Fast and Space-Efficient Construction of AVL Grammars from the LZ77 Parsing

Abstract

Grammar compression is, next to Lempel-Ziv (LZ77) and run-length Burrows-Wheeler transform (RLBWT), one of the most flexible approaches to representing and processing highly compressible strings. The main idea is to represent a text as a context-free grammar whose language is precisely the input string. This is called a straight-line grammar (SLG). An AVL grammar, proposed by Rytter [Theor. Comput. Sci., 2003] is a type of SLG that additionally satisfies the AVL-property: the heights of parse-trees for children of every nonterminal differ by at most one. In contrast to other SLG constructions, AVL grammars can be constructed from the LZ77 parsing in compressed time: O(z log n) where z is the size of the LZ77 parsing and n is the length of the input text. Despite these advantages, AVL grammars are thought to be too large to be practical.We present a new technique for rapidly constructing a small AVL grammar from an LZ77 or LZ77-like parse. Our algorithm produces grammars that are always at least five times smaller than those produced by the original algorithm, and never more than double the size of grammars produced by the practical Re-Pair compressor [Larsson and Moffat, Proc. IEEE, 2000]. Our algorithm also achieves low peak RAM usage. By combining this algorithm with recent advances in approximating the LZ77 parsing, we show that our method has the potential to construct a run-length BWT from an LZ77 parse in about one third of the time and peak RAM required by other approaches. Overall, we show that AVL grammars are surprisingly practical, opening the door to much faster construction of key compressed data structures.

Compressibility-Aware Quantum Algorithms on Strings

“…Obtaining a balanced CFG of size Õ(z) is similarly an application of previous results, and can be obtained by applying either the original LZ77 to balanced grammar conversion algorithm of Rytter [42], or more recent results for converting LZ77 encodings to grammars [25], and even balanced straight-line programs [22].…”