A Scalable Approach to Codesign of Topologies and Routing Algorithms for Families of Optimal Degree-Four Circulant Networks

Abstract

This paper presents a new approach to the joint construction of topologies of diameter-optimal circulant networks \( C(N; \pm 1, \pm s_2) \) and optimal routing algorithms of complexity \( O(1) \) implemented for them. New routing algorithms are based on the use of scalable parameters of \( L \)-shaped patterns in a dense packing of graphs on the plane for families of optimal networks. The scalability of the parameters of \( L \)-shaped templates for many families of optimal networks \( C(N; \pm 1, \pm s_2) \) has been proven, analytical formulas for the dependence of these parameters on the diameter of the graphs have been obtained, reducing the time for setting up the routing algorithm at the preliminary stage from \( O (\log N) \) to \( O(1) \). A comparison of the new routing algorithm with the optimal routing algorithm known in the literature showed its greater efficiency by an average of 10 percent in terms of time spent on routing in families of optimal graphs. Due to their good scalability and ease of routing, optimal degree-four circulant networks are of interest as efficient and reliable communication networks for networks-on-chip, multiprocessor supercomputer systems, telecommunications network structures, and neural communication networks.