How Packed Is It, Really?
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The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve π 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 π is c-packed. This paper presents a randomized (288+ε)-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 <cit.>, which runs in (roughly) 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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