Grammar Boosting: A New Technique for Proving Lower Bounds for Computation over Compressed Data
Abstract
Grammar compression is a general compression framework in which a string $T$ of length $N$ is represented as a contextfree grammar of size $n$ whose language contains only $T$. In this paper, we focus on studying the limitations of algorithms and data structures operating on strings in grammarcompressed form. Previous work focused on proving lower bounds for grammars constructed using algorithms that achieve the approximation ratio $\rho=\mathcal{O}(\text{polylog }N)$. Unfortunately, for the majority of grammar compressors, $\rho$ is either unknown or satisfies $\rho=\omega(\text{polylog }N)$. In their seminal paper, Charikar et al. [IEEE Trans. Inf. Theory 2005] studied seven popular grammar compression algorithms: RePair, Greedy, LongestMatch, Sequential, Bisection, LZ78, and $\alpha$Balanced. Only one of them ($\alpha$Balanced) is known to achieve $\rho=\mathcal{O}(\text{polylog }N)$. We develop the first technique for proving lower bounds for data structures and algorithms on grammars that is fully general and does not depend on the approximation ratio $\rho$ of the used grammar compressor. Using this technique, we first prove that $\Omega(\log N/\log \log N)$ time is required for random access on RePair, Greedy, LongestMatch, Sequential, and Bisection, while $\Omega(\log\log N)$ time is required for random access to LZ78. All these lower bounds hold within space $\mathcal{O}(n\text{ polylog }N)$ and match the existing upper bounds. We also generalize this technique to prove several conditional lower bounds for compressed computation. For example, we prove that unless the Combinatorial $k$Clique Conjecture fails, there is no combinatorial algorithm for CFG parsing on Bisection (for which it holds $\rho=\tilde{\Theta}(N^{1/2})$) that runs in $\mathcal{O}(n^c\cdot N^{3\epsilon})$ time for all constants $c>0$ and $\epsilon>0$. Previously, this was known only for $c<2\epsilon$.
 Publication:

arXiv eprints
 Pub Date:
 July 2023
 DOI:
 10.48550/arXiv.2307.08833
 arXiv:
 arXiv:2307.08833
 Bibcode:
 2023arXiv230708833D
 Keywords:

 Computer Science  Data Structures and Algorithms