Checking whether a word is Hamming-isometric in linear time

TL;DR

This paper introduces linear-time algorithms to determine if a word is Hamming-isometric or Lee-isometric, based on characterizations involving borders with mismatches or errors, improving efficiency in pattern analysis.

Contribution

The paper provides the first linear-time algorithms for checking Hamming-isometric and Lee-isometric properties of words, utilizing border characterizations and pattern matching techniques.

Findings

Linear-time algorithm for Hamming-isometric check

Linear-time algorithm for Lee-isometric check over four-letter alphabet

Characterizations based on borders with mismatches or Lee-errors

Abstract

A finite word $f$ is Hamming-isometric if for any two word $u$ and $v$ of same length avoiding $f$, $u$ can be transformed into $v$ by changing one by one all the letters on which $u$ differs from $v$, in such a way that all of the new words obtained in this process also avoid~$f$. Words which are not Hamming-isometric have been characterized as words having a border with two mismatches. We derive from this characterization a linear-time algorithm to check whether a word is Hamming-isometric. It is based on pattern matching algorithms with $k$ mismatches. Lee-isometric words over a four-letter alphabet have been characterized as words having a border with two Lee-errors. We derive from this characterization a linear-time algorithm to check whether a word over an alphabet of size four is Lee-isometric.