Streaming Diameter of High-Dimensional Points

2025-05-22

AI Summary

This paper addresses streaming computation of four fundamental geometric problems over dynamic point sets in high-dimensional Euclidean space: diameter approximation, farthest neighbor queries, minimum enclosing ball (MEB), and core-set construction. We propose the first deterministic streaming algorithm, built upon geometric pruning and hierarchical grid sampling, integrated with structural analysis of farthest-point pairs and core-set theory. Our algorithm achieves a (√2 + ε)-approximation guarantee while reducing space complexity to O(ε⁻² log(1/ε)), improving upon the prior SODA 2010 state-of-the-art by a factor of ε⁻¹. We further establish a tight Ω(ε⁻¹) lower bound on space, proving asymptotic optimality of our complexity. Crucially, a single unified framework supports all four query types, enhancing both space efficiency and theoretical completeness for geometric streaming.

Technology Category

• Data Mining & Knowledge Management: Data Stream Mining
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Security and Privacy: Large-scale security measurements

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.

Problem

• Improving space bounds for streaming diameter approximation
• Enhancing farthest neighbor and minimum enclosing ball queries
• Establishing lower bounds for high-dimensional approximation storage

Innovation

• Deterministic streaming algorithms for high-dimensional spaces
• Improved space bound for Diameter approximation
• Optimal point storage for Farthest Pair queries