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Applications of Chebyshev Polynomials to Low-Dimensional Computational Geometry

Author Timothy M. Chan

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Timothy M. Chan

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Timothy M. Chan. Applications of Chebyshev Polynomials to Low-Dimensional Computational Geometry. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.SoCG.2017.26

Abstract

We apply the polynomial method - specifically, Chebyshev polynomials - to obtain a number of new results on geometric approximation algorithms in low constant dimensions. For example, we give an algorithm for constructing epsilon-kernels (coresets for approximate width and approximate convex hull) in close to optimal time O(n + (1/epsilon)^{(d-1)/2}), up to a small near-(1/epsilon)^{3/2} factor, for any d-dimensional n-point set.  We obtain an improved data structure for Euclidean *approximate nearest neighbor search* with close to O(n log n + (1/epsilon)^{d/4} n) preprocessing time and O((1/epsilon)^{d/4} log n) query time.  We obtain improved approximation algorithms for discrete Voronoi diagrams, diameter, and bichromatic closest pair in the L_s-metric for any even integer constant s >= 2. The techniques are general and may have further applications.

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Keywords
  • diameter
  • coresets
  • approximate nearest neighbor search
  • the polynomial method
  • streaming

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