### Abstract

This paper describes a new approach for constructing the Voronoi diagram of n points in the plane in parallel. Our approach is based on a divide-and-conquer procedure where we implement the “marry” step by merging forests of free trees (to build the “contour” between the subproblem solutions) in O(log log n) time. This merging procedure is based an a √n -divide-and-merge technique reminiscent of the list-merging approach of Valiant. Our method also involves an optimal parallel method for computing the proximity envelope of a point set with respect to a given line. This structure facilitates the use of our fast mering procedure, for it allows the divide-and-conquer procedure to continue without needing to explicitly remove edges of recursively constructed diagrams that are not part of the final diagram. We use this approach to derive two results regarding the deterministic parallel construction of a Voronoi diagram. Specifically, we show that one can solve the Voronoi diagram problem in O(log n log log n) time and O(n log^{2}n) work (which improves the previous time bound while maintaining the same work bound) or, alternatively, in O(log^{2}n) time and O(n log n) work (which improves the previous work bound while maintaining the same time bound). Our model of computation is the CREW PRAM.

Original language | English (US) |
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Title of host publication | Automata, Languages and Programming - l7th International Colloquium, Proceedings |

Publisher | Springer Verlag |

Pages | 432-445 |

Number of pages | 14 |

Volume | 443 LNCS |

ISBN (Print) | 9783540528265 |

State | Published - 1990 |

Event | 17th International Colloquium on Automata, Languages and Programming, 1990 - Warwick, United Kingdom Duration: Jul 16 1990 → Jul 20 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 | 443 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 17th International Colloquium on Automata, Languages and Programming, 1990 |
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Country | United Kingdom |

City | Warwick |

Period | 7/16/90 → 7/20/90 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Automata, Languages and Programming - l7th International Colloquium, Proceedings*(Vol. 443 LNCS, pp. 432-445). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 443 LNCS). Springer Verlag.

**Merging free trees in parallel for efficient voronoi diagram construction.** / Cole, Richard; Goodrich, Michael T.; Dúnlaing, Colm.

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

*Automata, Languages and Programming - l7th International Colloquium, Proceedings.*vol. 443 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 443 LNCS, Springer Verlag, pp. 432-445, 17th International Colloquium on Automata, Languages and Programming, 1990, Warwick, United Kingdom, 7/16/90.

}

TY - GEN

T1 - Merging free trees in parallel for efficient voronoi diagram construction

AU - Cole, Richard

AU - Goodrich, Michael T.

AU - Dúnlaing, Colm

PY - 1990

Y1 - 1990

N2 - This paper describes a new approach for constructing the Voronoi diagram of n points in the plane in parallel. Our approach is based on a divide-and-conquer procedure where we implement the “marry” step by merging forests of free trees (to build the “contour” between the subproblem solutions) in O(log log n) time. This merging procedure is based an a √n -divide-and-merge technique reminiscent of the list-merging approach of Valiant. Our method also involves an optimal parallel method for computing the proximity envelope of a point set with respect to a given line. This structure facilitates the use of our fast mering procedure, for it allows the divide-and-conquer procedure to continue without needing to explicitly remove edges of recursively constructed diagrams that are not part of the final diagram. We use this approach to derive two results regarding the deterministic parallel construction of a Voronoi diagram. Specifically, we show that one can solve the Voronoi diagram problem in O(log n log log n) time and O(n log2n) work (which improves the previous time bound while maintaining the same work bound) or, alternatively, in O(log2n) time and O(n log n) work (which improves the previous work bound while maintaining the same time bound). Our model of computation is the CREW PRAM.

AB - This paper describes a new approach for constructing the Voronoi diagram of n points in the plane in parallel. Our approach is based on a divide-and-conquer procedure where we implement the “marry” step by merging forests of free trees (to build the “contour” between the subproblem solutions) in O(log log n) time. This merging procedure is based an a √n -divide-and-merge technique reminiscent of the list-merging approach of Valiant. Our method also involves an optimal parallel method for computing the proximity envelope of a point set with respect to a given line. This structure facilitates the use of our fast mering procedure, for it allows the divide-and-conquer procedure to continue without needing to explicitly remove edges of recursively constructed diagrams that are not part of the final diagram. We use this approach to derive two results regarding the deterministic parallel construction of a Voronoi diagram. Specifically, we show that one can solve the Voronoi diagram problem in O(log n log log n) time and O(n log2n) work (which improves the previous time bound while maintaining the same work bound) or, alternatively, in O(log2n) time and O(n log n) work (which improves the previous work bound while maintaining the same time bound). Our model of computation is the CREW PRAM.

UR - http://www.scopus.com/inward/record.url?scp=33646465247&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646465247&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9783540528265

VL - 443 LNCS

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

SP - 432

EP - 445

BT - Automata, Languages and Programming - l7th International Colloquium, Proceedings

PB - Springer Verlag

ER -