# Conversion from RLBWT to LZ77

## TL;DR

This paper introduces an improved algorithm for converting RLBWT to LZ77, reducing time complexity and maintaining linear space, by replacing dynamic data structures with more efficient ones.

## Contribution

The paper presents a novel conversion algorithm that enhances previous methods by optimizing data structures for faster RLBWT to LZ77 conversion.

## Key findings

- Achieves $O(n ext{ min } igrace ext{log log n, } ext{sqrt}(rac{ ext{log r}}{ ext{log log r}}) igrace)$ time complexity.
- Maintains $O(r)$ working space, improving efficiency over prior algorithms.
- Provides a practical approach for faster string compression conversions.

## Abstract

Converting a compressed format of a string into another compressed format without an explicit decompression is one of the central research topics in string processing. We discuss the problem of converting the run-length Burrows-Wheeler Transform (RLBWT) of a string to Lempel-Ziv 77 (LZ77) phrases of the reversed string. The first results with Policriti and Prezza's conversion algorithm [Algorithmica 2018] were $O(n \log r)$ time and $O(r)$ working space for length of the string $n$, number of runs $r$ in the RLBWT, and number of LZ77 phrases $z$. Recent results with Kempa's conversion algorithm [SODA 2019] are $O(n / \log n + r \log^{9} n + z \log^{9} n)$ time and $O(n / \log_{\sigma} n + r \log^{8} n)$ working space for the alphabet size $\sigma$ of the RLBWT. In this paper, we present a new conversion algorithm by improving Policriti and Prezza's conversion algorithm where dynamic data structures for general purpose are used. We argue that these dynamic data structures can be replaced and present new data structures for faster conversion. The time and working space of our conversion algorithm with new data structures are $O(n \min \{ \log \log n, \sqrt{\frac{\log r}{\log\log r}} \})$ and $O(r)$, respectively.