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Optimal Algorithm for the Planar Two-Center Problem with Tight Runtime Bound


Core Concepts
We present an O(n log n)-time deterministic algorithm for the planar two-center problem, which matches the known lower bound and resolves a longstanding open problem.
Abstract
The paper studies the planar two-center problem, which asks to find two smallest congruent disks whose union contains a given set S of n points in the plane. This problem has been extensively studied, with the previous best algorithm running in O(n log^2 n) time. The key contributions are: A new decision algorithm that can determine in O(n) time whether there exist two radius-r disks covering the point set S, after O(n log n) time preprocessing. This is achieved by introducing a new concept called "r-coverage" and proving several interesting properties about it. Using the decision algorithm and known techniques, an O(n log n)-time deterministic algorithm is presented for the planar two-center problem, which matches the known lower bound and resolves a longstanding open problem. The correctness analysis of the algorithm is technically involved, requiring a nontrivial combination of several novel insights into the problem, along with known observations in the literature.
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Key Insights Distilled From

by Kyungjin Cho... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2007.08784.pdf
Optimal Algorithm for the Planar Two-Center Problem

Deeper Inquiries

Can the techniques developed in this paper be extended to solve other variants or generalizations of the planar two-center problem, such as the discrete version or the outlier version

The techniques developed in this paper for the planar two-center problem could potentially be extended to solve other variants or generalizations of the problem. For example, in the discrete version of the two-center problem where the centers of the disks are required to be points in the input set, similar geometric insights and algorithmic techniques could be applied. By adapting the decision algorithm and preprocessing steps to handle the discrete constraints, it may be possible to devise an efficient algorithm for this variant. Additionally, for the outlier version of the problem where some points are considered outliers and not required to be covered by the disks, the concept of tight solutions and anchored points could be utilized to develop a solution strategy. By incorporating the principles of tightness and anchoring into the algorithm design, it may be feasible to address the outlier version of the planar two-center problem effectively.

Are there any applications or implications of the r-coverage concept introduced in this work beyond the two-center problem

The concept of r-coverage introduced in this work has potential applications and implications beyond the planar two-center problem. One possible application could be in the field of facility location or clustering, where the notion of covering a set of points with disks of a certain radius is relevant. The r-coverage concept could be utilized to determine the minimum number of disks of radius r required to cover a given point set efficiently, which is a fundamental problem in facility location analysis. Additionally, in the context of sensor networks or coverage optimization, the r-coverage concept could be employed to optimize the placement of sensors or devices to ensure complete coverage of a region with minimal overlap. By leveraging the properties and characteristics of r-coverage, it may be possible to enhance the efficiency and effectiveness of coverage optimization algorithms in various applications.

What other fundamental problems in computational geometry might benefit from a similar approach of combining novel geometric insights with known algorithmic techniques

Several fundamental problems in computational geometry could benefit from a similar approach of combining novel geometric insights with known algorithmic techniques. One such problem is the geometric clustering problem, where the goal is to partition a set of points into clusters based on their spatial proximity. By incorporating geometric properties and insights similar to those used in the planar two-center problem, novel clustering algorithms could be developed to improve the clustering quality and efficiency. Another potential application could be in geometric optimization problems, such as minimum enclosing circle or rectangle problems, where the integration of geometric insights with algorithmic techniques could lead to more efficient and accurate solutions. By exploring the intersection of geometric principles and algorithm design, advancements in solving fundamental problems in computational geometry can be achieved.
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