Map-Matching Queries under Fr\'echet Distance on Low-Density Spanners

TL;DR

This paper develops efficient data structures for map-matching GPS trajectories to low-density spanner graphs using the Fréchet distance, addressing a gap in previous work by enabling path reporting.

Contribution

It introduces a novel approach for preprocessing low-density spanner graphs to efficiently answer Fréchet distance queries and report optimal paths, extending prior research to more realistic network models.

Findings

Efficient data structures for Fréchet distance queries on low-density spanners.

Ability to report the actual path minimizing the Fréchet distance.

Improved applicability to real-world road networks.

Abstract

Map matching is a common task when analysing GPS tracks, such as vehicle trajectories. The goal is to match a recorded noisy polygonal curve to a path on the map, usually represented as a geometric graph. The Fr\'echet distance is a commonly used metric for curves, making it a natural fit. The map-matching problem is well-studied, yet until recently no-one tackled the data structure question: preprocess a given graph so that one can query the minimum Fr\'echet distance between all graph paths and a polygonal curve. Recently, Gudmundsson, Seybold, and Wong [SODA 2023, arXiv:2211.02951] studied this problem for arbitrary query polygonal curves and $c$-packed graphs. In this paper, we instead require the graphs to be $\lambda$-low-density $t$-spanners, which is significantly more representative of real-world networks. We also show how to report a path that minimises the distance efficiently rather than only returning the minimal distance, which was stated as an open problem in their paper.