Optimal Algorithm for the Planar Two-Center Problem

TL;DR

This paper presents a deterministic $O(n ext{log} n)$-time algorithm for the planar two-center problem, matching the theoretical lower bound and solving a longstanding open problem in computational geometry.

Contribution

The paper introduces the first $O(n ext{log} n)$ deterministic algorithm for the planar two-center problem, improving upon previous $O(n ext{log}^2 n)$ solutions.

Findings

Achieves optimal $O(n ext{log} n)$ runtime for the problem.

Resolves a decades-old open problem in computational geometry.

Provides a deterministic solution matching the lower bound.

Abstract

We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set $S$ of $n$ points in the plane and the goal is to find two smallest congruent disks whose union contains all points of $S$. A longstanding open problem has been to obtain an $O(n\log n)$-time algorithm for planar two-center, matching the $\Omega(n\log n)$ lower bound given by Eppstein [SODA'97]. Towards this, researchers have made a lot of efforts over decades. The previous best algorithm, given by Wang [SoCG'20], solves the problem in $O(n\log^2 n)$ time. In this paper, we present an $O(n\log n)$-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.