Random generation of combinatorial structures: Boltzmann samplers and beyond

International audienceThe Boltzmann model for the random generation of ''decomposable'' combinatorial structures is a set of techniques that allows for efficient random sampling algorithms for a large class of families of discrete objects. The usual requirement of sampling uniformly from the set of objects of a given size is somehow relaxed, though uniformity among objects of each size is still ensured. Generating functions, rather than the enumeration sequences they are based on, are the crucial ingredient. We give a brief description of the general theory, as well as a number of newer developments.Le modèle de Boltzmann pour la génération aléatoire de structures "décomposables" est un ensemble de techniques qui fournissent des algorithmes de tirage aléatoire pour une grande famille de classes d'objets discrets. L'exigence classique de génération uniforme parmi les objets d'une taille donnée est quelque peu relaxée, bien que l'équiprobabilité des objets de chaque taille soit préservée. Les séries génératrices, plutôt que les suites d'énumération sur lesquelles elles sont basées, sont l'ingrédient crucial. Nous donnons une brève description de la théorie générale, ainsi que quelques développements plus récents

Boys-and-girls birthdays and Hadamard products

Actes / Proceedings à paraître in Journal of Statistical Planning and Inference (special issue). http://www.journals.elsevier.com/journal-of-statistical-planning-and-inference/special-issues/International audienceBoltzmann models from statistical physics, combined with methods from analytic combinatorics, give rise to efficient and easy-to-write algorithms for the random generation of combinatorial objects. This paper proposes to extend Boltzmann generators to a new field of applications by uniformly sampling a Hadamard product. Under an abstract real-arithmetic computation model, our algorithm achieves approximate-size sampling in expected time ${\cal O}$(n${\sqrt n}$) or ${\cal O}$(nσ) depending on the objects considered, with σ the standard deviation of smallest order for the component object sizes. This makes it possible to generate random objects of large size on a standard computer. The analysis heavily relies on a variant of the so-called birthday paradox, which can be modelled as an occupancy urn problem