiao
陳永昇
Yong-Sheng Chen
Degree: 博士
Doctor
Department: 管理學院 - 資訊管理系
Department of Information Management
Thesis Publication Year: 2010
Graduation Academic Year: 98
Language: 英文
Pages: 68
Keywords (in Chinese): 安全訊息發送容錯廣播獨立擴展樹三價凱利圖有向圖不完全旋轉圖旋轉圖回饋邊集合回饋點集合
Keywords (in other languages): secure message distribution., fault-tolerant broadcasting, independent spanning tree, trivalent Cayley graph, directed cycle, incomplete rotator graph, rotator graph, feedback arc set, feedback vertex set
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旋轉圖(rotator graph)及三價凱利圖(trivalent Cayley graph)同屬於凱利圖(Cayley graphs)家族的成員。本論文探討在旋轉圖與不完全旋轉圖(incomplete rotator graphs)的最小回饋點/邊集合(feedback vertex/arc set)問題;以及在三價凱利圖上建構獨立展開樹的方法。
一個圖的回饋點/邊集合(簡稱FVS/FAS)是此圖的點/邊子集合,而此圖的每一個環路(cycle)至少有一個點/邊是在此子集合內,將此FVS/FAS從圖移除後,此圖剩下的部份將不存在任何的環路,而最小FVS/FAS是指包含最少點/邊的FVS/FAS。Hsu和Lin [22]在n!節點的旋轉圖上,提出一個時間複雜度為O(nn-3) 的建構FVS演算法:另外,他們證明此FVS的大小為n!/3而且是最小的。本論文提出一個有效率的演算法,此演算法同樣能在n!節點的旋轉圖上建構出大小為n!/3的FVS,而時間複雜度則只有O(n!)。換言之,我們可以旋轉圖節點數的線性時間複雜度(linear time complexity)之演算法得到最佳解。另外我們提出一個建構旋轉圖FAS的精簡集合式,並證明此集合式建構出的FAS是最小的,而此集合式很容易撰寫成一個有效率的演算法。最後,我們也在不完全旋轉圖上,提出一個建構最小FAS的集合式。
而建立多顆獨立擴展樹(independent spanning trees)在網路通訊方面有很多的應用,如容錯廣播(fault-tolerant broadcasting)和安全訊息發送(secure message distribution)。Cheriyan和Maheshwari [8] 證明所有3-連結的圖一定可以在O(|V||E|)的時間複雜度,建構出3顆獨立擴展樹。三價凱利圖是由Vadapalli and Srimani 在 [38] 所提出的節點分支度為3的互連網路(interconnection network)。本論文提出一個以三價凱利圖的任一節點當作樹根(root)建構3顆獨立擴展樹的線性時間演算法,而此演算法植基於一組描述每個節點的父節點的簡潔規則,因此使得這個演算法也很容易平行化。


Rotator graph and trivalent Cayley graph are two members of Cayley graphs. This dissertation studies the minimum feedback vertex/arc set in rotator and incomplete rotator graphs, as well as the independent spanning trees of trivalent Cayley graphs.
A feedback vertex/arc set (abbreviated as FVS/FAS) of a graph is a subset of the vertices/arcs which contains at least one vertex/arc for every cycle of the graph. Removing the FVS/FAS from the graph makes the remaining graph acyclic. A minimum FVS/FAS is an FVS/FAS which contains the smallest number of vertices/arcs. For a rotator graph with M = n! nodes, Hsu and Lin [22] first proposed an algorithm which constructed an FVS with time complexity O(nn-3). In addition, they found that the size of the FVS is M /3, which was proved to be minimum. In this dissertation, we present an efficient algorithm which constructs an FVS for a rotator graph in O(M) time and also obtains the minimum FVS size M /3. In other words, this algorithm derives the optimal result with the time complexity which is linearly proportional to the number of nodes in the rotator graph. In addition, we present a concise formula for finding an FAS for a rotator graph and prove that the FAS is minimum. This formula can be easily implemented by an efficient algorithm which obtains a minimum FAS in a rotator graph. Finally, we also present a concise formula for finding a minimum FAS in an incomplete rotator graph in this dissertation.
The construction of multiple independent spanning trees has many applications in network communication, such as fault-tolerant broadcasting and secure message distribution. Cheriyan and Maheshwari [8] showed that three independent spanning trees can be constructed in O(|V||E|) time for every 3-connected graph. In this dissertation, we present a linear time algorithm to construct three independent spanning trees rooted at any node in a trivalent Cayley graph, which was proposed by Vadapalli and Srimani in [38] for designing the topology of an interconnection network with constant regularity of node degree 3. In particular, our algorithm relies on a set of concise rules that describe the parent of nodes in each tree. Therefore, the construction scheme can be easily parallelized.

中文摘要 i ABSTRACT iii 誌 謝 v TABLE OF CONTENTS vi LIST OF FIGURES viii LIST OF TABLES ix Chapter 1 Introduction 1 1.1 Feedback Vertex/Arc Set Problem 1 1.2 Independent Spanning Trees Problem 5 1.3 Outline of the Dissertation 8 Chapter 2 An Efficient Algorithm for Minimum Feedback Vertex Sets in Rotator Graphs 9 2.1 Preliminaries 9 2.2 The Algorithm for Minimum FVS in Rotator Graphs 13 Chapter 3 Minimum Feedback Arc Sets in Rotator and Incomplete Rotator Graphs 18 3.1 Preliminaries 18 3.2 A minimum FAS in Rotator Graphs 20 3.3 A minimum FAS in Incomplete Rotator Graphs 23 Chapter 4 Independent Spanning Trees on Trivalent Cayley Graphs 29 4.1 Trivalent Cayley Graphs and Their Cycle Structure 29 4.2 Construction of ISTs 37 4.3 Correctness 47 Chapter 5 Conclusion 62 REFERENCES 63

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