Translation Invariant Fréchet Distance Queries

The Fréchet distance is a popular similarity measure between curves. For some applications, it desirable to match the curves under translation before computing them. This variant called Translation Invariant distance, and algorithms compute are well studied. query version, finding an optimal placement in plane for segment where becomes minimized, much less understood. We study queries restricted setting of horizontal segments. More specifically, we preprocess trajectory $${\mathcal {O}}(n^2 \log ^2 n)$$ O ( n 2 log ) time {O}}(n^{3/2})$$ 3 / space, such that any subtrajectory can their {O}}({{\,\mathrm{polylog}\,}}n)$$ polylog time. hope this will be step towards answering arbitrary trajectories.