On the closest pair of points problem

2026-01-09

arXiv.org

AI Summary

This work addresses the problem of efficiently finding the closest pair of points in large-scale point clouds by introducing, for the first time, the mathematical theory of optimal sphere packing into this domain. Two deterministic algorithms, cppMM and cppAPs, are proposed, with cppMM achieving linear time complexity on uniformly distributed point sets and offering straightforward implementation. Through a high-performance C++ implementation and extensive large-scale experiments, cppMM demonstrates superior empirical performance, significantly outperforming existing methods on datasets containing up to 33,554,432 points. The algorithm achieves the best observed running times in practice, highlighting its dual advantages of theoretical novelty and practical efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We introduce two novel algorithms for the problem of finding the closest pair in a cloud of $n$ points based on findings from mathematical optimal packing theory. Both algorithms are deterministic, show fast effective runtimes, and are very easy to implement. For our main algorithm, cppMM, we prove $O(n)$ time complexity for the case of uniformly distributed points. Our second algorithm, cppAPs, is almost as simple as the brute-force approach, but exhibits an extremely fast empirical running time, although its worst-case time complexity is also $O(n^2)$. We embed the new algorithms in a review of the most prominent contenders and empirically demonstrate their runtime behavior for problem sizes up to $n =$ 33,554,432 points observed in our C++ test environment. For large $n$, cppMM dominates the other algorithms under study.

Problem

• closest pair
• point cloud
• computational geometry
• nearest neighbor
• optimal packing

Innovation

• closest pair
• optimal packing theory
• linear time complexity
• deterministic algorithm
• empirical runtime