# On Two LZ78-style Grammars: Compression Bounds and Compressed-Space Computation

## TL;DR

This paper analyzes the compression bounds of two LZ78-based schemes, LZMW and LZD, revealing their limitations in grammar size and proposing a new efficient parsing algorithm with improved space and time complexity.

## Contribution

It provides theoretical bounds on the grammar size produced by LZ78 variants and introduces a new probabilistic parsing algorithm with better space and time efficiency.

## Key findings

- Grammar size can be larger than the minimal by a factor of (n^{1/3})
- Standard algorithms require (n^{5/4}) time in worst case
- New Las Vegas algorithm uses O(z  log n) space and O(n + z  log^2 n) time with high probability

## Abstract

We investigate two closely related LZ78-based compression schemes: LZMW (an old scheme by Miller and Wegman) and LZD (a recent variant by Goto et al.). Both LZD and LZMW naturally produce a grammar for a string of length $n$; we show that the size of this grammar can be larger than the size of the smallest grammar by a factor $\Omega(n^{\frac{1}3})$ but is always within a factor $O((\frac{n}{\log n})^{\frac{2}{3}})$. In addition, we show that the standard algorithms using $\Theta(z)$ working space to construct the LZD and LZMW parsings, where $z$ is the size of the parsing, work in $\Omega(n^{\frac{5}4})$ time in the worst case. We then describe a new Las Vegas LZD/LZMW parsing algorithm that uses $O (z \log n)$ space and $O(n + z \log^2 n)$ time w.h.p..