A discrete optimization problem in local networks and data alignment

Abstract

This paper presents the solution of the following optimization problem that appears in the design of double-loop structures for local networks and also in data memory, allocation and data alignment in SIMD processors. Consider the digraph on N vertices, labeled from 0 to N - 1, where every vertex i is adjacent to the vertices (i + a) mod Nand (i + b) mod N. How should a and b be chosen in order to minimize the diameter and/or the average distance between vertices of the digraph? The study shows that for every N there are several different solutions (a, b) that produce the minimum values of the diameter and average distance between vertices. These values are of the order of V3 and (5/9 )3N_, respectively. For most values of N there exists a solution with a = 1 that facilitates the implementation of a double-loop structure from a single-loop one. The geometrical approach used to characterize the optimal solutions greatly facilitates the study of routing, throughput, and reliability questions.