The Geodesic Fréchet Distance Between Two Curves Bounding a Simple Polygon

Abstract

The Fréchet distance is a popular similarity measure that is well-understood for polygonal curves in ${\mathbb{R}}^d$: near-quadratic time algorithms exist, and conditional lower bounds suggest that these results cannot be improved significantly, even in one dimension and when approximating with a factor less than three. We consider the special case where the curves bound a simple polygon and distances are measured via geodesics inside this simple polygon. Here the conditional lower bounds do not apply; Efrat $et$ $al.$ (2002) were able to give a near-linear time $2$-approximation algorithm. In this paper, we significantly improve upon their result: we present a $(1+\epsilon )$-approximation algorithm, for any $\epsilon >0$, that runs in $\mathcal{O}(\frac{1}{\epsilon }(n+m\log n)\log nm\log \frac{1}{\epsilon })$ time for a simple polygon bounded by two curves with $n$ and $m$ vertices, respectively. To do so, we show how to compute the reachability of specific groups of points in the free space at once, by interpreting the free space as one between separated one-dimensional curves. We solve this one-dimensional problem in near-linear time, generalizing a result by Bringmann and Künnemann (2015). Finally, we give a linear time exact algorithm if the two curves bound a convex polygon.

26 pages, 10 figures