An Algorithm to Find Optimal Double-Loop Networks with Non-unit Steps

Abstract

A double-loop network(DLN) G(N; r, s ) is a digraph with the vertex set V = {0, 1,..., N − 1} and the edge set $E=\{v \rightarrow v + r(\mod N)$ and $v \rightarrow v + s(\mod N)|v\in V$ }. Let D(N; r, s) be the diameter of G, D(N) =  min {D(N; r, s)|1 ≤ r < s < N and $\gcd(N; r, s ) = 1 \}$ and D ₁(N) =  min {D(N; 1, s)|1 < s < N }. Although the identity D(N) = D ₁(N) holds for infinite values of N, there are also another infinite set of integers with D(N) < D ₁(N). These other integral values of N are called non-unit step integers or nus integers. Xu and Aguil$\acute{o}$ et al. gave some infinite families of 0-tight nus integers with D ₁(N) − D(N) ≥ 1.

In this work, an algorithm is derived for finding nus integers. The running time complexity of the proposed algorithm is O(k ²)O(N ^1/4logN). It is verified by computer that the algorithm works extremely well. A new approach is also proposed for finding infinite families of nus integers. As an example, we present an infinite family of of 0-tight nus integers with D ₁(N) − D(N) = 4.