Computing Optimal Kernels in Two Dimensions

07/14/2022

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by   Pankaj K. Agarwal, et al.

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Let P be a set of n points in ℝ^2. A subset C⊆ P is an ε-kernel of P if the projection of the convex hull of C approximates that of P within (1-ε)-factor in every direction. The set C is a weak ε-kernel if its directional width approximates that of P in every direction. We present fast algorithms for computing a minimum-size ε-kernel as well as a weak ε-kernel. We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of ε-core, a convex polygon lying inside CH(P), prove that it is a good approximation of the optimal ε-kernel, present an efficient algorithm for computing it, and use it to compute an ε-kernel of small size.