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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

A λ-cover of a string S is a set of strings {C_i}₁^λ such that every index in S is contained in an occurrence of at least one string C_i. The existence of a 1-cover defines a well-known class of quasi-periodic strings. Quasi-periodicity can be decided in linear time, and all 1-covers of a string can be reported in linear time as well. Since in general it is NP-complete to decide whether a string has a λ-cover, the natural next step is the development of efficient algorithms for 2-covers. Radoszewski and Straszyński [ESA 2020] analysed the particular case where the strings in a 2-cover must be of the same length. They provided an algorithm that reports all such 2-covers of S in time near-linear in |S| and in the size of the output.
In this work, we consider 2-covers in full generality. Since every length-n string has Ω(n²) trivial 2-covers (every prefix and suffix of total length at least n constitute such a 2-cover), we state the reporting problem as follows: given a string S and a number m, report all 2-covers {C₁,C₂} of S with length |C₁|+|C₂| upper bounded by m. We present an Õ(n + output) time algorithm solving this problem, with output being the size of the output. This algorithm admits a simpler modification that finds a 2-cover of minimum length. We also provide an Õ(n) time construction of a 2-cover oracle which, given two substrings C₁,C₂ of S, reports in poly-logarithmic time whether {C₁,C₂} is a 2-cover of S.

Itai Boneh, Shay Golan, and Arseny Shur. String 2-Covers with No Length Restrictions. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 31:1-31:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boneh_et_al:LIPIcs.ESA.2024.31, author = {Boneh, Itai and Golan, Shay and Shur, Arseny}, title = {{String 2-Covers with No Length Restrictions}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {31:1--31:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.31}, URN = {urn:nbn:de:0030-drops-211029}, doi = {10.4230/LIPIcs.ESA.2024.31}, annote = {Keywords: Quasi-periodicity, String cover, Range query, Range stabbing} }

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**Published in:** LIPIcs, Volume 296, 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024)

We study a natural type of repetitions in 2-dimensional strings. Such a repetition, called a matching frame, is a rectangular substring of size at least 2× 2 with equal marginal rows and equal marginal columns. Matching frames first appeared in literature in the context of Wang tiles.
We present two algorithms finding a matching frame with the maximum perimeter in a given n× m input string. The first algorithm solves the problem exactly in Õ(n^{2.5}) time (assuming n ≥ m). The second algorithm finds a (1-ε)-approximate solution in Õ((nm)/ε⁴) time, which is near linear in the size of the input for constant ε. In particular, by setting ε = O(1) the second algorithm decides the existence of a matching frame in a given string in Õ(nm) time. Some technical elements and structural properties used in these algorithms can be of independent interest.

Itai Boneh, Dvir Fried, Shay Golan, Matan Kraus, Adrian Miclăuş, and Arseny Shur. Searching 2D-Strings for Matching Frames. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boneh_et_al:LIPIcs.CPM.2024.10, author = {Boneh, Itai and Fried, Dvir and Golan, Shay and Kraus, Matan and Micl\u{a}u\c{s}, Adrian and Shur, Arseny}, title = {{Searching 2D-Strings for Matching Frames}}, booktitle = {35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024)}, pages = {10:1--10:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-326-3}, ISSN = {1868-8969}, year = {2024}, volume = {296}, editor = {Inenaga, Shunsuke and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2024.10}, URN = {urn:nbn:de:0030-drops-201205}, doi = {10.4230/LIPIcs.CPM.2024.10}, annote = {Keywords: 2D string, matching frame, LCP, multidimensional range query} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

The order-preserving model (op-model, in short) was introduced quite recently but has already attracted significant attention because of its applications in data analysis. We introduce several types of periods in this setting (op-periods). Then we give algorithms to compute these periods in time O(n), O(n log log n), O(n log^2 log n/log log log n), O(n log n) depending on the type of periodicity. In the most general variant the number of different periods can be as big as Omega(n^2), and a compact representation is needed. Our algorithms require novel combinatorial insight into the properties of such periods.

Garance Gourdel, Tomasz Kociumaka, Jakub Radoszewski, Wojciech Rytter, Arseny Shur, and Tomasz Walen. String Periods in the Order-Preserving Model. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gourdel_et_al:LIPIcs.STACS.2018.38, author = {Gourdel, Garance and Kociumaka, Tomasz and Radoszewski, Jakub and Rytter, Wojciech and Shur, Arseny and Walen, Tomasz}, title = {{String Periods in the Order-Preserving Model}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {38:1--38:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.38}, URN = {urn:nbn:de:0030-drops-85064}, doi = {10.4230/LIPIcs.STACS.2018.38}, annote = {Keywords: order-preserving pattern matching, period, efficient algorithm} }

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**Published in:** LIPIcs, Volume 54, 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)

We consider computing a longest palindrome in the streaming model, where the symbols arrive one-by-one and we do not have random access to the input. While computing the answer exactly using sublinear space is not possible in such a setting, one can still hope for a good approximation guarantee. Our contribution is twofold. First, we provide lower bounds on the space requirements for randomized approximation algorithms processing inputs of length n. We rule out Las Vegas algorithms, as they cannot achieve sublinear space complexity. For Monte Carlo algorithms, we prove a lower bounds of Omega(M log min {|Sigma|, M}) bits of memory; here M=n/E for approximating the answer with additive error E, and M= log n / log (1 + epsilon) for approximating the answer with multiplicative error (1 + epsilon). Second, we design three real-time algorithms for this problem. Our Monte Carlo approximation algorithms for both additive and multiplicative versions of the problem use O(M) words of memory. Thus the obtained lower bounds are asymptotically tight up to a logarithmic factor. The third algorithm is deterministic and finds a longest palindrome exactly if it is short. This algorithm can be run in parallel with a Monte Carlo algorithm to obtain better results in practice. Overall, both the time and space complexity of finding a longest palindrome in a stream are essentially settled.

Pawel Gawrychowski, Oleg Merkurev, Arseny Shur, and Przemyslaw Uznanski. Tight Tradeoffs for Real-Time Approximation of Longest Palindromes in Streams. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gawrychowski_et_al:LIPIcs.CPM.2016.18, author = {Gawrychowski, Pawel and Merkurev, Oleg and Shur, Arseny and Uznanski, Przemyslaw}, title = {{Tight Tradeoffs for Real-Time Approximation of Longest Palindromes in Streams}}, booktitle = {27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)}, pages = {18:1--18:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-012-5}, ISSN = {1868-8969}, year = {2016}, volume = {54}, editor = {Grossi, Roberto and Lewenstein, Moshe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.18}, URN = {urn:nbn:de:0030-drops-60765}, doi = {10.4230/LIPIcs.CPM.2016.18}, annote = {Keywords: streaming algorithms, space lower bounds, real-time algorithms, palin- dromes, Monte Carlo algorithms} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Given a string s of length n over a general alphabet and an integer k, the problem is to decide whether s is a concatenation of k nonempty palindromes. Two previously known solutions for this problem work in time O(kn) and O(nlog n) respectively. Here we settle the complexity of this problem in the word-RAM model, presenting an O(n)-time online deciding algorithm. The algorithm simultaneously finds the minimum odd number of factors and the minimum even number of factors in a factorization of a string into nonempty palindromes. We also demonstrate how to get an explicit factorization of s into k palindromes with an O(n)-time offline postprocessing.

Mikhail Rubinchik and Arseny M. Shur. Palindromic k-Factorization in Pure Linear Time. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 81:1-81:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{rubinchik_et_al:LIPIcs.MFCS.2020.81, author = {Rubinchik, Mikhail and Shur, Arseny M.}, title = {{Palindromic k-Factorization in Pure Linear Time}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {81:1--81:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.81}, URN = {urn:nbn:de:0030-drops-127508}, doi = {10.4230/LIPIcs.MFCS.2020.81}, annote = {Keywords: stringology, palindrome, palindromic factorization, online algorithm} }

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**Published in:** LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

We consider two well-known related problems: Longest Repeated Substring (LRS) and Longest Repeated Reversed Substring (LRRS). Their streaming versions cannot be solved exactly; we show that only approximate solutions by Monte Carlo algorithms are possible, and prove a lower bound on consumed memory. For both problems, we present purely linear-time Monte Carlo algorithms working in O(E + n/E) space, where E is the additive approximation error. Within the same space bounds, we then present nearly real-time solutions, which require O(log n) time per symbol and O(n + n/E log n) time overall. The working space exactly matches the lower bound whenever E=O(n^{0.5}) and the size of the alphabet is Omega(n^{0.01}).

Oleg Merkurev and Arseny M. Shur. Searching Long Repeats in Streams. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{merkurev_et_al:LIPIcs.CPM.2019.31, author = {Merkurev, Oleg and Shur, Arseny M.}, title = {{Searching Long Repeats in Streams}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-103-0}, ISSN = {1868-8969}, year = {2019}, volume = {128}, editor = {Pisanti, Nadia and P. Pissis, Solon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.31}, URN = {urn:nbn:de:0030-drops-105029}, doi = {10.4230/LIPIcs.CPM.2019.31}, annote = {Keywords: Longest repeated substring, longest repeated reversed substring, streaming algorithm, Karp, Rabin fingerprint, suffix tree} }

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**Published in:** LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

Palindromic length of a string is the minimum number of palindromes whose concatenation is equal to this string. The problem of finding the palindromic length drew some attention, and a few O(n log n) time online algorithms were recently designed for it. In this paper we present the first linear time online algorithm for this problem.

Kirill Borozdin, Dmitry Kosolobov, Mikhail Rubinchik, and Arseny M. Shur. Palindromic Length in Linear Time. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 23:1-23:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{borozdin_et_al:LIPIcs.CPM.2017.23, author = {Borozdin, Kirill and Kosolobov, Dmitry and Rubinchik, Mikhail and Shur, Arseny M.}, title = {{Palindromic Length in Linear Time}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {23:1--23:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.23}, URN = {urn:nbn:de:0030-drops-73389}, doi = {10.4230/LIPIcs.CPM.2017.23}, annote = {Keywords: palindrome, palindromic length, palindromic factorization, online} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Lyndon factorization and Lempel-Ziv (LZ) factorization are both important tools for analysing the structure and complexity of strings, but their combinatorial structure is very different. In this paper, we establish the first direct connection between the two by showing that while the Lyndon factorization can be bigger than the non-overlapping LZ factorization (which we demonstrate by describing a new, non-trivial family of strings) it is always less than twice the size.

Juha Kärkkäinen, Dominik Kempa, Yuto Nakashima, Simon J. Puglisi, and Arseny M. Shur. On the Size of Lempel-Ziv and Lyndon Factorizations. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{karkkainen_et_al:LIPIcs.STACS.2017.45, author = {K\"{a}rkk\"{a}inen, Juha and Kempa, Dominik and Nakashima, Yuto and Puglisi, Simon J. and Shur, Arseny M.}, title = {{On the Size of Lempel-Ziv and Lyndon Factorizations}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {45:1--45:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.45}, URN = {urn:nbn:de:0030-drops-69878}, doi = {10.4230/LIPIcs.STACS.2017.45}, annote = {Keywords: Lempel-Ziv factorization, Lempel-Ziv parsing, LZ, Lyndon word, Lyndon factorization, Standard factorization} }

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