Geodesic Fr\'echet Distance Inside a Simple Polygon

We unveil an alluring alternative to parametric search that applies to both the non-geodesic and geodesic Fr\'echet optimization problems. This randomized approach is based on a variant of red-blue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search. We present the first algorithm for the geodesic Fr\'echet distance between two polygonal curves $A$ and $B$ inside a simple bounding polygon $P$. The geodesic Fr\'echet decision problem is solved almost as fast as its non-geodesic sibling and requires $O(N^{2\log k)$ time and $O(k+N)$ space after $O(k)$ preprocessing, where $N$ is the larger of the complexities of $A$ and $B$ and $k$ is the complexity of $P$. The geodesic Fr\'echet optimization problem is solved by a randomized approach in $O(k+N^{2\log kN\log N)$ expected time and $O(k+N^{2)$ space. This runtime is only a logarithmic factor larger than the standard non-geodesic Fr\'echet algorithm (Alt and Godau 1995). Results are also presented for the geodesic Fr\'echet distance in a polygonal domain with obstacles and the geodesic Hausdorff distance for sets of points or sets of line segments inside a simple polygon $P$.