# Inverse Lyndon words and Inverse Lyndon factorizations of words

## TL;DR

This paper introduces inverse Lyndon factorizations, a new string decomposition method that extends Lyndon factorizations, with a unique, efficiently computable canonical form that preserves key sorting properties.

## Contribution

It defines inverse Lyndon words, proves the existence of a unique canonical inverse Lyndon factorization (ICFL), and demonstrates its computational efficiency and sorting compatibility.

## Key findings

- ICFL(w) can be computed in linear time
- ICFL(w) is uniquely determined for each word w
- ICFL(w) preserves suffix sorting properties

## Abstract

Motivated by applications to string processing, we introduce variants of the Lyndon factorization called inverse Lyndon factorizations. Their factors, named inverse Lyndon words, are in a class that strictly contains anti-Lyndon words, that is Lyndon words with respect to the inverse lexicographic order. The Lyndon factorization of a nonempty word w is unique but w may have several inverse Lyndon factorizations. We prove that any nonempty word w admits a canonical inverse Lyndon factorization, named ICFL(w), that maintains the main properties of the Lyndon factorization of w: it can be computed in linear time, it is uniquely determined, it preserves a compatibility property for sorting suffixes. In particular, the compatibility property of ICFL(w) is a consequence of another result: any factor in ICFL(w) is a concatenation of consecutive factors of the Lyndon factorization of w with respect to the inverse lexicographic order.