## Abstract

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(n^{6}β(n) log n), where β(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 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 ≤ n^{2}, it can preprocess P in time O(n^{6}m^{1+δ}), for any δ > O, into a data structure of size O(n^{6}m^{1+δ}), so that a query can be answered in time O((√n/m^{1/4}) log n). If one query point always lies on an edge of P, the algorithm can be improved to use O(n^{5}m^{1+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.

Original language | English (US) |
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Pages (from-to) | 1689-1713 |

Number of pages | 25 |

Journal | SIAM Journal on Computing |

Volume | 26 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1997 |

## Keywords

- Convex polytopes
- Geodesics
- Shortest paths
- Star unfolding

## ASJC Scopus subject areas

- General Computer Science
- General Mathematics