### Abstract

In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E^{4}. Our algorithm runs in O((n + f) log^{2} f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f. By a standard lifting map, we obtain output-sensitive algorithms for the Voronoi diagram or Delaunay triangulation in E^{3} and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the "ultimate convex hull algorithm" of Kirkpatrick and Seidel in E^{2} and also leads to improved output-sensitive results on constructing convex hulls in E^{d} for any even constant d > 4.

Original language | English (US) |
---|---|

Pages (from-to) | 433-454 |

Number of pages | 22 |

Journal | Discrete and Computational Geometry |

Volume | 18 |

Issue number | 4 |

State | Published - 1997 |

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### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*,

*18*(4), 433-454.

**Primal dividing and dual pruning : Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams.** / Chan, T. M.; Snoeyink, J.; Yap, Chee.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 18, no. 4, pp. 433-454.

}

TY - JOUR

T1 - Primal dividing and dual pruning

T2 - Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams

AU - Chan, T. M.

AU - Snoeyink, J.

AU - Yap, Chee

PY - 1997

Y1 - 1997

N2 - In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E4. Our algorithm runs in O((n + f) log2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f. By a standard lifting map, we obtain output-sensitive algorithms for the Voronoi diagram or Delaunay triangulation in E3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the "ultimate convex hull algorithm" of Kirkpatrick and Seidel in E2 and also leads to improved output-sensitive results on constructing convex hulls in Ed for any even constant d > 4.

AB - In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E4. Our algorithm runs in O((n + f) log2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f. By a standard lifting map, we obtain output-sensitive algorithms for the Voronoi diagram or Delaunay triangulation in E3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the "ultimate convex hull algorithm" of Kirkpatrick and Seidel in E2 and also leads to improved output-sensitive results on constructing convex hulls in Ed for any even constant d > 4.

UR - http://www.scopus.com/inward/record.url?scp=0031522632&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031522632&partnerID=8YFLogxK

M3 - Article

VL - 18

SP - 433

EP - 454

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 4

ER -