Linear-time fitting of a k-step function

Abstract

Given a set of n weighted points on the x–y plane, we want to find a step function consisting of k steps parallel to the x axis such that the maximum weighted vertical distance from any point to a step is minimized. Using the prune and search technique, we solve this problem in O(2k2n) time, thus in linear time in n when k is a constant. Our approach can be applied directly or with small modifications to solve other related problems, such as the minmax error histogram problem and the line-constrained k-center problem, in O(n) time when k is a constant.