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 article, 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 article. 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.

地图匹配是分析车辆轨迹的常见预处理步骤。在理论界，最流行的地图匹配方法是计算道路网络上与轨迹在空间上最相似的路径，其中空间相似性是使用 Fréchet 距离来测量的。现有Fréchet距离下的地图匹配算法的一个缺点是，每次匹配轨迹时，都需要从头开始重新处理整个路网。一个悬而未决的问题是是否可以将道路网络预处理为一种数据结构，以便可以在亚线性时间内回答地图匹配查询。

在本文中，我们研究了 Fréchet 距离下的地图匹配查询。我们为几何平面图提供了负结果。我们表明，除非 SETH 失败，否则没有任何数据结构可以在多项式时间内构建，以在O((pq) ^1-δ ) 查询时间内回答地图匹配查询，对于任何 δ > 0，其中 p和 q是复杂度分别是几何平面图和查询轨迹。我们为实际输入图提供了积极的结果，我们将其视为本文的主要结果。我们证明，对于c压缩图，可以构造一个 \(\tilde{O}(cp)\) 大小的数据结构，该数据结构可以回答 \(\tilde{O} 中的 (1+ε) 近似地图匹配查询(c^4 q \log ^4 p)\) 时间，其中 \(\tilde{O}(\cdot)\) 隐藏低阶因子和对 ε 的依赖。