92 research outputs found
Analyzing Boltzmann Samplers for Bose-Einstein Condensates with Dirichlet Generating Functions
Boltzmann sampling is commonly used to uniformly sample objects of a
particular size from large combinatorial sets. For this technique to be
effective, one needs to prove that (1) the sampling procedure is efficient and
(2) objects of the desired size are generated with sufficiently high
probability. We use this approach to give a provably efficient sampling
algorithm for a class of weighted integer partitions related to Bose-Einstein
condensation from statistical physics. Our sampling algorithm is a
probabilistic interpretation of the ordinary generating function for these
objects, derived from the symbolic method of analytic combinatorics. Using the
Khintchine-Meinardus probabilistic method to bound the rejection rate of our
Boltzmann sampler through singularity analysis of Dirichlet generating
functions, we offer an alternative approach to analyze Boltzmann samplers for
objects with multiplicative structure.Comment: 20 pages, 1 figur
Uniform random sampling of planar graphs in linear time
This article introduces new algorithms for the uniform random generation of
labelled planar graphs. Its principles rely on Boltzmann samplers, as recently
developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the
Boltzmann framework, a suitable use of rejection, a new combinatorial bijection
found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic
description of the generating functions counting planar graphs, which was
recently obtained by Gim\'enez and Noy. This gives rise to an extremely
efficient algorithm for the random generation of planar graphs. There is a
preprocessing step of some fixed small cost. Then, the expected time complexity
of generation is quadratic for exact-size uniform sampling and linear for
approximate-size sampling. This greatly improves on the best previously known
time complexity for exact-size uniform sampling of planar graphs with
vertices, which was a little over .Comment: 55 page
Infinite Boltzmann Samplers and Applications to Branching Processes
National audienceIn this short note, we extend the Boltzmann model for combinatorial random sampling [8] to allow for infinite size objects; in particular, this extension now fully includes Galton-Watson processes. We then illustrate our idea with two examples, one of which is the generation of prefixes of infinite Cayley trees
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