Approximating the packedness of polygonal curves

Abstract

In 2012 Driemel et al. 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 Rd is c-packed. We present two approximation algorithms. The first algorithm is a 2-approximation algorithm and runs in O(dn2log⁡n) time. In the case d=2 we develop a faster algorithm that returns a (6+ε)-approximation and runs in O((n/ε3)4/3polylog(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.