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Fréchet距离下的若干多边形曲线问题。设 $\mathbb{X}_m^d$ 和 $\mathbb{X}_k^d$ 分别为 $\mathbb{R}^d$ 中具有 $m$ 个顶点和 $k$ 个顶点的多边形曲线空间，并假设 $k \leq m$。令 $\mathcal{R}^d_{k,m}$ 为 $\mathbb{X}_m^d$ 中由 $\mathbb{X}_k^d$ 内多边形曲线在Fr\'{e}chet距离下所有可能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维具有 $O(dk\log (km))$ 的近乎最优上界，将此前 $O(d^2k^2\log(dkm))$ 的上界改进至接近当前 $\Omega(dk\log k)$ 下界。该上界对于弱Fr\'{e}chet距离同样成立。此外，我们还在曲线简化、范围搜索、最近邻搜索及距离预言机等问题中获得了此前未知的精确解。