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

TL;DR

This paper improves the theoretical bounds for subtrajectory clustering under the continuous and discrete Fréchet distances, providing more efficient algorithms and establishing hardness results close to the conjectured limits.

Contribution

It introduces faster algorithms for subtrajectory clustering under both distances and refines the hardness bounds, nearly matching the conjectured computational limits.

Findings

Continuous case: $O(n^3 \, \log^2 n)$ time algorithm

Discrete case: $O(n^2 \, \log n)$ time algorithm

Hardness bounds close to SETH and 3OV assumptions

Abstract

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.