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We study several polygonal curve problems under the Fr\'{e}chet distance viaalgebraic geometric methods. Let $\mathbb{X}_m^d$ and $\mathbb{X}_k^d$ be thespaces 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 setof ranges in $\mathbb{X}_m^d$ for all possible metric balls of polygonal curvesin $\mathbb{X}_k^d$ under the Fr\'{e}chet distance. We prove a nearly optimalbound 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. Wealso obtain exact solutions that are hitherto unknown for curve simplification,range searching, nearest neighbor search, and distance oracle.

Solving Fr\'echet Distance Problems by Algebraic Geometric Methods