Detecting regularities on grammar-compressed strings

TL;DR

This paper presents algorithms for detecting regularities such as runs, squares, gapped-palindromes, periods, and covers in strings represented by straight line programs, optimizing for compressed string analysis.

Contribution

It introduces efficient algorithms for detecting various regularities directly on grammar-compressed strings, improving analysis of compressed data.

Findings

All runs and squares can be computed in O(n^3h) time

Gapped-palindromes are computable in O(n^3h + gnh log N) time

String periods and covers can be found in O(n^2 h) and O(nh(n+log^2 N)) time respectively.

Abstract

We solve the problems of detecting and counting various forms of regularities in a string represented as a Straight Line Program (SLP). Given an SLP of size $n$ that represents a string $s$ of length $N$, our algorithm compute all runs and squares in $s$ in $O(n^3h)$ time and $O(n^2)$ space, where $h$ is the height of the derivation tree of the SLP. We also show an algorithm to compute all gapped-palindromes in $O(n^3h + gnh\log N)$ time and $O(n^2)$ space, where $g$ is the length of the gap. The key technique of the above solution also allows us to compute the periods and covers of the string in $O(n^2 h)$ time and $O(nh(n+\log^2 N))$ time, respectively.