Unveiling the connection between the Lyndon factorization and the Canonical Inverse Lyndon factorization via a border property

TL;DR

This paper explores the relationship between Lyndon and inverse Lyndon factorizations, revealing a border property that leads to a linear-time algorithm for computing the canonical inverse Lyndon factorization from the classical Lyndon factorization.

Contribution

It introduces the border property as a key concept and establishes the uniqueness of the canonical inverse Lyndon factorization, along with an efficient algorithm to compute it.

Findings

The canonical inverse Lyndon factorization is unique and has the border property.

A linear-time algorithm is provided to compute ICFL from CFLin.

The border property characterizes the relationship between Lyndon and inverse Lyndon factorizations.

Abstract

The notion of Lyndon word and Lyndon factorization has shown to have unexpected applications in theory as well in developing novel algorithms on words. A counterpart to these notions are those of inverse Lyndon word and inverse Lyndon factorization. Differently from the Lyndon words, the inverse Lyndon words may be bordered. The relationship between the two factorizations is related to the inverse lexicographic ordering, and has only been recently explored. More precisely, a main open question is how to get an inverse Lyndon factorization from a classical Lyndon factorization under the inverse lexicographic ordering, named CFLin. In this paper we reveal a strong connection between these two factorizations where the border plays a relevant role. More precisely, we show two main results. We say that a factorization has the border property if a nonempty border of a factor cannot be a prefix of the next factor. First we show that there exists a unique inverse Lyndon factorization having the border property. Then we show that this unique factorization with the border property is the so-called canonical inverse Lyndon factorization, named ICFL. By showing that ICFL is obtained by compacting factors of the Lyndon factorization over the inverse lexicographic ordering, we provide a linear time algorithm for computing ICFL from CFLin.