An Algorithm for $L_\infty$ Approximation by Step Functions

TL;DR

This paper presents an efficient algorithm for optimal $L_$ approximation of weighted data using step functions, also applicable to isotonic regression and the 1D $k$-center problem, with proven time and space bounds.

Contribution

It introduces a novel algorithm that computes optimal $b$-step $L_$ approximation, reduced isotonic regression, and solves the 1D $k$-center problem efficiently.

Findings

Algorithm runs in $ heta(n + ext{log} n imes b(1+ ext{log} n/b))$ time.

Achieves $ heta(n)$ space complexity.

Effective for data presorted by the independent variable.

Abstract

An algorithm is given for determining an optimal $b$-step approximation of weighted data, where the error is measured with respect to the $L_\infty$ norm. For data presorted by the independent variable the algorithm takes $\Theta(n + \log n \cdot b(1+\log n/b))$ time and $\Theta(n)$ space. This is $\Theta(n \log n)$ in the worst case and $\Theta(n)$ when $b = O(n/\log n \log\log n)$. A minor change determines an optimal reduced isotonic regression in the same time and space bounds, and the algorithm also solves the $k$-center problem for 1-dimensional weighted data.