Optimal-Time Dictionary-Compressed Indexes

We describe the first self-indexes able to count and locate pattern occurrences in optimal time within a space bounded by the size of the most popular dictionary compressors. To achieve this result we combine several recent findings, including \emph{string attractors} --- new combinatorial objects encompassing most known compressibility measures for highly repetitive texts ---, and grammars based on \emph{locally-consistent parsing}. More in detail, let $\gamma$ be the size of the smallest attractor for a text $T$ of length $n$. The measure $\gamma$ is an (asymptotic) lower bound to the size of dictionary compressors based on Lempel--Ziv, context-free grammars, and many others. The smallest known text representations in terms of attractors use space $O(\gamma\log(n/\gamma))$, and our lightest indexes work within the same asymptotic space. Let $\epsilon>0$ be a suitably small constant fixed at construction time, $m$ be the pattern length, and $occ$ be the number of its text occurrences. Our index counts pattern occurrences in $O(m+\log^{2+\epsilon}n)$ time, and locates them in $O(m+(occ+1)\log^\epsilon n)$ time. These times already outperform those of most dictionary-compressed indexes, while obtaining the least asymptotic space for any index searching within $O((m+occ)\,\textrm{polylog}\,n)$ time. Further, by increasing the space to $O(\gamma\log(n/\gamma)\log^\epsilon n)$, we reduce the locating time to the optimal $O(m+occ)$, and within $O(\gamma\log(n/\gamma)\log n)$ space we can also count in optimal $O(m)$ time. No dictionary-compressed index had obtained this time before. All our indexes can be constructed in $O(n)$ space and $O(n\log n)$ expected time. As a byproduct of independent interest...