A Euclidean Algorithm for Binary Cycles with Minimal Variance

TL;DR

This paper introduces a Euclidean-like algorithm for arranging binary symbols in cycles to minimize variance in symbol distances, ensuring uniform distribution and optimality for applications in systems and information processing.

Contribution

A novel Euclidean algorithm for binary cycles that guarantees minimal variance in symbol spacing, with proven optimality and practical applications.

Findings

The algorithm produces cycles with minimal variance in symbol distances.

The cycle arrangement is characterized by uniform symbol spacing.

The method is computationally efficient and optimal for binary sequences.

Abstract

The problem is considered of arranging symbols around a cycle, in such a way that distances between different instances of a same symbol be as uniformly distributed as possible. A sequence of moments is defined for cycles, similarly to the well-known praxis in statistics and including mean and variance. Mean is seen to be invariant under permutations of the cycle. In the case of a binary alphabet of symbols, a fast, constructive, sequencing algorithm is introduced, strongly resembling the celebrated Euclidean method for greatest common divisor computation, and the cycle returned is characterized in terms of symbol distances. A minimal variance condition is proved, and the proposed Euclidean algorithm is proved to satisfy it, thus being optimal. Applications to productive systems and information processing are briefly discussed.