11 research outputs found
Radii minimal projections of polytopes and constrained optimization of symmetric polynomials
We provide a characterization of the radii minimal projections of polytopes
onto -dimensional subspaces in Euclidean space \E^n. Applied on simplices
this characterization allows to reduce the computation of an outer radius to a
computation in the circumscribing case or to the computation of an outer radius
of a lower-dimensional simplex. In the second part of the paper, we use this
characterization to determine the sequence of outer -radii of regular
simplices (which are the radii of smallest enclosing cylinders). This settles a
question which arose from the incidence that a paper by Wei{\ss}bach (1983) on
this determination was erroneous. In the proof, we first reduce the problem to
a constrained optimization problem of symmetric polynomials and then to an
optimization problem in a fixed number of variables with additional integer
constraints.Comment: Minor revisions. To appear in Advances in Geometr
Lines Tangent to 2n-2 spheres in R^n
We show that there are 3 \cdot 2^(n-1) complex common tangent lines to 2n-2
general spheres in R^n and that there is a choice of spheres with all common
tangents real.Comment: Minor revisions. Trans. AMer. Math. Soc., to appear. 15 pages, 3 .eps
figures; also a web page with computer code verifying the computations in the
paper and with additional picture
Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums
Approximation problems involving a single convex body in R^d have received a great deal of attention in the computational geometry community. In contrast, works involving multiple convex bodies are generally limited to dimensions d 0, we show how to independently preprocess two polytopes A,B subset R^d into data structures of size O(1/epsilon^{(d-1)/2}) such that we can answer in polylogarithmic time whether A and B intersect approximately. More generally, we can answer this for the images of A and B under affine transformations. Next, we show how to epsilon-approximate the Minkowski sum of two given polytopes defined as the intersection of n halfspaces in O(n log(1/epsilon) + 1/epsilon^{(d-1)/2 + alpha}) time, for any constant alpha > 0. Finally, we present a surprising impact of these results to a well studied problem that considers a single convex body. We show how to epsilon-approximate the width of a set of n points in O(n log(1/epsilon) + 1/epsilon^{(d-1)/2 + alpha}) time, for any constant alpha > 0, a major improvement over the previous bound of roughly O(n + 1/epsilon^{d-1}) time
Minimizing the error of linear separators on linearly inseparable data
Given linearly inseparable sets R of red points and B of blue points, we consider several
measures of how far they are from being separable. Intuitively, given a potential separator
(‘‘classifier’’), we measure its quality (‘‘error’’) according to how much work it would take
to move the misclassified points across the classifier to yield separated sets. We consider
several measures of work and provide algorithms to find linear classifiers that minimize
the error under these different measures.Ministerio de Educación y Ciencia MTM2008-05866-C03-0
How to Cover a Point Set with a V-Shape of Minimum Width
A balanced V-shape is a polygonal region in the plane contained in the union
of two crossing equal-width strips. It is delimited by two pairs of parallel
rays that emanate from two points x, y, are contained in the strip boundaries,
and are mirror-symmetric with respect to the line xy. The width of a balanced
V-shape is the width of the strips. We first present an O(n^2 log n) time
algorithm to compute, given a set of n points P, a minimum-width balanced
V-shape covering P. We then describe a PTAS for computing a
(1+epsilon)-approximation of this V-shape in time O((n/epsilon)log
n+(n/epsilon^(3/2))log^2(1/epsilon)). A much simpler constant-factor
approximation algorithm is also described.Comment: In Proceedings of the 12th International Symposium on Algorithms and
Data Structures (WADS), p.61-72, August 2011, New York, NY, US
Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms
We present a technical survey on the state of the art approaches in data reduction and the coreset framework. These include geometric decompositions, gradient methods, random sampling, sketching and random projections. We further outline their importance for the design of streaming algorithms and give a brief overview on lower bounding techniques