Computing Abelian regularities on RLE strings

TL;DR

This paper introduces efficient algorithms for computing Abelian regularities, such as Abelian squares, regular Abelian periods, and longest common Abelian factors, using run length encoding (RLE) of strings.

Contribution

It presents novel algorithms that compute Abelian regularities in strings efficiently by leveraging RLE, improving computational performance over previous methods.

Findings

Algorithms for Abelian squares in O(mn) time

Algorithms for regular Abelian periods in O(mn) time

Algorithms for longest common Abelian factors in O(m^2 n) time

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,...,v_s of strings such that v_1 ,..., v_{s-1} are all Abelian equivalent and vs 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+l-1] of a string w_1 and a substring w_2[j..j+l-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 l 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.