An Upper Bound and Linear-Space Queries on the LZ-End Parsing.

Abstract

Lempel-Ziv (LZ77) compression is the most commonly used lossless compression algorithm. The basic idea is to greedily break the input string into blocks (called "phrases"), every time forming as a phrase the longest prefix of the unprocessed part that has an earlier occurrence. In 2010, Kreft and Navarro introduced a variant of LZ77 called LZ-End, that additionally requires the previous occurrence of each phrase to end at the boundary of an already existing phrase. Due to its excellent practical performance as a compression algorithm and a compressed index, they conjectured that it achieves a compression that can be provably upper-bounded in terms of the LZ77 size. Despite the recent progress in understanding such relation for other compression algorithms (e.g., the run-length encoded Burrows-Wheeler transform), no such result is known for LZ-End. We prove that for any string of length $n$, the number $z_e$ of phrases in the LZ-End parsing satisfies $z_e=𝒪\left(z{log}^{2}n\right)$, where $z$ is the number of phrases in the LZ77 parsing. This is the first non-trivial upper bound on the size of LZ-End parsing in terms of LZ77, and it puts LZ-End among the strongest dictionary compressors. Using our techniques we also derive bounds for other variants of LZ-End and with respect to other compression measures. Our second contribution is a data structure that implements random access queries to the text in $𝒪\left(z_e\right)$ space and $𝒪(polylogn)$ time. This is the first linear-size structure on LZ-End that efficiently implements such queries. All previous data structures either incur a logarithmic penalty in the space or have slow queries. We also show how to extend these techniques to support longest-common-extension (LCE) queries.