eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
26:1
26:15
10.4230/LIPIcs.SoCG.2017.26
article
Applications of Chebyshev Polynomials to Low-Dimensional Computational Geometry
Chan, Timothy M.
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.26/LIPIcs.SoCG.2017.26.pdf
diameter
coresets
approximate nearest neighbor search
the polynomial method
streaming