A simple algorithm to find the steps of double-loop networks

Double-loop networks have been widely studied as architecture for local area networks and it is well-known that the minimum distance diagram of a double-loop network yields an L-shape. Given an N, it is desirable to find a double-loop network DL(N;s ₁,s ₂) with its diameter being the minimum among all double-loop networks with N stations. Since the diameter can be easily computed from an L-shape, one method is to start with a desirable L-shape and then asks whether there exist s ₁ and s ₂ (also called the steps of the double-loop network) to realize it. In this paper, we propose a simple and efficient algorithm to find s ₁ and s ₂, which is based on the Smith normalization method of Aguiló, Esqué and Fiol.