Cubic upper and lower bounds for subtrajectory clustering under the continuous Fr\'echet distance

Detecting commuting patterns or migration patterns in movement data is an important problem in computational movement analysis. Given a trajectory, or set of trajectories, this corresponds to clustering similar subtrajectories. We study subtrajectory clustering under the continuous and discrete Fr\'echet distances. The most relevant theoretical result is by Buchin et al. (2011). They provide, in the continuous case, an $O(n^5)$ time algorithm and a 3SUM-hardness lower bound, and in the discrete case, an $O(n^3)$ time algorithm. We show, in the continuous case, an $O(n^3 \log^2 n)$ time algorithm and a 3OV-hardness lower bound, and in the discrete case, an $O(n^2 \log n)$ time algorithm and a quadratic lower bound. Our bounds are almost tight unless SETH fails.