The infinite families of optimal double loop networks

Abstract

Let N, s be integers, 1 < s < N. The double loop network (DLN) G(N; s) is a digraph with N nodes 0, 1,. ..,N 1 and 2N arcs {i +i+I,i+s:i=O,l,..., N-l),where nodes are always represented by residues modulo N. DLN’s have been widely studied lately as practical models in the design of local area networks and parallel processing architectures [ 1,3,5]. Let d(N; s) be the diameter of G(N; s), and let d(N) = min{d(N;s): 1 < s < N}. The network G(N;s) is said to be optimal if d(N; s) = d(N). Wong and Coppersmith initiated the studies of finding optimal G(N; s) for every N 2 4 and they established a good lower bound lb(N) = L fiN A2 for d(N) [S]. A network G(N;s) is said to be tight optimal if d(N;s) = lb(N). Obviously, a tight optimal DLN is certainly optimal but the converse is not true. For example, if N = 3(t + 1)’ then lb(N) = 3t + 1 but d(N) = d(N;3t + 5) = 3t + 2 for every t 2 1. In accordance with this situation, a network G(N; s) is said to be nearly tight optimal if d(N;s) = d(N) = lb(N) + 1. A lot of infinite families of tight optimal and only one infinite family of nearly tight optimal DLN’s, namely {G(3(t + l)*; 3t + 2: t 2 11, have been found by several authors since 1987. However the union of all the known infinite families of optimal DLN’s can not even include a DLN with N nodes for every N I 50 [l, 2, 43. In this note we list explicitly 69 infinite families of tight optimal and 33 infinite families of nearly tight optimal DLN’s such that for every N, 4 I N 5 300, at least one of our families contain an optimal G(N; s) (see Tables 1,2,3).