### Abstract

This paper studies the geodesic diameter of polygonal domains having h holes and n corners. For simple polygons (i.e., h = 0), the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time, as shown by Hershberger and Suri. For general polygonal domains with h ≥ 1, however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time O(n^{7.73}) or O(n^{7}(log n + h)). The main difficulty unlike the simple polygon case relies on the following observation revealed in this paper: two interior points can determine the geodesic diameter and in that case there exist at least five distinct shortest paths between the two.

Original language | English |
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Pages (from-to) | 306-329 |

Number of pages | 24 |

Journal | Discrete and Computational Geometry |

Volume | 50 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2013 Sep 1 |

Externally published | Yes |

### Keywords

- Convex function
- Exact algorithm
- Geodesic diameter
- Lower envelope
- Polygonal domain
- Shortest path

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Discrete and Computational Geometry*,

*50*(2), 306-329. https://doi.org/10.1007/s00454-013-9527-8