Polynomial tuning of multiparametric combinatorial samplers

Boltzmann samplers and the recursive method are prominent algorithmic frameworks for the approximate-size and exact-size random generation of large combinatorial structures, such as maps, tilings, RNA sequences or various tree-like structures. In their multiparametric variants, these samplers allow to control the profile of expected values corresponding to multiple combinatorial parameters. One can control, for instance, the number of leaves, profile of node degrees in trees or the number of certain subpatterns in strings. However, such a flexible control requires an additional non-trivial tuning procedure. In this paper, we propose an efficient polynomial-time, with respect to the number of tuned parameters, tuning algorithm based on convex optimisation techniques. Finally, we illustrate the efficiency of our approach using several applications of rational, algebraic and P\'olya structures including polyomino tilings with prescribed tile frequencies, planar trees with a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures, colours. Implementation and examples are available at [1] https://github.com/maciej-bendkowski/boltzmann-brain [2] https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler

Boltzmann samplers for first-order differential specifications

15 pagesInternational audienceIn the framework of analytic combinatorics, Boltzmann models give rise to efficient algorithms for the random generation of combinatorial objects. This paper proposes an efficient Boltzmann sampler for ordered structures defined by first-order differential specifications. Under an abstract real-arithmetic computation model, our algorithm is of linear complexity for free generation; in addition, for many classical structures, the complexity is also linear when a small tolerance is allowed on the size of the generated object. The resulting implementation makes it possible to generate very large random objects, such as increasing trees, in a few seconds on a standard machine