Finding Complex Patterns in Trajectory Data via Geometric Set Cover

Clustering trajectories is a central challenge when confronted with large amounts of movement data such as full-body motion data or GPS data. We study a clustering problem that can be stated as a geometric set cover problem: Given a polygonal curve of complexity $n$, find the smallest number $k$ of representative trajectories of complexity at most $l$ such that any point on the input trajectories lies on a subtrajectory of the input that has Fr\'echet distance at most $\Delta$ to one of the representative trajectories. This problem was first studied by Akitaya et al. (2021) and Br\"uning et al. (2022). They present a bicriteria approximation algorithm that returns a set of curves of size $O(kl\log(kl))$ which covers the input with a radius of $11\Delta$ in time $\widetilde{O}((kl)^2n + kln^3)$, where $k$ is the smallest number of curves of complexity $l$ needed to cover the input with a distance of $\Delta$. The representative trajectories computed by their algorithm are always line segments. In applications however, one is usually interested in representative curves of higher complexity which consist of several edges. We present a new approach that builds upon the works of Br\"uning et al. (2022) computing a set of curves of size $O(k\log(n))$ in time $\widetilde{O}(l^2n^4 + kln^4)$ with the same distance guarantee of $11\Delta$, where each curve may consist of curves of complexity up to the given complexity parameter $l$. To validate our approach, we conduct experiments on different types of real world data: high-dimensional full-body motion data and low-dimensional GPS-tracking data.