We study range spaces, where the ground set consists of polygonal curves and the ranges are balls defined by an elastic distance measure. Such range spaces appear in various applications like classification, range counting, density estimation and clustering when the instances are trajectories or time series. The Vapnik-Chervonenkis dimension (VC-dimension) plays an important role when designing algorithms for these range spaces. We show for the Fr\'echet distance and the Hausdorff distance 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 curve 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. Our approach rests on an argument that was first used by Goldberg and Jerrum and later improved by Anthony and Bartlett. The idea is to interpret the ranges as combinations of sign values of polynomials and to bound the growth function via the number of connected components in an arrangement of zero sets of polynomials.

翻译：我们研究一类范围空间，其中基础集由多边形曲线组成，范围是由弹性距离度量定义的球体。当实例为轨迹或时间序列时，此类范围空间出现在分类、范围计数、密度估计和聚类等多种应用中。Vapnik-Chervonenkis维数（VC维）在设计这类范围空间的算法中起着重要作用。我们针对Fréchet距离和Hausdorff距离证明，VC维的上界为$O(dk \log(km))$，其中$k$是球心的复杂度，$m$是基础集中曲线的复杂度，$d$是环境维度。对于$d \geq 4$，该界在参数$d$、$k$和$m$上各自是紧的。我们的方法基于Goldberg和Jerrum首次使用、后由Anthony和Bartlett改进的论证。其核心思想是将范围解释为多项式符号值的组合，并通过多项式零点集排列中的连通分量数量来限制生长函数。