Solving Proximity Problems for Polygonal Shapes via Polynomials

Abstract

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 distance measures, such as the Hausdorff distance, the Fr\'echet distance and the dynamic time warping distance. These range spaces appear in various applications like classification, simplification, range searching, density estimation and clustering when the instances are trajectories, time series or polygons. 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 each 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 a variant of dynamic time warping distance of polygonal curves using squared distances, our analysis directly yields an upper-bound of $O(\min(dk^2\log(m),dkm\log(k)))$. We also obtain exact solutions that are hitherto unknown for simplification, range searching, nearest neighbor search, and distance oracle.