2,267 research outputs found
Every graph admits an unambiguous bold drawing
Let r and w be fixed positive numbers, w < r. In a bold drawing of a graph, every vertex is represented by a disk of radius r, and every edge by a narrow rectangle of width w. We solve a problem of van Kreveld [10] by showing that every graph admits a bold drawing in which the region occupied by the union of the disks and rectangles representing the vertices and edges does not contain any disk of radius r other than the ones representing the vertices. © 2015, Brown University. All rights reserved
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
Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces
The Subgraph Isomorphism problem asks, given a host graph G on n vertices and
a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P.
The restriction of this problem to planar graphs has often been considered.
After a sequence of improvements, the current best algorithm for planar graphs
is a linear time algorithm by Dorn (STACS '10), with complexity .
We generalize this result, by giving an algorithm of the same complexity for
graphs that can be embedded in surfaces of bounded genus. At the same time, we
simplify the algorithm and analysis. The key to these improvements is the
introduction of surface split decompositions for bounded genus graphs, which
generalize sphere cut decompositions for planar graphs. We extend the algorithm
for the problem of counting and generating all subgraphs isomorphic to P, even
for the case where P is disconnected. This answers an open question by Eppstein
(SODA '95 / JGAA '99)
Limit cycles in piecewise-affine gene network models with multiple interaction loops
In this paper we consider piecewise affine differential equations modeling
gene networks. We work with arbitrary decay rates, and under a local hypothesis
expressed as an alignment condition of successive focal points. The interaction
graph of the system may be rather complex (multiple intricate loops of any
sign, multiple thresholds...). Our main result is an alternative theorem
showing that, if a sequence of region is periodically visited by trajectories,
then under our hypotheses, there exists either a unique stable periodic
solution, or the origin attracts all trajectories in this sequence of regions.
This result extends greatly our previous work on a single negative feedback
loop. We give several examples and simulations illustrating different cases
Dynamics of Black Hole Pairs I: Periodic Tables
Although the orbits of comparable mass, spinning black holes seem to defy
simple decoding, we find a means to decipher all such orbits. The dynamics is
complicated by extreme perihelion precession compounded by spin-induced
precession. We are able to quantitatively define and describe the fully three
dimensional motion of comparable mass binaries with one black hole spinning and
expose an underlying simplicity. To do so, we untangle the dynamics by
capturing the motion in the orbital plane. Our results are twofold: (1) We
derive highly simplified equations of motion in a non-orthogonal orbital basis,
and (2) we define a complete taxonomy for fully three-dimensional orbits. More
than just a naming system, the taxonomy provides unambiguous and quantitative
descriptions of the orbits, including a determination of the zoom-whirliness of
any given orbit. Through a correspondence with the rationals, we are able to
show that zoom-whirl behavior is prevalent in comparable mass binaries in the
strong-field regime. A first significant conclusion that can be drawn from this
analysis is that all generic orbits in the final stages of inspiral under
gravitational radiation losses are characterized by precessing clovers with few
leaves and that no orbit will behave like the tightly precessing ellipse of
Mercury. The gravitational waveform produced by these low-leaf clovers will
reflect the natural harmonics of the orbital basis -- harmonics that,
importantly, depend only on radius. The significance for gravitational wave
astronomy will depend on the number of windings the pair executes in the
strong-field regime and could be more conspicuous for intermediate mass pairs
than for stellar mass pairs.Comment: 19 pages, lots of figure
A network approach to topic models
One of the main computational and scientific challenges in the modern age is
to extract useful information from unstructured texts. Topic models are one
popular machine-learning approach which infers the latent topical structure of
a collection of documents. Despite their success --- in particular of its most
widely used variant called Latent Dirichlet Allocation (LDA) --- and numerous
applications in sociology, history, and linguistics, topic models are known to
suffer from severe conceptual and practical problems, e.g. a lack of
justification for the Bayesian priors, discrepancies with statistical
properties of real texts, and the inability to properly choose the number of
topics. Here we obtain a fresh view on the problem of identifying topical
structures by relating it to the problem of finding communities in complex
networks. This is achieved by representing text corpora as bipartite networks
of documents and words. By adapting existing community-detection methods --
using a stochastic block model (SBM) with non-parametric priors -- we obtain a
more versatile and principled framework for topic modeling (e.g., it
automatically detects the number of topics and hierarchically clusters both the
words and documents). The analysis of artificial and real corpora demonstrates
that our SBM approach leads to better topic models than LDA in terms of
statistical model selection. More importantly, our work shows how to formally
relate methods from community detection and topic modeling, opening the
possibility of cross-fertilization between these two fields.Comment: 22 pages, 10 figures, code available at https://topsbm.github.io
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