Static and Streaming Data Structures for Fréchet Distance Queries

Abstract

Given a curve P with points in ℝ d in a streaming fashion, and parameters ɛ > 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 ℝ d returns in \(\tilde{O}(kd)\) time a 1+ɛ 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.