Figure 1: Clustering using the Fréchet distance (left), dynamic time warping (middle), and our new approach called continuous dynamic time warping (right), where the color denotes the clusters and the bold trajectories are the cluster centers. While the Fréchet distance shows a strong influence of outliers and dynamic time warping shows discretization issues, clustering via continuous dynamic time warping gives arguably the most natural results. Map data © OpenStreetMap contributors.

In 2012 Driemel et al. [18] 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.

One approach to studying the Fréchet distance is to consider curves that satisfy realistic assumptions. By now, the most popular realistic assumption for curves is c-packedness. Existing algorithms for computing the Fréchet distance between c-packed curves require both curves to be c-packed. In this paper, we only require one of the two curves to be c-packed. Our result is a nearly-linear time algorithm that (1 + ε)-approximates the Fréchet distance between a c-packed curve and a general curve in R d , for constant values of ε, d and c.

We study three covering problems in the plane. Our original motivation for these problems come from trajectory analysis. The first is to decide whether a given set of line segments can be covered by up to four unit-sized, axis-parallel squares. The second is to build a data structure on a trajectory to efficiently answer whether any query subtrajectory is coverable by up to three unit-sized axis-parallel squares. The third problem is to compute a longest subtrajectory of a given trajectory that can be covered by up to two unit-sized axis-parallel squares.

The yolk is an important concept in spatial voting games: the yolk center generalises the equilibrium and the yolk radius bounds the uncovered set. We present near-linear time algorithms for computing the yolk in the plane. To the best of our knowledge our algorithm is the first that does not precompute median lines, and hence is able to break the best known upper bound of O(n 4/3 ) on the number of limiting median lines. We avoid this requirement by carefully applying Megiddo's parametric search technique, which is a powerful framework that could lead to faster algorithms for other spatial voting problems.