Computing MEMs and Relatives on Repetitive Text Collections

Abstract

We consider the problem of computing the Maximal Exact Matches (MEMs) of a given pattern $P[\mathrm{1..}m]$ on a large repetitive text collection $T[\mathrm{1..}n]$, which is represented as a (hopefully much smaller) run-length context-free grammar of size $g_{rl}$. We show that the problem can be solved in time $O(m^2{\log }^{\epsilon }n)$, for any constant $\epsilon >0$, on a data structure of size $O(g_{rl})$. Further, on a locally consistent grammar of size $O(\delta \log \frac{n}{\delta })$, the time decreases to $O(m\log m(\log m+{\log }^{\epsilon }n))$. The value $\delta$ is a function of the substring complexity of $T$ and $\Omega (\delta \log \frac{n}{\delta })$ is a tight lower bound on the compressibility of repetitive texts $T$, so our structure has optimal size in terms of $n$ and $\delta$. We extend our results to several related problems, such as finding $k$-MEMs, MUMs, rare MEMs, and applications.