Better approximation algorithm for point-set diameter

TL;DR

This paper introduces a faster approximation algorithm for computing the diameter of a point set in fixed-dimensional Euclidean space, achieving near-linear time complexity with controlled accuracy.

Contribution

It presents a new $(1+O( ext{epsilon}))$-approximation algorithm with improved running time for the point-set diameter problem in fixed dimensions.

Findings

Achieves $O(n + 1/ ext{epsilon}^{(d-1)/2})$ running time

Provides a $(1+O( ext{epsilon}))$-approximation

Improves upon previous algorithms' efficiency

Abstract

We propose a new $(1+O(\varepsilon))$-approximation algorithm with $O(n+ 1/\varepsilon^{\frac{(d-1)}{2}})$ running time for computing the diameter of a set of $n$ points in the $d$-dimensional Euclidean space for a fixed dimension $d$, where $0 < \varepsilon\leqslant 1$. This result provides some improvements in the running time of this problem in comparison with previous algorithms.