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

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 |

### Fingerprint

### Keywords

- Computational geometry
- Convex sets
- Voronoi diagrams

### ASJC Scopus subject areas

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

### Cite this

*International Journal of Computational Geometry and Applications*,

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

**Optimal voronoi diagram construction with n convex sites in three dimensions.** / Harrington, Paul; Dúnlaing, C. Ó; Yap, Chee.

Research output: Contribution to journal › Article

*International Journal of Computational Geometry and Applications*, vol. 17, no. 6, pp. 555-593. https://doi.org/10.1142/S0218195907002483

}

TY - JOUR

T1 - Optimal voronoi diagram construction with n convex sites in three dimensions

AU - Harrington, Paul

AU - Dúnlaing, C. Ó

AU - Yap, Chee

PY - 2007/12

Y1 - 2007/12

N2 - 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 Sharir25,30 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(n2), but sharp estimates do not seem to be available for other classes of site.

AB - 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 Sharir25,30 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(n2), but sharp estimates do not seem to be available for other classes of site.

KW - Computational geometry

KW - Convex sets

KW - Voronoi diagrams

UR - http://www.scopus.com/inward/record.url?scp=37749029444&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37749029444&partnerID=8YFLogxK

U2 - 10.1142/S0218195907002483

DO - 10.1142/S0218195907002483

M3 - Article

AN - SCOPUS:37749029444

VL - 17

SP - 555

EP - 593

JO - International Journal of Computational Geometry and Applications

JF - International Journal of Computational Geometry and Applications

SN - 0218-1959

IS - 6

ER -