Streaming Diameter of High-Dimensional Points

TL;DR

This paper presents improved deterministic streaming algorithms for high-dimensional geometric problems like Diameter and Farthest Neighbor, reducing space complexity and establishing necessary bounds for approximation accuracy.

Contribution

It introduces simpler, more space-efficient streaming algorithms for high-dimensional geometric problems and proves lower bounds for approximation requirements.

Findings

Deterministic algorithms store O(ε^{-2} log(1/ε)) points for Diameter approximation.

Improved previous space bounds by a factor of ε^{-1}.

Necessary storage of Ω(ε^{-1}) points for certain approximation guarantees.

Abstract

We improve the space bound for streaming approximation of Diameter but also of Farthest Neighbor queries, Minimum Enclosing Ball and its Coreset, in high-dimensional Euclidean spaces. In particular, our deterministic streaming algorithms store $\mathcal{O}(\varepsilon^{-2}\log(\frac{1}{\varepsilon}))$ points. This improves by a factor of $\varepsilon^{-1}$ the previous space bound of Agarwal and Sharathkumar (SODA 2010), while offering a simpler and more complete argument. We also show that storing $\Omega(\varepsilon^{-1})$ points is necessary for a $(\sqrt{2}+\varepsilon)$-approximation of Farthest Pair or Farthest Neighbor queries.