Abstract
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 I similar to(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 I similar to(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 (2)log n) algorithm for the case where h outliers are allowed. The running time of all our algorithms is independent of k.