Computing MEMs and Relatives on Repetitive Text Collections

We consider the problem of computing the Maximal Exact Matches (MEMs) of a given pattern $P[1 .. m]$ on a large repetitive text collection $T[1 .. n]$, 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 $O(m^2 \log^εn)$, for any constant $ε> 0$, on a data structure of size $O(g_{rl})$. Further, on a locally consistent grammar of size $O(δ\log\frac{n}δ)$, the time decreases to $O(m\log m(\log m + \log^εn))$. The value $δ$ is a function of the substring complexity of $T$ and $Ω(δ\log\frac{n}δ)$ is a tight lower bound on the compressibility of repetitive texts $T$, so our structure has optimal size in terms of $n$ and $δ$. We extend our results to several related problems, such as finding $k$-MEMs, MUMs, rare MEMs, and applications.