# The VC Dimension of Metric Balls under Fr\'echet and Hausdorff Distances

## TL;DR

This paper investigates the VC dimension of metric balls under Fréchet and Hausdorff distances for sets of polygonal curves, providing bounds that facilitate efficient sampling-based algorithms in computational geometry.

## Contribution

It derives upper and lower bounds on the VC dimension for set systems of polygonal curves with respect to these metrics, advancing understanding of their complexity.

## Key findings

- Upper bounds are near-quadratic or near-linear in curve complexity.
- Bounds are logarithmic in the complexity of the ground set curves.
- Results enable improved sampling bounds for large sets of simple curves.

## Abstract

The Vapnik-Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set $X$ is a set of polygonal curves in $\mathbb{R}^d$ and the sets $\mathcal{R}$ are metric balls defined by curve similarity metrics, such as the Fr\'echet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.