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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
CitacióFiol, M. A. [et al.]. A discrete optimization problem in local networks and data alignment. "IEEE transactions on computers", Juny 1987, vol. C-36, núm. 6, p. 702-713.