Solving Fr\'echet Distance Problems by Algebraic Geometric Methods

TL;DR

This paper applies algebraic geometric methods to analyze polygonal curve problems under the Fréchet distance, providing new bounds on VC dimension and exact solutions for various geometric problems.

Contribution

It introduces a nearly optimal VC dimension bound for range spaces of polygonal curves under the Fréchet distance and offers exact solutions for key geometric problems.

Findings

Established a VC dimension bound of O(dk log (km)) for range spaces.

Improved previous VC bounds from O(d^2k^2 log(dkm)).

Provided exact algorithms for curve simplification and nearest neighbor search.

Abstract

We study several polygonal curve problems under the Fr\'{e}chet distance via algebraic geometric methods. Let $\mathbb{X}_m^d$ and $\mathbb{X}_k^d$ be the spaces of all polygonal curves of $m$ and $k$ vertices in $\mathbb{R}^d$, respectively. We assume that $k \leq m$. Let $\mathcal{R}^d_{k,m}$ be the set of ranges in $\mathbb{X}_m^d$ for all possible metric balls of polygonal curves in $\mathbb{X}_k^d$ under the Fr\'{e}chet distance. We prove a nearly optimal bound of $O(dk\log (km))$ on the VC dimension of the range space $(\mathbb{X}_m^d,\mathcal{R}_{k,m}^d)$, improving on the previous $O(d^2k^2\log(dkm))$ upper bound and approaching the current $\Omega(dk\log k)$ lower bound. Our upper bound also holds for the weak Fr\'{e}chet distance. We also obtain exact solutions that are hitherto unknown for curve simplification, range searching, nearest neighbor search, and distance oracle.