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Optimal Algorithm for the Planar Two-Center Problem

Kyungjin Cho, Eunjin Oh, Haitao Wang, Jie Xue

TL;DR

This paper presents an O(n\log n)-time algorithm for the planar two-center problem that matches the best known lower bound of $\Omega(n\)log n$ as well as improving the previously best known algorithms which takes $O( n\log^2 n)$ time.

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 $Ω(n\log n)$ 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.

Optimal Algorithm for the Planar Two-Center Problem

TL;DR

This paper presents an O(n\log n)-time algorithm for the planar two-center problem that matches the best known lower bound of as well as improving the previously best known algorithms which takes time.

Abstract

We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set of points in the plane and the goal is to find two smallest congruent disks whose union contains all points of . A longstanding open problem has been to obtain an -time algorithm for planar two-center, matching the 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 time. In this paper, we present an -time (deterministic) algorithm for planar two-center, which completely resolves this open problem.

Paper Structure

This paper contains 19 sections, 15 theorems, 1 equation, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our result.
  3. Other related work.
  4. Outline.
  5. Preliminaries
  6. Basic notions.
  7. Circular hulls.
  8. Solutions for the two-center problem.
  9. r-Coverage
  10. Algorithm of Theorem \ref{['thm-main']}
  11. Preprocessing.
  12. Decision procedure.
  13. Time analysis.
  14. Correctness of the algorithm
  15. Geometric lemmas
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $S$ be a set of $n$ points in the plane. After a preprocessing step on $S$ in $O(n \log n)$ time, for any given $r \geq 0$, one can compute in $O(n)$ time two congruent disks of radius $r$ in the plane that together covers $S$, or decide the nonexistence of two such disks.

Figures (4)

  • Figure 4: (a) Illustrating the $r$-circular hull $\alpha_r(Q)$, which is bounded by the solid arcs. The radius of the dashed circle is $r$. (b) Illustrating the $r$-coverage $\mathcal{CR}_r(Q)$ (bounded by the outer cycle of solid arcs) for the set of five black points, while the inner cycle of solid arcs bounds $\alpha_r(Q)$. The larger dashed arc is of radius $2r$ while the smaller dashed circle is of radius $r$.
  • Figure 5: The two solid arcs connecting $v,u,w$ are $e(u,v)$ and $e(u,w)$, respectively. The red dashed arc is $\tau(u)$. The two dotted circles contain $e(u,w)$ and $e(u,v)$, respectively. The two blue solid arcs on the two circles are $\tau(e(u,v))$ and $\tau(e(u,w))$, respectively, and they share endpoints $\hat{u}_v$ and $\hat{u}_w$ with $\tau(u)$.
  • Figure 7: Illustration of the proof of Lemma \ref{['lem-equiv']}.
  • Figure 10: Illustration of $u,v, s, \hat{s}, \vec{\gamma}, \vec{\eta}$. For convenience, only the directions of $\vec{\gamma},\vec{\eta}$ are presented.

Theorems & Definitions (16)

  • Theorem 1.1
  • corollary 1.2
  • Lemma 2.1: Theorem 6.7 in wang2022planar
  • Lemma 2.2: Choi and Ahn choi2021efficient
  • definition 3.1
  • Lemma 3.2
  • corollary 3.3
  • Lemma 3.4
  • corollary 3.5
  • Lemma 3.6
  • ...and 6 more