16,371 research outputs found
Counting, generating and sampling tree alignments
Pairwise ordered tree alignment are combinatorial objects that appear in RNA
secondary structure comparison. However, the usual representation of tree
alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce
identical sets of matches between identical pairs of trees. This ambiguity is
uninformative, and detrimental to any probabilistic analysis.In this work, we
consider tree alignments up to equivalence. Our first result is a precise
asymptotic enumeration of tree alignments, obtained from a context-free grammar
by mean of basic analytic combinatorics. Our second result focuses on
alignments between two given ordered trees and . By refining our grammar
to align specific trees, we obtain a decomposition scheme for the space of
alignments, and use it to design an efficient dynamic programming algorithm for
sampling alignments under the Gibbs-Boltzmann probability distribution. This
generalizes existing tree alignment algorithms, and opens the door for a
probabilistic analysis of the space of suboptimal RNA secondary structures
alignments.Comment: ALCOB - 3rd International Conference on Algorithms for Computational
Biology - 2016, Jun 2016, Trujillo, Spain. 201
Approximately Sampling Elements with Fixed Rank in Graded Posets
Graded posets frequently arise throughout combinatorics, where it is natural
to try to count the number of elements of a fixed rank. These counting problems
are often -complete, so we consider approximation algorithms for
counting and uniform sampling. We show that for certain classes of posets,
biased Markov chains that walk along edges of their Hasse diagrams allow us to
approximately generate samples with any fixed rank in expected polynomial time.
Our arguments do not rely on the typical proofs of log-concavity, which are
used to construct a stationary distribution with a specific mode in order to
give a lower bound on the probability of outputting an element of the desired
rank. Instead, we infer this directly from bounds on the mixing time of the
chains through a method we call .
A noteworthy application of our method is sampling restricted classes of
integer partitions of . We give the first provably efficient Markov chain
algorithm to uniformly sample integer partitions of from general restricted
classes. Several observations allow us to improve the efficiency of this chain
to require space, and for unrestricted integer partitions,
expected time. Related applications include sampling permutations
with a fixed number of inversions and lozenge tilings on the triangular lattice
with a fixed average height.Comment: 23 pages, 12 figure
Liquid crystal films on curved surfaces: An entropic sampling study
The confining effect of a spherical substrate inducing anchoring (normal to
the surface) of rod-like liquid crystal molecules contained in a thin film
spread over it has been investigated with regard to possible changes in the
nature of the isotropic-to-nematic phase transition as the sample is cooled.
The focus of these Monte Carlo simulations is to study the competing effects of
the homeotropic anchoring due to the surface inducing orientational ordering in
the radial direction and the inherent uniaxial order promoted by the
intermolecular interactions. By adopting entropic sampling procedure, we could
investigate this transition with a high temperature precision, and we studied
the effect of the surface anchoring strength on the phase diagram for a
specifically chosen geometry. We find that there is a threshold anchoring
strength of the surface below which uniaxial nematic phase results, and above
which the isotropic fluid cools to a radially ordered nematic phase, besides of
course expected changes in the phase transition temperature with the anchoring
strength. In the vicinity of the threshold anchoring strength we observe a
bistable region between these two structures, clearly brought out by the
characteristics of the corresponding microstates constituting the entropic
ensemble.Comment: 14 pages, 5 figure
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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