Computing MEMs and Relatives on Repetitive Text Collections

TL;DR

This paper introduces efficient algorithms for computing Maximal Exact Matches (MEMs) on large, repetitive text collections using grammar-based compression, achieving near-optimal size and improved time complexities.

Contribution

It presents novel algorithms for MEM computation on grammar-compressed texts with improved time bounds and optimal size in terms of text repetitiveness.

Findings

Time complexity for general case: $O(m^2 \, \log^\epsilon n)$

Time complexity for locally consistent grammar: $O(m \log m (\log m + \log^\epsilon n))$

Structure size is optimal relative to text repetitiveness measure \(\delta\)

Abstract

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 $O(m^2 \log^\epsilon n)$, 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.