On Practical Nearest Sub-Trajectory Queries under the Fréchet Distance

Abstract

We study the problem of sub-trajectory nearest-neighbor queries on polygonal curves under the continuous Fréchet 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 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 (HST) 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 Fréchet decision problem using 𝒪(n+m) space and 𝒪O(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.