Fitting a step function to a point set

We consider the problem of fitting a step function to a set of points. More precisely, given an integer k and a set P of n points in the plane, our goal is to find a step function f with k steps that minimizes the maximum vertical distance between f and all the points in P. We first give an optimal Θ(nlog∈n) algorithm for the general case. In the special case where the points in P are given in sorted order according to their x-coordinates, we give an optimal Θ(n) time algorithm. Then, we show how to solve the weighted version of this problem in time O(nlog∈ ⁴ n). Finally, we give an O(nh ²log∈n) algorithm for the case where h outliers are allowed. The running time of all our algorithms is independent of k.