Approximating the packedness of polygonal curves

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 $(1+\epsilon )$-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 $O(d{n}^2\log n)$ time. In the case $d=2$ we develop a faster algorithm that returns a $(6+\epsilon )$-approximation and runs in $O\left({(n/{\epsilon }^3)}^{4/3}\mathrm{polylog}\right(n/\epsilon \left)\right))$ 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.