Map matching queries on realistic input graphs under the Fréchet distance

Abstract

Map matching is a common preprocessing step for analysing vehicle trajectories. In the theory community, the most popular approach for map matching is to compute a path on the road network that is the most spatially similar to the trajectory, where spatial similarity is measured using the Fréchet distance. A shortcoming of existing map matching algorithms under the Fréchet distance is that every time a trajectory is matched, the entire road network needs to be reprocessed from scratch. An open problem is whether one can preprocess the road network into a data structure, so that map matching queries can be answered in sublinear time. In this paper, we investigate map matching queries under the Fréchet distance. We provide a negative result for geometric planar graphs. We show that, unless SETH fails, there is no data structure that can be constructed in polynomial time that answers map matching queries in O (( pq ) 1 − δ ) query time for any δ > 0, where  p and  q are the complexities of the geometric planar graph and the query trajectory, respectively. We provide a positive result for realistic input graphs, which we regard as the main result of this paper. We show that for c -packed graphs, one can construct a data structure of \(\tilde{O}(cp) \) size that can answer (1 + ε)-approximate map matching queries in \(\tilde{O}(c^4 q \log ^4 p) \) time, where \(\tilde{O}(\cdot) \) hides lower-order factors and dependence on ε.