Space-efficient conversions from SLPs

TL;DR

This paper introduces space-efficient algorithms for converting straight-line programs into Lempel-Ziv parses and locally consistent grammars, optimizing for space and time complexity.

Contribution

It presents novel algorithms that efficiently convert SLPs into LZ parses and optimal size grammars using minimal space and improved time bounds.

Findings

LZ parse construction in $O(g)$ space and $O(n ext{log}^2 n)$ time.

Construction of an optimal size locally consistent grammar in $O(g+g_{lc})$ space and $O(n ext{log} g)$ time.

Conversion from locally consistent grammar to LZ parse in $O(g_{lc})$ space and $O(z ext{log}^2 n ext{log}^2(n/z))$ time.

Abstract

We give algorithms that, given a straight-line program (SLP) with $g$ rules that generates (only) a text $T [1..n]$, builds within $O(g)$ space the Lempel-Ziv (LZ) parse of $T$ (of $z$ phrases) in time $O(n\log^2 n)$ or in time $O(gz\log^2(n/z))$. We also show how to build a locally consistent grammar (LCG) of optimal size $g_{lc} = O(\delta\log\frac{n}{\delta})$ from the SLP within $O(g+g_{lc})$ space and in $O(n\log g)$ time, where $\delta$ is the substring complexity measure of $T$. Finally, we show how to build the LZ parse of $T$ from such a LCG within $O(g_{lc})$ space and in time $O(z\log^2 n \log^2(n/z))$. All our results hold with high probability.