Abstract. The paradigm of coresets has recently emerged as a powerful tool for efficiently approximating various extent measures of a point set P. Using this paradigm, one quickly computes a small subset Q of P, called a coreset, that approximates the original set P and and then solves the problem on Q using a relatively inefficient algorithm. The solution for Q is then translated to an approximate solution to the original point set P. This paper describes the ways in which this paradigm has been successfully applied to various optimization and extent measure problems. 1.

The notion of ε-kernel was introduced by Agarwal et al. [5] to set up a unified framework for computing various extent measures of a point set P approximately. Roughly speaking, a subset Q ⊆ P is an ε-kernel of P if for every slab W containing Q, the expanded slab (1 + ε)W contains P. They illustrated the significance of ε-kernel by showing that it yields approximation algorithms for a wide range of geometric optimization problems. We present a simpler and more practical algorithm for computing the ε-kernel of a set P of points in R d. We demonstrate the practicality of our algorithm by showing its empirical performance on various inputs. We then describe an incremental algorithm for fitting various shapes and use the ideas of our algorithm for computing ε-kernels to analyze the performance of this algorithm. We illustrate the versatility and practicality of this technique by implementing approximation algorithms for minimum enclosing cylinder, minimum-volume bounding box, and minimum-width annulus. Finally, we show that ε-kernels can be effectively used to expedite the algorithms for maintaining extents of moving points. 1

We study the problem of computing a (1 + #)-approximation to the minimum volume enclosing ellipsoid of a given point set , p . Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's first-order algorithm. Our analysis leads to a slightly improved complexity bound of O(nd (0, 1). As a byproduct, our algorithm returns a core set with the property that the minimum volume enclosing ellipsoid of provides a good approximation to that of S.

Abstract. We study the problem of maintaining a (1+ffl)-factor approximation of the diameter of a stream of points under the sliding window model. In one dimension, we give a simple algorithm that only needs to store O( 1 ffl log R) points at any time, where the parameter R denotes the "spread " of the point set. This bound is optimal and improves Feigenbaum, Kannan, and Zhang's recent solution by two logarithmic factors. We then extend our one-dimensional algorithm to higher constant dimensions and, at the same time, correct an error in the previous solution. In high nonconstant dimensions, we also observe a constant-factor approximation algorithm that requires sublinear space. Related optimization problems, such as the width, are also considered in the two-dimensional case.

Given a point set P ⊆ R², a subset Q ⊆ P is an ε-kernel of P if for every slab W containing Q, the (1+ε)-expansion of W also contains P. We present a data-stream algorithm for maintaining an ε-kernel of a stream of points in R² that uses O(1/√ε) space and takes O(log(1/ε)) amortized time to process each point. This is the first space-optimal data-stream algorithm for this problem. As a consequence, we obtain improved data-stream approximation algorithms for other extent measures, such as width, robust kernels, as well as ε-kernels in higher dimensions.

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We study the problem of computing a (1+ɛ)-approximation to the minimum volume covering ellipsoid of a given set S of the convex hull of m full-dimensional ellipsoids in R n. We extend the first-order algorithm of Kumar and Yıldırım that computes an approximation to the minimum volume covering ellipsoid of a finite set of points in R n, which, in turn, is a modification of Khachiyan’s algorithm. For fixed ɛ> 0, we establish a polynomial-time complexity, which is linear in the number of ellipsoids m. In particular, the iteration complexity of our algorithm is identical to that for a set of m points. The main ingredient in our analysis is the extension of polynomialtime complexity of certain subroutines in the algorithm from a set of points to a set of ellipsoids. As a byproduct, our algorithm returns a finite “core ” set X ⊆ S with the property that the minimum volume covering ellipsoid of X provides a good approximation to that of S. Furthermore, the size of X depends only on the dimension n and ɛ, but not on the number of ellipsoids m. We also discuss the extent to which our algorithm can be used to compute the minimum volume covering ellipsoid of the convex hull of other sets in R n. We adopt the real number model of computation in our analysis.

Let P be a set of n points in R d. A subset S of P is called a (k, ε)-kernel if for every direction, the direction width of S ε-approximates that of P, when k “outliers ” can be ignored in that direction. We show that a (k, ε)-kernel of P of size O(k/ε (d−1)/2) can be computed in time O(n+k 2 /ε d−1). The new algorithm works by repeatedly “peeling” away (0, ε)-kernels from the point set. We also present a simple ε-approximation algorithm for fitting various shapes through a set of points with at most k outliers. The algorithm is incremental and works by repeatedly “grating ” critical points into a working set, till the working set provides the required approximation. We prove that the size of the working set is independent of n, and thus results in a simple and practical, near-linear ε-approximation algorithm for shape fitting with outliers in low dimensions. We demonstrate the practicality of our algorithms by showing their empirical performance on various inputs and problems. 1

Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in P, and the smallest such set S ′ that contains P. More precisely, for any ε> 0, we find an axially symmetric convex polygon Q ⊂ C with area |Q |> (1 − ε)|S | and we find an axially symmetric convex polygon Q ′ containing C with area |Q ′ | < (1 + ε)|S ′ |. We assume that C is given in a data structure that allows to answer the following two types of query in time TC: given a direction u, find an extreme point of C in direction u, and given a line ℓ, find C ∩ ℓ. For instance, if C is a convex n-gon and its vertices are given in a sorted array, then TC = O(log n). Then we can find Q in time O(TCε −1/2 + ε −3/2) and we can find Q ′ in time O(TCε −1/2 + ε −3/2 log(ε −1)). Using these techniques, we can also find approximations to the perimeter, area, diameter, width, smallest enclosing restangle and smallest enclosing circle of C in time O(TCε −1/2). 1

We present a general framework for computing two-dimensional Voronoi diagrams of different site classes under various distance functions. The computation of the diagrams employs the Cgal software for constructing envelopes of surfaces in 3-space, which implements a divide-and-conquer algorithm. A straightforward application of the divide-andconquer approach for Voronoi diagrams yields highly inefficient algorithms. We show that through randomization, the expected running time is near-optimal (in a worst-case sense). We believe this result, which also holds for general envelopes, to be of independent interest. We describe the interface between the construction of the diagrams and the underlying construction of the envelopes, together with methods we have applied to speed up the (exact) computation. We then present results, where a variety of diagrams are constructed with our implementation, including power diagrams, Apollonius diagrams, diagrams of line segments, Voronoi diagrams on a sphere, and more. In all cases the implementation is exact and can handle degenerate input.