On the Combinatorial Complexity of Approximating Polytopes

Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body $K$ of diameter $\mathrm{diam}(K)$ is given in Euclidean $d$-dimensional space, where $d$ is a constant. Given an error parameter $\varepsilon > 0$, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from $K$ is at most $\varepsilon \cdot \mathrm{diam}(K)$. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that $O(1/\varepsilon^{(d-1)/2})$ facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is $\tilde{O}(1/\varepsilon^{(d-1)/2})$, where $\tilde{O}$ conceals a polylogarithmic factor in $1/\varepsilon$. This is a significant improvement upon the best known bound, which is roughly $O(1/\varepsilon^{d-2})$. Our result is based on a novel combination of both old and new ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of B\'{a}r\'{a}ny and Larman's economical cap covering. Finally, we use a deterministic adaptation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.Comment: In Proceedings of the 32nd International Symposium Computational Geometry (SoCG 2016) and accepted to SoCG 2016 special issue of Discrete and Computational Geometr

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

Optimal Area-Sensitive Bounds for Polytope Approximation

Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Given a convex body $K$ of diameter $\Delta$ in $\mathbb{R}^d$ for fixed $d$, the objective is to minimize the number of vertices (alternatively, the number of facets) of an approximating polytope for a given Hausdorff error $\varepsilon$. The best known uniform bound, due to Dudley (1974), shows that $O((\Delta/\varepsilon)^{(d-1)/2})$ facets suffice. While this bound is optimal in the case of a Euclidean ball, it is far from optimal for ``skinny'' convex bodies. A natural way to characterize a convex object's skinniness is in terms of its relationship to the Euclidean ball. Given a convex body $K$, define its surface diameter $\Delta_{d-1}$ to be the diameter of a Euclidean ball of the same surface area as $K$. It follows from generalizations of the isoperimetric inequality that $\Delta \geq \Delta_{d-1}$. We show that, under the assumption that the width of the body in any direction is at least $\varepsilon$, it is possible to approximate a convex body using $O((\Delta_{d-1}/\varepsilon)^{(d-1)/2})$ facets. This bound is never worse than the previous bound and may be significantly better for skinny bodies. The bound is tight, in the sense that for any value of $\Delta_{d-1}$, there exist convex bodies that, up to constant factors, require this many facets. The improvement arises from a novel approach to sampling points on the boundary of a convex body. We employ a classical concept from convexity, called Macbeath regions. We demonstrate that Macbeath regions in $K$ and $K$'s polar behave much like polar pairs. We then apply known results on the Mahler volume to bound their number

Approximate Nearest Neighbor Searching with Non-Euclidean and Weighted Distances

We present a new approach to approximate nearest-neighbor queries in fixed dimension under a variety of non-Euclidean distances. We are given a set $S$ of $n$ points in $\mathbb{R}^d$, an approximation parameter $\varepsilon > 0$, and a distance function that satisfies certain smoothness and growth-rate assumptions. The objective is to preprocess $S$ into a data structure so that for any query point $q$ in $\mathbb{R}^d$, it is possible to efficiently report any point of $S$ whose distance from $q$ is within a factor of $1+\varepsilon$ of the actual closest point. Prior to this work, the most efficient data structures for approximate nearest-neighbor searching in spaces of constant dimensionality applied only to the Euclidean metric. This paper overcomes this limitation through a method called convexification. For admissible distance functions, the proposed data structures answer queries in logarithmic time using $O(n \log (1 / \varepsilon) / \varepsilon^{d/2})$ space, nearly matching the best known bounds for the Euclidean metric. These results apply to both convex scaling distance functions (including the Mahalanobis distance and weighted Minkowski metrics) and Bregman divergences (including the Kullback-Leibler divergence and the Itakura-Saito distance)

Avaliação do conteúdo de carotenoides totais e compostos cianogênicos em híbridos de mandioca das gerações 2007 e 2008.

De origem brasileira, a cultura da mandioca (Manihot esculenta Crantz.) apresenta grande importância socioeconômica para o Brasil e para o mundo (FOLEGATTI et al., 2005), sendo base energética para mais de 700 milhões de pessoas, em vários países tropicais e subtropicais (MARCON et al., 2007). No Brasil, o consumo in natura é responsável pela alimentação das populações mais carentes, sendo o teor de compostos cianogênicos contido nas raízes um dos fatores que definem a finalidade de uso da mandioca (PONTE, 2008). É com base na concentração de compostos cianogênicos, que as variedades são classificadas em 'mansas' e 'bravas'. As mandiocas mansas, destinadas ao consumo fresco, apresentam menos de 100 ppm de ácido cianídrico na polpa crua das raízes e são denominadas como 'mandioca de mesa', 'macaxeira', 'aipim' ou 'mandioca doce'. As bravas apresentam mais de 100 ppm de ácido cianídrico na polpa crua das raízes e devem ser processadas antes do consumo, sendo designadas como 'mandioca amarga' e destinam-se à industrialização (PAZINATO et al., 2003).Melhoramento genético. Resumo n. 228