Weighted closest pairs

Abstract

In this paper we study the following scaling problem: Given a set of planar starshaped objects with centerpoints (in the kernel), determine the maximal scaling factor δmax, such that the objects scaled by δmax about their centerpoints are pairwise disjoint. We describe a method to compute the maximal scaling factor for n disks with different radii in optimal O(n log n) time. In this case the problem can be viewed as computing the closest pair of a set of weighted points. We indicate how to extend the method to a broader class of objects, including disks generated by Lp-norms (1 ≤ p ≤ ∞). A different approach, using the parametric search technique is taken to solve the scaling problem for an even wider class, namely starshaped, x-monotone objects. This method runs in O(n log2 n) time. As a corollary of this result we can compute the maximal scaling factor of a set of starshaped polygons (not necessarily z-monotone) with a total number of n edges in O(n log2 n) time.