44,857 research outputs found
Combinatorial generation via permutation languages
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations.
This approach provides a unified view on many known results and allows us to prove many new ones.
In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an -element set by adjacent transpositions; the binary reflected Gray code to generate all -bit strings by flipping a single bit in each step; the Gray code for generating all -vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an -element ground set by element exchanges due to Kaye.
We present two distinct applications for our new framework:
The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others.
We thus also obtain new Gray code algorithms for the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into rectangles subject to certain restrictions.
The second main application of our framework are lattice congruences of the weak order on the symmetric group~.
Recently, Pilaud and Santos realized all those lattice congruences as -dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc.
Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian.
We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope
Efficient generation of random derangements with the expected distribution of cycle lengths
We show how to generate random derangements efficiently by two different
techniques: random restricted transpositions and sequential importance
sampling. The algorithm employing restricted transpositions can also be used to
generate random fixed-point-free involutions only, a.k.a. random perfect
matchings on the complete graph. Our data indicate that the algorithms generate
random samples with the expected distribution of cycle lengths, which we
derive, and for relatively small samples, which can actually be very large in
absolute numbers, we argue that they generate samples indistinguishable from
the uniform distribution. Both algorithms are simple to understand and
implement and possess a performance comparable to or better than those of
currently known methods. Simulations suggest that the mixing time of the
algorithm based on random restricted transpositions (in the total variance
distance with respect to the distribution of cycle lengths) is
with and the length of the
derangement. We prove that the sequential importance sampling algorithm
generates random derangements in time with probability of
failing.Comment: This version corrected and updated; 14 pages, 2 algorithms, 2 tables,
4 figure
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
Using parallel computation to improve Independent Metropolis--Hastings based estimation
In this paper, we consider the implications of the fact that parallel
raw-power can be exploited by a generic Metropolis--Hastings algorithm if the
proposed values are independent. In particular, we present improvements to the
independent Metropolis--Hastings algorithm that significantly decrease the
variance of any estimator derived from the MCMC output, for a null computing
cost since those improvements are based on a fixed number of target density
evaluations. Furthermore, the techniques developed in this paper do not
jeopardize the Markovian convergence properties of the algorithm, since they
are based on the Rao--Blackwell principles of Gelfand and Smith (1990), already
exploited in Casella and Robert (1996), Atchade and Perron (2005) and Douc and
Robert (2010). We illustrate those improvements both on a toy normal example
and on a classical probit regression model, but stress the fact that they are
applicable in any case where the independent Metropolis-Hastings is applicable.Comment: 19 pages, 8 figures, to appear in Journal of Computational and
Graphical Statistic
A methodological approach for algorithmic composition systems' parameter spaces aesthetic exploration
Algorithmic composition is the process of creating musical material by means of formal methods. As a consequence of its design, algorithmic composition systems are (explicitly or implicitly) described in terms of parameters. Thus, parameter space exploration plays a key role in learning the system's capabilities. However, in the computer music field, this task has received little attention. This is due in part, because the produced changes on the human perception of the outputs, as a response to changes on the parameters, could be highly nonlinear, therefore models with strongly predictable outputs are needed. The present work describes a methodology for the human perceptual (or aesthetic) exploration of generative systems' parameter spaces. As the systems' outputs are intended to produce an aesthetic experience on humans, audition plays a central role in the process. The methodology starts from a set of parameter combinations which are perceptually evaluated by the user. The sampling process of such combinations depends on the system under study and possible on heuristic considerations. The evaluated set is processed by a compaction algorithm able to generate linguistic rules describing the distinct perceptions (classes) of the user evaluation. The semantic level of the extracted rules allows for interpretability, while showing great potential in describing high and low-level musical entities. As the resulting rules represent discrete points in the parameter space, further possible extensions for interpolation between points are also discussed. Finally, some practical implementations and paths for further research are presented.Peer ReviewedPostprint (author's final draft
Random and exhaustive generation of permutations and cycles
In 1986 S. Sattolo introduced a simple algorithm for uniform random
generation of cyclic permutations on a fixed number of symbols. This algorithm
is very similar to the standard method for generating a random permutation, but
is less well known.
We consider both methods in a unified way, and discuss their relation with
exhaustive generation methods. We analyse several random variables associated
with the algorithms and find their grand probability generating functions,
which gives easy access to moments and limit laws.Comment: 9 page
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