LZ-ABT: A Practical Algorithm for $\alpha$-Balanced Grammar Compression

Abstract

We propose a new LZ78-style grammar compression algorithm, named LZ-ABT, which is a simple online algorithm to create, given a string of length N over an alphabet of size $\sigma$, an $\alpha$-balanced grammar in $O(N \log N \log \sigma )$ time and O(n) space in addition to the input string, where n is the grammar size to output. LZ-ABT can avoid the lower-bound of $\varOmega (N^{5/4})$ time of the naive algorithms for LZMW and LZD, other LZ78-style compression algorithms, which was observed in [Badkobeh et al. SPIRE 2017, pp. 51–67]. We also show that the algorithm can be executed in compressed space, i.e., without storing the whole input string explicitly in memory: in $O(N \log ^2 N \log \sigma )$ time and O(n) space, or $O(N \log N \log \sigma )$ time and $O(n \log ^{*} N)$ space. We implement LZ-ABT running in $O(N \log N \log \sigma )$ time and O(N) space and empirically show that its performance is competitive to LZD. This is the first practical implementation of $\alpha$-balanced grammar compression to the best of our knowledge.