Computing Abelian String Regularities Based on RLE

Abstract

Two strings x and y are said to be Abelian equivalent if x is a permutation of y, or vice versa. If a string z satisfies $z = xy$ with x and y being Abelian equivalent, then z is said to be an Abelian square. If a string w can be factorized into a sequence $v_1, \ldots , v_s$ of strings such that $v_1$, ..., $v_{s-1}$ are all Abelian equivalent and $v_s$ is a substring of a permutation of $v_1$, then w is said to have a regular Abelian period (p, t) where $p = |v_1|$ and $t = |v_s|$. If a substring $w_1[i..i+\ell -1]$ of a string $w_1$ and a substring $w_2[j..j+\ell -1]$ of another string $w_2$ are Abelian equivalent, then the substrings are said to be a common Abelian factor of $w_1$ and $w_2$ and if the length $\ell$ is the maximum of such then the substrings are said to be a longest common Abelian factor of $w_1$ and $w_2$. We propose efficient algorithms which compute these Abelian regularities using the run length encoding (RLE) of strings. For a given string w of length n whose RLE is of size m, we propose algorithms which compute all Abelian squares occurring in w in O(mn) time, and all regular Abelian periods of w in O(mn) time. For two given strings $w_1$ and $w_2$ of total length n and of total RLE size m, we propose an algorithm which computes all longest common Abelian factors in $O(m^2n)$ time.