Linear-Time Fitting of a $k$-Step Function

TL;DR

This paper presents a linear-time algorithm for fitting a $k$-step function to weighted points, minimizing the maximum vertical distance, using prune-and-search, with applications to related problems.

Contribution

It introduces a prune-and-search based linear-time algorithm for $k$-step function fitting, applicable to multiple geometric optimization problems.

Findings

Algorithm runs in O(n) time for fixed k

Applicable to line-constrained k-center problem

Effective for size-k histogram construction

Abstract

Given a set of $n$ weighted points on the $x$-$y$ plane, we want to find a step function consisting of $k$ horizontal steps such that the maximum vertical weighted distance from any point to a step is minimized. We solve this problem in $O(n)$ time when $k$ is a constant. Our approach relies on the prune-and-search technique, and can be adapted to design similar linear time algorithms to solve the line-constrained k-center problem and the size-$k$ histogram construction problem as well.