Map matching queries on realistic input graphs under the Fréchet distance

TL;DR

This work addresses map matching under the Fréchet distance in two regimes. It proves a conditional hardness result for geometric planar graphs, showing that, under SETH, no polynomial-time preprocessing can yield truly subquadratic query time for map matching. It then delivers a positive, near-linear-space data structure for realistic input graphs (specifically $c$-packed graphs) that computes a $(1+\varepsilon)$-approximate map-matching path in polylogarithmic time with respect to the graph size, using a three-stage pipeline built from straightest-path queries, segment queries, and full map-matching queries. The results collectively reconcile worst-case hardness with practical performance on realistic road networks and provide a framework (SSPD-based, low-density, and hierarchical clustering techniques) that could guide future geometric-query data-structure work. The practical impact lies in enabling efficient, approximate trajectory-pathing on large networks while clarifying fundamental limits in worst-case graph models.

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 $(1+\varepsilon)$-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 $\varepsilon$.

Figures (19)

• Figure 1: A road network (black), a noisy trajectory (red), and its matched path (blue).
• Figure 2: Given a pair of query vertices $u,v$ (red), the data structure in Subproblem \ref{['problem:straightest_path_queries']} returns $\min_{\pi} d_F(\pi, uv)$ (orange) where $\pi$ ranges over all paths between $u$ and $v$ (blue) in the graph $P$ (black).
• Figure 3: Given a query segment $Q$ (red), the data structure in Subproblem \ref{['problem:map_matching_segment_queries']} returns $\min_{\pi} d_F(\pi, Q)$ (orange) where $\pi$ ranges over all paths (blue) in the graph $P$ (black).
• Figure 4: Given a query trajectory $Q$ (red), the data structure in Problem \ref{['problem:mapmatchingqueriesproblem']} returns $\min_{\pi} d_F(\pi, Q)$ (orange) where $\pi$ ranges over all paths (blue) in the graph $P$ (black).
• Figure 5: A semi-separated pair $A_i$ (red) and $B_i$ (blue). The circles $D_1$ and $D_2$ (orange) are centred at a vertex $a_0 \in A_i$, and have radius $\mathop{\mathrm{diameter}}\nolimits(A_i)$ and $2 \cdot \mathop{\mathrm{diameter}}\nolimits(A_i)$ respectively. The value of the max-flow/min-cut in the figure is $\ell = 4$, so $|C_i| = 4$ (grey).

Theorems & Definitions (48)

• Theorem 2
• Theorem 4
• Theorem 6
• Theorem 6
• Theorem 7
• Definition 8: SSPD
• Lemma 10
• proof
• Lemma 11
• proof