Point to face shortest paths in simple polytopes with applications in structural proteomics

We study the following problem. Given a simple polytope S in R
³
, with a total of n edges, and a query point s on S, find a shortest path from s to the boundary of the convex hull, CH(S), of S, that does not go through the interior of S. The problem appears in structural proteomics in the computation of shape descriptors for measuring the depth of a point on a surface. We present an algorithm with running time O(n
³
(λ(n)log(n/∈)/∈
⁴
+log(nρ) log(n log ρ))), that can find a path from s to the boundary of CH(S) that has length at most (1+∈) times the length of a shortest path from s to the boundary of CH(S).