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Author: 王皓正
Hao-cheng Wang
Thesis Title: 在一般性弦環圖上找邊互斥展開樹之研究
Finding Edge-Disjoint Spanning Trees in General Chordal Rings
Advisor: 徐俊傑
Chiun-chieh Hsu
Committee: 賴源正
Yuan-cheng Lai
Da-ren Chen
Degree: 碩士
Department: 管理學院 - 資訊管理系
Department of Information Management
Thesis Publication Year: 2011
Graduation Academic Year: 99
Language: 中文
Pages: 37
Keywords (in Chinese): 獨立展開樹點互斥展開樹邊互斥展開樹弦環圖連結網路容錯
Keywords (in other languages): independent spanning trees, node-disjoint spanning trees, edge-disjoint spanning trees, chordal rings, interconnection network, fault-tolerant
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在分散式的網路架構下,獨立展開樹(independent spanning trees)的建構能讓根節點便於從中找到數條互斥的路徑。這些互斥路徑,可使訊息廣播(broadcasting)具有容錯性,容許多個節點失效時仍可完成廣播程序;並且可對同一節點進行平行傳輸,達到分散負載及縮短完成時間的目的;此外,互斥路徑的平行傳輸方式,也可被應用於訊息安全分送。
近年,有許多學者進行互斥展開樹的研究,其中在1989年Zehavi和Itai兩位學者猜測在k-連結圖(k-connected graph)任意節點為根節點r的情況下,都存在k棵互斥展開樹,但目前仍只有k≤4的情況下被證實,至於k≥5的情況仍是未被解開的問題。
本文將探討一般性弦環圖(general chordal rings)互斥展開樹的建構問題。在環形圖(circulant graph)家族裡,分支度為四的弦環圖(chordal ring)及遞迴環形圖(recursive circulant graph)已經找到點互斥展開樹之建構方法;由於一般性弦環圖需要比前兩者建構更多的互斥展開樹,因此建構的難度也隨之提高。本文將更進一步分析一般性弦環圖基本路徑和延伸路徑的組成,以及以弦將節點切割的情況,提出在一般性弦環圖建構與分支度(δ_m)相同數量的邊互斥展開樹,並證明其演算法之正確性。

In a distributed network, independent spanning trees can be constructed by joining several disjoint paths rooted on the same node. These disjoint paths can be used for fault-tolerant message broadcasting, which can guarantee successful transmission of data even if some nodes and links fail. Another application of independent spanning trees is concurrent transmission of data from one node along disjoint paths to another, which can distribute the transmission load and reduce transmission time. In addition, we can utilize the disjoin paths to accomplish security message distribution.
In 1989, Zehavi and Itai conjectured that, for any node r in the k-connected graph G, there exist k disjoint spanning trees of G rooted at r. This conjecture has been proved only for k-connected graph with k ≤ 4, and it is still open for arbitrary k-connected graph when k ≥ 5.
In this thesis, we will propose a method of constructing multiple disjoint spanning trees on general chordal rings. Four-degree chordal rings and recursive circulant graphs are two kinds of circulant graphs, for which node-disjoint spanning trees have been confirmed. Since the number of disjoint spanning trees needed to construct on the general chordal ring is larger than those on the two graphs, it is more difficult to construct the disjoint spanning trees on the general chordal ring. First we explore topological properties of the general chordal ring. Then we use the proposed topological properties to find disjoint elementary paths and extend paths on the general chordal ring. Based on these paths, an algorithm is presented for constructing edge-disjoint spanning trees and the correctness of its algorithm is proved, where the number of edge-disjoint spanning trees is just the same as the connectivity of the general chordal ring.

第一章 緒論 1 1.1 研究動機與目的 1 1.2 論文架構 3 第二章 相關研究探討 4 2.1 弦環圖(Chordal Rings) 4 2.2 遞迴環形圖(Recursive Circulant Graphs) 8 第三章 邊互斥展開樹之演算法 13 3.1 建構路徑分類表 13 3.2 建構邊互斥展開樹 20 3.3 驗證邊互斥展開樹 22 第四章 結論與未來發展 32

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