A Euclidean Algorithm for Binary Cycles with Minimal Variance

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.