On Abelian Longest Common Factor with and without RLE

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.Comment: Submitted to a journa