Loopless Algorithms to Generate Maximum Length Gray Cycles wrt. k-Character Substitution

TL;DR

This paper introduces efficient loopless algorithms for generating maximum length Gray cycles based on k-character substitution, advancing the construction of Gray codes with specific Hamming distance constraints.

Contribution

It presents the first loopless and constant amortized time algorithms for constructing maximum length Gray cycles under k-character substitution rules.

Findings

Algorithms operate in loopless and constant amortized time.

Successfully generate maximum length Gray cycles for specified k-character substitutions.

Provides theoretical bounds on the complexity measure λ for these cycles.

Abstract

Given a binary word relation $\tau$ onto A * and a finite language X $\subseteq$ A * , a $\tau$-Gray cycle over X consists in a permutation w [i] 0$\le$i$\le$|X|--1 of X such that each word w [i] is an image under $\tau$ of the previous word w [i--1]. We define the complexity measure $\lambda$A,$\tau$ (n), equal to the largest cardinality of a language X having words of length at most n, and st. some $\tau$-Gray cycle over X exists. The present paper is concerned with $\tau$ = $\sigma$ k , the so-called k-character substitution, st. (u, v) $\in$ $\sigma$ k holds if, and only if, the Hamming distance of u and v is k. We present loopless (resp., constant amortized time) algorithms for computing specific maximum length $\sigma$ k-Gray cycles.