Geometric k-Center Problems with Centers Constrained to Two Lines

TL;DR

This paper introduces efficient algorithms for the geometric k-center problem with centers constrained to two lines, achieving optimal time complexity for both weighted and unweighted cases.

Contribution

It provides the first $O(n\log^2 n)$ algorithms for k-center problems with centers on two lines, including perpendicular and parallel configurations.

Findings

Algorithms run in $O(n\log^2 n)$ time for both weighted and unweighted cases.

Centers constrained to two lines can be optimally placed to minimize maximum distance.

The approach extends to different orientations of the two lines.

Abstract

We consider the $k$-center problem in which the centers are constrained to lie on two lines. Given a set of $n$ weighted points in the plane, we want to locate up to $k$ centers on two parallel lines. We present an $O(n\log^2 n)$ time algorithm, which minimizes the weighted distance from any point to a center. We then consider the unweighted case, where the centers are constrained to be on two perpendicular lines. Our algorithms run in $O(n\log^2 n)$ time also in this case.