Lempel-Ziv: a "one-bit catastrophe" but not a tragedy

TL;DR

This paper demonstrates that adding a single bit in front of an infinite word can significantly alter its LZ'78 compression ratio, showing a positive answer to the 'one-bit catastrophe' question and analyzing the behavior on finite words.

Contribution

It provides the first positive example of the 'one-bit catastrophe' for LZ'78 and establishes bounds on how compression ratios change with added bits.

Findings

Existence of an infinite word with zero upper compression ratio but positive lower ratio after prepending a bit.

A universal constant bounds the increase in compression ratio when adding a letter, for words with low initial ratios.

The bounds are tight, with examples showing the ratio can jump from near zero to a constant.

Abstract

The so-called "one-bit catastrophe" for the compression algorithm LZ'78 asks whether the compression ratio of an infinite word can change when a single bit is added in front of it. We answer positively this open question raised by Lutz and others: we show that there exists an infinite word $w$ such that $\rho_{sup}(w)=0$ but $\rho_{inf}(0w)>0$, where $\rho_{sup}$ and $\rho_{inf}$ are respectively the $\limsup$ and the $\liminf$ of the compression ratios $\rho$ of the prefixes. To that purpose we explore the behaviour of LZ'78 on finite words and show the following results: - There is a constant $C>0$ such that, for any finite word $w$ and any letter $a$, $\rho(aw)\leq C\sqrt{\rho(w)\log|w|}$. Thus, sufficiently compressible words ($\rho(w)=o(1/\log|w|)$) remain compressible with a letter in front; - The previous result is tight up to a multiplicative constant for any compression ratio $\rho(w)=O(1/\log|w|)$. In particular, there are infinitely many words $w$ satisfying $\rho(w)=O(1/\log|w|)$ but $\rho(0w)=\Omega(1)$.