Efficient Planar Two-Center Algorithms

TL;DR

This paper introduces efficient algorithms for the planar two-center problem, achieving optimal $O(n ext{log} n)$ time in specific cases, improving upon previous bounds.

Contribution

It provides the first $O(n ext{log} n)$ algorithms for the two-center problem when centers are close or points are convex, enhancing previous results.

Findings

Algorithms run in $O(n ext{log} n)$ time for specific cases.

Improves previous $O(n ext{log} n ext{log} ext{log} n)$ bound.

Applicable to points in convex position and overlapping centers.

Abstract

We consider the planar Euclidean two-center problem in which given $n$ points in the plane we are to find two congruent disks of the smallest radius covering the points. We present a deterministic $O(n \log n)$-time algorithm for the case that the centers of the two optimal disks are close to each other, that is, the overlap of the two optimal disks is a constant fraction of the disk area. We also present a deterministic $O(n\log n)$-time algorithm for the case that the input points are in convex position. Both results improve the previous best $O(n\log n\log\log n)$ bound on the problems.