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系統識別號 U0002-0706200601030000
中文論文名稱 模擬具正規性的網路拓蹼結構至具有容錯能力之超立方體及可逐步擴充超立方體結構圖
英文論文名稱 Fault-tolerant Simulation of a Class of Regular Graphs in Hypercubes and Incrementally Extensible Hypercubes
校院名稱 淡江大學
系所名稱(中) 資訊工程學系博士班
系所名稱(英) Department of Computer Science and Information Engineering
學年度 94
學期 2
出版年 95
研究生中文姓名 武士戎
研究生英文姓名 Shih-Jung Wu
學號 890190084
學位類別 博士
語文別 英文
口試日期 2006-06-01
論文頁數 109頁
口試委員 指導教授-葛煥昭
委員-王亦凡
委員-謝楠楨
委員-蔣定安
委員-施國琛
中文關鍵字 容錯  超立方體  可逐步擴充超立方體  費伯納茲立方體 
英文關鍵字 Embedding  Hypercube  Incrementally Extensible Hypercube  IEH Fibonacci cube  Fault-tolerant 
學科別分類 學科別應用科學資訊工程
中文摘要 在大型平行計算機器中,超立方體結構圖(hypercube)是最為廣泛使用的。其結構上的正規性(regularity)可使得電腦系統能較容易地被建立起來,而平行演算法上的容錯性使得一般與網路拓蹼結構相關的平行演算法不須經過大幅修改就能容易地被移植到超立方體結構電腦系統上來執行。可逐步擴充超立方體(Incrementally Extensible Hypercube)是超立方體的變形結構,它可以有任意的節點個數,並且保證其直徑(diameter)為節點個數和的對數值,以及每一個節點的鏈結分支度(degree)相差最多為1。本論文中,我們討論將一些正規的網路拓蹼,例如,環路(ring)、網格(mesh)及樹(tree),嵌入至有容錯能力的可逐步擴充超立方體結構的演算法。
而費伯納茲立方體結構(Fibonacci Cube),也是近年熱門的超立方體變形結構,本論文也一並討論將它嵌入至有容錯能力之超立方體結構的方法。在這些議題中,我們利用位元移動的方法,提出了一些容錯演算法,並且得到了重要的結果。在可逐步擴充超立方體有節點發生故障的情形下,可將環路結構完全嵌入,並達到延展度4、擴張度N、壅塞度1及負載度1。容錯程度達到O(n*log2m)。網格與樹的嵌入也有O(n2-(r+s)2)及O(n2-h2)的容錯程度。此外我們也對於費伯納茲立方體結構嵌入至有錯誤節點的超立方體結構的演算法作一討論,並達到延展度3、擴張度N、壅塞度1及負載度1及O(m2-n2)的容錯程度。
英文摘要 The hypercube is a widely-used interconnection architecture in the parallel machine. The Incrementally Extensible Hypercube (IEH), which is derived from the hypercube, is a generalization of interconnection network. Unlike the hypercube, the IEH can be constructed for any number of nodes. In other words, the IEH is incrementally expandable. In this thesis, the problem of embedding and reconfiguring some regular structures is considered in an IEH with faulty nodes. In recent years, the Fibonacci cube is a new interconnection architecture derived from hypercube. It also has some properties differ from hypercube. Thus we discuss the embedding of Fibonacci cube into the faulty hypercube. Some fault-tolerant embedding algorithms are proposed in this thesis. First, the algorithm in the present study enables us to obtain the good embedding of a ring into a faulty IEH with 2-expansion. Such result can be tolerated up to (n+1) faults with congestion 1, load 1, and dilation 3. When we allow unbounded expansion, the result of embedding of a ring into a faulty IEH can be tolerated up to O(n*log2m) faults with congestion 1, load 1, and dilation 4. The embedding methods in the study are mainly optimized for balancing the processor loads, under the situation of minimizing dilation and congestion as far as possible. Next we consider embedding of mesh into faulty IEH. In 2-expansion, it can be tolerated (n+1) faults with dilation 3, congestion 1, and load 1. Moreover, it can be tolerated up to O(n2-(r+s)2) in unbounded expansion. We discuss embedding of a complete binary tree into faulty IEH in the third. The cost is dilation 4, congestion 1, and load 1. In 2-expansion and unbounded expansion, embedding of a complete binary tree into faulty IEH can be tolerated (n+1) and O(n2-h2) faults. Finally, embedding of Fibonacci cube into faulty hypercube with dilation 3, congestion 2, load 1, unbounded expansion and O(m2-n2) faults can be tolerated, induced by our algorithm.
論文目次 Contents IV
List of Figures V
Chapter 1 Introduction 1
1.1 Motivations 1
1.2 Outline of the Dissertation 7
Chapter 2 Related works and preliminaries 8
2.1 Definitions and notations 8
2.2 Hypercubes 10
2.3 Incremental Extensible Hypercube 14
2.4 Rings and linear arrays 18
2.5 Mesh 26
2.6 Trees 27
2.7 Fibonacci cube 42
Chapter 3 Embedding applications of IEH graphs 45
3.1 Embedding of rings and linear arrays 45
3.2 Embedding of mesh 49
3.3 Embedding of complete binary tree 52
Chapter 4 Fault-tolerant embedding applications of IEH graphs 54
4.1 Ring can be embedded into faulty IEH graph 54
4.2 Mesh can be embedded into faulty IEH graph 69
4.3 Complete binary tree can be embedded into faulty IEH graph 80
Chapter 5 Fibonacci cube can be embedded into faulty hypercube 90
Chapter 6 Conclusions and future works 97
6.1 Conclusions 97
6.2 Future works 98
Bibliography 100
Appendix A. Publication List 106
List of Figures
Figure 1: Examples of hypercubes with dimension 1,2, and 3. 12
Figure 2: The IEH graph contains 14 nodes 17
Figure 3: The Hamiltonian cycle of G3(10) 25
Figure 4: 22x22 2-dimensional mesh 26
Figure 5: 3-dimensional mesh 27
Figure 6: A double-rooted complete binary tree contains 2d node. 29
Figure 7: The process of the transformation (1) 29
Figure 8: The process of the transformation (2) 29
Figure 9: The process of the transformation (3) 30
Figure 10: A double-rooted complete binary tree with 2d+1 nodes 30
Figure 11: A double-rooted with 4 nodes can be embedded into a 2-cube 31
Figure 12: The transformation of mapping (1) 31
Figure 13: The transformation of mapping (2) 32
Figure 14: The transformation of mapping (3) 32
Figure 15: A double-rooted tree with 8 nodes can be embedded into a 3-cube 32
Figure 16: A complete binary tree T3 can be embedded into F2 37
Figure 17: T4 is embedded into F3 39
Figure 18: T5 is embedded into F4 40
Figure 19: Fibonacci cube 43
Figure 20: can be embedded into H4 44
Figure 21: A ring with 9 nodes can be embedded into an IEH with 12 nodes 49
Figure 22: 2-dimensional BRGC for an 22×21 mesh 50
Figure 23: 22×21 mesh can be embedded into G3(13) with 2-expansion 52
Figure 24: Three steps of T3 embedding to G3(11) 53
Figure 25: R8 can be embedded into a H3 of F8 (case1) 57
Figure 26: R8 can be embedded into a H3 of F8 (case2) 58
Figure 27: R6 can be embedded into a H3 of G3(11) with dilation 1 65
Figure 28: R5 can be embedded into a H3 of G3(11) with dilation 1 66
Figure 29: R5 can be embedded into a H3 of G3(12) with dilation 2 66
Figure 30: 4 2 mesh (with 8 nodes) can be embedded into faulty G3(15) with 2-expansion. 72
Figure 31: 21×21 mesh can be embedded into G3(13) 74
Figure 32: 21×21 mesh can be embedded into faulty G3(15) 78
Figure 33: T3 can be embedded into faulty G3(11) 82
Figure 34: T2 can be embedded into faulty G3(15) 88
Figure 35: be embedded into faulty H4 94



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