Article

# Weighted multi-connected loop networks

Department of Mathematics, University of Bergen, Allégt. 55, N-5007 Bergen, Norway
(Impact Factor: 0.56). 01/1996; 148(1-3):161-173. DOI: 10.1016/0012-365X(94)00239-F
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ABSTRACT

Given relatively prime integers N, a1,…,ak, a multi-connected loop network is defined as the directed graph with vertex set , and directed edges i → r ≡ i + aj (mod N). If each edge i → i + aj is given a positive real weight wj for j = I,…,k, then we have a weighted multi-connected loop network. The weight of a path is the sum of weights on its edges. The distance from a vertex to another is the minimum weight of all paths from the first vertex to the second. The diameter of the network is the maximum distance, and the average diameter is the average distance in the network. In this paper we study the diameter and the average diameter of a weighted multi-connected loop network. We give a unified and generalized presentation of several results in the literature, and also some new results are obtained.

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• "D DL (N) √ 3N + 2(3N) 1/4 + 5 for N 6348. (2) In [16] "
##### Article: Improved upper and lower bounds on the optimization of mixed chordal ring networks
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ABSTRACT: Recently, Chen, Hwang and Liu [S.K. Chen, F.K. Hwang, Y.C. Liu, Some combinatorial properties of mixed chordal rings, J. Interconnection Networks 1 (2003) 3–16] introduced the mixed chordal ring network as a topology for interconnection networks. In particular, they showed that the amount of hardware and the network structure of the mixed chordal ring network are very comparable to the (directed) double-loop network, yet the mixed chordal ring network can achieve a better diameter than the double-loop network. More precisely, the mixed chordal ring network can achieve diameter about 2N as compared to 3N for the (directed) double-loop network, where N is the number of nodes in the network. One of the most important questions in interconnection networks is, for a given number of nodes, how to find an optimal network (a network with the smallest diameter) and give the construction of such a network. Chen et al. [S.K. Chen, F.K. Hwang, Y.C. Liu, Some combinatorial properties of mixed chordal rings, J. Interconnection Networks 1 (2003) 3–16] gave upper and lower bounds for such an optimization problem on the mixed chordal ring network. In this paper, we improve the upper and lower bounds as 2⌈N/2⌉+1 and ⌈2N−3/2⌉, respectively. In addition, we correct some deficient contexts in [S.K. Chen, F.K. Hwang, Y.C. Liu, Some combinatorial properties of mixed chordal rings, J. Interconnection Networks 1 (2003) 3–16].
Information Processing Letters 06/2009; 109(13):757-762. DOI:10.1016/j.ipl.2009.03.017 · 0.55 Impact Factor
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• "They showed that the two L-shapes can be obtained from each other through a 3-rectangle transformation as shown in Fig. 6. RR odseth [28] gave an equivalence theorem for the multiloop. Its double-loop version is as follows: "
##### Article: A complementary survey on double-loop networks
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ABSTRACT: We give a survey on double-loop networks with emphasis on new development since the surveys in 1986, 1991 and 1995.
Theoretical Computer Science 07/2001; 263(s 1–2):211–229. DOI:10.1016/S0304-3975(00)00243-7 · 0.66 Impact Factor
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##### Article: Equivalent double-loop networks
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ABSTRACT: F. K. Hwang and Y. H. Xu [Discrete Math. 66, 109-118 (1987; Zbl 0614.90035)] defined equivalent double-loop networks and gave one such result showing that the L-shapes of the two equivalent networks are recombinations of three rectangles. Recently, O. J. Rödseth [Discrete Math. 148, No. 1-3, 161-173 (1996; Zbl 0842.05039)] gave an elegant algebraic theorem for equivalent multi-loop networks. We show that its double-loop version yields equivalent network of the 3-rectangle version. We also show that other seemingly different geometric recombinations also all turn out to be special cases of the 3-rectangle version.
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