2022/07/14 by Agarwal, Pankaj K., Har-Peled, Sariel
#Computational Geometry (cs.CG) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2207.07211
Let P be a set of n points in \Re2. For a parameter ε∈ (0,1), 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 of P if its directional width approximates that of P in every direction. Let kε(P) (resp. kwε(P)) denote the minimum-size of an ε-kernel (resp. weak ε-kernel) of P. We present an O(nkε(P)log n)-time algorithm for computing an ε-kernel of P of size kε(P), and an O(n2log n)-time algorithm for computing a weak ε-kernel of P of size kwε(P). 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.