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 Fr\'echet distance. A shortcoming of existing map matching algorithms under the Fr\'echet 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 Fr\'echet 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 $O((pq)^{1-\delta})$ query time for any $\delta > 0$, 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 $(1+\varepsilon)$-approximate map matching queries in $\tilde O(c^4 q \log^4 p)$ time, where $\tilde O(\cdot)$ hides lower-order factors and dependence of $\varepsilon$.

翻译：地图匹配是分析车辆轨迹时常见的预处理步骤。在理论界，最流行的地图匹配方法是在道路网络上计算一条与轨迹空间相似度最高的路径，其中空间相似度由Fréchet距离衡量。现有基于Fréchet距离的地图匹配算法的一个缺点是，每次匹配轨迹时都需要从头重新处理整个道路网络。一个开放问题是，是否可以对道路网络进行预处理，构建一个数据结构，使得地图匹配查询能够在亚线性时间内完成。本文研究基于Fréchet距离的地图匹配查询。对于几何平面图，我们给出了一个否定结果。我们证明，除非SETH假设不成立，否则不存在可以在多项式时间内构建的数据结构，使得对于任何δ > 0，地图匹配查询时间能达到O((pq)^{1-δ})，其中p和q分别是几何平面图和查询轨迹的复杂度。对于真实输入图，我们给出了一个肯定结果，这被视为本文的主要成果。我们证明，对于c-packed图，可以构建一个大小为Õ(cp)的数据结构，在Õ(c^4 q log^4 p)时间内回答(1+ε)-近似的地图匹配查询，其中Õ(·)隐藏了低阶因子和对ε的依赖。