A note on the longest common Abelian factor problem

TL;DR

This paper improves the space efficiency and time complexity of algorithms solving the longest common Abelian factor problem in string matching, making them more practical for larger datasets.

Contribution

It reduces the space complexity by a factor of the alphabet size while maintaining the same time complexity, and also improves time complexity with additional space for larger alphabets.

Findings

Space complexity reduced by a factor of 

Time complexity preserved with space reduction

Improved time complexity with superlinear space for larger alphabets

Abstract

Abelian string matching problems are becoming an object of considerable interest in last years. Very recently, Alatabbi et al. \cite{AILR2015} presented the first solution for the longest common Abelian factor problem for a pair of strings, reaching $O(\sigma n^2)$ time with $O(\sigma n \log n)$ bits of space, where $n$ is the length of the strings and $\sigma$ is the alphabet size. In this note we show how the time complexity can be preserved while the space is reduced by a factor of $\sigma$, and then how the time complexity can be improved, if the alphabet is not too small, when superlinear space is allowed.