How Packed Is It, Really?

TL;DR

This paper introduces a randomized near-linear time algorithm that approximates the congestion of a curve within a factor of 42, improving computational efficiency over previous methods.

Contribution

It presents the first near-linear time randomized approximation algorithm for congestion of curves, with a significant speed-up over prior algorithms.

Findings

Runs in O(n log^2 n) time with high probability

Achieves a 42-approximation factor for congestion

Improves over previous O~(n^{4/3}) time algorithms

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.