On Practical Nearest Sub-Trajectory Queries under the Fr\'echet Distance

We study the problem of sub-trajectory nearest-neighbor queries on polygonal curves under the continuous Fr\'echet distance. Given an $n$ vertex trajectory $P$ and an $m$ vertex query trajectory $Q$, we seek to report a vertex-aligned sub-trajectory $P'$ of $P$ that is closest to $Q$, i.e. $P'$ must start and end on contiguous vertices of $P$. Since in real data $P$ typically contains a very large number of vertices, we focus on answering queries, without restrictions on $P$ or $Q$, using only precomputed structures of ${\mathcal{O}}(n)$ size. We use three baseline algorithms from straightforward extensions of known work, however they have impractical performance on realistic inputs. Therefore, we propose a new Hierarchical Simplification Tree data structure and an adaptive clustering based query algorithm that efficiently explores relevant parts of $P$. The core of our query methods is a novel greedy-backtracking algorithm that solves the Fr\'echet decision problem using ${\cal O}(n+m)$ space and ${\cal O}(nm)$ time in the worst case. Experiments on real and synthetic data show that our heuristic effectively prunes the search space and greatly reduces computations compared to baseline approaches.