### Abstract

This paper presents a worst-case optimal algorithm for constructing the Voronoi diagram for n disjoint convex and rounded semi-algebraic sites in 3 dimensions. Rather than extending optimal 2-dimensional methods, ^{32,16,20,2} we base our method on a suboptimal 2-dimensional algorithm, outlined by Lee and Drysdale and modified by Sharir^{25,30} for computing the diagram of circular sites. For complexity considerations, we assume the sites have bounded complexity, i.e., the algebraic degree is bounded as is the number of algebraic patches making up the site. For the sake of simplicity we assume that the sites are what we call rounded. This assumption simplifies the analysis, though it is probably unnecessary. Our algorithm runs in time O(C(n)) where C(n) is the worst-case complexity of an n-site diagram. For spherical sites C(n) is θ(n^{2}), but sharp estimates do not seem to be available for other classes of site.

Original language | English (US) |
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Pages (from-to) | 555-593 |

Number of pages | 39 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 17 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2007 |

### Keywords

- Computational geometry
- Convex sets
- Voronoi diagrams

### ASJC Scopus subject areas

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

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## Cite this

*International Journal of Computational Geometry and Applications*,

*17*(6), 555-593. https://doi.org/10.1142/S0218195907002483