8,947 research outputs found
Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space
We present a logspace algorithm that constructs a canonical intersection
model for a given proper circular-arc graph, where `canonical' means that
models of isomorphic graphs are equal. This implies that the recognition and
the isomorphism problems for this class of graphs are solvable in logspace. For
a broader class of concave-round graphs, that still possess (not necessarily
proper) circular-arc models, we show that those can also be constructed
canonically in logspace. As a building block for these results, we show how to
compute canonical models of circular-arc hypergraphs in logspace, which are
also known as matrices with the circular-ones property. Finally, we consider
the search version of the Star System Problem that consists in reconstructing a
graph from its closed neighborhood hypergraph. We solve it in logspace for the
classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio
Learning Latent Tree Graphical Models
We study the problem of learning a latent tree graphical model where samples
are available only from a subset of variables. We propose two consistent and
computationally efficient algorithms for learning minimal latent trees, that
is, trees without any redundant hidden nodes. Unlike many existing methods, the
observed nodes (or variables) are not constrained to be leaf nodes. Our first
algorithm, recursive grouping, builds the latent tree recursively by
identifying sibling groups using so-called information distances. One of the
main contributions of this work is our second algorithm, which we refer to as
CLGrouping. CLGrouping starts with a pre-processing procedure in which a tree
over the observed variables is constructed. This global step groups the
observed nodes that are likely to be close to each other in the true latent
tree, thereby guiding subsequent recursive grouping (or equivalent procedures)
on much smaller subsets of variables. This results in more accurate and
efficient learning of latent trees. We also present regularized versions of our
algorithms that learn latent tree approximations of arbitrary distributions. We
compare the proposed algorithms to other methods by performing extensive
numerical experiments on various latent tree graphical models such as hidden
Markov models and star graphs. In addition, we demonstrate the applicability of
our methods on real-world datasets by modeling the dependency structure of
monthly stock returns in the S&P index and of the words in the 20 newsgroups
dataset
Experimental demonstration of a directionally-unbiased linear-optical multiport
All existing optical quantum walk approaches are based on the use of
beamsplitters and multiple paths to explore the multitude of unitary
transformations of quantum amplitudes in a Hilbert space. The beamsplitter is
naturally a directionally biased device: the photon cannot travel in reverse
direction. This causes rapid increases in optical hardware resources required
for complex quantum walk applications, since the number of options for the
walking particle grows with each step. Here we present the experimental
demonstration of a directionally-unbiased linear-optical multiport, which
allows reversibility of photon direction. An amplitude-controllable probability
distribution matrix for a unitary three-edge vertex is reconstructed with only
linear-optical devices. Such directionally-unbiased multiports allow direct
execution of quantum walks over a multitude of complex graphs and in tensor
networks. This approach would enable simulation of complex Hamiltonians of
physical systems and quantum walk applications in a more efficient and compact
setup, substantially reducing the required hardware resources
Reconstructing polygons from scanner data
A range-finding scanner can collect information about the shape of an (unknown) polygonal room in which it is placed. Suppose that a set of scanners returns not only a set of points, but also additional information, such as the normal to the plane when a scan beam detects a wall. We consider the problem of reconstructing the floor plan of a room from different types of scan data. In particular, we present algorithmic and hardness results for reconstructing two-dimensional polygons from point-wall pairs, point-normal pairs, and visibility polygons. The polygons may have restrictions on topology (e.g., to be simply connected) or geometry (e.g., to be orthogonal). We show that this reconstruction problem is NP-hard under most models, but that some restrictive assumptions do allow polynomial-time reconstruction algorithms
Nonlinear dynamics on branched structures and networks
Nonlinear dynamics on graphs has rapidly become a topical issue with many
physical applications, ranging from nonlinear optics to Bose-Einstein
condensation. Whenever in a physical experiment a ramified structure is
involved, it can prove useful to approximate such a structure by a metric
graph, or network. For the Schroedinger equation it turns out that the sixth
power in the nonlinear term of the energy is critical in the sense that below
that power the constrained energy is lower bounded irrespectively of the value
of the mass (subcritical case). On the other hand, if the nonlinearity power
equals six, then the lower boundedness depends on the value of the mass: below
a critical mass, the constrained energy is lower bounded, beyond it, it is not.
For powers larger than six the constrained energy functional is never lower
bounded, so that it is meaningless to speak about ground states (supercritical
case). These results are the same as in the case of the nonlinear Schrodinger
equation on the real line. In fact, as regards the existence of ground states,
the results for systems on graphs differ, in general, from the ones for systems
on the line even in the subcritical case: in the latter case, whenever the
constrained energy is lower bounded there always exist ground states (the
solitons, whose shape is explicitly known), whereas for graphs the existence of
a ground state is not guaranteed. For the critical case, our results show a
phenomenology much richer than the analogous on the line.Comment: 47 pages, 44 figure. Lecture notes for a course given at the Summer
School "MMKT 2016, Methods and Models of Kinetic Theory, Porto Ercole, June
5-11, 2016. To be published in Riv. Mat. Univ. Parm
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