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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Every language recognized by a non-deterministic finite automaton can be recognized by a deterministic automaton, at the cost of a potential increase of the number of states, which in the worst case can go from n states to 2ⁿ states. In this article, we investigate this classical result in a probabilistic setting where we take a deterministic automaton with n states uniformly at random and add just one random transition. These automata are almost deterministic in the sense that only one state has a non-deterministic choice when reading an input letter. In our model each state has a fixed probability to be final. We prove that for any d ≥ 1, with non-negligible probability the minimal (deterministic) automaton of the language recognized by such an automaton has more than n^d states; as a byproduct, the expected size of its minimal automaton grows faster than any polynomial. Our result also holds when each state is final with some probability that depends on n, as long as it is not too close to 0 and 1, at distance at least Ω(1/√n) to be precise, therefore allowing models with a sublinear number of final states in expectation.

Arnaud Carayol, Philippe Duchon, Florent Koechlin, and Cyril Nicaud. One Drop of Non-Determinism in a Random Deterministic Automaton. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{carayol_et_al:LIPIcs.STACS.2023.19, author = {Carayol, Arnaud and Duchon, Philippe and Koechlin, Florent and Nicaud, Cyril}, title = {{One Drop of Non-Determinism in a Random Deterministic Automaton}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.19}, URN = {urn:nbn:de:0030-drops-176719}, doi = {10.4230/LIPIcs.STACS.2023.19}, annote = {Keywords: non-deterministic automaton, powerset construction, probabilistic analysis} }

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

A Lyndon word is a word that is lexicographically smaller than all of its non-trivial rotations (e.g. ananas is a Lyndon word; banana is not a Lyndon word due to its smaller rotation abanan). The Lyndon forest (or equivalently Lyndon table) identifies maximal Lyndon factors of a word, and is of great combinatoric interest, e.g. when finding maximal repetitions in words. While optimal linear time algorithms for computing the Lyndon forest are known, none of them work in an online manner. We present algorithms that compute the Lyndon forest of a word in a reverse online manner, processing the input word from back to front. We assume a general ordered alphabet, i.e. the only elementary operations on symbols are comparisons of the form less-equal-greater. We start with a naive algorithm and show that, despite its quadratic worst-case behaviour, it already takes expected linear time on words drawn uniformly at random. We then introduce a much more sophisticated algorithm that takes linear time in the worst case. It borrows some ideas from the offline algorithm by Bille et al. (ICALP 2020), combined with new techniques that are necessary for the reverse online setting. While the back-to-front approach for this computation is rather natural (see Franek and Liut, PSC 2019), the steps required to achieve linear time are surprisingly intricate. We envision that our algorithm will be useful for the online computation of maximal repetitions in words.

Golnaz Badkobeh, Maxime Crochemore, Jonas Ellert, and Cyril Nicaud. Back-To-Front Online Lyndon Forest Construction. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 13:1-13:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{badkobeh_et_al:LIPIcs.CPM.2022.13, author = {Badkobeh, Golnaz and Crochemore, Maxime and Ellert, Jonas and Nicaud, Cyril}, title = {{Back-To-Front Online Lyndon Forest Construction}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {13:1--13:23}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.13}, URN = {urn:nbn:de:0030-drops-161404}, doi = {10.4230/LIPIcs.CPM.2022.13}, annote = {Keywords: Lyndon factorisation, Lyndon forest, Lyndon table, Lyndon array, right Lyndon tree, Cartesian tree, standard factorisation, online algorithms} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We investigate the connection between properties of formal languages and properties of their generating series, with a focus on the class of holonomic power series. We first prove a strong version of a conjecture by Castiglione and Massazza: weakly-unambiguous Parikh automata are equivalent to unambiguous two-way reversal bounded counter machines, and their multivariate generating series are holonomic. We then show that the converse is not true: we construct a language whose generating series is algebraic (thus holonomic), but which is inherently weakly-ambiguous as a Parikh automata language. Finally, we prove an effective decidability result for the inclusion problem for weakly-unambiguous Parikh automata, and provide an upper-bound on its complexity.

Alin Bostan, Arnaud Carayol, Florent Koechlin, and Cyril Nicaud. Weakly-Unambiguous Parikh Automata and Their Link to Holonomic Series. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 114:1-114:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bostan_et_al:LIPIcs.ICALP.2020.114, author = {Bostan, Alin and Carayol, Arnaud and Koechlin, Florent and Nicaud, Cyril}, title = {{Weakly-Unambiguous Parikh Automata and Their Link to Holonomic Series}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {114:1--114:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.114}, URN = {urn:nbn:de:0030-drops-125212}, doi = {10.4230/LIPIcs.ICALP.2020.114}, annote = {Keywords: generating series, holonomicity, ambiguity, reversal bounded counter machine, Parikh automata} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

In this article, we question the relevance of uniform random models for algorithms that use expressions as inputs. Using a general framework to describe expressions, we prove that if there is a subexpression that is absorbing for a given operator, then, after repeatedly applying the induced simplification to a uniform random expression of size n, we obtain an equivalent expression of constant expected size. This proves that uniform random expressions lack expressivity, as soon as there is an absorbing pattern. For instance, (a+b)^* is absorbing for the union for regular expressions on {a,b}, hence random regular expressions can be drastically reduced using the induced simplification.

Florent Koechlin, Cyril Nicaud, and Pablo Rotondo. Uniform Random Expressions Lack Expressivity. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{koechlin_et_al:LIPIcs.MFCS.2019.51, author = {Koechlin, Florent and Nicaud, Cyril and Rotondo, Pablo}, title = {{Uniform Random Expressions Lack Expressivity}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {51:1--51:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.51}, URN = {urn:nbn:de:0030-drops-109957}, doi = {10.4230/LIPIcs.MFCS.2019.51}, annote = {Keywords: Random expressions, simplification algorithms, analytic combinatorics} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in R^3. We show that this question, which is equivalent to deciding the emptiness of certain semi-algebraic sets bounded by cubic polynomials, can be "lifted" to a purely combinatorial problem. We propose an effective algorithm for that problem, and use it to gain new insights into the structure of geometric permutations.

Xavier Goaoc, Andreas Holmsen, and Cyril Nicaud. An Experimental Study of Forbidden Patterns in Geometric Permutations by Combinatorial Lifting. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{goaoc_et_al:LIPIcs.SoCG.2019.40, author = {Goaoc, Xavier and Holmsen, Andreas and Nicaud, Cyril}, title = {{An Experimental Study of Forbidden Patterns in Geometric Permutations by Combinatorial Lifting}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {40:1--40:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.40}, URN = {urn:nbn:de:0030-drops-104442}, doi = {10.4230/LIPIcs.SoCG.2019.40}, annote = {Keywords: Geometric permutation, Emptiness testing of semi-algebraic sets, Computer-aided proof} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

TimSort is an intriguing sorting algorithm designed in 2002 for Python, whose worst-case complexity was announced, but not proved until our recent preprint. In fact, there are two slightly different versions of TimSort that are currently implemented in Python and in Java respectively. We propose a pedagogical and insightful proof that the Python version runs in O(n log n). The approach we use in the analysis also applies to the Java version, although not without very involved technical details. As a byproduct of our study, we uncover a bug in the Java implementation that can cause the sorting method to fail during the execution. We also give a proof that Python's TimSort running time is in O(n + n log rho), where rho is the number of runs (i.e. maximal monotonic sequences), which is quite a natural parameter here and part of the explanation for the good behavior of TimSort on partially sorted inputs.

Nicolas Auger, Vincent Jugé, Cyril Nicaud, and Carine Pivoteau. On the Worst-Case Complexity of TimSort. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 4:1-4:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{auger_et_al:LIPIcs.ESA.2018.4, author = {Auger, Nicolas and Jug\'{e}, Vincent and Nicaud, Cyril and Pivoteau, Carine}, title = {{On the Worst-Case Complexity of TimSort}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {4:1--4:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.4}, URN = {urn:nbn:de:0030-drops-94678}, doi = {10.4230/LIPIcs.ESA.2018.4}, annote = {Keywords: Sorting algorithms, Merge sorting algorithms, TimSort, Analysis of algorithms} }

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

We give a probabilistic analysis of parameters related to alpha-gapped repeats and palindromes in random words, under both uniform and memoryless distributions (where letters have different probabilities, but are drawn independently). More precisely, we study the expected number of maximal alpha-gapped patterns, as well as the expected length of the longest alpha-gapped pattern in a random word.

Philippe Duchon, Cyril Nicaud, and Carine Pivoteau. Gapped Pattern Statistics. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 21:1-21:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{duchon_et_al:LIPIcs.CPM.2017.21, author = {Duchon, Philippe and Nicaud, Cyril and Pivoteau, Carine}, title = {{Gapped Pattern Statistics}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {21:1--21:12}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.21}, URN = {urn:nbn:de:0030-drops-73309}, doi = {10.4230/LIPIcs.CPM.2017.21}, annote = {Keywords: combinatorics on words, alpha-gapped repeats, random words, memoryless sources, analytic combinatorics} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

A synchronizing word for an automaton is a word that brings that automaton into one and the same state, regardless of the starting position. Cerny conjectured in 1964 that if a $n$-state deterministic automaton has a synchronizing word, then it has a synchronizing word of length at most (n-1)^2. Berlinkov recently made a breakthrough in the probabilistic analysis of synchronization: he proved that, for the uniform distribution on deterministic automata with n states, an automaton admits a synchronizing word with high probability. In this article, we are interested in the typical length of the smallest synchronizing word, when such a word exists: we prove that a random automaton admits a synchronizing word of length O(n log^{3}n) with high probability. As a consequence, this proves that most automata satisfy the Cerny conjecture.

Cyril Nicaud. Fast Synchronization of Random Automata. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 43:1-43:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{nicaud:LIPIcs.APPROX-RANDOM.2016.43, author = {Nicaud, Cyril}, title = {{Fast Synchronization of Random Automata}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {43:1--43:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.43}, URN = {urn:nbn:de:0030-drops-66665}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.43}, annote = {Keywords: random automata, synchronization, the \v{C}ern\'{y} conjecture} }

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

In this article we propose an alternative way to prove some recent results on statistics on words, such as the expected number of runs or the expected sum of the run exponents. Our approach consists in designing a general framework, based on the symbolic method developped in analytic combinatorics. The descriptions obtained in this framework are built in such a way that the degree of ambiguity of an object O (i.e., the number of different descriptions corresponding to O) is exactly the value of the statistic under study for O. The asymptotic estimation of the expectation is then done using classical techniques from analytic combinatorics. To show the generality of our method, we not only apply it to obtain new proofs of known results but also extend them from the uniform distribution to any memoryless distribution.

Cyril Nicaud. Estimating Statistics on Words Using Ambiguous Descriptions. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{nicaud:LIPIcs.CPM.2016.9, author = {Nicaud, Cyril}, title = {{Estimating Statistics on Words Using Ambiguous Descriptions}}, booktitle = {27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)}, pages = {9:1--9:12}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.9}, URN = {urn:nbn:de:0030-drops-60859}, doi = {10.4230/LIPIcs.CPM.2016.9}, annote = {Keywords: random words, runs, symbolic method, analytic combinatorics} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Most modern processors are heavily parallelized and use predictors to guess the outcome of conditional branches, in order to avoid costly stalls in their pipelines. We propose predictor-friendly versions of two classical algorithms: exponentiation by squaring and binary search in a sorted array. These variants result in less mispredictions on average, at the cost of an increased number of operations. These theoretical results are supported by experimentations that show that our algorithms perform significantly better than the standard ones, for primitive data types.

Nicolas Auger, Cyril Nicaud, and Carine Pivoteau. Good Predictions Are Worth a Few Comparisons. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{auger_et_al:LIPIcs.STACS.2016.12, author = {Auger, Nicolas and Nicaud, Cyril and Pivoteau, Carine}, title = {{Good Predictions Are Worth a Few Comparisons}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.12}, URN = {urn:nbn:de:0030-drops-57135}, doi = {10.4230/LIPIcs.STACS.2016.12}, annote = {Keywords: branch misses, binary search, exponentiation by squaring, Markov chains} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

We study the distribution of the number of accessible states in deterministic and complete automata with n states over a k-letters alphabet. We show that as n tends to infinity and for a fixed alphabet size, the distribution converges in law toward a Gaussian centered around vk n and of standard deviation equivalent to sk n^(1/2), for some explicit constants vk and sk. Using this characterization, we give a simple algorithm for random uniform generation of accessible deterministic and complete automata of size n of expected complexity O(n^(3/2)), which matches the best methods known so far. Moreover, if we allow a variation around n in the size of the output automaton, our algorithm is the first solution of linear expected complexity. Finally we show how this work can be used to study accessible automata (which are difficult to apprehend from a combinatorial point of view) through the prism of the simpler deterministic and complete automata. As an example, we show how the average complexity in O(n log log n) for Moore's minimization algorithm obtained by David for deterministic and complete automata can be extended to accessible automata.

Arnaud Carayol and Cyril Nicaud. Distribution of the number of accessible states in a random deterministic automaton. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 194-205, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{carayol_et_al:LIPIcs.STACS.2012.194, author = {Carayol, Arnaud and Nicaud, Cyril}, title = {{Distribution of the number of accessible states in a random deterministic automaton}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {194--205}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.194}, URN = {urn:nbn:de:0030-drops-34422}, doi = {10.4230/LIPIcs.STACS.2012.194}, annote = {Keywords: finite automata, random sampling, average complexity} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We study the average number of transitions in Glushkov automata built from random regular expressions. This statistic highly depends on the probabilistic distribution set on the expressions. A recent work shows that, under the uniform distribution, regular expressions lead to automata with a linear number of transitions. However, uniform regular expressions are not necessarily a satisfying model. Therefore, we rather focus on an other model, inspired from random binary search trees (BST), which is widely used, in particular for testing. We establish that, in this case, the average number of transitions becomes quadratic according to the size of the regular expression.

Cyril Nicaud, Carine Pivoteau, and Benoît Razet. Average Analysis of Glushkov Automata under a BST-Like Model. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 388-399, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{nicaud_et_al:LIPIcs.FSTTCS.2010.388, author = {Nicaud, Cyril and Pivoteau, Carine and Razet, Beno\^{i}t}, title = {{Average Analysis of Glushkov Automata under a BST-Like Model}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {388--399}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.388}, URN = {urn:nbn:de:0030-drops-28809}, doi = {10.4230/LIPIcs.FSTTCS.2010.388}, annote = {Keywords: Glushkov automata, random binary search tree, regular expression} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

We prove that, for any arbitrary finite alphabet and for the uniform distribution over deterministic and accessible automata with $n$ states, the average complexity of Moore's state minimization algorithm is in $\mathcal{O}(n \log n)$. Moreover this bound is tight in the case of unary automata.

Frederique Bassino, Julien David, and Cyril Nicaud. On the Average Complexity of Moore's State Minimization Algorithm. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 123-134, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{bassino_et_al:LIPIcs.STACS.2009.1822, author = {Bassino, Frederique and David, Julien and Nicaud, Cyril}, title = {{On the Average Complexity of Moore's State Minimization Algorithm}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {123--134}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1822}, URN = {urn:nbn:de:0030-drops-18222}, doi = {10.4230/LIPIcs.STACS.2009.1822}, annote = {Keywords: Finite automata, State minimization, Moore’s algorithm, Average complexity} }

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