On Abelian Longest Common Factor with and without RLE

TL;DR

This paper introduces new algorithms for the Abelian longest common factor problem, improving efficiency for both uncompressed and run-length encoded strings, with significant theoretical advancements over previous methods.

Contribution

It presents the first sub-quadratic algorithm for uncompressed strings with constant alphabet size and two improved algorithms for RLE-compressed strings, surpassing prior complexity bounds.

Findings

O(n) space and o(n^2) time algorithm for uncompressed strings with constant alphabet

An O(m^2 σ^2 log^3 m) time algorithm for RLE-compressed strings

An O(m^3) time algorithm for RLE-compressed strings that improves previous bounds

Abstract

We consider the Abelian longest common factor problem in two scenarios: when input strings are uncompressed and are of size $n$, and when the input strings are run-length encoded and their compressed representations have size at most $m$. The alphabet size is denoted by $\sigma$. For the uncompressed problem, we show an $o(n^2)$-time and $\Oh(n)$-space algorithm in the case of $\sigma=\Oh(1)$, making a non-trivial use of tabulation. For the RLE-compressed problem, we show two algorithms: one working in $\Oh(m^2\sigma^2 \log^3 m)$ time and $\Oh(m (\sigma^2+\log^2 m))$ space, which employs line sweep, and one that works in $\Oh(m^3)$ time and $\Oh(m)$ space that applies in a careful way a sliding-window-based approach. The latter improves upon the previously known $\Oh(nm^2)$-time and $\Oh(m^4)$-time algorithms that were recently developed by Sugimoto et al.\ (IWOCA 2017) and Grabowski (SPIRE 2017), respectively.