Computing Instance-Optimal Kernels in Two Dimensions

Let P be a set of n points in R2. For a parameter (Equation presented), a subset C Ď P is an ε-kernel of P if the projection of the convex hull of C approximates that of P within (Equation presented)-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 (Equation presented) (resp. (Equation presented)) denote the minimum-size of an ε-kernel (resp. weak ε-kernel) of P. We present an (Equation presented)-time algorithm for computing an ε-kernel of P of size (Equation presented), and an (Equation presented)-time algorithm for computing a weak ε-kernel of P of size (Equation presented). 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.

Duke Scholars

Author Pankaj K. Agarwal Computer Science