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Streaming Diameter of High-Dimensional Points

Magnús M. Halldórsson, Nicolaos Matsakis, Pavel Veselý

TL;DR

This work advances streaming algorithms for high-dimensional extent problems by introducing the Guarded Ball Cover, a simpler data structure that uses $O\left(\varepsilon^{-2}\log\left(\frac{1}{\varepsilon}\right)\right)$ stored points. By representing each ball with a center, radius, and a guard point, and by leveraging $(1+\varepsilon)$-expansions and proxying deleted points, the authors achieve tighter space bounds and straightforward correctness arguments, extending to Farthest Neighbor queries, Farthest Pair, Diameter, the Minimum Enclosing Ball, and its coreset. The approach yields $(\sqrt{2}+\varepsilon)$-approximate FN and FP with the same space, a $(1.22+\varepsilon)$-approximate MEC, and a $(\sqrt{2}+\varepsilon)$-approximate MEC-coreset, while proving a near-matching lower bound of $\Omega(\varepsilon^{-1})$ points for similar approximations. Overall, the Guarded Ball Cover provides tighter, uniform guarantees across multiple high-dimensional streaming extent problems with provable space savings.

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 $Ω(\varepsilon^{-1})$ points is necessary for a $(\sqrt{2}+\varepsilon)$-approximation of Farthest Pair or Farthest Neighbor queries.

Streaming Diameter of High-Dimensional Points

TL;DR

This work advances streaming algorithms for high-dimensional extent problems by introducing the Guarded Ball Cover, a simpler data structure that uses stored points. By representing each ball with a center, radius, and a guard point, and by leveraging -expansions and proxying deleted points, the authors achieve tighter space bounds and straightforward correctness arguments, extending to Farthest Neighbor queries, Farthest Pair, Diameter, the Minimum Enclosing Ball, and its coreset. The approach yields -approximate FN and FP with the same space, a -approximate MEC, and a -approximate MEC-coreset, while proving a near-matching lower bound of points for similar approximations. Overall, the Guarded Ball Cover provides tighter, uniform guarantees across multiple high-dimensional streaming extent problems with provable space savings.

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 points. This improves by a factor of the previous space bound of Agarwal and Sharathkumar (SODA 2010), while offering a simpler and more complete argument. We also show that storing points is necessary for a -approximation of Farthest Pair or Farthest Neighbor queries.

Paper Structure

This paper contains 7 sections, 10 theorems, 4 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Related Work
  4. The Guarded Ball Cover
  5. Applications
  6. Lower Bound for Farthest Pair and Farthest Neighbor Queries
  7. A Note on Lemma 2 in AS15j

Key Result

Lemma 1

If $B=B(c,r)$ is the MEB of a set $X$ of points, then any closed half-space containing $c$ also contains a point of $X$ on the boundary of $B$.

Figures (1)

  • Figure 1: Left: Case $\lVert c_i c_{i+1}\rVert \le 5\varepsilon r_{i}/6$, Right: Case $\lVert c_i c_{i+1}\rVert > 5\varepsilon r_{i}/6$ of Lemma \ref{['lm:expon']}. The dashed ball is the MEB of all points, assuming that $q'$ is deleted before $B_{i+1}$ is created.

Theorems & Definitions (15)

  • Lemma 1: Lemma 2.2 in BadoiuHI02
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • ...and 5 more