臺灣博碩士論文加值系統

隨著電腦科技的進步與網際網路（Internet）的成長，將數以千百計之處理器或通訊單元連結在同一個網路的需求正在不斷地增加。對於這樣的一個互連網路而言，網路的拓撲架構（Network Topology）是一個最重要的考量因素，因為網路的拓撲架構決定了這個網路的效能。在相關的參考文獻中，許多不同的網路拓撲架構曾被提出。在這些不同的拓撲架構中，超方體（Hypercube）因其擁有良好的性質，是最有名且最為廣泛探討的一個拓撲架構。
在與超方體相關的參考文獻中，許多不同的變形超方體曾被提出來以改善超方體的性質，其中非常重要的一種稱為「扭曲超方體」（Twisted Cube）。扭曲超方體是經由特定的法則來改變超方體內某些邊（Edge）的連結而得。雖然在某些性質上有改善，但是在原著中，只提出了建構奇數維度（Odd Dimension）扭曲超方體的建構法。這個缺點嚴重地限制了它的應用。
在本論文中，架構於奇數維度的基礎上，我們首先提出建構偶數維度(Even Dimension) 扭曲超方體的建構法則。然後，我們提出一個具有可漸進延伸性質的網路拓撲架構，這個拓撲架構我們稱之為「可漸進延伸之扭曲超方體」（Incrementally Extensible Twisted Cubes, IETQ）。藉由探討IETQ的幾種重要性質，我們證明IETQ繼承了扭曲超方體良好的性質，可是卻沒有扭曲超方體的一些限制。
環型 (Cycle, Ring, or Hamiltonian Cycle) 結構由於其良好的特性，也常使用於互連網路中。而探討環型結構是否可嵌入某特定之網路拓撲架構，一直以來都是研究的重點問題。一個具有可嵌入環型結構的網路 (A Pancyclic Network)，被定義為在這個網路內必須可嵌入包含長度由4開始到節點總數之所有的環型。因此，探討環型結構是否可嵌入某特定網路拓撲架構的問題，其實就是探討這個特定網路的拓撲架構是否具有環型結構可嵌入性。
除了扭曲超方體之外，「交錯超方體」 (Crossed Cube) 與「Möbius超方體」也是非常重要的兩個變形超方體，因為除了建構法則不同之外，三者重要的網路性質均十分相近。在本論文中，我們證明這三種超方體的變形均具有環型結構可嵌入性。
超方體之另一個重要的變形是「單向超方體」（Uni-directional Hypercube）。單向超方體是藉由更改在超方體內原本為無向性的邊為有向性而得。在使用光纖的大型互連網路中，由於將線路與傳輸設備加倍的費用極為高昂，因而廣泛使用單向超方體。在本論文中，我們將探討單向超方體幾項在文獻中尚未被探討過的性質。

With the continuing growth of computer technology and the Internet,
the demand for connecting hundreds or thousands processing or communication elements
in a network is increasing rapidly.
Network topology is one of the most crucial factors for interconnection networks
since it determines the performance of a network.
Many different topologies of interconnection networks have been proposed in the literature.
Among these topologies,
hypercube is the most famous and well studied one because of its nice properties.
A few variants have been proposed to enhance hypercube.
One of the most important variants is called twisted cube,
which is derived by changing the connection
of some links of a hypercube according to some specific rules.
Despite of the enhancements, a twisted cube can only be built with an odd number of dimension.
This drawback has seriously limited the applicability of a twisted cube.
In this Dissertation,
based on the odd case of a twisted cube,
we first propose a construction method for the even case of a twisted cube.
Then, a new family of interconnection networks which we call
the Incrementally Extensible Twisted Cube (\ietq) is proposed to eliminate
the limitation of incremental extensibility.
The proposed \ietq\ inherits all the merits but without the limitations owned by a twisted cube.
Several important combinatorial properties of the \ietq\ network
are studied in this Dissertation.
Because of its nice properties,
a ring structure (hamiltonian cycle) is widely used in interconnection networks.
The problem of embedding a ring structure into different topologies of networks
is always an important topic for research.
A pancyclic network is defined to be a network that
contains cycles of all lengths from 4 to
the total number of nodes $N$ in the network.
Therefore, to determine whether a ring can be embedded into a specific topology of network
is to determine if this topology of network is a pancyclic network or not.
A crossed cube can be constructed recursively from lower-dimension crossed cubes.
By joining every vertex of the first crossed cube of dimension $n-1$
to the vertex of the second crossed cube of the same dimension
following a specific designed relation, called ''pair related relation'''',
we can construct a crossed cube of dimension $n$.
Similar to the cases of a crossed cube and a twisted cube,
we can use one 0-M\"obius cube of dimension $n-1$
together with one 1-M\"obius cube of the same dimension
to create a 0-M\"obius cube or 1-M\"obius cube of dimension $n$.
Crossed cube and M\"obius cube together with twisted cube
are the most important three variants of hypercube
as they all have a diameter that is only half of the size
that a hypercube with the same number of nodes has.
Though a hypercube is not a pancyclic network,
we prove in this Dissertation that these three variants of hypercube
are all pancyclic networks by induction.
A \ud\ hypercube ($UQ$) is also one of the most important variants of hypercube.
$UQ$ is derived by replacing the links in a hypercube with simplex \linebreak \ud\ links;
thus, it can be used in a network where the transmission between nodes
is simplex and \ud\ only.
This $UQ$ topology is well adopted especially in large scale interconnection networks
built with optical fiber where double the links and transmission equipments cannot be afforded.
In this Dissertation, we also investigate several important combinatorial properties
of $UQ$ which have never been studied in the literature before.

Cover
Abstract
Contents
List of Tables
List of Figures
List of Algorithms
1 Introduction
1.1 Hypercube
1.2 Definitions and Performance Problems
1.3 Variants of Hypercube
1.4 Organization of this Dissertation
2 Incrementally Extensible Twisted Cubes
2.1 Twisted Cube
2.2 Even Case of a Twisted Cube
2.3 Construction of an IETQn(N)
2.4 Short Roution Path Algorithm
3 Combinatorial Properties of an IETQn(N)
3.1 Properties of an IETQn(N)
3.2 Pancyclic Property
3.3 Properties Comparison with Other Topologies of Networks
4 Pancyclic Property of Crossed Cube, Mobius Cube,and Twisted Cube
4.1 Crossed Cube
4.2 Mobius Cube
4.3 Pancyclic Property of a TQn
5 Combinatorial Properties of a Uni-directional Hypercube
5.1 Basic Properties of a UQn
5.2 Shortest Routing Path Algorithm
5.3 Containers of a UQn
5.4 k-wide Diameter and Fault Diameter
6 Conclusion and Future Work

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TITLE = "{Fault-Tolerant Ring Embedding in Faulty Crossed Cubes}",
BOOKTITLE = "Proceeding of World Multiconf. Systems, Cybernetics and Information SCI''2000",
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AUTHOR = "P. Cull and S. M. Larson",
TITLE = "{The M\"obius Cubes}",
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VOLUME = ൴",
PAGES = 螋--702"}
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AUTHOR = "S. Latifi and S. Q. Zheng and N. Bagherzadeh",
TITLE = "{Optimal ring embedding in hypercubes with faulty links}",
BOOKTITLE = "Proceedings of the IEEE Symposium on Fault-Tolerant Computing",
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PAGES = 薂--184"}
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AUTHOR = "C. H. Chou and D. H. C. Du",
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