111 research outputs found
Multi-dimensional Boltzmann Sampling of Languages
This paper addresses the uniform random generation of words from a
context-free language (over an alphabet of size ), while constraining every
letter to a targeted frequency of occurrence. Our approach consists in a
multidimensional extension of Boltzmann samplers \cite{Duchon2004}. We show
that, under mostly \emph{strong-connectivity} hypotheses, our samplers return a
word of size in and exact frequency in
expected time. Moreover, if we accept tolerance
intervals of width in for the number of occurrences of each
letters, our samplers perform an approximate-size generation of words in
expected time. We illustrate these techniques on the
generation of Tetris tessellations with uniform statistics in the different
types of tetraminoes.Comment: 12p
Stochastic Flips on Two-letter Words
This paper introduces a simple Markov process inspired by the problem of
quasicrystal growth. It acts over two-letter words by randomly performing
\emph{flips}, a local transformation which exchanges two consecutive different
letters. More precisely, only the flips which do not increase the number of
pairs of consecutive identical letters are allowed. Fixed-points of such a
process thus perfectly alternate different letters. We show that the expected
number of flips to converge towards a fixed-point is bounded by in the
worst-case and by in the average-case, where denotes the
length of the initial word.Comment: ANALCO'1
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
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