Geodesic Fréchet distance inside a simple polygon

We present an alternative to parametric search that applies to both the nongeodesic and geodesic Fréchet 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 introduce the first algorithm to compute the geodesic Fréchet distance between two polygonal curves A and B inside a simple bounding polygon P. The geodesic Fréchet decision problem is solved almost as fast as its nongeodesic sibling in O(N2 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 Fréchet optimization problem is solved by a randomized approach in O(k+N2 log kN log N) expected time and O(k+N2) space. This runtime is only a logarithmic factor larger than the standard nongeodesic Fréchet algorithm [Alt and Godau 1995]. Results are also presented for the geodesic Fréchet 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.