Approximating the packedness of polygonal curves
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In 2012 Driemel et al. <cit.> introduced the concept of c-packed curves as a realistic input model. In the case when c is a constant they gave a near linear time (1+ε)-approximation algorithm for computing the Fréchet distance between two c-packed polygonal curves. Since then a number of papers have used the model. In this paper we consider the problem of computing the smallest c for which a given polygonal curve in ℝ^d is c-packed. We present two approximation algorithms. The first algorithm is a 2-approximation algorithm and runs in O(dn^2 log n) time. In the case d=2 we develop a faster algorithm that returns a (6+ε)-approximation and runs in O((n/ε^3)^4/3 polylog (n/ε))) time. We also implemented the first algorithm and computed the approximate packedness-value for 16 sets of real-world trajectories. The experiments indicate that the notion of c-packedness is a useful realistic input model for many curves and trajectories.
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