Algorithms for Longest Common Abelian Factors

Alatabbi, Ali; Iliopoulos, Costas S.; Langiu, Alessio; Rahman, M. Sohel

doi:10.1142/s0129054116500143

Cited by 5 publications

(16 citation statements)

References 9 publications

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“…A pair (s 1 , s 2 ) of a substring s 1 of w 1 and a substring s 2 of w 2 is said to be a common Abelian factor of w 1 and w 2 , iff s 1 and s 2 are Abelian equivalent. Alatabbi et al [1] proposed an O(σn 2 )-time and O(σn)space algorithm to solve Problem 3 of computing all longest common Abelian factors (LCAFs) of two given strings of total length n. Later, Grabowski [10] showed an algorithm which finds all LCAFs in O(σn 2 ) time with O(n) space. 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 .…”

Section: Our Problems and Previous Resultsmentioning

confidence: 99%

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Computing Abelian String Regularities Based on RLE

Sugimoto

Noda

Inenaga

et al. 2018

Lecture Notes in Computer Science

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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 v1, . . . , vs of strings such that v1, . . . , vs−1 are all Abelian equivalent and vs is a substring of a permutation of v1, then w is said to have a regular Abelian period (p, t) where p = |v1| and t = |vs|. If a substring w1[i..i+ℓ−1] of a string w1 and a substring w2[j..j +ℓ−1] of another string w2 are Abelian equivalent, then the substrings are said to be a common Abelian factor of w1 and w2 and if the length ℓ is the maximum of such then the substrings are said to be a longest common Abelian factor of w1 and w2. 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 w1 and w2 of total length n and of total RLE size m, we propose an algorithm which computes all longest common Abelian factors in O(m 2 n) time.

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Section: Our Problems and Previous Resultsmentioning

confidence: 99%

“…Solution (2) is faster than the O(n(log log n+log σ))-time solution by Kociumaka et al [11] for highly RLE-compressible strings with log log n = ω(m) 1 .…”

Section: Our Contributionmentioning

confidence: 90%

Computing Abelian String Regularities Based on RLE

Sugimoto

Noda

Inenaga

et al. 2018

Lecture Notes in Computer Science

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“…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.…”

Section: Algorithm For Rle-lcaf Over Large Alphabetmentioning

confidence: 99%

“…We have presented efficient algorithms for the LCAF and RLE-LCAF problems: For LCAF over a constant-sized alphabet, we have obtained an over-logarithmic speedup comparing to a naive O(n 2 )-time solution. Let us recall that over the binary alphabet, LCAF can be solved much more efficiently, in O(n 1.859 ) time [2,10]. An open question is to design an O(n 2−ε )-time algorithm (ε > 0) for LCAF for alphabet of any constant size, e.g., for σ = 3.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

“…Subquadratic-time and space constructions of a jumbled index for any constant-sized alphabet were proposed in [19,10].Binary jumbled indexing was also considered in the case that the text is given as its RLE representation of length m. Constructions of the index working in O(n+m 2 log m) time [3,4] and in O(n+m 2 ) time [13,14] were proposed.As for other Abelian stringology problems, subquadratic-time algorithms for computing Abelian squares, Abelian periods, Abelian runs, Abelian covers, and Abelian borders over a constant-sized alphabet were designed in [20,21]. Computation of Abelian borders, Abelian periods, and Abelian squares on strings specified by their RLE representations was considered in [3,24].Longest common Abelian factor In the special case of a binary alphabet, the LCAF problem reduces in linear time to binary jumbled indexing [2]. Indeed, it suffices to construct jumbled indexes of each of the strings and then to check, for each length ℓ, if both strings contain an Abelian factor of length ℓ containing the same number of ones.…”

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On Abelian Longest Common Factor with and without RLE

Grabowski

Kociumaka

Radoszewski

2018

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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 σ. For the uncompressed problem, we show an o(n 2 )-time and O(n)-space algorithm in the case of σ = O(1), making a non-trivial use of tabulation. For the RLE-compressed problem, we show two algorithms: one working in O(m 2 σ 2 log 3 m) time and O(m(σ 2 + log 2 m)) space, which employs line sweep, and one that works in O(m 3 ) time and O(m) space that applies in a careful way a sliding-window-based approach. The latter improves upon the previously known O(nm 2 )-time and O(m 4 )-time algorithms that were recently developed by Sugimoto et al. (IWOCA 2017) and Grabowski (SPIRE 2017), respectively.Keywords: Abelian longest common factor problem, jumbled pattern matching, run-length encoding (RLE) After a series of works of Burcsi et al. [8,9] and Moosa and Rahman [22,23], the construction of a binary jumbled index was improved to O( n 2 (log n) 2 ). Furthermore, Hermelin et al. [18] reduced binary jumbled indexing to all-pairs shortest paths problem and obtained preprocessing time of O( n 2 2 Ω((log n/ log log n) 0.5 ) ) (a similar reduction was shown by Bremner at el. [7]). Finally, Chan and Lewenstein [10] used techniques from additive combinatorics to improve the construction time of the binary index to O(n 1.859 ). Subquadratic-time and space constructions of a jumbled index for any constant-sized alphabet were proposed in [19,10].Binary jumbled indexing was also considered in the case that the text is given as its RLE representation of length m. Constructions of the index working in O(n+m 2 log m) time [3,4] and in O(n+m 2 ) time [13,14] were proposed.As for other Abelian stringology problems, subquadratic-time algorithms for computing Abelian squares, Abelian periods, Abelian runs, Abelian covers, and Abelian borders over a constant-sized alphabet were designed in [20,21]. Computation of Abelian borders, Abelian periods, and Abelian squares on strings specified by their RLE representations was considered in [3,24].Longest common Abelian factor In the special case of a binary alphabet, the LCAF problem reduces in linear time to binary jumbled indexing [2]. Indeed, it suffices to construct jumbled indexes of each of the strings and then to check, for each length ℓ, if both strings contain an Abelian factor of length ℓ containing the same number of ones. 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 i...

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Longest Common Abelian Factors and Large Alphabets

Badkobeh

Gagie

Grabowski

et al. 2016

String Processing and Information Retrieval

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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).

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Algorithms for Longest Common Abelian Factors

Cited by 5 publications

References 9 publications

Computing Abelian String Regularities Based on RLE

Computing Abelian String Regularities Based on RLE

On Abelian Longest Common Factor with and without RLE

Longest Common Abelian Factors and Large Alphabets

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