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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

For three decades binary decision diagrams, a data structure efficiently representing Boolean functions, have been widely used in many distinct contexts like model verification, machine learning, cryptography and also resolution of combinatorial problems. The most famous variant, called reduced ordered binary decision diagram (robdd for short), can be viewed as the result of a compaction procedure on the full decision tree. A useful property is that once an order over the Boolean variables is fixed, each Boolean function is represented by exactly one robdd. In this paper we aim at computing the {exact distribution of the Boolean functions in k variables according to the robdd size}, where the robdd size is equal to the number of decision nodes of the underlying directed acyclic graph (dag) structure. Recall the number of Boolean functions with k variables is equal to 2^{2^k}, which is of double exponential growth with respect to the number of variables. The maximal size of a robdd with k variables is M_k ≈ 2^k / k. Apart from the natural combinatorial explosion observed, another difficulty for computing the distribution according to size is to take into account dependencies within the dag structure of robdds. In this paper, we develop the first polynomial algorithm to derive the distribution of Boolean functions over k variables with respect to robdd size denoted by n. The algorithm computes the (enumerative) generating function of robdds with k variables up to size n. It performs O(k n⁴) arithmetical operations on integers and necessitates storing O((k+n) n²) integers with bit length O(nlog n). Our new approach relies on a decomposition of robdds layer by layer and on an inclusion-exclusion argument.

Julien Clément and Antoine Genitrini. An Iterative Approach for Counting Reduced Ordered Binary Decision Diagrams. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{clement_et_al:LIPIcs.MFCS.2023.36, author = {Cl\'{e}ment, Julien and Genitrini, Antoine}, title = {{An Iterative Approach for Counting Reduced Ordered Binary Decision Diagrams}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {36:1--36:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.36}, URN = {urn:nbn:de:0030-drops-185702}, doi = {10.4230/LIPIcs.MFCS.2023.36}, annote = {Keywords: Boolean Function, Reduced Ordered Binary Decision Diagram (\{robdd\}), Enumerative Combinatorics, Directed Acyclic Graph} }

Document

**Published in:** LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

The paper studies the behaviour of selection algorithms that are based on dichotomy principles. On the entry formed by an ordered list L and a searched element x not in L, they return the interval of the list L the element x belongs to. We focus here on the case of words, where dichotomy principles lead to a selection algorithm designed by Crochemore, Hancart and Lecroq, which appears to be "quasi-optimal". We perform a probabilistic analysis of this algorithm that exhibits its quasi-optimality on average.

Ali Akhavi, Julien Clément, Dimitri Darthenay, Loïck Lhote, and Brigitte Vallée. Dichotomic Selection on Words: A Probabilistic Analysis. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akhavi_et_al:LIPIcs.CPM.2019.19, author = {Akhavi, Ali and Cl\'{e}ment, Julien and Darthenay, Dimitri and Lhote, Lo\"{i}ck and Vall\'{e}e, Brigitte}, title = {{Dichotomic Selection on Words: A Probabilistic Analysis}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {19:1--19:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-103-0}, ISSN = {1868-8969}, year = {2019}, volume = {128}, editor = {Pisanti, Nadia and P. Pissis, Solon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.19}, URN = {urn:nbn:de:0030-drops-104903}, doi = {10.4230/LIPIcs.CPM.2019.19}, annote = {Keywords: dichotomic selection, text algorithms, analysis of algorithms, average case analysis of algorithms, trie, suffix array, lcp-array, information theory, numeration process, sources, entropy, coincidence, analytic combinatorics, depoissonization techniques} }

Document

**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

We describe a general framework for realistic analysis of sorting and searching algorithms, and we apply it to the average-case analysis of five basic algorithms: three sorting algorithms (QuickSort, InsertionSort, BubbleSort) and two selection algorithms (QuickMin and SelectionMin). Usually, the analysis deals with the mean number of key comparisons, but, here, we view keys as words produced by the same source, which are compared via their symbols in the lexicographic order. The "realistic" cost of the algorithm is now the total number of symbol comparisons performed by the algorithm, and, in this context, the average-case analysis aims to providee stimates for the mean number of symbol comparisons used by the algorithm. For sorting algorithms, and with respect to key comparisons, the average-case complexity of QuickSort is asymptotic to 2n log n, InsertionSort to n^2/4 and BubbleSort to n^2/2. With respect to symbol comparisons, we prove that their average-case complexity becomes Theta(n log^2n), Theta(n^2), Theta (n^2 log n). For selection algorithms, and with respect to key comparisons, the average-case complexity of QuickMin is asymptotic to 2n, of SelectionMin is n - 1. With respect to symbol comparisons, we prove that their average-case complexity remains Theta(n). In these five cases, we describe the dominant constants which exhibit the probabilistic behaviour of the source (namely, entropy, and various notions of coincidence) with respect to the algorithm.

Julien Clément, Thu Hien Nguyen Thi, and Brigitte Vallée. A general framework for the realistic analysis of sorting and searching algorithms. Application to some popular algorithms. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 598-609, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{clement_et_al:LIPIcs.STACS.2013.598, author = {Cl\'{e}ment, Julien and Nguyen Thi, Thu Hien and Vall\'{e}e, Brigitte}, title = {{A general framework for the realistic analysis of sorting and searching algorithms. Application to some popular algorithms}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {598--609}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.598}, URN = {urn:nbn:de:0030-drops-39681}, doi = {10.4230/LIPIcs.STACS.2013.598}, annote = {Keywords: Probabilistic analysis of algorithms, Sorting and searching algorithms, Pattern matching, Permutations, Information theory, Rice formula, Asymptotic e} }

Document

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

The Prefix table of a string reports for each position the maximal length of its prefixes starting here. The Prefix table and its dual Suffix table are basic tools used in the design of the most efficient string-matching and pattern extraction algorithms. These tables can be computed in linear time independently of the alphabet size.
We give an algorithmic characterisation of a Prefix table (it can be adapted to a Suffix table). Namely, the algorithm tests if an integer table of size $n$ is the Prefix table of some word and, if successful, it constructs the lexicographically smallest string having it as a Prefix table. We show that the alphabet of the string can be bounded to $\log_2 n$ letters. The overall algorithm runs in $O(n)$ time.

Julien Clement, Maxime Crochemore, and Giuseppina Rindone. Reverse Engineering Prefix Tables. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 289-300, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{clement_et_al:LIPIcs.STACS.2009.1825, author = {Clement, Julien and Crochemore, Maxime and Rindone, Giuseppina}, title = {{Reverse Engineering Prefix Tables}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {289--300}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1825}, URN = {urn:nbn:de:0030-drops-18258}, doi = {10.4230/LIPIcs.STACS.2009.1825}, annote = {Keywords: Design and analysis of algorithms, Algorithms on strings, Pattern matching, String matching, Combinatorics on words, Prefix table, Suffix table} }

Document

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

For three decades binary decision diagrams, a data structure efficiently representing Boolean functions, have been widely used in many distinct contexts like model verification, machine learning, cryptography and also resolution of combinatorial problems. The most famous variant, called reduced ordered binary decision diagram (robdd for short), can be viewed as the result of a compaction procedure on the full decision tree. A useful property is that once an order over the Boolean variables is fixed, each Boolean function is represented by exactly one robdd. In this paper we aim at computing the {exact distribution of the Boolean functions in k variables according to the robdd size}, where the robdd size is equal to the number of decision nodes of the underlying directed acyclic graph (dag) structure. Recall the number of Boolean functions with k variables is equal to 2^{2^k}, which is of double exponential growth with respect to the number of variables. The maximal size of a robdd with k variables is M_k ≈ 2^k / k. Apart from the natural combinatorial explosion observed, another difficulty for computing the distribution according to size is to take into account dependencies within the dag structure of robdds. In this paper, we develop the first polynomial algorithm to derive the distribution of Boolean functions over k variables with respect to robdd size denoted by n. The algorithm computes the (enumerative) generating function of robdds with k variables up to size n. It performs O(k n⁴) arithmetical operations on integers and necessitates storing O((k+n) n²) integers with bit length O(nlog n). Our new approach relies on a decomposition of robdds layer by layer and on an inclusion-exclusion argument.

Julien Clément and Antoine Genitrini. An Iterative Approach for Counting Reduced Ordered Binary Decision Diagrams. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{clement_et_al:LIPIcs.MFCS.2023.36, author = {Cl\'{e}ment, Julien and Genitrini, Antoine}, title = {{An Iterative Approach for Counting Reduced Ordered Binary Decision Diagrams}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {36:1--36:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.36}, URN = {urn:nbn:de:0030-drops-185702}, doi = {10.4230/LIPIcs.MFCS.2023.36}, annote = {Keywords: Boolean Function, Reduced Ordered Binary Decision Diagram (\{robdd\}), Enumerative Combinatorics, Directed Acyclic Graph} }

Document

**Published in:** LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

The paper studies the behaviour of selection algorithms that are based on dichotomy principles. On the entry formed by an ordered list L and a searched element x not in L, they return the interval of the list L the element x belongs to. We focus here on the case of words, where dichotomy principles lead to a selection algorithm designed by Crochemore, Hancart and Lecroq, which appears to be "quasi-optimal". We perform a probabilistic analysis of this algorithm that exhibits its quasi-optimality on average.

Ali Akhavi, Julien Clément, Dimitri Darthenay, Loïck Lhote, and Brigitte Vallée. Dichotomic Selection on Words: A Probabilistic Analysis. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akhavi_et_al:LIPIcs.CPM.2019.19, author = {Akhavi, Ali and Cl\'{e}ment, Julien and Darthenay, Dimitri and Lhote, Lo\"{i}ck and Vall\'{e}e, Brigitte}, title = {{Dichotomic Selection on Words: A Probabilistic Analysis}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {19:1--19:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-103-0}, ISSN = {1868-8969}, year = {2019}, volume = {128}, editor = {Pisanti, Nadia and P. Pissis, Solon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.19}, URN = {urn:nbn:de:0030-drops-104903}, doi = {10.4230/LIPIcs.CPM.2019.19}, annote = {Keywords: dichotomic selection, text algorithms, analysis of algorithms, average case analysis of algorithms, trie, suffix array, lcp-array, information theory, numeration process, sources, entropy, coincidence, analytic combinatorics, depoissonization techniques} }

Document

**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

We describe a general framework for realistic analysis of sorting and searching algorithms, and we apply it to the average-case analysis of five basic algorithms: three sorting algorithms (QuickSort, InsertionSort, BubbleSort) and two selection algorithms (QuickMin and SelectionMin). Usually, the analysis deals with the mean number of key comparisons, but, here, we view keys as words produced by the same source, which are compared via their symbols in the lexicographic order. The "realistic" cost of the algorithm is now the total number of symbol comparisons performed by the algorithm, and, in this context, the average-case analysis aims to providee stimates for the mean number of symbol comparisons used by the algorithm. For sorting algorithms, and with respect to key comparisons, the average-case complexity of QuickSort is asymptotic to 2n log n, InsertionSort to n^2/4 and BubbleSort to n^2/2. With respect to symbol comparisons, we prove that their average-case complexity becomes Theta(n log^2n), Theta(n^2), Theta (n^2 log n). For selection algorithms, and with respect to key comparisons, the average-case complexity of QuickMin is asymptotic to 2n, of SelectionMin is n - 1. With respect to symbol comparisons, we prove that their average-case complexity remains Theta(n). In these five cases, we describe the dominant constants which exhibit the probabilistic behaviour of the source (namely, entropy, and various notions of coincidence) with respect to the algorithm.

Julien Clément, Thu Hien Nguyen Thi, and Brigitte Vallée. A general framework for the realistic analysis of sorting and searching algorithms. Application to some popular algorithms. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 598-609, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{clement_et_al:LIPIcs.STACS.2013.598, author = {Cl\'{e}ment, Julien and Nguyen Thi, Thu Hien and Vall\'{e}e, Brigitte}, title = {{A general framework for the realistic analysis of sorting and searching algorithms. Application to some popular algorithms}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {598--609}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.598}, URN = {urn:nbn:de:0030-drops-39681}, doi = {10.4230/LIPIcs.STACS.2013.598}, annote = {Keywords: Probabilistic analysis of algorithms, Sorting and searching algorithms, Pattern matching, Permutations, Information theory, Rice formula, Asymptotic e} }

Document

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

The Prefix table of a string reports for each position the maximal length of its prefixes starting here. The Prefix table and its dual Suffix table are basic tools used in the design of the most efficient string-matching and pattern extraction algorithms. These tables can be computed in linear time independently of the alphabet size.
We give an algorithmic characterisation of a Prefix table (it can be adapted to a Suffix table). Namely, the algorithm tests if an integer table of size $n$ is the Prefix table of some word and, if successful, it constructs the lexicographically smallest string having it as a Prefix table. We show that the alphabet of the string can be bounded to $\log_2 n$ letters. The overall algorithm runs in $O(n)$ time.

Julien Clement, Maxime Crochemore, and Giuseppina Rindone. Reverse Engineering Prefix Tables. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 289-300, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{clement_et_al:LIPIcs.STACS.2009.1825, author = {Clement, Julien and Crochemore, Maxime and Rindone, Giuseppina}, title = {{Reverse Engineering Prefix Tables}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {289--300}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1825}, URN = {urn:nbn:de:0030-drops-18258}, doi = {10.4230/LIPIcs.STACS.2009.1825}, annote = {Keywords: Design and analysis of algorithms, Algorithms on strings, Pattern matching, String matching, Combinatorics on words, Prefix table, Suffix table} }

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