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

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.