On Abelian Longest Common Factor with and without RLE

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 $σ=\Oh(1)$, making a non-trivial use of tabulation. For the RLE-compressed problem, we show two algorithms: one working in $\Oh(m^2σ^2 \log^3 m)$ time and $\Oh(m (σ^2+\log^2 m))$ 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.

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