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

  • Authors:

  • Chiuyuan Chen;James K. Lan;Wen-Shiang Tang

  • Affiliations:

  • Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan;Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan;Department of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2006

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Abstract

Double-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(log N) time. In this paper, we will provide a simple and efficient O(log N)-time algorithm for solving this problem.