$(1+\varepsilon)$-ANN Data Structure for Curves via Subspaces of Bounded Doubling Dimension

TL;DR

This paper develops a data structure for approximate nearest neighbor search among polygonal curves under the Fréchet distance by identifying a subspace with bounded doubling dimension, enabling efficient querying.

Contribution

It introduces a novel approach to handle the unbounded doubling dimension in curve spaces by focusing on a subspace with bounded doubling dimension and small Gromov-Hausdorff distance.

Findings

Achieves efficient $(1+ ext{varepsilon})$-ANN queries for polygonal curves.

Provides bounds on preprocessing time, space, and query time based on curve parameters.

Extends results to $c$-packed curves with improved bounds for small $c$.

Abstract

We consider the $(1+\varepsilon)$-Approximate Nearest Neighbour (ANN) Problem for polygonal curves in $d$-dimensional space under the Fr\'echet distance and ask to what extent known data structures for doubling spaces can be applied to this problem. Initially, this approach does not seem viable, since the doubling dimension of the target space is known to be unbounded -- even for well-behaved polygonal curves of constant complexity in one dimension. In order to overcome this, we identify a subspace of curves which has bounded doubling dimension and small Gromov-Hausdorff distance to the target space. We then apply state-of-the-art techniques for doubling spaces and show how to obtain a data structure for the $(1+\varepsilon)$-ANN problem for any set of parametrized polygonal curves. The expected preprocessing time needed to construct the data-structure is $F(d,k,S,\varepsilon)n\log n$ and the space used is $F(d,k,S,\varepsilon)n$, with a query time of $F(d,k,S,\varepsilon)\log n + F(d,k,S,\varepsilon)^{-\log(\varepsilon)}$, where $F(d,k,S,\varepsilon)=O\left(2^{O(d)}k\Phi(S)\varepsilon^{-1}\right)^k$ and $\Phi(S)$ denotes the spread of the set of vertices and edges of the curves in $S$. We extend these results to the realistic class of $c$-packed curves and show improved bounds for small values of $c$.