Geodesic Fréchet Distance Inside a Simple Polygon

Abstract

We unveil an alluring alternative to parametric search that applies
   to both the non-geodesic and geodesic Fr{'\e}chet 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{'\e}chet distance
   between two polygonal curves $A$ and $B$ inside a simple bounding
   polygon $P$.  The geodesic Fr{'\e}chet decision problem is solved
   almost as fast as its non-geodesic sibling and requires $O(N^{2log
   k)$ 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{'\e}chet optimization problem is
   solved by a randomized approach in $O(k+N^{2log kNlog N)$
   expected time and $O(k+N^{2)$ space.  This runtime is only a
   logarithmic factor larger than the standard non-geodesic Fr{'\e}chet
   algorithm (Alt and Godau 1995).  Results are also presented for the
   geodesic Fr{'\e}chet 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 $P$.