Applications of Chebyshev Polynomials to Low-Dimensional Computational Geometry

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 ε-kernels (coresets for approximate width and approximate convex hull) in close to optimal time O(n + (1/ε)(d−1)/2), up to a small near-(1/ε)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 logn + (1/ε)d/4n) preprocessing time and O((1/ε)d/4 logn) query time. We obtain improved approximation algorithms for discrete Voronoi diagrams, diameter, and bichromatic closest pair in the Ls-metric for any even integer constant s ≥ 2. The techniques are general and may have further applications. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems