On computing the diameter of a point set in high dimensional Euclidean space

We consider the problem of computing the diameter of a set of $n$ points in $d$-dimensional Euclidean space under Euclidean distance function. We describe an algorithm that in time $O(dnlog n +n^{2})$ finds with high probability an arbitrarily close approximation of the diameter. For large values of $d$ the complexity bound of our algorithm is a substantial improvement over the complexity bounds of previously known exact algorithms. Computing and approximating the diameter are fundamental primitives in high dimensional computational geometry and find practical application, for example, in clustering operations for image databases

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

Differentially Private Approximations of a Convex Hull in Low Dimensions

We give the first differentially private algorithms that estimate a variety of geometric features of points in the Euclidean space, such as diameter, width, volume of convex hull, min-bounding box, min-enclosing ball, etc. Our work relies heavily on the notion of Tukey-depth. Instead of (non-privately) approximating the convex-hull of the given set of points P, our algorithms approximate the geometric features of D_{P}(?) - the ?-Tukey region induced by P (all points of Tukey-depth ? or greater). Moreover, our approximations are all bi-criteria: for any geometric feature ? our (?,?)-approximation is a value "sandwiched" between (1-?)?(D_P(?)) and (1+?)?(D_P(?-?)). Our work is aimed at producing a (?,?)-kernel of D_P(?), namely a set ? such that (after a shift) it holds that (1-?)D_P(?) ? CH(?) ? (1+?)D_P(?-?). We show that an analogous notion of a bi-critera approximation of a directional kernel, as originally proposed by [Pankaj K. Agarwal et al., 2004], fails to give a kernel, and so we result to subtler notions of approximations of projections that do yield a kernel. First, we give differentially private algorithms that find (?,?)-kernels for a "fat" Tukey-region. Then, based on a private approximation of the min-bounding box, we find a transformation that does turn D_P(?) into a "fat" region but only if its volume is proportional to the volume of D_P(?-?). Lastly, we give a novel private algorithm that finds a depth parameter ? for which the volume of D_P(?) is comparable to the volume of D_P(?-?). We hope our work leads to the further study of the intersection of differential privacy and computational geometry

On the expected diameter, width, and complexity of a stochastic convex-hull

We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of $n$ points in $\mathbb{R}^d$ each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both $n$ and $d$. For width, two approximation algorithms are provided: a deterministic $O(1)$-approximation running in $O(n^{d+1} \log n)$ time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact $O(n^d)$-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest