Efficient Representation and Counting of Antipower Factors in Words

Kociumaka, Tomasz; Radoszewski, Jakub; Rytter, Wojciech; Straszyński, Juliusz; Waleń, Tomasz; Zuba, Wiktor

doi:10.1007/978-3-030-13435-8_31

Cited by 8 publications

(3 citation statements)

References 14 publications

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“…Let us call this grid G q . A chain Chain q can be conveniently represented in the grid G q using the following lemma; it was stated in [28] and its proof can be found in the full version of that paper [27]. Lemma 5 can be used to compute a union of interval chains.…”

Section: Simple Geometry Of Arithmetic Sequences Of Intervalsmentioning

confidence: 99%

Circular Pattern Matching with k Mismatches

Charalampopoulos

Kociumaka

Pissis

et al. 2019

Fundamentals of Computation Theory

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The k-mismatch problem consists in computing the Hamming distance between a pattern P of length m and every length-m substring of a text T of length n, if this distance is no more than k. In many real-world applications, any cyclic shift of P is a relevant pattern, and thus one is interested in computing the minimal distance of every length-m substring of T and any cyclic shift of P . This is the circular pattern matching with k mismatches (k-CPM) problem. A multitude of papers have been devoted to solving this problem but, to the best of our knowledge, only average-case upper bounds are known. In this paper, we present the first non-trivial worst-case upper bounds for the k-CPM problem. Specifically, we show an O(nk)-time algorithm and an O(n + n m k 5 )-time algorithm. The latter algorithm applies in an extended way a technique that was very recently developed for the k-mismatch problem [Bringmann et al., SODA 2019].

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Section: Simple Geometry Of Arithmetic Sequences Of Intervalsmentioning

confidence: 99%

Circular Pattern Matching with k Mismatches

Charalampopoulos

Kociumaka

Pissis

et al. 2019

Fundamentals of Computation Theory

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“…For example, 011000 is a 3-antipower, as 01, 10, 00 are pairwise distinct. A variety of papers have been produced on the subject in the following years including Badkobeh et al (2018); Burcroff (2018); Defant (2017); Fici et al (2019); Gaetz (2021); Kociumaka et al (2019); Narayanan (2020), with Defant (2017); Gaetz (2021); Narayanan (2020) finding bounds on antipower lengths in the Thue-Morse word.…”

Section: Introductionmentioning

confidence: 99%

Antipowers in Uniform Morphic Words and the Fibonacci Word

Garg¹

2021

Discrete Mathematics &Amp; Theoretical Computer Science

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Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a word composed of $k$ pairwise distinct, concatenated words of equal length. Berger and Defant conjecture that for any sufficiently well-behaved aperiodic morphic word $w$, there exists a constant $c$ such that for any $k$ and any index $i$, a $k$-antipower with block length at most $ck$ starts at the $i$th position of $w$. They prove their conjecture in the case of binary words, and we extend their result to alphabets of arbitrary finite size and characterize those words for which the result does not hold. We also prove their conjecture in the specific case of the Fibonacci word.

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“…Specifically, they prove that the number of such factors over an alphabet of any size is Ω(n 2 /k), and provide an algorithm that finds them all in O(n 2 /k) time and linear space. The latter results are improved in [21], where the authors give an algorithm that counts and reports the number C of substrings of a word w of length n that are k-antipowers, in O(nk log k) and O(nk log k + C) time, respectively. Moreover, they are also able to test whether a factor w[i, j] is a k-antipower (i.e, answering an antipower query (i, j, k)) in O(r) time, by constructing a data structure of size O(n 2 /r) in O(n 2 /r) time, for any r ∈ {1, .…”

Section: Introductionmentioning

confidence: 99%

Online Algorithms on Antipowers and Antiperiods

Alzamel

Conte

Greco

et al. 2019

String Processing and Information Retrieval

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The definition of antipower introduced by Fici et al. (ICALP 2016) captures the notion of being the opposite of a power : a sequence of k pairwise distinct blocks of the same length. Recently, Alamro et al. (CPM 2019) defined a string to have an antiperiod if it is a prefix of an antipower, and gave complexity bounds for the offline computation of the minimum antiperiod and all the antiperiods of a word. In this paper, we address the same problems in the online setting. Our solutions rely on new arrays that compactly and incrementally store antiperiods and antipowers as the word grows, obtaining in the process this information for all the word's prefixes. We show how to compute those arrays online in O(n log n) space, O(n log n) time, and o(n) delay per character, for any constant > 0. Running times are worst-case and hold with high probability. We also discuss more space-efficient solutions returning the correct result with high probability, and small data structures to support random access to those arrays.

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Efficient Representation and Counting of Antipower Factors in Words

Cited by 8 publications

References 14 publications

Circular Pattern Matching with k Mismatches

Circular Pattern Matching with k Mismatches

Antipowers in Uniform Morphic Words and the Fibonacci Word

Online Algorithms on Antipowers and Antiperiods

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