From the Lyndon factorization to the Canonical Inverse Lyndon factorization: back and forth
Paola Bonizzoni, Clelia De Felice, Rocco Zaccagnino, Rosalba Zizza
TL;DR
The paper addresses how the canonical inverse Lyndon factorization $\mathop{\mathrm{ICFL}}(w)$ relates to the inverse-Lyndon Lyndon factorization $\text{CFL}_{in}(w)$ by introducing and exploiting the grouping concept. It proves that $\mathop{\mathrm{ICFL}}(w)$ is a grouping of $\text{CFL}_{in}(w)$ under the inverse-prefix order and provides constructive methods to translate between the two via PMCs and the canonical pair $(p,\overline{p})$, enabling linear-time computation in both directions. Key contributions include a full characterization of the CFL$_{in}$–ICFL relationship through grouping, a local computation framework on non-increasing maximal chains, and explicit algorithms to obtain one factorization from the other, supported by detailed proofs and examples. The results open avenues for inverse-word based variants of the Burrows–Wheeler Transform and for extending Lyndon-factorization results to the inverse-Lyndon setting, with future work on a grouping-based, direct definition of ICFL.
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.