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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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**Published in:** LIPIcs, Volume 191, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)

The k-mappability problem has two integers parameters m and k. For every subword of size m in a text S, we wish to report the number of indices in S in which the word occurs with at most k mismatches.
The problem was lately tackled by Alzamel et al. [Mai Alzamel et al., 2018]. For a text with constant alphabet Σ and k ∈ O(1), they present an algorithm with linear space and O(nlog^{k+1}n) time. For the case in which k = 1 and a constant size alphabet, a faster algorithm with linear space and O(nlog(n)log log(n)) time was presented in [Mai Alzamel et al., 2020].
In this work, we enhance the techniques of [Mai Alzamel et al., 2020] to obtain an algorithm with linear space and O(n log(n)) time for k = 1. Our algorithm removes the constraint of the alphabet being of constant size. We also present linear algorithms for the case of k = 1, |Σ| ∈ O(1) and m = Ω(√n).

Amihood Amir, Itai Boneh, and Eitan Kondratovsky. The k-Mappability Problem Revisited. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 5:1-5:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{amir_et_al:LIPIcs.CPM.2021.5, author = {Amir, Amihood and Boneh, Itai and Kondratovsky, Eitan}, title = {{The k-Mappability Problem Revisited}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {5:1--5:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-186-3}, ISSN = {1868-8969}, year = {2021}, volume = {191}, editor = {Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2021.5}, URN = {urn:nbn:de:0030-drops-139566}, doi = {10.4230/LIPIcs.CPM.2021.5}, annote = {Keywords: Pattern Matching, Hamming Distance, Suffix Tree, Suffix Array} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

The Suffix Array SA(S) of a string S[1 … n] is an array containing all the suffixes of S sorted by lexicographic order. The suffix array is one of the most well known indexing data structures, and it functions as a key tool in many string algorithms.
In this paper, we present a data structure for maintaining the Suffix Array of a dynamic string. For every 1 ≤ k ≤ n, our data structure reports SA[i] in 𝒪̃(n/k) time and handles text modification in 𝒪̃(k) time. Additionally, our data structure enables the same query time for reporting iSA[i], with iSA being the Inverse Suffix Array of S[1 … n].
Our data structure can be used to construct sub-linear dynamic variants of static strings algorithms or data structures that are based on the Suffix Array and the Inverse Suffix Array.

Amihood Amir and Itai Boneh. Update Query Time Trade-Off for Dynamic Suffix Arrays. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 63:1-63:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{amir_et_al:LIPIcs.ISAAC.2020.63, author = {Amir, Amihood and Boneh, Itai}, title = {{Update Query Time Trade-Off for Dynamic Suffix Arrays}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {63:1--63:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.63}, URN = {urn:nbn:de:0030-drops-134070}, doi = {10.4230/LIPIcs.ISAAC.2020.63}, annote = {Keywords: String Algorithms, Dynamic Algorithms, Suffix Array, Inverse Suffix Array} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

The recovery problem is the problem whose input is a corrupted text T that was originally periodic, and where one wishes to recover its original period. The algorithm’s input is T without any information about either the period’s length or the period itself. An algorithm that solves this problem is called a recovery algorithm. In order to make recovery possible, there must be some assumption that not "too many" errors corrupted the initial periodic string. This is called the error bound. In previous recovery algorithms, it was shown that a given error bound of n/((2+ε)p) can lead to O(log_{1+ε} n) period candidates, that are guaranteed to include the original period, where p is the length of the original period (unknown by the algorithm) and ε > 0 is an arbitrary constant.
This paper provides the first analysis of the relationship between the error bound and the number of candidates, as well as identification of the error parameters that still guarantee recovery. We improve the previously known upper error bound on the number of corruptions, n/((2+ε)p), that outputs O(log_{1+ε} n) period candidates. We show how to (1) remove ε from the bound, (2) relax the error bound to allow more errors while keeping the candidates set of size O(log n). It turns out that this relaxation on the previously known upper bound is quite challenging.
To achieve this result we provide what, to our knowledge, is the first known non-trivial lower bound on the Hamming distance between two periodic strings. This proof leads to an error bound, that produces a family of period candidates of size 2log₃ n. We show that this result is tight and further provide a compact representation of the period candidates. We call this representation the canonic period seed.
In addition to providing less restrictive error bounds that guarantee a smaller candidate set, we also provide a hierarchy of more restrictive upper error bounds that asymptotically reduces the size of the potential period candidate set.

Amihood Amir, Itai Boneh, Michael Itzhaki, and Eitan Kondratovsky. Analysis of the Period Recovery Error Bound. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 5:1-5:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{amir_et_al:LIPIcs.ESA.2020.5, author = {Amir, Amihood and Boneh, Itai and Itzhaki, Michael and Kondratovsky, Eitan}, title = {{Analysis of the Period Recovery Error Bound}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {5:1--5:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.5}, URN = {urn:nbn:de:0030-drops-128717}, doi = {10.4230/LIPIcs.ESA.2020.5}, annote = {Keywords: Period Recovery, Period Recovery Hierarchy, Hamming Distance} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

A string UU for a non-empty string U is called a square. Squares have been well-studied both from a combinatorial and an algorithmic perspective. In this paper, we are the first to consider the problem of maintaining a representation of the squares in a dynamic string S of length at most n. We present an algorithm that updates this representation in n^o(1) time. This representation allows us to report a longest square-substring of S in O(1) time and all square-substrings of S in O(output) time. We achieve this by introducing a novel tool - maintaining prefix-suffix matches of two dynamic strings.
We extend the above result to address the problem of maintaining a representation of all runs (maximal repetitions) of the string. Runs are known to capture the periodic structure of a string, and, as an application, we show that our representation of runs allows us to efficiently answer periodicity queries for substrings of a dynamic string. These queries have proven useful in static pattern matching problems and our techniques have the potential of offering solutions to these problems in a dynamic text setting.

Amihood Amir, Itai Boneh, Panagiotis Charalampopoulos, and Eitan Kondratovsky. Repetition Detection in a Dynamic String. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 5:1-5:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{amir_et_al:LIPIcs.ESA.2019.5, author = {Amir, Amihood and Boneh, Itai and Charalampopoulos, Panagiotis and Kondratovsky, Eitan}, title = {{Repetition Detection in a Dynamic String}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {5:1--5:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.5}, URN = {urn:nbn:de:0030-drops-111265}, doi = {10.4230/LIPIcs.ESA.2019.5}, annote = {Keywords: string algorithms, dynamic algorithms, squares, repetitions, runs} }

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**Published in:** LIPIcs, Volume 105, 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)

There has been recent interest in dynamic string algorithms, i.e. string problems where the input changes dynamically. One such problem is the longest common factor (LCF) problem. It is well known that the LCF of two strings S and D of length n over a fixed constant-sized alphabet Sigma can be computed in time linear in n. Recently, a new challenge was introduced - finding the LCF of two strings in a dynamic setting. The problem is the fully dynamic one sided LCF (FDOS-LCF) problem. In the FDOS-LCF problem we get q consecutive queries of the form <i,alpha >, where each such query means: "replace D[i] by alpha, alpha in Sigma and output the LCF of S and (the updated) D. The goal is to initially preprocess S and D so that we do not need O(n) time to compute an LCF for each such query.
The state-of-the-art is an algorithm that preprocesses the two strings S and D in time O(n log^4 n). Subsequently, the algorithm answers in time O(log^3 n) a single query of the form: Given a position i on D and a letter alpha, return an LCF of S and D', where D' is the string resulting from D after substituting D[i] with alpha. That algorithm is not extendable to multiple queries. In this paper we present a tool - Locally Maximal Common Factors (LMCF) - that proves to be quite useful in solving some restricted versions of the FDOS-LCF problem . The versions we solve are the Decremental FDOS-LCS problem, where every change <i,alpha> is of the form <i,omega>, omega !in Sigma, and the Periodic FDOS-LCS problem, where S is a periodic string with period length p.
For the decremental problem we provide an algorithm with linear time preprocessing and O(log log n) time per query. For the periodic problem our preprocessing time is linear and the query time is O(p log log n).

Amihood Amir and Itai Boneh. Locally Maximal Common Factors as a Tool for Efficient Dynamic String Algorithms. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 11:1-11:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{amir_et_al:LIPIcs.CPM.2018.11, author = {Amir, Amihood and Boneh, Itai}, title = {{Locally Maximal Common Factors as a Tool for Efficient Dynamic String Algorithms}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {11:1--11:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-074-3}, ISSN = {1868-8969}, year = {2018}, volume = {105}, editor = {Navarro, Gonzalo and Sankoff, David and Zhu, Binhai}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2018.11}, URN = {urn:nbn:de:0030-drops-86983}, doi = {10.4230/LIPIcs.CPM.2018.11}, annote = {Keywords: Dynamic Algorithms, Periodicity, Longest Common Factor, Priority Queue Data Structures, Suffix Tree, Balanced Search Tree, Range Maximum Queries} }

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