How Packed Is It, Really?
DOI:
https://doi.org/10.57717/cgt.v4i1.65Abstract
The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve p 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 log2 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 runs in (roughly) O(n4/3) time. We carefully combine new ideas with known techniques to obtain our new near-linear time algorithm.
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Copyright (c) 2025 Sariel Har-Peled, Timothy Zhou
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