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.

翻译：我们通过代数几何方法研究了Fréchet距离下的若干多边形曲线问题。令$\mathbb{X}_m^d$和$\mathbb{X}_k^d$分别表示$\mathbb{R}^d$中具有$m$个和$k$个顶点的所有多边形曲线构成的空间。假设$k \leq m$。设$\mathcal{R}^d_{k,m}$表示$\mathbb{X}_m^d$中所有可能的、由$\mathbb{X}_k^d$中多边形曲线在Fréchet距离下定义的度量球所构成的集合族。我们证明了范围空间$(\mathbb{X}_m^d,\mathcal{R}_{k,m}^d)$的VC维数具有接近最优的上界$O(dk\log (km))$，改进了之前$O(d^2k^2\log(dkm))$的上界，并逼近当前的$\Omega(dk\log k)$下界。该上界对弱Fréchet距离同样成立。此外，我们还获得了曲线简化、范围搜索、最近邻搜索和距离预言问题中此前未知的精确解。