Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

Kociumaka, Tomasz; Radoszewski, Jakub; Rytter, Wojciech

doi:10.1007/978-3-642-40450-4_53

Cited by 30 publications

(27 citation statements)

References 15 publications

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“…The problem has also been extended to graphs and trees [22,15], to the streaming model [27], and to approximate indexes [16]. There is also interest in the non-binary variant [20,17,11,14,7,8,26], as well as in reconstruction from the Parikh multi-set of a string [1]. Applications in computational biology include SNP discovery, alignment, gene clusters, pattern discovery, and mass spectrometry data interpretation [4,3,5,19,33].…”

Section: Introductionmentioning

confidence: 99%

On prefix normal words and prefix normal forms

Burcsi

Fici

Lipták

et al. 2017

Theoretical Computer Science

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A 1-prefix normal word is a binary word with the property that no factor has more 1s than the prefix of the same length; a 0-prefix normal word is defined analogously. These words arise in the context of indexed binary jumbled pattern matching, where the aim is to decide whether a word has a factor with a given number of 1s and 0s (a given Parikh vector). Each binary word has an associated set of Parikh vectors of the factors of the word. Using prefix normal words, we provide a characterization of the equivalence class of binary words having the same set of Parikh vectors of their factors.We prove that the language of prefix normal words is not context-free and is strictly contained in the language of pre-necklaces, which are prefixes of powers of Lyndon words. We give enumeration results on pnw(n), the number of prefix normal words of length n, showing that, for sufficiently large n,For fixed density (number of 1s), we show that the ordinary generating function of the number of prefix normal words of length n and density d is a rational function. Finally, we give experimental results on pnw(n), discuss further properties, and state open problems.

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Section: Introductionmentioning

confidence: 99%

On prefix normal words and prefix normal forms

Burcsi

Fici

Lipták

et al. 2017

Theoretical Computer Science

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“…A problem worth considering in this context, and that was recently considered in the context of abelian pattern matching [15,6], is that of building index structures for k-abelian pattern matching. Basically, now we are given a positive integer k and a text T , and we want to preprocess the text such that we can answer quickly queries in which we are given a pattern P and have to report whether T has a factor that is k-abelian to P .…”

Section: Index Structuresmentioning

confidence: 99%

k-Abelian pattern matching

Ehlers

Manea

Mercaş

et al. 2015

Journal of Discrete Algorithms

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Two words are called k-abelian equivalent, if they share the same multiplicities for all factors of length at most k. We present an optimal linear time algorithm for identifying all occurrences of factors in a text that are k-abelian equivalent to some pattern P . Moreover, an optimal algorithm for finding the largest k for which two words are k-abelian equivalent is given. Solutions for online versions of the k-abelian pattern matching problem are also proposed.

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“…In a single stage, we have O(nσ) packed lists with O(nb) entries in total, and the universe size is N = b σ . By [19,Lemma 5.3], the F irstOcc bitmasks can be computed in O(nσ + nb log 2 (b σ )/w) = O(nσ + nbσ 2 log 2 b/w) time, while the space complexity is O(nσ + nσb log b/w) words. Across all stages, the overall running time becomes O(n 2 σ 2 /b + n 2 σ 2 log 2 b/w), whereas the space consumption is O(nσ 2 + nσb log b/w) words.…”

Section: O(n 2 / Log 1/σ N) Timementioning

confidence: 99%

“…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].…”

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confidence: 99%

On Abelian Longest Common Factor with and without RLE

Grabowski

Kociumaka

Radoszewski

2018

Self Cite

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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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Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

Cited by 30 publications

References 15 publications

On prefix normal words and prefix normal forms

On prefix normal words and prefix normal forms

k-Abelian pattern matching

On Abelian Longest Common Factor with and without RLE

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