### Abstract

We define the notion of a “star unfolding” of the surface P of a convex polytope with n vertices and use it to construct an algorithm for computing a small superset of the set of all sequences of edges traversed by shortest paths on P. It requires O(n^{6}) time and produces O(n^{8}) sequences, thereby improving an earlier algorithm of Sharir that in O(n^{8} log n) time produces O(n^{7}) sequences, A variant of our algorithm runs in O(n^{5} log n) time and produces a more compact representation of size O(n^{5}) for the same set of O(n^{6}) sequences. In addition, we describe an O(n^{10}) time procedure for computing the geodesic diameter of P, which is the maximum possible separation of two points on P, with the distance measured along P, improving an earlier O(n^{14} log n) algorithm of O’Rourke and Schevon.

Original language | English (US) |
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Title of host publication | SWAT 1990 - 2nd Scandinavian Workshop on Algorithm Theory, Proceedings |

Publisher | Springer Verlag |

Pages | 251-263 |

Number of pages | 13 |

Volume | 447 LNCS |

ISBN (Print) | 9783540528463 |

DOIs | |

State | Published - 1990 |

Event | 2nd Scandinavian Workshop on Algorithm Theory, SWAT 1990 - Bergen, Norway Duration: Jul 11 1990 → Jul 14 1990 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 447 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 2nd Scandinavian Workshop on Algorithm Theory, SWAT 1990 |
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Country | Norway |

City | Bergen |

Period | 7/11/90 → 7/14/90 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*SWAT 1990 - 2nd Scandinavian Workshop on Algorithm Theory, Proceedings*(Vol. 447 LNCS, pp. 251-263). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 447 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-52846-6_94

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*SWAT 1990 - 2nd Scandinavian Workshop on Algorithm Theory, Proceedings.*vol. 447 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 447 LNCS, Springer Verlag, pp. 251-263, 2nd Scandinavian Workshop on Algorithm Theory, SWAT 1990, Bergen, Norway, 7/11/90. https://doi.org/10.1007/3-540-52846-6_94

}

TY - GEN

T1 - Star unfolding of a polytope with applications

AU - Agarwal, Pankaj K.

AU - Aronov, Boris

AU - O’Rourke, Joseph

AU - Schevon, Catherine A.

PY - 1990

Y1 - 1990

N2 - We define the notion of a “star unfolding” of the surface P of a convex polytope with n vertices and use it to construct an algorithm for computing a small superset of the set of all sequences of edges traversed by shortest paths on P. It requires O(n6) time and produces O(n8) sequences, thereby improving an earlier algorithm of Sharir that in O(n8 log n) time produces O(n7) sequences, A variant of our algorithm runs in O(n5 log n) time and produces a more compact representation of size O(n5) for the same set of O(n6) sequences. In addition, we describe an O(n10) time procedure for computing the geodesic diameter of P, which is the maximum possible separation of two points on P, with the distance measured along P, improving an earlier O(n14 log n) algorithm of O’Rourke and Schevon.

AB - We define the notion of a “star unfolding” of the surface P of a convex polytope with n vertices and use it to construct an algorithm for computing a small superset of the set of all sequences of edges traversed by shortest paths on P. It requires O(n6) time and produces O(n8) sequences, thereby improving an earlier algorithm of Sharir that in O(n8 log n) time produces O(n7) sequences, A variant of our algorithm runs in O(n5 log n) time and produces a more compact representation of size O(n5) for the same set of O(n6) sequences. In addition, we describe an O(n10) time procedure for computing the geodesic diameter of P, which is the maximum possible separation of two points on P, with the distance measured along P, improving an earlier O(n14 log n) algorithm of O’Rourke and Schevon.

UR - http://www.scopus.com/inward/record.url?scp=85031907378&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85031907378&partnerID=8YFLogxK

U2 - 10.1007/3-540-52846-6_94

DO - 10.1007/3-540-52846-6_94

M3 - Conference contribution

SN - 9783540528463

VL - 447 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 251

EP - 263

BT - SWAT 1990 - 2nd Scandinavian Workshop on Algorithm Theory, Proceedings

PB - Springer Verlag

ER -