Arrangement graphs: a class of generalized star graphs

Day, Khaled; Tripathi, Anand

doi:10.1016/0020-0190(92)90030-y

Cited by 179 publications

(89 citation statements)

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“…These k symbols are denoted as X x 1 x 2 Á Á Á x k X Refer x i as the ith element of X. The (nY k)-arrangement graph, denoted as A nYk , de®ned in [6] is an undirected graph V Y E as follows:…”

Section: à ámentioning

confidence: 99%

“…For example, in A 4Y2 Y the node p 41 is connected to the nodes 42, 43, 21 and 31. An edge of A nYk connecting two arrangements p and q, which dier only in position i, is called an i-edgeX For all values of n and k, A nYk is a regular graph on n3an À k3 nodes that is regular of degree kn À k, and a diameter 3a2k [6]. For an…”

Section: à ámentioning

confidence: 99%

“…The A nYk is with recursive structure [6] i.e., an A nYk can be partitioned into n copies of A nÀ1YkÀ1Y each embedded A nÀ1YkÀ1 is conveniently denoted by hÃ kÀ1 ai nYk , where a P f1Y 2Y F F F Y ngY (where Ã represents a Generally, A nYk can be partitioned into n3an À p3 node-disjoint copies of A nÀpYkÀp in n3ap3n À p3 dierent ways and that in total A nXk contains k p n3an À p3 copies of A nÀpYkÀp , for 1 T p T k À 1. As a conclusion, the drawback of star graph [10,12,13] is limited with respect to its number of nodes, which is equal to n3.…”

Section: à ámentioning

confidence: 99%

“…This implies what the low scalability will be. The proposed arrangement graph [6] is motivated by having the high scalability. The arrangement graph is more¯exible than the star graph in terms of choosing major design parameters, i.e.…”

Section: à ámentioning

confidence: 99%

“…member of vertices, degree and diameter, while preserving most of the excellent properties of the star graph. However, the potential troubles of the arrangement graph [6] is the high-degree problem, which is kn À k. This problem leads to troublesome development of a VLSI layout for the real machine.…”

Section: à ámentioning

confidence: 99%

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Congestion-free embedding of 2(n−k) spanning trees in an arrangement graph

Chen

Juang

Shen³

2001

Journal of Systems Architecture

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The arrangement graph A nYk is not only a generalization of star graph (n À k 1), but also more¯exible. In this investigation, we elucidate the problem of embedding of multiple spanning trees in an arrangement graph with the objective of congestion-free. This result is to report how to exploit 2n À k edge disjoint spanning trees in an arrangement graph, where each congestion-free spanning tree's height is 2k À 1. Our scheme is based on a subgraphpartitioning scheme. First, we construct 2n À k base spanning trees in every A nÀk2Y2 . Then, we recursively construct 2n À k spanning trees from every A nÀk2Y2 up to A nYk by a bottom-up approach. This is a near-optimal result since all of possible edges in the base subarrangement A nÀk2Y2 are fully utilized. Ó

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Section: à ámentioning

confidence: 99%

Section: à ámentioning

confidence: 99%

Section: à ámentioning

confidence: 99%

Section: à ámentioning

confidence: 99%

Section: à ámentioning

confidence: 99%

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Congestion-free embedding of 2(n−k) spanning trees in an arrangement graph

Chen

Juang

Shen³

2001

Journal of Systems Architecture

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Hypercomplete: A pancyclic recursive topology for large-scale distributed multicomputer systems

Chen¹,

Fu²,

Fang³

2000

Networks

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This paper introduces a pancyclic recursive topology, named the hypercomplete, for building large‐scale distributed multicomputer systems. The hypercomplete is maximally fault‐tolerant and has a logarithmic diameter. In addition to being pancyclic, the hypercomplete is Hamiltonian‐connected. Consequently, the hypercomplete can embed rings of all possible lengths and a longest linear array between arbitrary two nodes. Besides, the hypercomplete can embed tori and trees with constant dilation and constant congestion. Some algorithms are developed for the hypercomplete. An algorithm is proposed to determine a shortest path between arbitrary two nodes. A broadcasting algorithm without redundant messages is proposed, whose resulting spanning tree has its height bounded above by the diameter. The class of ascend/descend algorithms can be executed efficiently. © 2000 John Wiley & Sons, Inc.

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Embedding longest fault-free paths in arrangement graphs with faulty vertices

Chen

2001

Networks

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The arrangement graph is a generalization of the star graph. Recent work by Hsieh et al. showed that an arrangement graph with faulty vertices contains a longest fault‐free cycle. In this paper, we generalize Hsieh et al.'s result by constructing a longest fault‐free path between two arbitrary distinct healthy vertices. Moreover, our result can tolerate more faulty vertices than can Hsieh et al.'s result. The situation of simultaneous faulty vertices and faulty edges is also discussed. © 2001 John Wiley & Sons, Inc.

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Arrangement graphs: a class of generalized star graphs

Cited by 179 publications

References 1 publication

Congestion-free embedding of 2(n−k) spanning trees in an arrangement graph

Congestion-free embedding of 2(n−k) spanning trees in an arrangement graph

Hypercomplete: A pancyclic recursive topology for large-scale distributed multicomputer systems

Embedding longest fault-free paths in arrangement graphs with faulty vertices

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