2023/08/11 by Frederik Brüning, Anne Driemel, Brüning, Frederik +1 · 1 citation
Computer Science · #Anomaly Detection Techniques and Applications #Computational Geometry (cs.CG) #Data Management and Algorithms #FOS: Computer and information sciences #Time Series Analysis and Forecasting
paper · pdf · doi:10.48550/arxiv.2308.05998
openalex publication_date 2023/08/11 · openalex created_date 2023/08/15 · openalex updated_date 2026/07/28
We study range spaces, where the ground set consists of either polygonal curves in ℝd 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 Fréchet 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 Fréchet distance of polygonal curves and the Hausdorff distance of polygonal curves and planar polygonal regions that the VC-dimension is upper-bounded by O(dklog(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 ≥ 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(dk2log(m),dkmlog(k))).