Author :
Xiaolin Wang ; Bian, Zhenpeng ; Wang, Peng ; Zhou, J. ; Zhenxu Bian ; Pengfei Wang ; Jianqin Zhou
Author_Institution :
Dept. of Comput. Sci., Anhui Univ. of Technol., Ma´anshan, China
Abstract :
Double-loop networks have been widely studied as architecture for local area networks. A double-loop digraph G(N; s1, s2) has N vertices 0, 1,..., N-l and 2N edges of two types: s1-edge: i¿1 + s (mod N); i=0, 1,..., N-l, and s2-edge: i¿ s + s2 (modN); i=0, 1,..., N-l, for some fixed steps 1¿ s1<s2<N with gcd(N; s1, s2) = 1. Let D(N; s1, s2) be the diameter of G and let us define D(N) = min{ D(N; s1, s2) | 1¿s1<s2<N and gcd(N; s1, s2) = 1} and D(N) = min{D(N; 1, s)| 1<s<N}. Given a fixed number of vertices N, the general problem is to find steps s1 and s2, such that the digraph G(N; s1, s2) has minimum diameter D(N). A lower bound of this diameter is known to be lb(N)= [¿(3N)]-2. In this work, we give a simple and efficient algorithmic solution of the problem by using a geometrical approach. Given n, the algorithm has outputs s1, s2 and the minimum integer k=k(N), such that D(N; s1, s2 )=D(N)=lb(N)+k. The running time complexity of the algorithm is 0(k2 )0(N1/4 logN). Also, some flaws in the bibliography are corrected.
Keywords :
computational complexity; directed graphs; local area networks; double-loop digraph; fast algorithm; local area networks; optimal double-loop networks; running time complexity; Bibliographies; Computational intelligence; Computer architecture; Computer science; Computer security; Local area networks; Tiles; Diameter; L-shaped tile; algorithm; double-loop network; tight optimal;
