On the Approximate Nearest Neighbor Queries among Curves under the Fr\'echet Distance

Approximate near-neighbors search (\textsc{ANNS}) is a long-studied problem in computational geometry. %that has received considerable attention by researchers in the community. In this paper, we revisit the problem and propose the first data structure for curves under the (continuous) Fr\'echet distance in $\Reals^d$. Given a set $\P$ of $n$ curves of size at most $m$ each in $\Reals^d$, and a real fixed $\delta>0$, we aim to preprocess $\P$ into a data structure so that for any given query curve $Q$ of size $k$, we can efficiently report all curves in $\P$ whose Fr\'echet distances to $Q$ are at most $\delta$. In the case that $k$ is given in the preprocessing stage, for any $\eps>0$ we propose a deterministic data structure whose space is $n \cdot O\big(\max\big\{\big(\frac{\sqrt{d}}{\eps}\big)^{kd}, \big(\frac{\D\sqrt{d}}{\eps^2}\big)^{kd}\big\}\big)$ that can answer \textsc{$(1+\eps)\delta$-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 $n \cdot O\big(\frac{1}{\eps}\big)^{md}$ with $O(kd)$ query time within an approximation factor of $5+\eps$. 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. [8]. We also revisit the time-window data structure for spatial density maps in [6]. Given $\theta>0$, and $n$ time-stamped points spread over $m$ regions in a map, for any query window $W$, we propose a data structure of size $O(n/\eps^2)$ and construction time $O((n+m)/\eps^2)$ that can approximately return the regions containing at least $\theta$ points whose times are within $W$ in $O(1)$ query time.