Fast Approximation and Randomized Algorithms for Diameter

TL;DR

This paper introduces fast approximation algorithms for the diameter of point sets in high dimensions, including a randomized method, demonstrating superior performance and accuracy on large datasets up to one million points.

Contribution

It presents a new randomized algorithm for diameter approximation, improving efficiency and accuracy over existing methods, especially in high-dimensional and large-scale data.

Findings

Algorithms achieve solutions within 10^{-4} absolute error.

Performance surpasses many existing algorithms on large datasets.

Effective in high dimensions with minimal iterations.

Abstract

We consider approximation of diameter of a set $S$ of $n$ points in dimension $m$. E$\tilde{g}$ecio$\tilde{g}$lu and Kalantari \cite{kal} have shown that given any $p \in S$, by computing its farthest in $S$, say $q$, and in turn the farthest point of $q$, say $q'$, we have ${\rm diam}(S) \leq \sqrt{3} d(q,q')$. Furthermore, iteratively replacing $p$ with an appropriately selected point on the line segment $pq$, in at most $t \leq n$ additional iterations, the constant bound factor is improved to $c_*=\sqrt{5-2\sqrt{3}} \approx 1.24$. Here we prove when $m=2$, $t=1$. This suggests in practice a few iterations may produce good solutions in any dimension. Here we also propose a randomized version and present large scale computational results with these algorithm for arbitrary $m$. The algorithms outperform many existing algorithms. On sets of data as large as $1,000,000$ points, the proposed algorithms compute solutions to within an absolute error of $10^{-4}$.