In this paper, we present simple randomized multi-pass streaming algorithms for fundamental computational geometry problems of finding the skyline (maximal) points and the extreme points of the convex hull. For the skyline problem, one of our algorithm occupies O(h) space and performs O(log n) passes, where h is the number of skyline points. This improves the space bound of the currently best known result by Das Sarma, Lall, Nanongkai, and Xu [VLDB'09] by a logarithmic factor. For the extreme points problem, we present the first non-trivial result for any constant dimension greater than two: an O(h log^{O(1)}n) space and O(log^d n) pass algorithm, where h is the number of extreme points. Finally, we argue why randomization seems unavoidable for these problems, by proving lower bounds on the performance of deterministic algorithms for a related problem of finding maximal elements in a poset.
Introduced by Agarwal, Har-Peled, and Varadarajan (2004), an epsilon-kernel of a point set is a coreset that can be used to approximate the width, minimum enclosing cylinder, minimum bounding box, and solve various related geometric optimization problems. Such coresets form one of the most important tools in the design of linear-time approximation algorithms in computational geometry, as well as efficient insertion-only streaming algorithms and dynamic (non-streaming) data structures. In this paper, we continue the theme and explore dynamic streaming algorithms (in the so-called turnstile model).
Andoni and Nguyen [SODA'12] described a dynamic streaming algorithm for maintaining a (1+epsilon)-approximation of the width using O(polylog U) space and update time for a point set in [U]^d for any constant dimension d and any constant epsilon > 0. Their sketch, based on a polynomial method, does not explicitly maintain an epsilon-kernel. We extend their method to maintain an epsilon-kernel, and at the same time reduce some of logarithmic factors. As an application, we obtain the first randomized dynamic streaming algorithm for the width problem (and related geometric optimization problems) that supports k outliers, using poly(k, log U) space and time.
At SODA'10, Agarwal and Sharathkumar presented a streaming algorithm for approximating the minimum enclosing ball of a set of points in d-dimensional Euclidean space. Their algorithm requires one pass, uses O(d) space, and was shown to have approximation factor at most (1+sqrt{3})/2 + eps ~ 1.3661. We prove that the same algorithm has approximation factor less than 1.22, which brings us much closer to a (1+sqrt{2})/2 ~ 1.207 lower bound given by Agarwal and Sharathkumar.
We also apply this technique to the dynamic version of the minimum enclosing ball problem (in the non-streaming setting). We give an O(dn)-space data structure that can maintain a 1.22-approximate minimum enclosing ball in O(d log n) expected amortized time per insertion/deletion.
We analyze an extremely simple approximation algorithm for computing the minimum enclosing ball (or the 1-center) of a set of points in high dimensions. We prove that this algorithm computes a 3/2-factor approximation in any dimension using minimum space in just one pass over the data points.
We initiate the study of exact geometric algorithms that require limited storage and make only a small number of passes over the input. Fundamental problems such as low-dimensional linear programming and convex hulls are considered.
We study the problem of maintaining a (1+epsilon)-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/epsilon) 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.
We speed up previous (1+epsilon)-factor approximation algorithms for a number of geometric optimization problems in fixed dimensions: diameter, width, minimum-radius enclosing cylinder, minimum-width annulus, minimum-volume bounding box, minimum-width cylindrical shell, etc. Linear time bounds were known before; we further improve the dependence of the "constants" in terms of epsilon.
We next consider the data stream model and present new (1+epsilon)-factor approximation algorithms that need only constant space for all of the above problems in any fixed dimension. Previously, such a result was known only for diameter.
Both sets of results are obtained using the core-set framework recently proposed by Agarwal, Har-Peled, and Varadarajan.
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