Star Unfolding of a Polytope with Applications

Abstract

We introduce the notion of a star unfolding of the surface ${\cal P}$ of a three-dimensional convex polytope with n vertices, and use it to solve several problems related to shortest paths on ${\cal P}$. The first algorithm computes the edge sequences traversed by shortest paths on ${\cal P}$ in time $O(n^6 \beta (n) \log n)$, where $\beta (n)$ is an extremely slowly growing function. A much simpler $O(n^6)$ time algorithm that finds a small superset of all such edge sequences is also sketched. The second algorithm is an $O(n^{8}\log n)$ time procedure for computing the geodesic diameter of ${\cal P}$: the maximum possible separation of two points on ${\cal P}$ with the distance measured along ${\cal P}$. Finally, we describe an algorithm that preprocesses ${\cal P}$ 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 $1 \le m \le n^2$, it can preprocess ${\cal P}$ in time $O(n^6 m^{1+\delta})$, for any $\delta > 0$, into a data structure of size $O(n^6m^{1+\delta})$, so that a query can be answered in time $O((\sqrt{n}/m^{1/4}) \log n)$. If one query point always lies on an edge of ${\cal P}$, the algorithm can be improved to use $O(n^5 m^{1+\delta})$ preprocessing time and storage and guarantee $O((n/m)^{1/3} \log n)$ query time for any choice of m between 1 and n.