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**Published in:** LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

This paper focuses on the concept of partial permutations and their use in algorithmic tasks. A partial permutation over Σ is a bijection π_{par}: Σ₁↦Σ₂ mapping a subset Σ₁ ⊂ Σ to a subset Σ₂ ⊂ Σ, where |Σ₁| = |Σ₂| (|Σ| denotes the size of a set Σ). Intuitively, two partial permutations agree if their mapping pairs do not form conflicts. This notion, which is formally defined in this paper, enables a consistent as well as informatively rich comparison between partial permutations. We formalize the Partial Permutations Agreement problem (PPA), as follows. Given two sets A₁, A₂ of partial permutations over alphabet Σ, each of size n, output all pairs (π_i, π_j), where π_i ∈ A₁, π_j ∈ A₂ and π_i agrees with π_j. The possibility of having a data structure for efficiently maintaining a dynamic set of partial permutations enabling to retrieve agreement of partial permutations is then studied, giving both negative and positive results. Applying our study enables to point out fruitful versus futile methods for efficient genes sequences comparison in database or automatic color transformation data augmentation technique for image processing through neural networks. It also shows that an efficient solution of strict Parameterized Dictionary Matching with One Gap (PDMOG) over general dictionary alphabets is not likely, unless the Strong Exponential Time Hypothesis (SETH) fails, thus negatively answering an open question posed lately.

Avivit Levy, Ely Porat, and B. Riva Shalom. Partial Permutations Comparison, Maintenance and Applications. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 10:1-10:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{levy_et_al:LIPIcs.CPM.2022.10, author = {Levy, Avivit and Porat, Ely and Shalom, B. Riva}, title = {{Partial Permutations Comparison, Maintenance and Applications}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {10:1--10:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-234-1}, ISSN = {1868-8969}, year = {2022}, volume = {223}, editor = {Bannai, Hideo and Holub, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.10}, URN = {urn:nbn:de:0030-drops-161376}, doi = {10.4230/LIPIcs.CPM.2022.10}, annote = {Keywords: Partial permutations, Partial words, Genes comparison, Color transformation, Dictionary matching with gaps, Parameterized matching, SETH hypothesis} }

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

Tracing regularities plays a key role in data analysis for various areas of science, including coding and automata theory, formal language theory, combinatorics, molecular biology and many others. Part of the scientific process is understanding and explaining these regularities. A common notion to describe regularity in a string T is a cover or quasi-period, which is a string C for which every letter of T lies within some occurrence of C. In many applications finding exact repetitions is not sufficient, due to the presence of errors. In this paper we initiate the study of quasi-periodicity persistence under mismatch errors, and our goal is to characterize situations where a given quasi-periodic string remains quasi-periodic even after substitution errors have been introduced to the string. Our study results in proving necessary conditions as well as a theorem stating sufficient conditions for quasi-periodicity persistence. As an application, we are able to close the gap in understanding the complexity of Approximate Cover Problem (ACP) relaxations studied by [Amir 2017a, Amir 2017b] and solve an open question.

Amihood Amir, Avivit Levy, and Ely Porat. Quasi-Periodicity Under Mismatch Errors. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 4:1-4:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{amir_et_al:LIPIcs.CPM.2018.4, author = {Amir, Amihood and Levy, Avivit and Porat, Ely}, title = {{Quasi-Periodicity Under Mismatch Errors}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {4:1--4:15}, 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.2018.4}, URN = {urn:nbn:de:0030-drops-87054}, doi = {10.4230/LIPIcs.CPM.2018.4}, annote = {Keywords: Periodicity, Quasi-Periodicity, Cover, Approximate Cover} }

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

Data analysis typically involves error recovery and detection of regularities as two different key tasks. In this paper we show that there are data types for which these two tasks can be powerfully combined. A common notion of regularity in strings is that of a cover. Data describing measures of a natural coverable phenomenon may be corrupted by errors caused by the measurement process, or by the inexact features of the phenomenon itself. Due to this reason, different variants of approximate covers have been introduced, some of which are NP-hard to compute. In this paper we assume that the Hamming distance metric measures the amount of corruption experienced, and study the problem of recovering the correct cover from data corrupted by mismatch errors, formally defined as the cover recovery problem (CRP). We show that for the Hamming distance metric, coverability is a powerful property allowing detecting the original cover and correcting the data, under suitable conditions.
We also study a relaxation of another problem, which is called the approximate cover problem (ACP). Since the ACP is proved to be NP-hard [Amir,Levy,Lubin,Porat, CPM 2017], we study a relaxation, which we call the candidate-relaxation of the ACP, and show it has a polynomial time complexity. As a result, we get that the ACP also has a polynomial time complexity in many practical situations. An important application of our ACP relaxation study is also a polynomial time algorithm for the cover recovery problem (CRP).

Amihood Amir, Avivit Levy, Moshe Lewenstein, Ronit Lubin, and Benny Porat. Can We Recover the Cover?. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 25:1-25:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{amir_et_al:LIPIcs.CPM.2017.25, author = {Amir, Amihood and Levy, Avivit and Lewenstein, Moshe and Lubin, Ronit and Porat, Benny}, title = {{Can We Recover the Cover?}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {25:1--25:15}, 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.2017.25}, URN = {urn:nbn:de:0030-drops-73190}, doi = {10.4230/LIPIcs.CPM.2017.25}, annote = {Keywords: periodicity, quasi-periodicity, cover, approximate cover, data recovery} }

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

Regularities in strings arise in various areas of science, including coding and automata theory, formal language theory, combinatorics, molecular biology and many others. A common notion to describe regularity in a string T is a cover, which is a string C for which every letter of T lies within some occurrence of C. The alignment of the cover repetitions in the given text is called a tiling. In many applications finding exact repetitions is not sufficient, due to the presence of errors. In this paper, we use a new approach for handling errors in coverable phenomena and define the approximate cover problem (ACP), in which we are given a text that is a sequence of some cover repetitions with possible mismatch errors, and we seek a string that covers the text with the minimum number of errors. We first show that the ACP is NP-hard, by studying the cover-size relaxation of the ACP, in which the requested size of the approximate cover is also given with the input string. We show this relaxation is already NP-hard. We also study another two relaxations of the ACP, which we call the partial-tiling relaxation of the ACP and the full-tiling relaxation of the ACP, in which a tiling of the requested cover is also given with the input string. A given full tiling retains all the occurrences of the cover before the errors, while in a partial tiling there can be additional occurrences of the cover that are not marked by the tiling. We show that the partial-tiling relaxation has a polynomial time complexity and give experimental evidence that the full-tiling also has polynomial time complexity. The study of these relaxations, besides shedding another light on the complexity of the ACP, also involves a deep understanding of the properties of covers, yielding some key lemmas and observations that may be helpful for a future study of regularities in the presence of errors.

Amihood Amir, Avivit Levy, Ronit Lubin, and Ely Porat. Approximate Cover of Strings. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 26:1-26:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{amir_et_al:LIPIcs.CPM.2017.26, author = {Amir, Amihood and Levy, Avivit and Lubin, Ronit and Porat, Ely}, title = {{Approximate Cover of Strings}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {26:1--26:14}, 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.2017.26}, URN = {urn:nbn:de:0030-drops-73189}, doi = {10.4230/LIPIcs.CPM.2017.26}, annote = {Keywords: periodicity, quasi-periodicity, cover, approximate cover} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

We examine the complexity of the online Dictionary Matching with One Gap Problem (DMOG) which is the following. Preprocess a dictionary D of d patterns, where each pattern contains a special gap symbol that can match any string, so that given a text that arrives online, a character at a time, we can report all of the patterns from D that are suffixes of the text that has arrived so far, before the next character arrives. In more general versions the gap symbols are associated with bounds determining the possible lengths of matching strings. Online DMOG captures the difficulty in a bottleneck procedure for cyber-security, as many digital signatures of viruses manifest themselves as patterns with a single gap.
In this paper, we demonstrate that the difficulty in obtaining efficient solutions for the DMOG problem, even in the offline setting, can be traced back to the infamous 3SUM conjecture. We show a conditional lower bound of Omega(delta(G_D)+op) time per text character, where G_D is a bipartite graph that captures the structure of D, delta(G_D) is the degeneracy of this graph, and op is the output size. Moreover, we show a conditional lower bound in terms of the magnitude of gaps for the bounded case, thereby showing that some known offline upper bounds are essentially optimal.
We also provide matching upper-bounds (up to sub-polynomial factors), in terms of the degeneracy, for the online DMOG problem. In particular, we introduce algorithms whose time cost depends linearly on delta(G_D). Our algorithms make use of graph orientations, together with some additional techniques. These algorithms are of practical interest since although delta(G_D) can be as large as sqrt(d), and even larger if G_D is a multi-graph, it is typically a very small constant in practice. Finally, when delta(G_D) is large we are able to obtain even more efficient solutions.

Amihood Amir, Tsvi Kopelowitz, Avivit Levy, Seth Pettie, Ely Porat, and B. Riva Shalom. Mind the Gap: Essentially Optimal Algorithms for Online Dictionary Matching with One Gap. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 12:1-12:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{amir_et_al:LIPIcs.ISAAC.2016.12, author = {Amir, Amihood and Kopelowitz, Tsvi and Levy, Avivit and Pettie, Seth and Porat, Ely and Shalom, B. Riva}, title = {{Mind the Gap: Essentially Optimal Algorithms for Online Dictionary Matching with One Gap}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {12:1--12:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.12}, URN = {urn:nbn:de:0030-drops-67841}, doi = {10.4230/LIPIcs.ISAAC.2016.12}, annote = {Keywords: Pattern matching, Dictionary matching, 3SUM, Triangle reporting} }

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