112,232 research outputs found
Probabilistic models for pattern statistics
In this work we study some probabilistic models for the random generation of words over a given alphabet used in the literature in connection with pattern statistics. Our goal is to compare models based on Markovian processes (where the occurrence of a symbol in a given position only depends on a finite number of previous occurrences) and the stochastic models that can generate a word of given length from a regular language under uniform distribution. We present some results that show the differences between these two stochastic models and their relationship with the rational probabilistic measures
Controlled non uniform random generation of decomposable structures
Consider a class of decomposable combinatorial structures, using different
types of atoms \Atoms = \{\At_1,\ldots ,\At_{|{\Atoms}|}\}. We address the
random generation of such structures with respect to a size and a targeted
distribution in of its \emph{distinguished} atoms. We consider two
variations on this problem. In the first alternative, the targeted distribution
is given by real numbers \TargFreq_1, \ldots, \TargFreq_k such that 0 <
\TargFreq_i < 1 for all and \TargFreq_1+\cdots+\TargFreq_k \leq 1. We
aim to generate random structures among the whole set of structures of a given
size , in such a way that the {\em expected} frequency of any distinguished
atom \At_i equals \TargFreq_i. We address this problem by weighting the
atoms with a -tuple \Weights of real-valued weights, inducing a weighted
distribution over the set of structures of size . We first adapt the
classical recursive random generation scheme into an algorithm taking
\bigO{n^{1+o(1)}+mn\log{n}} arithmetic operations to draw structures from
the \Weights-weighted distribution. Secondly, we address the analytical
computation of weights such that the targeted frequencies are achieved
asymptotically, i. e. for large values of . We derive systems of functional
equations whose resolution gives an explicit relationship between \Weights
and \TargFreq_1, \ldots, \TargFreq_k. Lastly, we give an algorithm in
\bigO{k n^4} for the inverse problem, {\it i.e.} computing the frequencies
associated with a given -tuple \Weights of weights, and an optimized
version in \bigO{k n^2} in the case of context-free languages. This allows
for a heuristic resolution of the weights/frequencies relationship suitable for
complex specifications. In the second alternative, the targeted distribution is
given by a natural numbers such that
where is the number of undistinguished atoms.
The structures must be generated uniformly among the set of structures of size
that contain {\em exactly} atoms \At_i (). We give
a \bigO{r^2\prod_{i=1}^k n_i^2 +m n k \log n} algorithm for generating
structures, which simplifies into a \bigO{r\prod_{i=1}^k n_i +m n} for
regular specifications
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
Optimal Discrete Uniform Generation from Coin Flips, and Applications
This article introduces an algorithm to draw random discrete uniform
variables within a given range of size n from a source of random bits. The
algorithm aims to be simple to implement and optimal both with regards to the
amount of random bits consumed, and from a computational perspective---allowing
for faster and more efficient Monte-Carlo simulations in computational physics
and biology. I also provide a detailed analysis of the number of bits that are
spent per variate, and offer some extensions and applications, in particular to
the optimal random generation of permutations.Comment: first draft, 22 pages, 5 figures, C code implementation of algorith
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
Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties
We consider the problem of uniform sampling of points on an algebraic
variety. Specifically, we develop a randomized algorithm that, given a small
set of multivariate polynomials over a sufficiently large finite field,
produces a common zero of the polynomials almost uniformly at random. The
statistical distance between the output distribution of the algorithm and the
uniform distribution on the set of common zeros is polynomially small in the
field size, and the running time of the algorithm is polynomial in the
description of the polynomials and their degrees provided that the number of
the polynomials is a constant
Uniform Random Sampling of Traces in Very Large Models
This paper presents some first results on how to perform uniform random walks
(where every trace has the same probability to occur) in very large models. The
models considered here are described in a succinct way as a set of
communicating reactive modules. The method relies upon techniques for counting
and drawing uniformly at random words in regular languages. Each module is
considered as an automaton defining such a language. It is shown how it is
possible to combine local uniform drawings of traces, and to obtain some global
uniform random sampling, without construction of the global model
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