On the closest pair of points problem

TL;DR

This paper presents two new algorithms for the closest pair of points problem, leveraging mathematical packing theory, with one achieving linear time for uniform distributions and both demonstrating superior empirical performance.

Contribution

Introduces two deterministic algorithms based on packing theory, with proven linear time for uniform distributions and fast empirical runtimes, improving over existing methods.

Findings

cppMM achieves O(n) time for uniform distributions

cppAPs has near-brute-force simplicity with fast empirical performance

cppMM outperforms other algorithms for large datasets

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.