An efficient approximation for point-set diameter in higher dimensions

TL;DR

This paper introduces a new efficient approximation algorithm for computing the diameter of point sets in higher-dimensional Euclidean spaces, improving simplicity and data structure efficiency over previous methods.

Contribution

It presents a novel $(1+ ext{}\varepsilon)$-approximation algorithm with improved time complexity for high-dimensional point-set diameter computation.

Findings

Achieves $O(n+ 1/ ext{ ext}varepsilon^{d-1})$ time complexity

Provides a modified $(1+O( ext{ ext}varepsilon))$-approximation with $O(n+ 1/ ext{ ext}varepsilon^{rac{2d}{3}-rac{1}{3}})$ time

Offers improvements in simplicity and data structure over existing algorithms

Abstract

In this paper, we study the problem of computing the diameter of a set of $n$ points in $d$-dimensional Euclidean space for a fixed dimension $d$, and propose a new $(1+\varepsilon)$-approximation algorithm with $O(n+ 1/\varepsilon^{d-1})$ time and $O(n)$ space, where $0 < \varepsilon\leqslant 1$. We also show that the proposed algorithm can be modified to a $(1+O(\varepsilon))$-approximation algorithm with $O(n+ 1/\varepsilon^{\frac{2d}{3}-\frac{1}{3}})$ running time. These results provide some improvements in comparison with existing algorithms in terms of simplicity and data structure.