We study range spaces, where the ground set consists of either polygonal curves in $\mathbb{R}^d$ or polygonal regions in the plane that may contain holes and the ranges are balls defined by an elastic distance measure, such as the Hausdorff distance, the Fr\'echet distance and the dynamic time warping distance. The range spaces appear in various applications like classification, range counting, density estimation and clustering when the instances are trajectories, time series or polygons. The Vapnik-Chervonenkis dimension (VC-dimension) plays an important role when designing algorithms for these range spaces. We show for the Fr\'echet distance of polygonal curves and the Hausdorff distance of polygonal curves and planar polygonal regions that the VC-dimension is upper-bounded by $O(dk\log(km))$ where $k$ is the complexity of the center of a ball, $m$ is the complexity of the polygonal curve or region in the ground set, and $d$ is the ambient dimension. For $d \geq 4$ this bound is tight in each of the parameters $d, k$ and $m$ separately. For the dynamic time warping distance of polygonal curves, our analysis directly yields an upper-bound of $O(\min(dk^2\log(m),dkm\log(k)))$.

翻译：我们研究了范围空间，其中基础集由 $\mathbb{R}^d$ 中的多边形曲线或可能包含孔的平面多边形区域组成，范围由弹性距离度量（如豪斯多夫距离、弗雷歇距离和动态时间规整距离）定义的球构成。这些范围空间出现在各种应用中，如分类、范围计数、密度估计和聚类，当实例是轨迹、时间序列或多边形时。Vapnik-Chervonenkis维数（VC维）在为这些范围空间设计算法时起着重要作用。我们证明，对于多边形曲线的弗雷歇距离以及多边形曲线和平面多边形区域的豪斯多夫距离，VC维的上界为 $O(dk\log(km))$，其中 $k$ 是球中心的复杂度，$m$ 是基础集中多边形曲线或区域的复杂度，$d$ 是环境维度。对于 $d \geq 4$，该界在参数 $d$、$k$ 和 $m$ 上分别是最紧的。对于多边形曲线的动态时间规整距离，我们的分析直接得出上界 $O(\min(dk^2\log(m),dkm\log(k)))$。