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

Document Type : Research Paper


1 Yazd University, Iran

2 Department of Mathematical Sciences, Yazd University, Yazd, Iran


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 $\mathcal{O}(n\log{}n)$ preprocessing time, $\mathcal{O}(n)$ space, and $\mathcal{O}(\log{} n)$ query time. & & \vspace{0.2cm}