Practical Methods for Approximating Shortest Paths on a Convex Polytope in R3

We propose a n extremely simple approximation scheme for computing shortest paths on the surface of a convex polytope in three dimensions. Given a convex polytope P with n vertices and two points p,q on its surface, let d{subscript p}(p,q) denote the shortest path distance between p and q on the surface of P. Our algorithm produces a path of length at most 2*d{subscript p}(p,q) in time O(n). Extending this results, we can also compute ana pproximation of the shortest path tree rooted at an arbitrary point x {lying in} P in time O(nlogn). In the approximate tree, the distance between a vertex v {lying in} P and x is at most c*d{subscript p}(x,v), where c = 2.38(1+{epsilon}) for any fixed {epsilon} > 0. The best algorithm for computing an exact shortest path on a convex polytopt take {omega}(n squared) tiem in the worst case; in addition, they are too complicated to be suitable in practice. We can also get a weak approximation result in the general case of k disjoint convex polyhedra: in... Read complete abstract on page 2.