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

We revisit the problem of multiplying two square matrices over the (min, +) semi-ring, where all entries are integers from a bounded range [-M : M] ∪ {∞}. The current state of the art for this problem is a simple O(M n^{ω} log M) time algorithm by Alon, Galil and Margalit [JCSS'97], where ω is the exponent in the runtime of the fastest matrix multiplication (FMM) algorithm. We design a new simple algorithm whose runtime is O(M n^ω + M n² log M), thereby removing the logM factor in the runtime if ω > 2 or if n^ω = Ω (n²log n).

Dvir Fried, Tsvi Kopelowitz, and Ely Porat. Removing the log Factor from (min,+)-Products on Bounded Range Integer Matrices. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 57:1-57:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fried_et_al:LIPIcs.ESA.2024.57, author = {Fried, Dvir and Kopelowitz, Tsvi and Porat, Ely}, title = {{Removing the log Factor from (min,+)-Products on Bounded Range Integer Matrices}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {57:1--57:6}, 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.2024.57}, URN = {urn:nbn:de:0030-drops-211283}, doi = {10.4230/LIPIcs.ESA.2024.57}, annote = {Keywords: FMM, (min , +)-product, FFT} }

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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 296, 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024)

Hairpin completion, derived from the hairpin formation observed in DNA biochemistry, is an operation applied to strings, particularly useful in DNA computing. Conceptually, a right hairpin completion operation transforms a string S into S⋅ S' where S' is the reverse complement of a prefix of S. Similarly, a left hairpin completion operation transforms a string S into S'⋅ S where S' is the reverse complement of a suffix of S. The hairpin completion distance from S to T is the minimum number of hairpin completion operations needed to transform S into T. Recently Boneh et al. [Itai Boneh et al., 2023] showed an O(n²) time algorithm for finding the hairpin completion distance between two strings of length at most n. In this paper we show that for any ε > 0 there is no O(n^{2-ε})-time algorithm for the hairpin completion distance problem unless the Strong Exponential Time Hypothesis (SETH) is false. Thus, under SETH, the time complexity of the hairpin completion distance problem is quadratic, up to sub-polynomial factors.

Itai Boneh, Dvir Fried, Shay Golan, and Matan Kraus. Hairpin Completion Distance Lower Bound. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boneh_et_al:LIPIcs.CPM.2024.11, author = {Boneh, Itai and Fried, Dvir and Golan, Shay and Kraus, Matan}, title = {{Hairpin Completion Distance Lower Bound}}, booktitle = {35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024)}, pages = {11:1--11:16}, 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.2024.11}, URN = {urn:nbn:de:0030-drops-201215}, doi = {10.4230/LIPIcs.CPM.2024.11}, annote = {Keywords: Fine-grained complexity, Hairpin completion, LCS} }

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**Published in:** LIPIcs, Volume 259, 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)

Hairpin completion is an operation on formal languages that has been inspired by hairpin formation in DNA biochemistry and has many applications especially in DNA computing. Consider s to be a string over the alphabet {A, C, G, T} such that a prefix/suffix of it matches the reversed complement of a substring of s. Then, in a hairpin completion operation the reversed complement of this prefix/suffix is added to the start/end of s forming a new string.
In this paper we study two problems related to the hairpin completion. The first problem asks the minimum number of hairpin operations necessary to transform one string into another, number that is called the hairpin completion distance. For this problem we show an algorithm of running time O(n²), where n is the maximum length of the two strings. Our algorithm improves on the algorithm of Manea (TCS 2010), that has running time O(n² log n).
In the minimum distance common hairpin completion ancestor problem we want to find, for two input strings x and y, a string w that minimizes the sum of the hairpin completion distances to x and y. Similarly, we present an algorithm with running time O(n²) that improves by a O(log n) factor the algorithm of Manea (TCS 2010).

Itai Boneh, Dvir Fried, Adrian Miclăuş, and Alexandru Popa. Faster Algorithms for Computing the Hairpin Completion Distance and Minimum Ancestor. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{boneh_et_al:LIPIcs.CPM.2023.5, author = {Boneh, Itai and Fried, Dvir and Micl\u{a}u\c{s}, Adrian and Popa, Alexandru}, title = {{Faster Algorithms for Computing the Hairpin Completion Distance and Minimum Ancestor}}, booktitle = {34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)}, pages = {5:1--5:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-276-1}, ISSN = {1868-8969}, year = {2023}, volume = {259}, editor = {Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023.5}, URN = {urn:nbn:de:0030-drops-179592}, doi = {10.4230/LIPIcs.CPM.2023.5}, annote = {Keywords: dynamic programming, incremental trees, exact algorithm} }

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