On Constructions of Optimal Double-Loop Networks

Abstract

A double-loop digraph G(N; s ₁, s ₂ ) has N vertices 0, 1,...,N − 1 and 2N edges of two types: s ₁-edge: $v\rightarrow v + s_1 (\mod N)$, and s ₂-edge: $v\rightarrow v + s_2 (\mod N), v=0, 1,\ldots , N-1$, for some fixed steps 1 ≤ s ₁ < s ₂ < N with $\gcd(N; s_1 , s_2 ) = 1$. Let D(N; s ₁ , s ₂) be the diameter of G and let us define: D(N) =  min {D(N; s ₁, s ₂)| 1 ≤ s ₁ < s ₂ < N and $\gcd(N; s_1 , s_2 ) = 1$ }. Given a fixed number of vertices N, the general problem is to find steps s ₁ and s ₂, such that the digraph G(N; s ₁, s ₂ ) has minimum diameter D(N). A lower bound of this diameter is known to be $lb(N)= \lceil \sqrt{3N} \ \rceil-2.$ In this work, we give a simple and efficient algorithmic solution of the problem by using a geometrical approach. Given N, the algorithm find the minimum integer k = k(N), such that D(N) = lb(N) + k. The running time complexity of the algorithm is O(k ²)O(N ^1/4logN). With a new approach, we prove that infinite families of k-tight optimal double-loop networks can be constructed for any k ≥ 0.