Static and Streaming Data Structures for Fréchet Distance Queries

Given a curve P with points in \(\mathbb {R}^d \) in a streaming fashion, and parameters ε > 0 and k, we construct a distance oracle that uses \(O(\frac{1}{\varepsilon })^{kd}\log \varepsilon ^{-1} \) space, and given a query curve Q with k points in \(\mathbb {R}^d \), returns in \(\tilde{O}(kd) \) time a 1 + ε approximation of the discrete Fréchet distance between Q and P. In addition, we construct simplifications in the streaming model, oracle for distance queries to a sub-curve (in the static setting), and introduce the zoom-in problem. Our algorithms work in any dimension d, and therefore we generalize some useful tools and algorithms for curves under the discrete Fréchet distance to work efficiently in high dimensions.