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Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums

Authors Sunil Arya, Guilherme D. da Fonseca, David M. Mount



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Sunil Arya
  • Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Guilherme D. da Fonseca
  • Université Clermont Auvergne, LIMOS, and INRIA Sophia Antipolis, France
David M. Mount
  • Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA

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Sunil Arya, Guilherme D. da Fonseca, and David M. Mount. Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 3:1-3:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ESA.2018.3

Abstract

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 <= 3 and/or do not consider approximation. In this paper, we consider approximations to two natural problems involving multiple convex bodies: detecting whether two polytopes intersect and computing their Minkowski sum. Given an approximation parameter epsilon > 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Minkowski sum
  • convex intersection
  • width
  • approximation

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