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arXiv · 2105.10776

How Packed Is It, Really?

Sariel Har-Peled·Timothy Zhou
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Abstract

The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve $\pi$ 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 $\pi$ 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.

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Sariel Har-Peled, Timothy Zhou. 2021-05-22. How Packed Is It, Really?. https://arxiv.org/abs/2105.10776

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