Longest Common Abelian Factors and Large Alphabets

Abstract

Two strings X and Y are considered Abelian equal if the letters of X can be permuted to obtain Y (and vice versa). Recently, Alatabbi et al. (2015) considered the longest common Abelian factor problem in which we are asked to find the length of the longest Abelian-equal factor present in a given pair of strings. They provided an algorithm that uses O(σn 2) time and O(σn) space, where n is the length of the pair of strings and σ is the alphabet size. In this paper we describe an algorithm that uses O(n 2 log 2 n log * n) time and O(n log 2 n) space, significantly improving Alatabbi et al.'s result unless the alphabet is small. Our algorithm makes use of techniques for maintaining a dynamic set of strings under split, join, and equality testing (Melhorn et al., Algorithmica 17(2), 1997).

Computing Abelian String Regularities Based on RLE

“…Compute regular Abelian periods of a given string. 3. Compute longest common Abelian factors of two given strings.…”

“…He also presented an O(( σ k + log σ)n 2 log n)-time O(kn)-space algorithm for a parameter k ≤ σ log σ . Recently, Badkobeh et al [3] proposed an O(n log 2 n log * n)-time O(n log 2 n)-space algorithm for finding all LCAFs.…”

“…Our solution (3) is faster than the O(σn 2 )-time solution by Grabowski [10] when σn = ω(m 2 ), and is faster than the fastest variant of the other solution by Grabowski [10] (choosing k = σ log σ ) when √ n log n log σ = ω(m). Also, solution (3) is faster than the O(n log 2 n log * n)-time solution by Badkobeh et al [3] when log n log * n = ω(m). The time bounds of our algorithms are all deterministic.…”

On Abelian Longest Common Factor with and without RLE

“…We have P(R 1 ) = (5, ⋆, 4, ⋆, 3) and P(R 2 ) = (⋆, 2, 4, ⋆, 3). Here the set ∆(R 1 , R 2 ) stores the intervals {5}, [1,3], and [1,6] from R 1 and [4,5], {2}, and [2,5] from R 2 that correspond to the coordinates 1, 2, and 4.…”

“…Thus binary LCAF can be solved in O(n 1.859 ) time using using the best known jumbled index [10]. Moreover, binary RLE-LCAF can be solved in O(n + m 2 ) time and O(n) space by applying an efficient binary jumbled index for an RLE representation of the text [13,14].Over a general alphabet, for the LCAF problem the fastest known algorithms work in O(n 2 σ) time and O(n) space, and in O(n 2 log 2 n log * n) time and O(n log 2 n) space [5].Known solutions for the RLE-LCAF problem (for arbitrary σ) work in O(nm 2 ) [24], in O(m 4 ), and in O(n 3/2 σ √ m log n) (provided that m = O(n/ log n)) time [15], respectively.…”