A New Approach to Regular & Indeterminate Strings

TL;DR

This paper introduces a new model for regular and indeterminate strings, providing linear time algorithms for their classification, and explores their connection to graph transitive closure and palindrome arrays, opening new research directions.

Contribution

It proposes a novel string representation model, efficient algorithms for regularity testing and lexicographic minimization, and links string regularity to graph transitive closure, advancing the theoretical understanding of indeterminate strings.

Findings

Linear time algorithm for regularity detection

Connection between string regularity and graph transitive closure

Construction of strings from feasible palindrome arrays

Abstract

In this paper we propose a new, more appropriate definition of regular and indeterminate strings. A regular string is one that is "isomorphic" to a string whose entries all consist of a single letter, but which nevertheless may itself include entries containing multiple letters. A string that is not regular is said to be indeterminate. We begin by proposing a new model for the representation of strings, regular or indeterminate, then go on to describe a linear time algorithm to determine whether or not a string $x = x[1..n]$ is regular and, if so, to replace it by a lexicographically least (lex-least) string $y$ whose entries are all single letters. Furthermore, we connect the regularity of a string to the transitive closure problem on a graph, which in our special case can be efficiently solved. We then introduce the idea of a feasible palindrome array MP of a string, and prove that every feasible MP corresponds to some (regular or indeterminate) string. We describe an algorithm that constructs a string $x$ corresponding to given feasible MP, while ensuring that whenever possible $x$ is regular and if so, then lex-least. A final section outlines new research directions suggested by this changed perspective on regular and indeterminate strings.