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