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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

We study the problem of recognizing regular languages in a variant of the streaming model of computation, called the sliding window model. In this model, we are given a size of the sliding window n and a stream of symbols. At each time instant, we must decide whether the suffix of length n of the current stream ("the active window") belongs to a given regular language.
Recent works [Moses Ganardi et al., 2018; Moses Ganardi et al., 2016] showed that the space complexity of an optimal deterministic sliding window algorithm for this problem is either constant, logarithmic or linear in the window size n and provided natural language theoretic characterizations of the space complexity classes. Subsequently, [Moses Ganardi et al., 2018] extended this result to randomized algorithms to show that any such algorithm admits either constant, double logarithmic, logarithmic or linear space complexity.
In this work, we make an important step forward and combine the sliding window model with the property testing setting, which results in ultra-efficient algorithms for all regular languages. Informally, a sliding window property tester must accept the active window if it belongs to the language and reject it if it is far from the language. We show that for every regular language, there is a deterministic sliding window property tester that uses logarithmic space and a randomized sliding window property tester with two-sided error that uses constant space.

Moses Ganardi, Danny Hucke, Markus Lohrey, and Tatiana Starikovskaya. Sliding Window Property Testing for Regular Languages. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ganardi_et_al:LIPIcs.ISAAC.2019.6, author = {Ganardi, Moses and Hucke, Danny and Lohrey, Markus and Starikovskaya, Tatiana}, title = {{Sliding Window Property Testing for Regular Languages}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {6:1--6:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.6}, URN = {urn:nbn:de:0030-drops-115023}, doi = {10.4230/LIPIcs.ISAAC.2019.6}, annote = {Keywords: Streaming algorithms, approximation algorithms, regular languages} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

A sliding window algorithm receives a stream of symbols and has to output at each time instant a certain value which only depends on the last n symbols. If the algorithm is randomized, then at each time instant it produces an incorrect output with probability at most epsilon, which is a constant error bound. This work proposes a more relaxed definition of correctness which is parameterized by the error bound epsilon and the failure ratio phi: a randomized sliding window algorithm is required to err with probability at most epsilon at a portion of 1-phi of all time instants of an input stream. This work continues the investigation of sliding window algorithms for regular languages. In previous works a trichotomy theorem was shown for deterministic algorithms: the optimal space complexity is either constant, logarithmic or linear in the window size. The main results of this paper concerns three natural settings (randomized algorithms with failure ratio zero and randomized/deterministic algorithms with bounded failure ratio) and provide natural language theoretic characterizations of the space complexity classes.

Moses Ganardi, Danny Hucke, and Markus Lohrey. Randomized Sliding Window Algorithms for Regular Languages. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 127:1-127:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ganardi_et_al:LIPIcs.ICALP.2018.127, author = {Ganardi, Moses and Hucke, Danny and Lohrey, Markus}, title = {{Randomized Sliding Window Algorithms for Regular Languages}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {127:1--127:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.127}, URN = {urn:nbn:de:0030-drops-91317}, doi = {10.4230/LIPIcs.ICALP.2018.127}, annote = {Keywords: sliding windows, regular languages, randomized complexity} }

Document

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

In a recent paper we analyzed the space complexity of streaming algorithms whose goal is to decide membership of a sliding window to a fixed language. For the class of regular languages we proved a space trichotomy theorem: for every regular language the optimal space bound is either constant, logarithmic or linear. In this paper we continue this line of research: We present natural characterizations for the constant and logarithmic space classes and establish tight relationships to the concept of language growth. We also analyze the space complexity with respect to automata size and prove almost matching lower and upper bounds. Finally, we consider the decision problem whether a language given by a DFA/NFA admits a sliding window algorithm using logarithmic/constant space.

Moses Ganardi, Danny Hucke, Daniel König, Markus Lohrey, and Konstantinos Mamouras. Automata Theory on Sliding Windows. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ganardi_et_al:LIPIcs.STACS.2018.31, author = {Ganardi, Moses and Hucke, Danny and K\"{o}nig, Daniel and Lohrey, Markus and Mamouras, Konstantinos}, title = {{Automata Theory on Sliding Windows}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {31:1--31:14}, 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.2018.31}, URN = {urn:nbn:de:0030-drops-84851}, doi = {10.4230/LIPIcs.STACS.2018.31}, annote = {Keywords: regular languages, sliding window algorithms} }

Document

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

The circuit evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or multiplicative identity. The following dichotomy is shown: If a finite semiring R (i) has a solvable multiplicative semigroup and (ii) does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 != 0, then its circuit evaluation problem is in the complexity class DET (which is contained in NC^2). In all other cases, the circuit evaluation problem is P-complete.

Moses Ganardi, Danny Hucke, Daniel König, and Markus Lohrey. Circuit Evaluation for Finite Semirings. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 35:1-35:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ganardi_et_al:LIPIcs.STACS.2017.35, author = {Ganardi, Moses and Hucke, Danny and K\"{o}nig, Daniel and Lohrey, Markus}, title = {{Circuit Evaluation for Finite Semirings}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {35:1--35:14}, 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.2017.35}, URN = {urn:nbn:de:0030-drops-69978}, doi = {10.4230/LIPIcs.STACS.2017.35}, annote = {Keywords: circuit value problem, finite semirings, circuit complexity} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

We study the space complexity of querying regular languages over data streams in the sliding window model. The algorithm has to answer at any point of time whether the content of the sliding window belongs to a fixed regular language. A trichotomy is shown: For every regular language the optimal space requirement is either in Theta(n), Theta(log(n)), or constant, where $n$ is the size of the sliding window.

Moses Ganardi, Danny Hucke, and Markus Lohrey. Querying Regular Languages over Sliding Windows. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ganardi_et_al:LIPIcs.FSTTCS.2016.18, author = {Ganardi, Moses and Hucke, Danny and Lohrey, Markus}, title = {{Querying Regular Languages over Sliding Windows}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.18}, URN = {urn:nbn:de:0030-drops-68539}, doi = {10.4230/LIPIcs.FSTTCS.2016.18}, annote = {Keywords: streaming algorithms, regular languages, space complexity} }

Document

**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

It is shown that every tree of size n over a fixed set of sigma different ranked symbols can be decomposed into O(n/log_sigma(n)) = O((n * log(sigma))/ log(n)) many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size O(n/log_sigma(n)), which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammar-based tree compressors were not analyzed for the worst-case size of the computed grammar, except for the top dag of Bille et al., for which only the weaker upper bound of O(n/log^{0.19}(n)) for unranked and unlabelled trees has been derived. The main result is used to show that every arithmetical formula of size n, in which only m <= n different variables occur, can be transformed (in time O(n * log(n)) into an arithmetical circuit of size O((n * log(m))/log(n)) and depth O(log(n)). This refines a classical result of Brent, according to which an arithmetical formula of size n can be transformed into a logarithmic depth circuit of size O(n).

Danny Hucke, Markus Lohrey, and Eric Noeth. Constructing Small Tree Grammars and Small Circuits for Formulas. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 457-468, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{hucke_et_al:LIPIcs.FSTTCS.2014.457, author = {Hucke, Danny and Lohrey, Markus and Noeth, Eric}, title = {{Constructing Small Tree Grammars and Small Circuits for Formulas}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {457--468}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.457}, URN = {urn:nbn:de:0030-drops-48639}, doi = {10.4230/LIPIcs.FSTTCS.2014.457}, annote = {Keywords: grammar-based compression, tree compression, arithmetical circuits} }

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