Simplified and Improved Bounds on the VC-Dimension for Elastic Distance Measures

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.