On the approximation ratio of LZ-End to LZ77

TL;DR

This paper investigates the relationship between LZ-End and LZ77 factorizations, demonstrating that for the period-doubling sequence over a binary alphabet, the approximation ratio approaches 2, highlighting limitations in compression efficiency.

Contribution

The paper provides the first analysis of the approximation ratio of LZ-End to LZ77 for the period-doubling sequence over a binary alphabet.

Findings

Approximation ratio approaches 2 for the period-doubling sequence.

LZ-End can be nearly twice as large as LZ77 for certain sequences.

Analysis extends understanding of LZ-End's efficiency on specific sequences.

Abstract

A family of Lempel-Ziv factorizations is a well-studied string structure. The LZ-End factorization is a member of the family that achieved faster extraction of any substrings (Kreft & Navarro, TCS 2013). One of the interests for LZ-End factorizations is the possible difference between the size of LZ-End and LZ77 factorizations. They also showed families of strings where the approximation ratio of the number of LZ-End phrases to the number of LZ77 phrases asymptotically approaches 2. However, the alphabet size of these strings is unbounded. In this paper, we analyze the LZ-End factorization of the period-doubling sequence. We also show that the approximation ratio for the period-doubling sequence asymptotically approaches 2 for the binary alphabet.