Fr{'e}chet and Hausdorff Queries on $x$-Monotone Trajectories

vspace{0.2cm}In this paper, we design a data structure for the following problem. Let $pi$ be an $x$-monotone trajectory with $n$ vertices in the plane and $epsilon >0$. We show how to preprocess $pi$ and $epsilon$ into a data structure such that for any horizontal query segment $Q$ in the plane, one can quickly determine the minimal continuous fraction of $pi$ whose Fr{'e}chet and Hausdorff distance to the horizontal query segment $Q$ is at most some threshold value $epsilon$. We present a data structure for this query that needs $mathcal{O}(nlog{}n)$ preprocessing time, $mathcal{O}(n)$ space, and $mathcal{O}(log{} n)$ query time. & & vspace{0.2cm}