在平行計算機器上，嵌入法是一項很重要的應用。在每一項平行的應用中，具有其個別的傳輸架構。這一些傳輸架構，可被映射成為多處理器架構下的拓撲，因此與其相符合的應用將可被執行。本篇論文在具有故障節點的超立方體圖或超立方體衍生圖中，提出一項新的正規圖形階層演算法，包括漢米爾頓路徑(Hamiltonian path)，漢米爾頓迴路(Hamiltonian cycle)，環路(ring)，網格(mesh)，完全二元樹(complete binary tree)，以及費伯納茲立方體(Fibonacci cube)。
首先，在具有故障的節點圖中，找出一個可取代故障節點的點，當n個故障節點出現時，容錯可達擴張度2，延展度3，壅塞度1及負載度1。此外，這種方法也可擴展至超立方體或超立方體衍生圖中。不同於許多目前所見的演算法只能嵌入單一類型的圖，這種演算法以一致的方法嵌入到上述的圖型中。藉由這個結果，我們可以輕鬆的將平行演算法發展到超立方體或超立方體衍生圖的正規圖型架構。此一嵌入的方法非常適合使用於高速的平行電腦中。

Embedding is of great importance in the applications of parallel computing. Every parallel application has its intrinsic communication pattern. The communication pattern graph is mapped into the topology of multiprocessor structures so that the corresponding application can be executed. This thesis proposes a novel algorithm for emulation a class of regular graphs in the faulty hypercube or hypercube derivatives, including the Hamiltonian path, the Hamiltonian cycle, the linear array, the ring, the mesh, the complete binary tree, and the Fibonacci cube.
First, to obtain the replaceable node of the faulty node, n faults can be tolerated with expansion 2, dilation 3, congestion 1, and load 1. Furthermore, our method is also extending the distributed fault-tolerant emulation of a class of regular graphs in hypercube or hypercube derivatives. Unlike many existing algorithms which are capable of embedding only type of graphs, our algorithm embeds the above graphs in a unified way. By the results, we can easily port the parallel algorithms developed for the structure of a class of regular graphs to hypercube or hypercube derivatives. This methodology of embedding enables extremely high-speed parallel computation.

1	Introduction	1
1.1	Motivations	1
1.2	Outline of the Dissertation	6
2	Related works and preliminaries	7
2.1	Definitions and notations	7
2.2	Hypercubes	9
2.3    Rings and linear arrays	13
2.4    Mesh	16
2.5    Trees	18
2.6    Fibonacci cube	21
3	Embedding applications of Hypercube	23
3.1	Embedding of rings and linear arrays	23
3.2	Embedding of mesh	25
3.3	Embedding of complete binary tree	26
3.4	Embedding of Fibonacci cube	33
4	Fault-Avoiding Methods for Mapping Symmetric Graphs on a Faulty Hypercube Derivative	34
4.1	Embedding Rings on a Faulty Hypercube Derivative	34
4.2	Embedding Meshes on a Faulty Hypercube Derivative	40
4.3	Embedding Complete binary trees on a Faulty Hypercube Derivative	43
4.4	Embedding the Fibonacci cube on the Faulty Hypercube	46
5	Conclusions and Future Works	54
5.1	Conclusions	54
5.2	Future works	55
Bibliography	56
Appendix A."Fault Tolerant Emulation of a Class of Regular Graphs in 
Hypercubes" WSEAS TRANSATIONS ON SYSTEMS	61
Appendix B.	"Embedding the Fibonacci Cube into the Faulty Hypercube" 
WSEAS TRANSACTIONS ON COMPUTERS	72
List of Figures
Figure 2.1 One-dimensional cube	9
Figure 2.2 Two-dimensional cube	9
Figure 2.3 Three-dimensional cube	10
Figure 2.4 The Four-dimensional hypercube with 16 nodes	10
Figure 2.5 An example of Ring topology	13
Figure 2.6 An example of linear arrays topology	13
Figure 2.7 A 3-bits ring of length 8	14
Figure 2.8 A Hamiltonian cycle on the hypercube	14
Figure 2.9 mesh network	16
Figure 2.10 Example of 2-dimensional mesh	17
Figure 2.11 Example of d-dimensional mesh	17
Figure 2.12 An example of the tree	18
Figure 2.13 An example of the complete binary tree	18
Figure 2.14 Emulation DTh in H3	20
Figure 2.15 Fibonacci cube	22
Figure 3.1 The transformation of double-rooted complete binary tree	26
Figure 3.2 A double-rooted complete binary tree contains 2d node	27
Figure 3.3 The process of the transformation(1)	27
Figure 3.4 The process of the transformation(2)	28
Figure 3.5 The process of the transformation(3)	28
Figure 3.6 A double-rooted complete binary tree with 2d+1 nodes	29
Figure 3.7 A double-rooted complete binary tree with 4 nodes can be embedded 
          on a hypercube with 4 nodes	30
Figure 3.8 The transformation of mapping(1)	30
Figure 3.9 The transformation of mapping(2)	30
Figure 3.10 The transformation of mapping(3)	31
Figure 3.11 A double-rooted complete binary tree with 8 nodes	31
Figure 3.12 A double-rooted complete binary tree with 8 nodes can be embedded 
            on a hypercube with 8 nodes	31
Figure 3.13   can be embedded to 	33
Figure 4.1 The searching sequence of the replaceable node of the faulty node	36
Figure 4.2 Fault-tolerant emulation of a class of regular graph in a hypercube	45
Figure 4.3   can be embedded on faulty 	52

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