On the Size of Lempel-Ziv and Lyndon Factorizations

TL;DR

This paper explores the relationship between Lyndon and Lempel-Ziv factorizations, establishing that Lyndon factorization can be larger but is always at most twice the size of the LZ factorization, revealing new combinatorial insights.

Contribution

The paper provides the first direct connection between Lyndon and Lempel-Ziv factorizations, including bounds on their relative sizes and a new family of strings demonstrating these properties.

Findings

Lyndon factorization can be larger than LZ factorization

Lyndon factorization is never more than twice the size of LZ factorization

Introduces a new family of strings illustrating the relationship

Abstract

Lyndon factorization and Lempel-Ziv (LZ) factorization are both important tools for analysing the structure and complexity of strings, but their combinatorial structure is very different. In this paper, we establish the first direct connection between the two by showing that while the Lyndon factorization can be bigger than the non-overlapping LZ factorization (which we demonstrate by describing a new, non-trivial family of strings) it is never more than twice the size.