Random-bit optimal uniform sampling for rooted planar trees with given sequence of degrees and Applications

In this paper, we redesign and simplify an algorithm due to Remy et al. for the generation of rooted planar trees that satisfies a given partition of degrees. This new version is now optimal in terms of random bit complexity, up to a multiplicative constant. We then apply a natural process "simulate-guess-and-proof" to analyze the height of a random Motzkin in function of its frequency of unary nodes. When the number of unary nodes dominates, we prove some unconventional height phenomenon (i.e. outside the universal square root behaviour.)Comment: 19 page

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