A deterministic algorithm for fitting a step function to a weighted point-set

Given a set of n points in the plane, each point having a positive weight, and an integer k>0, we present an optimal O(nlogn)-time deterministic algorithm to compute a step function with k steps that minimizes the maximum weighted vertical distance to the input points. It matches the expected time bound of the best known randomized algorithm for this problem. Our approach relies on Coles improved parametric searching technique. As a direct application, our result yields the first O(nlogn)-time algorithm for computing a k-center of a set of n weighted points on the real line. © 2012 Elsevier B.V.