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Net and Prune: A Linear Time Algorithm for Euclidean Distance Problems

Sariel Har-Peled, Banjamin Raichel

Abstract

We provide a general framework for getting expected linear time constant factor approximations (and in many cases FPTASs) to several well-known problems in Computational Geometry, such as $k$-center clustering and farthest nearest neighbor. The new approach is robust to variations in the input problem, and yet it is simple, elegant, and practical. In particular, many of these well-studied problems, which fit easily into our framework, either previously had no linear time approximation algorithm, or required rather involved algorithms and analysis. A short list of the problems we consider includes farthest nearest neighbor, $k$-center clustering, smallest disk enclosing $k$ points, Hausdorff distance, $k$th largest distance, $k$th smallest $m$-nearest neighbor distance, $k$th heaviest edge in the MST, and other spanning-forest type problems, problems involving upward closed set systems, and more. Finally, we show how to extend our framework such that the linear running time bound holds with high probability.

Net and Prune: A Linear Time Algorithm for Euclidean Distance Problems

Abstract

We provide a general framework for getting expected linear time constant factor approximations (and in many cases FPTASs) to several well-known problems in Computational Geometry, such as -center clustering and farthest nearest neighbor. The new approach is robust to variations in the input problem, and yet it is simple, elegant, and practical. In particular, many of these well-studied problems, which fit easily into our framework, either previously had no linear time approximation algorithm, or required rather involved algorithms and analysis. A short list of the problems we consider includes farthest nearest neighbor, -center clustering, smallest disk enclosing points, Hausdorff distance, th largest distance, th smallest -nearest neighbor distance, th heaviest edge in the MST, and other spanning-forest type problems, problems involving upward closed set systems, and more. Finally, we show how to extend our framework such that the linear running time bound holds with high probability.

Paper Structure

This paper contains 59 sections, 38 theorems, 6 equations, 6 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Linear time algorithms.
  3. Nets.
  4. Outline of the new technique.
  5. Significance of results.
  6. Significance of the netting step.
  7. Framework and results.
  8. High probability.
  9. What can be done without the net & prune framework?
  10. Paper organization.
  11. Preliminaries
  12. Basic definitions
  13. Computing nets quickly for a point set in Rd
  14. Identifying close and far points
  15. Approximating nice distance problems
  16. ...and 44 more sections

Key Result

Lemma 2.2.1

Given a point set $\mathsf{P}\subseteq {\rm I\!\space R}^d$ of size $n$ and a parameter $r>0$, one can compute an $r$-net for $\mathsf{P}$ in $O(n)$ time.

Figures (6)

  • Figure 2.1:
  • Figure 3.1: The approximation algorithm. The implicit target value being approximated is $f = f\mleft({ \mathsf{W}, \Gamma}\mright)$, where $f$ is a $\varphi$-NDP. That is, there is a $\varphi$-decider for $f$, denoted by decider, and the only access to the function $f$ is via this procedure.
  • Figure 4.1:
  • Figure 5.1: Algorithm for computing a constant factor approximation to a value in the interval $[ \mathtt{d}^{\delta}\mleft({\mathsf{P}}\mright), \mathtt{d}^{1- \delta}\mleft({\mathsf{P}}\mright)]$, for some fixed constant $\delta \in (0,1)$.
  • Figure 5.2: Roughly estimating $\mathtt{d}_{O(\log n)}^{1/4} \mleft({\mathsf{P}}\mright)$. Here $c_5$ and $c_6$ are sufficiently large constants.
  • ...and 1 more figures

Theorems & Definitions (63)

  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Lemma 2.2.1
  • Corollary 2.2.2
  • Lemma 2.3.1
  • Definition 3.1.1
  • Definition 3.1.2
  • Definition 3.1.3: $\varphi$-NDP
  • Remark 3.1.4: $(1+\varepsilon)$-NDP
  • ...and 53 more