Small diameter symmetric networks from linear groups

Campbell, L. Andrew; Carlsson, Gunnar; Dinneen, Michael J.; Faber, Vance; Fellows, Michael R.; Langston, Micheal A.; Moore, James W.; Mullhaupt, Andrew P.; Sexton, Harlan

doi:10.1109/12.123397

Cited by 31 publications

(17 citation statements)

References 6 publications

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“…Consider the maximum number g(D, k) of nodes in a network of Large networks with a given diameter and maximum degree have applications in parallel processor or communimaximum degree D and diameter at most k. There is an easily derived upper bound (the Moore bound) (cf. [3]): cation networks where there is a need for each pair of nodes to communicate or exchange data efficiently, but where it is also impractical to directly connect each pair In [7], we remarked that for larger (fixed) values of k we have p (D, k) …”

Section: Introductionmentioning

confidence: 99%

Constructions of large planar networks with given degree and diameter

1998

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Section: Introductionmentioning

confidence: 99%

Constructions of large planar networks with given degree and diameter

1998

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“…There has been an extensive literature on the use of Cayley graphs to design interconnection networks [1,2,5,6,7,12]. Cayley graphs have been used extensively to study point-to-point routing, and they have been particularly attractive for the degree-diameter problem [3,5,7].…”

Section: Introductionmentioning

confidence: 99%

“…Cayley graphs have been used extensively to study point-to-point routing, and they have been particularly attractive for the degree-diameter problem [3,5,7]. Nevertheless, other patterns of routing, such as broadcast, permutation routing, and general many-to-one routing are at least as important for parallel computing.…”

Section: Introductionmentioning

confidence: 99%

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Permutation routing via Cayley graphs with an example for bus interconnection networks

Cooperman¹,

Finkelstein²

1995

DIMACS Series in Discrete Mathematics And Theoretical Computer Science

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Abstract. Cayley graphs have been used extensively to design interconnection networks and provide a natural setting for studying point-to-point routing [1,2,3,5,6,7,12]. The extension of these techniques to the more important problem of permutation routing on interconnection networks presents fundamental problems. This is due to the potentially explosive growth in both the size of the graph and the number of generating permutations, referred to as one-step permutation routes, used to define the underlying graph. This paper describes a technique for moderating that growth so that the techniques in [8] can be applied for finding optimal permutation routes. In a particularly striking example, a bus interconnection architecture involving 1.0 × 10 17 permutations (nodes of the Cayley graph) is reduced to a computation on a graph with only 3,950 nodes. Further, it is shown how many of the 58,624 generators (directed edges labelled by one-step permutation routes) at each node of the graph may be eliminated as locally redundant.

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“…Most importantly, the relationship between the generators and the diameter of the graph is unknown. Currently, identification of "good" generators are achieved through random or extensive systematic search of all possibilities [16]. In an effort to resolve this problem, we investigate the parameters of Borel Cayley graphs.…”

Section: Conclusion and Futureworkmentioning

confidence: 99%

Dense and Symmetric Graph Formulation and Generation for Wireless Information Networks

Kim

Tang

2009

2009 International Conference on Wireless Networks and Information Systems

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Abstract-Dense and symmetric graphs are useful for modeling fast information distribution in wireless information networks. In this paper, we focus on a specific family of dense and symmetric graphs, the Borel Cayley graphs. More specifically, we investigate the various parameters in the original formulation of Borel Cayley graphs defined in the matrix domain. By eliminating redundant parameters, we propose a new and simpler formulation of Borel Cayley graphs. This new formulation is defined in the integer domain and the group operation resembles the generation of pseudorandom numbers, hence the name pseudo-random formulation. Through the establishment of propositions and corollaries, we proved that certain parameters do not affect the diameter under a specific condition. This result provides a guideline in choosing appropriate generators and thus reducing the computation time in the search of good or bad generators. Using this new formulation, we also show that Borel Cayley graphs are isomorphic to a sub-class of Cayley graphs proposed by Dinneen. Finally, some guidelines for choosing generators to avoid disconnected graphs are also provided.

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Small diameter symmetric networks from linear groups

Cited by 31 publications

References 6 publications

Constructions of large planar networks with given degree and diameter

Constructions of large planar networks with given degree and diameter

Permutation routing via Cayley graphs with an example for bus interconnection networks

Dense and Symmetric Graph Formulation and Generation for Wireless Information Networks

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