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

TL;DR

This paper introduces an optimal deterministic algorithm that efficiently computes a k-step function fitting a weighted point set, minimizing maximum weighted vertical distance, matching randomized algorithm performance.

Contribution

It presents the first deterministic O(n log n) algorithm for fitting a step function to weighted points, improving reliability over randomized methods.

Findings

Achieves optimal O(n log n) runtime.

Matches the performance of the best randomized algorithms.

Uses Cole's parametric searching technique effectively.

Abstract

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(n \log n)-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 Cole's improved parametric searching technique.