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 on $\varepsilon$.

翻译：地图匹配是分析车辆轨迹时常用的预处理步骤。在理论界，最常用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，其中空间相似度通过 Fréchet 距离度量。现有基于 Fréchet 距离的地图匹配算法的一个缺点在于，每次匹配轨迹时都需要从头重新处理整个道路网络。一个未解决的问题是：是否可以对道路网络进行预处理，构建一个数据结构，从而支持在亚线性时间内回答地图匹配查询。本文研究了基于 Fréchet 距离的地图匹配查询。我们给出了几何平面图上的一个否定性结果：除非 SETH 失效，否则不存在可在多项式时间内构建的数据结构，使得对于任意 δ > 0，查询时间达到 O((pq)^{1-δ})，其中 p 和 q 分别为几何平面图和查询轨迹的复杂度。我们给出了一个针对真实输入图的肯定性结果，这被认为是本文的主要贡献。我们证明，对于 c-密集图（c-packed graphs），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。