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

AbstractDouble-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;s1,s2) 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 s1 and s2 (also called the steps of the double-loop network) to realize it. In this paper, we propose a simple and efficient algorithm to find s1 and s2, which is based on the Smith normalization method of Aguiló, Esqué and Fiol

An efficient algorithm to find a double-loop network that realizes a given L-shape

AbstractDouble-loop networks have been widely studied as an architecture for local area networks. It is well known that the minimum distance diagram of a double-loop network yields an L-shape. Given a positive integer N, it is desirable to find a double-loop network with its diameter being the minimum among all double-loop networks with N nodes. Since the diameter of a double-loop network can be easily computed from its L-shape, one method is to start with a desirable L-shape and then find a double-loop network to realize it. This is a problem discussed by many authors [F. Aguiló, M.A. Fiol, An efficient algorithm to find optimal double loop networks, Discrete Math. 138 (1995) 15–29, R.C. Chan, C.Y. Chen, Z.X. Hong, A simple algorithm to find the steps of double-loop networks, Discrete Appl. Math. 121 (2002) 61–72, C.Y. Chen, F.K. Hwang, The minimum distance diagram of double-loop networks, IEEE Trans. Comput. 49 (2000) 977–979, P. Esqué, F. Aguiló, M.A. Fiol, Double commutative-step diagraphs with minimum diameters, Discrete Math. 114 (1993) 147–157] and it has been open for a long time whether this problem can be solved in O(logN) time. In this paper, we will provide a simple and efficient O(logN)-time algorithm for solving this problem