From the Lyndon factorization to the Canonical Inverse Lyndon factorization: back and forth

TL;DR

This paper explores the relationship between Lyndon and inverse Lyndon factorizations, introducing methods to convert between them, which enhances understanding of their structural properties and potential applications.

Contribution

It establishes a method to derive one factorization from the other using grouping, linking Lyndon and inverse Lyndon factorizations in a novel way.

Findings

A method to convert between Lyndon and inverse Lyndon factorizations.

Insights into the structural relationship between these factorizations.

Potential applications in data transformations like Burrows Wheeler Transform.

Abstract

The notion of inverse Lyndon word is related to the classical notion of Lyndon word. More precisely, inverse Lyndon words are all and only the nonempty prefixes of the powers of the anti-Lyndon words, where an anti-Lyndon word with respect to a lexicographical order is a classical Lyndon word with respect to the inverse lexicographic order. Each word $w$ admits a factorization in inverse Lyndon words, named the canonical inverse Lyndon factorization $\ICFL(w)$, which maintains the main properties of the Lyndon factorization of $w$. Although there is a huge literature on the Lyndon factorization, the relation between the Lyndon factorization $\CFL_{in}$ with respect to the inverse order and the canonical inverse Lyndon factorization $\ICFL$ has not been thoroughly investigated. In this paper, we address this question and we show how to obtain one factorization from the other via the notion of grouping. This result naturally opens new insights in the investigation of the relationship between $\ICFL$ and other notions, e.g., variants of Burrows Wheeler Transform, as already done for the Lyndon factorization.