How Packed Is It, Really?
Sariel Har-Peled, Timothy Zhou

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 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.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Complexity and Algorithms in Graphs · Digital Image Processing Techniques
