On approximate near-neighbors search under the (continuous) Fréchet distance in higher dimensions

Abstract

Previous studies on Approximate Near-Neighbors Search (ANNS) among curves are either focused on curves in R1 or under the discrete Fréchet distance. In this paper, we propose the first data structure for curves under the (continuous) Fréchet distance in higher dimensions. Given a set P of n curves each with number of vertices at most m in Rd, and a fixed δ>0, we aim to preprocess P into a data structure so that for any given query curve Q with k vertices, we can efficiently report all curves in P whose Fréchet distances to Q are at most δ. In the case that k is given in the preprocessing stage, for any ε>0 we propose a deterministic data structure whose space is n⋅O(max⁡{(dε)kd,(Ddε2)kd}) that can answer (1+ε)δ-ANNS queries in O(kd) query time, where D is the diameter of P. Considering k as part of the query slightly changes the space to n⋅O(1/ε)md with O(kd) query time within an approximation factor of 5+ε. Moreover, we show that our generic data structure for ANNS can give an alternative treatment of the approximate subtrajectory range searching problem studied by de Berg et al. [1].