Team : APR
Departure date : 09/30/2012
Supervision : Michèle SORIA
Générateur aléatoire de strutures ordonnées par le modèle de Boltzmann
Uniform random generation is a central issue in combinatorics. Indeed, random sampling is virtually connected to all parts of combinatorics, whether to exact or asymptotic enumeration, or to the experimental verification of conjectures. Various methods have been developed in order to efficiently solve that issue. Boltzmann model is among them. This method, relaxing some constraints about the size of the object being currently generated, ensures a linear complexity in many actual cases, and can easily be automatized for various combinatorial classes.
This thesis aims at enlarging the set of such admissible classes, while keeping the nice properties of linear complexity and ease of automation. The first part is devoted to the presentation of the Boltzmann model and existing Boltzmann samplers, and the study of their properties and mathematical foundations. In the second part, we introduce our idea of biasing those samplers in order to enlarge their range of validity. Firstly, we present a general extension, and then specialize it to several combinatorial operations such as the derivation, the shuffle product or the unpointing operation. Finally, we present a uniform random sampler for the Hadamard product.
We highlight our algorithms through this thesis with examples and experimental results, illustrating the efficiency of our methods.
Defence : 09/25/2012 - 14h - Site Jussieu 25-26/105 Jury members : Philippe Duchon
A. Darrasse, K. Panagiotou, O. Roussel, M. Soria : “Biased Boltzmann samplers and generation of extended linear languages with shuffle”, 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'12), vol. DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), DMTCS Proceedings, Montreal, Canada, pp. 125-140, (Discrete Mathematics and Theoretical Computer Science) (2012)
O. Bodini, D. Gardy, O. Roussel : “Boys-and-girls birthdays and Hadamard products”, 7th International Conference on Lattice Path Combinatorics and Applications, vol. 117 (1-4), Fundamenta Informaticae, Siena, Italy, pp. 85-101, (IOS Press) (2011)
O. Roussel, M. Soria : “Boltzmann sampling of ordered structures”, LAGOS 2009 - 5th Latin-American Algorithms, Graphs and Optimization Symposium, vol. 35, Electronic Notes in Discrete Mathematics, Rio Grande do Sul, Brazil, pp. 305-310, (Elsevier) (2009)