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How Packed Is It, Really?

Sariel Har-Peled, Timothy Zhou

TL;DR

This work addresses the problem of estimating the congestion of planar curves, a measure of local zigzagging defined via $c$-packedness. It introduces a randomized $42$-approximation algorithm that runs in $O(n\log^3 n)$ time with high probability, significantly speeding beyond the prior $\tilde{O}(n^{4/3})$-time algorithms. The approach partitions the input into short and long segments and reduces the problem to a small number of quadtrees, using a combination of square-based representations, dynamic programming on the short part, and an exponential-search with random sampling to bound the long part. The result yields near-linear-time, constant-factor congestion estimates useful for efficient Fréchet-distance approximations and related problems, with potential extensions to higher dimensions and practical implementations.

Abstract

The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve $π$ is $c$-packed if the length of the curve lying inside any ball is at most $c$ times the radius of the ball, and its congestion is the minimum $c$ for which $π$ is $c$-packed. This paper presents a randomized $42$-approximation algorithm for computing the congestion of a curve (or any set of segments in the plane). It runs in $O( n \log^2 n)$ time and succeeds with high probability. Although the approximation factor is large, the running time improves over the previous fastest constant approximation algorithm, which took $\widetilde{O}(n^{4/3})$ time. We carefully combine new ideas with known techniques to obtain our new near-linear time algorithm.

How Packed Is It, Really?

TL;DR

This work addresses the problem of estimating the congestion of planar curves, a measure of local zigzagging defined via -packedness. It introduces a randomized -approximation algorithm that runs in time with high probability, significantly speeding beyond the prior -time algorithms. The approach partitions the input into short and long segments and reduces the problem to a small number of quadtrees, using a combination of square-based representations, dynamic programming on the short part, and an exponential-search with random sampling to bound the long part. The result yields near-linear-time, constant-factor congestion estimates useful for efficient Fréchet-distance approximations and related problems, with potential extensions to higher dimensions and practical implementations.

Abstract

The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve is -packed if the length of the curve lying inside any ball is at most times the radius of the ball, and its congestion is the minimum for which is -packed. This paper presents a randomized -approximation algorithm for computing the congestion of a curve (or any set of segments in the plane). It runs in time and succeeds with high probability. Although the approximation factor is large, the running time improves over the previous fastest constant approximation algorithm, which took time. We carefully combine new ideas with known techniques to obtain our new near-linear time algorithm.

Paper Structure

This paper contains 19 sections, 12 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard tools
  4. Congestion
  5. The algorithm: The long and short of it
  6. Reduction to quadtrees
  7. First steps towards approximating the congestion of a quadtree
  8. A naıve exact algorithm for the congestion of a quadtree
  9. The long and the short of it
  10. Registering the segments
  11. Computing the congestion of the short segments
  12. Approximating the maximum load of the long segments in a quadtree
  13. A naıve algorithm for computing the $\alpha$-load
  14. From long congestion to load, and back
  15. Exponential search for maximum $\alpha$-load
  16. ...and 4 more sections

Key Result

Lemma 2.6

Given a set $\mathcal{S}$ of $n$ segments in the plane, and a parameter ${\varepsilon} \in (0,1)$, one can compute, in $O(n \log n + n / {\varepsilon}^2)$ time, a set ${\mathcal{G}}_\mathcal{S}$ of $O(n/{\varepsilon}^2)$ axis-aligned squares, such that $\mathcalb{c}\mleft({\mathcal{S}}\mright) \geq

Figures (2)

  • Figure 3.1: Computing a maximal set of $\alpha$-long segments for a segment $s$ (see \ref{['lemma:register']}). In this case, the segment has length $5$, and each grid cell is of radius $0.5$, so the segment is $10$-long for all the cells of the grid it intersects.
  • Figure 4.1: An $\alpha$-long segment for a square $\square = \square\mleft({p, r}\mright)$ intersects the square $\square' = \square\mleft({p, (1+\alpha)r}\mright)$ with a segment of length at least $\alpha r$ (here, $\alpha=3$).

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6: Lemma 12 in gsw-appc-20
  • Remark 2.7
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 13 more