Solving Fréchet Distance Problems by Algebraic Geometric Methods

Abstract: We study several polygonal curve problems under the Fr\'{e}chet distance via algebraic geometric methods. Let $\mathbb{X}<em>m<sup>d$ and $\mathbb{X}_k<sup>d$ be the spaces of all polygonal curves of $m$ and $k$ vertices in $\mathbb{R}<sup>d$, respectively. We assume that $k \leq m$. Let $\mathcal{R}^d</em>{k,m}$ be the set of ranges in $\mathbb{X}<em>m<sup>d$ for all possible metric balls of polygonal curves in $\mathbb{X}_k<sup>d$ 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 $$(\mathbb{X}_m^{d,\mathcal{R}}</em>{k,m}<sup>d)$$, improving on the previous $O(d<sup>2k<sup>2\log(dkm))$ 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.