Static and Streaming Data Structures for Fr\'echet Distance Queries

TL;DR

This paper develops efficient data structures and algorithms for approximating the discrete Fréchet distance between curves in high-dimensional spaces, applicable in streaming and static models, with applications to curve simplification and sub-curve queries.

Contribution

It introduces new streaming and static data structures for approximate Fréchet distance queries in high dimensions, generalizing previous methods to work efficiently in any dimension.

Findings

Space complexity is $O((1/\varepsilon)^{kd}\log\varepsilon^{-1})$ for the distance oracle.

Queries return a $1+\varepsilon$ approximation in $\tilde{O}(kd)$ time.

Algorithms work efficiently in high-dimensional spaces.

Abstract

Given a curve $P$ with points in $\mathbb{R}^d$ in a streaming fashion, and parameters $\varepsilon>0$ 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 $1+\varepsilon$ approximation of the discrete Fr\'echet 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 Fr\'echet distance to work efficiently in high dimensions.