Regular Abelian Periods and Longest Common Abelian Factors on Run-Length Encoded Strings

Grabowski, Szymon

doi:10.1007/978-3-319-67428-5_17

Cited by 1 publication

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“…• O(m 3 ) time and O(m) space (see Section 6). The latter improves upon the time complexities of the algorithms of Sugimoto et al [24] (O(nm 2 )) and Grabowski [15] (O(m 4 )). In the case of constant σ, we obtain the following improved versions of the former:…”

Section: Our Resultsmentioning

confidence: 80%

“…In Section 4 we show a reduction of RLE-LCAF to a problem of intersecting rectangles in Z σ . This allows us to develop two solutions to RLA-LCAF, that work in: The latter improves upon the time complexities of the algorithms of Sugimoto et al [24] (O(nm 2 )) and Grabowski [15] (O(m 4 )). In the case of constant σ, we obtain the following improved versions of the former:…”

Section: Our Resultsmentioning

confidence: 99%

“…Known solutions for the RLE-LCAF problem (for arbitrary σ) work in O(nm 2 ) [24], in O(m 4 ), and in [15], respectively.…”

Section: Previous Resultsmentioning

confidence: 99%

“…Point (a) of the observation is straightforward. Point (b) boils down to simple arithmetics (see also [15]). Observation 6.3.…”

Section: Algorithm For Rle-lcaf Over Large Alphabetmentioning

confidence: 99%

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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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Section: Our Resultsmentioning

confidence: 80%

Section: Our Resultsmentioning

confidence: 99%