Clustering Subtrajectories

1. Clustering Subtrajectories Kevin Buchin, MaikeBuchin, Joachim Gudmundsson, Maarten Löffler, Jun Luo

2. Motivation • Wildlife migration patterns • Commuting patterns • Sports analysis • … Problem: Given a single or a set of trajectories, find clusters of subtrajectories of the given trajectories.

3. Problem Definition • Given a trajectory T, a subtrajectory cluster of T consists of subtrajectories of T such that the time intervals of any two subtrajectories overlap in at most one point. • A subtrajectory cluster has • size m if it consists of at least m subtrajectories • length l if at least one subtrajectory has length l • distance d if the “distance” between any two subtrajectories is at most d.

4. Subtrajectory Similarity – which measure? • A good option: Fréchet Distance • continuous matching of points along the trajectories • allows for different speed development • Even better: Constrained Free Space Diagrams • points on the trajectories are matched fulfilling • constraints on matched points • time (e.g., similar or consecutive times) • distance (e.g., Euclidean distance) • direction (e.g., similar directions of movement) • attributes • constraints on the matching • ratio of time • ratio of travelled distance

5. Our results • Hardness: • Finding a distance approximation of the max-length subtrajectory problem is 3SUM-hard. • The decision version of the subtrajectory problem is NP-complete. • Approximation Algorithms: • A 2-distance approximation of the max-length subtrajectoryproblem can be computed in O(n2+nml) time and O(nl) space using the discrete Fréchet distance. • A 2-distance approximation of the max-length subtrajectory problem can be computed in time O(n3m 2(n/m)(log2 n+m)) using the continuous Fréchet distance.

6. Approximation algorithm Sweep the (constrained) free space diagram from left to right with two vertical lines (L and R) At each event point decide if there are m monotone curves between L and R R L

7. Many subtrajectories

8. Experimental Results (discrete) Experiment environment: Microsoft Windows XP operated PC with an Intel Pentium-4 3.0GHz processor and 1GB of RAM

9. Experimental Evaluation (cont’d)