210 research outputs found
How to Walk Your Dog in the Mountains with No Magic Leash
We describe a -approximation algorithm for computing the
homotopic \Frechet distance between two polygonal curves that lie on the
boundary of a triangulated topological disk. Prior to this work, algorithms
were known only for curves on the Euclidean plane with polygonal obstacles.
A key technical ingredient in our analysis is a -approximation
algorithm for computing the minimum height of a homotopy between two curves. No
algorithms were previously known for approximating this parameter.
Surprisingly, it is not even known if computing either the homotopic \Frechet
distance, or the minimum height of a homotopy, is in NP
Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics
Using the geodesic distance on the -dimensional sphere, we study the
expected radius function of the Delaunay mosaic of a random set of points.
Specifically, we consider the partition of the mosaic into intervals of the
radius function and determine the expected number of intervals whose radii are
less than or equal to a given threshold. Assuming the points are not contained
in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of
the convex hull in , so we also get the expected number of
faces of a random inscribed polytope. We find that the expectations are
essentially the same as for the Poisson-Delaunay mosaic in -dimensional
Euclidean space. As proved by Antonelli and collaborators, an orthant section
of the -sphere is isometric to the standard -simplex equipped with the
Fisher information metric. It follows that the latter space has similar
stochastic properties as the -dimensional Euclidean space. Our results are
therefore relevant in information geometry and in population genetics
Shortest Path Problems on a Polyhedral Surface
We develop algorithms to compute shortest path edge sequences, Voronoi diagrams, the Fréchet distance, and the diameter for a polyhedral surface
- …