LIPIcs.STACS.2021.48.pdf
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Absorbing Patterns in BST-Like Expression-Trees
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38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
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- Publication Date: 2021-03-10
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Acknowledgements
The authors would like to thank Arnaud Carayol and Cyril Nicaud for their very useful advice and remarks in the writing of this paper.
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Florent Koechlin and Pablo Rotondo. Absorbing Patterns in BST-Like Expression-Trees. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.48
Abstract
In this article we study the effect of simple semantic reductions on random BST-like expression-trees. Such random unary-binary expression-trees are often used in benchmarks for model-checking tools. We consider the reduction induced by an absorbing pattern for some given operator ⊛, which we apply bottom-up, producing an equivalent (and smaller) tree-expression. Our main result concerns the expected size of a random tree, of given input size n → ∞, after reduction. We show that there are two different thresholds, leading to a total of five regimes, ranging from no significant reduction at all, to almost complete reduction. These regimes are completely characterized according to the probability of the absorbing operator. Our results prove that random BST-like trees have to be considered with care, and that they offer a richer range of behaviours than uniform random trees.
Subject Classification
ACM Subject Classification
- Mathematics of computing → Discrete mathematics
Keywords
- BST trees
- absorbing pattern
- reduction
- analytic combinatorics
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References
-
T.M. Apostol. Calculus. Vol. II: Multi-variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability. Blaisdell international textbook series. Xerox College Publ., 1969.
- Nicolas Broutin and Cécile Mailler. And/or trees: A local limit point of view. Random Struct. Algorithms, 53(1):15-58, 2018. URL: https://doi.org/10.1002/rsa.20758.
- Brigitte Chauvin, Danièle Gardy, and Cécile Mailler. The growing tree distribution on boolean functions. In 2011 Proceedings of the Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pages 45-56, 2011. URL: https://doi.org/10.1137/1.9781611973013.5.
-
Marco Daniele, Fausto Giunchiglia, and Moshe Y. Vardi. Improved automata generation for linear temporal logic. In Nicolas Halbwachs and Doron Peled, editors, Computer Aided Verification, pages 249-260, Berlin, Heidelberg, 1999. Springer Berlin Heidelberg.
- Luc Devroye. A note on the height of binary search trees. J. ACM, 33(3):489–498, May 1986. URL: https://doi.org/10.1145/5925.5930.
- Alexandre Duret-Lutz, Alexandre Lewkowicz, Amaury Fauchille, Thibaud Michaud, Etienne Renault, and Laurent Xu. Spot 2.0 - a framework for LTL and ω-automata manipulation. In Proceedings of the 14th International Symposium on Automated Technology for Verification and Analysis (ATVA'16), volume 9938 of Lecture Notes in Computer Science, pages 122-129. Springer, October 2016. URL: https://doi.org/10.1007/978-3-319-46520-3_8.
- Philippe Flajolet, Xavier Gourdon, and Conrado Martinez. Patterns in random binary search trees. Random Struct. Algorithms, 11(3):223-244, 1997. URL: https://doi.org/10.1002/(SICI)1098-2418(199710)11:3<223::AID-RSA2>3.0.CO;2-2.
- Philippe Flajolet and Andrew M. Odlyzko. The average height of binary trees and other simple trees. J. Comput. Syst. Sci., 25(2):171-213, 1982. URL: https://doi.org/10.1016/0022-0000(82)90004-6.
-
Philippe Flajolet and Robert Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009.
- Antoine Genitrini, Bernhard Gittenberger, Veronika Kraus, and Cécile Mailler. Associative and commutative tree representations for boolean functions. Theoretical Computer Science, 570:70-101, 2015. URL: https://doi.org/10.1016/j.tcs.2014.12.025.
-
Florent Koechlin, Cyril Nicaud, and Pablo Rotondo. Uniform random expressions lack expressivity. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, volume 138 of LIPIcs, pages 51:1-51:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.
-
Florent Koechlin, Cyril Nicaud, and Pablo Rotondo. On the degeneracy of random expressions specified by systems of combinatorial equations. In Natasa Jonoska and Dmytro Savchuk, editors, Developments in Language Theory - 24th International Conference, DLT 2020, Tampa, FL, USA, May 11-15, 2020, Proceedings, volume 12086 of Lecture Notes in Computer Science, pages 164-177. Springer, 2020.
-
Cyril Nicaud. On the Average Size of Glushkov’s Automata. In Adrian-Horia Dediu, Armand-Mihai Ionescu, and Carlos Martín-Vide, editors, Language and Automata Theory and Applications, Third International Conference, LATA 2009, Tarragona, Spain, April 2-8, 2009. Proceedings, volume 5457 of Lecture Notes in Computer Science, pages 626-637. Springer, 2009.
- Cyril Nicaud, Carine Pivoteau, and Benoît Razet. Average Analysis of Glushkov Automata under a BST-Like Model. In Kamal Lodaya and Meena Mahajan, editors, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010), volume 8 of Leibniz International Proceedings in Informatics (LIPIcs), pages 388-399, Dagstuhl, Germany, 2010. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2010.388.
- Carine Pivoteau, Bruno Salvy, and Michèle Soria. Algorithms for combinatorial structures: Well-founded systems and newton iterations. J. Comb. Theory, Ser. A, 119(8):1711-1773, 2012. URL: https://doi.org/10.1016/j.jcta.2012.05.007.
- Richard P. Stanley. Differentiably finite power series. Eur. J. Comb., 1(2):175-188, 1980. URL: https://doi.org/10.1016/S0195-6698(80)80051-5.
-
Heikki Tauriainen. Automated testing of Büchi automata translators for linear temporal logic. Research Report A66, Helsinki University of Technology, Laboratory for Theoretical Computer Science, Espoo, Finland, December 2000.
- W. Wasow. Asymptotic Expansions for Ordinary Differential Equations. Dover Books on Mathematics. Dover Publications, 2018. URL: https://books.google.fr/books?id=NQNKDwAAQBAJ.