### 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) |
---|---|

Pages (from-to) | 1689-1713 |

Number of pages | 25 |

Journal | SIAM Journal on Computing |

Volume | 26 |

Issue number | 6 |

State | Published - Dec 1997 |

### Fingerprint

### Keywords

- Convex polytopes
- Geodesics
- Shortest paths
- Star unfolding

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics
- Theoretical Computer Science

### Cite this

*SIAM Journal on Computing*,

*26*(6), 1689-1713.

**Star unfolding of a polytope with applications.** / Agarwal, Pankaj K.; Aronov, Boris; O'Rourke, Joseph; Schevon, Catherine A.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 26, no. 6, pp. 1689-1713.

}

TY - JOUR

T1 - Star unfolding of a polytope with applications

AU - Agarwal, Pankaj K.

AU - Aronov, Boris

AU - O'Rourke, Joseph

AU - Schevon, Catherine A.

PY - 1997/12

Y1 - 1997/12

N2 - 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(n6β(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(n8 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 ≤ n2, it can preprocess P in time O(n6m1+δ), for any δ > O, 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(n5m1+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.

AB - 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(n6β(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(n8 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 ≤ n2, it can preprocess P in time O(n6m1+δ), for any δ > O, 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(n5m1+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.

KW - Convex polytopes

KW - Geodesics

KW - Shortest paths

KW - Star unfolding

UR - http://www.scopus.com/inward/record.url?scp=0013013983&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0013013983&partnerID=8YFLogxK

M3 - Article

VL - 26

SP - 1689

EP - 1713

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 6

ER -