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Please use this identifier to cite or link to this item: http://hdl.handle.net/2031/4379

Title: Edge-pancyclicity, path-embeddability, and t/k-diagnosability in interconnection networks
Other Titles: Hu lian wang luo zhong de bian-fan quan xing, lu jing-ke qian ru xing, ji t/k-ke zhen duan xing
互連網絡中的邊-泛圈性, 路徑-可嵌入性, 及 t/k-可診斷性
Authors: Fan, Jianxi (樊建席)
Department: Dept. of Computer Science
Degree: Doctor of Philosophy
Issue Date: 2006
Publisher: City University of Hong Kong
Subjects: Computer networks
Parallel computers
Notes: CityU Call Number: QA76.58.F36 2006
Includes bibliographical references (leaves 154-161)
Thesis (Ph.D.)--City University of Hong Kong, 2006
xiii, 163 leaves : ill. ; 30 cm.
Type: Thesis
Abstract: Interconnection networks play important roles in parallel computing systems. Am- ong many interconnection networks, the hypercube is one of the most popular structures. By changing some connections between the processors in hypercubes, three variants of hypercubes|crossed cubes, twisted cubes, and Möbius cubes were proposed in the literature. Their diameters are about half those of the hypercubes with the same dimensions. Recently, a large family of variants of hypercubes|bijective connection graphs (In brief, BC graphs), which include hy- percubes, crossed cubes, twisted cubes, and Möbius cubes, were proposed. Embeddability is an important property of interconnection networks. Graph embedding is a technique in parallel computing that maps a guest graph into a host graph (usually an interconnection network). There are many applications of graph embedding, such as architecture simulation, processor allocation, VLSI layout, etc.. An important performance metric of embedding is dilation. The dilation of an embedding measures the communication delay that the host graph simulates the guest graph. Because cycles and paths are popular structures, a lot of embeddings take cycles and paths as guest graphs. There is a kind of enhanced embedding of cycles--edge-pancyclicity, which refer to such a cycle embedding that a series of cycles of different lengths can be embedded in a given host graph and all these cycles pass an arbitrarily given edge in the host graph. Diagnosability is another important property of interconnection networks. Di- agnosis is a process of identifying the faulty processors of the system. The maximal number of faulty nodes that a system can guarantee to diagnose is called the degree of diagnosability of the system. Pursuing high degrees of diagnosability has been an important goal in system-level diagnosis. Various system diagnosis strategies are based on the well-known PMC diagnostic model. There are three diagnosis strategies--the precise diagnosis strategy, the pessimistic diagnosis strategy, and the t/k-diagnosis strategy, based on the diagnostic model. The fault-set identi- fied under the precise diagnosis strategy does not contain any fault-free node, the fault-set identified under the pessimistic diagnosis strategy contains at most one fault-free nodes, while the fault-set identified under t/k-diagnosis strategy can con- tain many fault-free nodes. In general, a system has higher degree of diagnosability under t/k-diagnosis strategy than under the other two strategies. In this thesis, we study the edge-pancyclicity, path-embeddability, and t/k- diagnosability of BC graphs, which include crossed cubes, twisted cubes, and Möbius cubes, etc.. For a given graph G and any two nodes u and v in G, we use dist(G, u, v) to denote the distance between u and v in G and V (G) and E(G) to denoted the node set and edge set of G, respectively. And, we use Qn, CQn, TQn, and MQn to denote the n-dimensional hypercube, crossed cube, twisted cube, and Möbius cube, respectively. The original contributions of this thesis are as follows: - We study the path-embeddability and edge-pancyclicity of crossed cubes. We first prove that CQn is edge-pancyclic (n≥2). According to the edge-pancyclicity of CQn, we then prove that a path of length l can be embedded between x and y with dilation 1 in CQn for any two distinct nodes x and y in CQn and any integer l with [n+1/2] + 1≤ l ≤2n - 1 (n≥3). Furthermore, we give another style of path embedding in crossed cubes--a path of length l can be embedded between x and y with dilation 1 in CQn for any two distinct nodes x and y in CQn and any integer l with dist(CQn, x, y) + 2≤ l≤ 2n - 1 (n≥3). We also show the diifferences between the two styles of path embeddings in CQn. Moreover, we prove that there exist two nodes x, y2 V (CQn) such that dist(CQn, x, y) = l and any path of length l + 1 cannot be embedded between x and y with dilation 1 in CQn for any integer l with 1≤ l ≤[n+1/2] - 1 (n≥3). By this result, we can conclude that [n+1/2]+1 and dist(CQn, x, y)+2 are the tight bounds on the path lengths l in the first style of path embedding and the second style of path embedding for the case 1≤dist(CQn, x, y) ≤ [n+1/2] - 1, respectively. - We study the path-embeddability and edge-pancyclicity of twisted cubes. We first prove that TQn is edge-pancyclic (n≥3); and by the constructive proof of edge-pancyclicity of twisted cubes, we give a polynomial-time algorithm for cycle embedding with edge-pancyclic in twisted cubes. According to the edge- pancyclicity of TQn, we then prove that a path of length l can be embedded between x and y with dilation 1 in TQn for any two distinct nodes x and y in CQn and any integer l with dist(TQn, x, y)+2≤ l≤2n - 1 (n≥3). Moreover, we prove that the bound on path lengths l is tight: There exist two nodes x, y Є V (TQn) such that dist(TQn, x, y) = l and any path of length l+1 cannot be embedded between x and y with dilation 1 in TQn for any integer l with 1≤ l≤[n+1/2] - 1 (n≥3). We also show that the edge-pancyclicity and path-embeddability of twisted cubes are similar to those of crossed cubes, although the proof methods are di®erent from each other. - We propose a more general method to study the edge-pancyclicity and path- embeddability of a family of BC graphs. First, we prove that a path of length l with dist(Xn, x, y) + 2≤ l≤2n - 1 can be embedded between x and y with dilation 1 in Xn for any two distinct nodes x and y in Xn, where Xn (n≥4) is an n-dimensional BC graphs satisfying the three specific conditions. Furthermore, by this result, we can claim that Xn is edge-pancyclic. Moreover, we show that these results can be applied to not only crossed cubes and Möbius cubes, but also some other BC graphs. - We propose a much more general method to study the t/k-diagnosability of all the BC graphs, which include hypercubes, crossed cubes, Möbius cubes, and twisted cubes, etc.. We prove that any n-dimensional BC graph is t(n, k)/k- diagnosable when n≥4 and 0≤ k≤ n, where t(n, k) = (k + 1)n – 1/2 (k + 1)(k + 2) + 1. Since CQn, TQn, and MQn are specific examples of the n-dimensional BC graphs, they all have the same t/k-diagnosability as Qn. As a result, the algorithms developed for diagnosis on hypercubes may also be used to diagnose the multiprocessor systems whose network topologies are based on BC graphs. We point out that all the considered embeddings are optimal in the sense that all of them have the smallest dilation 1.
Online Catalog Link: http://lib.cityu.edu.hk/record=b2147040
Appears in Collections:CS - Doctor of Philosophy

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