Random Access to Grammar-Compressed Strings and Trees

Grammar-based compression, where one replaces a long string by a small context-free grammar that generates the string, is a simple and powerful paradigm that captures (sometimes with slight reduction in efficiency) many of the popular compression schemes, including the Lempel--Ziv family, run-length encoding, byte-pair encoding, Sequitur, and Re-Pair. In this paper, we present a novel grammar representation that allows efficient random access to any character or substring without decompressing the string. Let $S$ be a string of length $N$ compressed into a context-free grammar $\mathcal{S}$ of size $n$. We present two representations of $\mathcal{S}$ achieving $O(\log N)$ random access time, and either $O(n\cdot\alpha_k(n))$ construction time and space on the pointer machine model, or $O(n)$ construction time and space on the RAM. Here, $\alpha_k(n)$ is the inverse of the $k$th row of Ackermann's function. Our representations also efficiently support decompression of any substring in $S$: we can decompres...