6 research outputs found
Exact-size Sampling for Motzkin Trees in Linear Time via Boltzmann Samplers and Holonomic Specification
International audienceBoltzmann samplers are a kind of random samplers; in 2004, Duchon, Flajolet, Louchard and Schaeffer showed that given a combinatorial class and a combinatorial specification for that class, one can automatically build a Boltzmann sampler. In this paper, we introduce a Boltzmann sampler for Motzkin trees built from a holonomic specification, that is, a specification that uses the pointing operator. This sampler is inspired by R茅my's algorithm on binary trees. We show that our algorithm gives an exact size sampler with a linear time and space complexity in average
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
Combinatorics of explicit substitutions
is an extension of the -calculus which
internalises the calculus of substitutions. In the current paper, we
investigate the combinatorial properties of focusing on the
quantitative aspects of substitution resolution. We exhibit an unexpected
correspondence between the counting sequence for -terms and
famous Catalan numbers. As a by-product, we establish effective sampling
schemes for random -terms. We show that typical
-terms represent, in a strong sense, non-strict computations
in the classic -calculus. Moreover, typically almost all substitutions
are in fact suspended, i.e. unevaluated, under closures. Consequently, we argue
that is an intrinsically non-strict calculus of explicit
substitutions. Finally, we investigate the distribution of various redexes
governing the substitution resolution in and investigate the
quantitative contribution of various substitution primitives