Star Unfolding of a Polytope with Applications

We introduce the notion of a star unfolding of the surface P of a three-dimensional convex polytope with n vertices, and use it to solve several problems related to shortest paths on P. The first algorithm computes the edge sequences traversed by shortest paths on P in time O(n⁶ β (n) log n), where β (n) is an extremely slowly growing function. A much simpler O(n⁶) time algorithm that finds a small superset of all such edge sequences is also sketched. The second algorithm is an O(n⁸log n) time procedure for computing the geodesic diameter of P: the maximum possible separation of two points on P with the distance measured along P. Finally, we describe an algorithm that preprocesses P into a data structure that can efficiently answer the queries of the following form: "Given two points, what is the length of the shortest path connecting them?" Given a parameter 1 ≤ m ≤ n², it can preprocess P in time O(n⁶ m1+δ), for any δ > 0, into a data structure of size O(n⁶m1+δ), so that a query can be answered in time O((√n/m1/4) log n). If one query point always lies on an edge of P, the algorithm can be improved to use O(n⁵ m1+δ) preprocessing time and storage and guarantee O((n/m)1/3 log n) query time for any choice of m between 1 and n.