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# How Packed Is It, Really?

$\newcommand{\tldO}{{\widetilde{O}}}% \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\| {#1} - {#2} \right\|} \newcommand{\ptq}{q} \newcommand{\pts}{s}$
Sariel Har-Peled and Timothy Zhou.
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 maximum $c$ for which $\pi$ is $c$-packed. This paper presents a randomized $(288+\eps)$-approximation algorithm for computing the congestion of a curve (or any set of segments in constant dimension). 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 \cite{gsw-appc-20}, which took $\tldO(n^{4/3})$ time. We carefully combine new ideas with known techniques to obtain our new, near-linear time algorithm.
PDF : arXiv.