Compact q-gram Profiling of Compressed Strings

TL;DR

This paper introduces a space-efficient algorithm for computing the q-gram profile of compressed strings using a novel q-gram graph, achieving optimal space and matching the best known time bounds.

Contribution

It presents a new algorithm that computes q-gram profiles of grammar-compressed strings with optimal space and expected time, improving previous space bounds.

Findings

Algorithm runs in expected O(N-α) time

Uses space O(n+q+κ_q), which is asymptotically optimal

Introduces the q-gram graph for efficient structure capturing

Abstract

We consider the problem of computing the q-gram profile of a string \str of size $N$ compressed by a context-free grammar with $n$ production rules. We present an algorithm that runs in $O(N-\alpha)$ expected time and uses $O(n+q+\kq)$ space, where $N-\alpha\leq qn$ is the exact number of characters decompressed by the algorithm and $\kq\leq N-\alpha$ is the number of distinct q-grams in $\str$. This simultaneously matches the current best known time bound and improves the best known space bound. Our space bound is asymptotically optimal in the sense that any algorithm storing the grammar and the q-gram profile must use $\Omega(n+q+\kq)$ space. To achieve this we introduce the q-gram graph that space-efficiently captures the structure of a string with respect to its q-grams, and show how to construct it from a grammar.