72,188 research outputs found
Floer cohomology of torus fibers and real lagrangians in Fano toric manifolds
In this article, we consider the Floer cohomology (with coefficients)
between torus fibers and the real Lagrangian in Fano toric manifolds. We first
investigate the conditions under which the Floer cohomology is defined, and
then develop a combinatorial description of the Floer complex based on the
polytope of the toric manifold. We show that if the Floer cohomology is
defined, and the Floer cohomology of the torus fiber is non-zero, then the
Floer cohomology of the pair is non-zero. We use this result to develop some
applications to non-displaceability and the minimum number of intersection
points under Hamiltonian isotopy.Comment: v2: Modified exposition and new corollary adde
Two-Level Rectilinear Steiner Trees
Given a set of terminals in the plane and a partition of into
subsets , a two-level rectilinear Steiner tree consists of a
rectilinear Steiner tree connecting the terminals in each set
() and a top-level tree connecting the trees . The goal is to minimize the total length of all trees. This problem
arises naturally in the design of low-power physical implementations of parity
functions on a computer chip.
For bounded we present a polynomial time approximation scheme (PTAS) that
is based on Arora's PTAS for rectilinear Steiner trees after lifting each
partition into an extra dimension. For the general case we propose an algorithm
that predetermines a connection point for each and
().
Then, we apply any approximation algorithm for minimum rectilinear Steiner
trees in the plane to compute each and independently.
This gives us a -factor approximation with a running time of
suitable for fast practical computations. The
approximation factor reduces to by applying Arora's approximation scheme
in the plane
Hyperorthogonal well-folded Hilbert curves
R-trees can be used to store and query sets of point data in two or more
dimensions. An easy way to construct and maintain R-trees for two-dimensional
points, due to Kamel and Faloutsos, is to keep the points in the order in which
they appear along the Hilbert curve. The R-tree will then store bounding boxes
of points along contiguous sections of the curve, and the efficiency of the
R-tree depends on the size of the bounding boxes---smaller is better. Since
there are many different ways to generalize the Hilbert curve to higher
dimensions, this raises the question which generalization results in the
smallest bounding boxes. Familiar methods, such as the one by Butz, can result
in curve sections whose bounding boxes are a factor larger
than the volume traversed by that section of the curve. Most of the volume
bounded by such bounding boxes would not contain any data points. In this paper
we present a new way of generalizing Hilbert's curve to higher dimensions,
which results in much tighter bounding boxes: they have at most 4 times the
volume of the part of the curve covered, independent of the number of
dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.Comment: Manuscript submitted to Journal of Computational Geometry. An
abstract appeared in the 31st Int Symp on Computational Geometry (SoCG 2015),
LIPIcs 34:812-82
Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions
Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processin
Polychromatic Coloring for Half-Planes
We prove that for every integer , every finite set of points in the plane
can be -colored so that every half-plane that contains at least
points, also contains at least one point from every color class. We also show
that the bound is best possible. This improves the best previously known
lower and upper bounds of and respectively. We also show
that every finite set of half-planes can be colored so that if a point
belongs to a subset of at least of the half-planes then
contains a half-plane from every color class. This improves the best previously
known upper bound of . Another corollary of our first result is a new
proof of the existence of small size \eps-nets for points in the plane with
respect to half-planes.Comment: 11 pages, 5 figure
Centroidal localization game
One important problem in a network is to locate an (invisible) moving entity
by using distance-detectors placed at strategical locations. For instance, the
metric dimension of a graph is the minimum number of detectors placed
in some vertices such that the vector
of the distances between the detectors and the entity's location
allows to uniquely determine . In a more realistic setting, instead
of getting the exact distance information, given devices placed in
, we get only relative distances between the entity's
location and the devices (for every , it is provided
whether , , or to ). The centroidal dimension of a
graph is the minimum number of devices required to locate the entity in
this setting.
We consider the natural generalization of the latter problem, where vertices
may be probed sequentially until the moving entity is located. At every turn, a
set of vertices is probed and then the relative distances
between the vertices and the current location of the entity are
given. If not located, the moving entity may move along one edge. Let be the minimum such that the entity is eventually located, whatever it
does, in the graph .
We prove that for every tree and give an upper bound
on in cartesian product of graphs and . Our main
result is that for any outerplanar graph . We then prove
that is bounded by the pathwidth of plus 1 and that the
optimization problem of determining is NP-hard in general graphs.
Finally, we show that approximating (up to any constant distance) the entity's
location in the Euclidean plane requires at most two vertices per turn
- …