### Abstract

Many facility location problems are concerned with minimizing operation and transportation costs by partitioning territory into regions of similar size, each of which is served by a facility. For many optimization problems, the overall cost can be reduced by means of a partitioning into balanced subsets, especially in those cases where the cost associated with a subset is superlinear in its size. In this paper, we consider the problem of generating a Voronoi partition of a discrete graph so as to achieve balance conditions on the region sizes. Through experimentation, we first establish that the region sizes of randomly-generated graph Voronoi diagrams vary greatly in practice.We then show how to achieve a balanced partition of a graph via Voronoi site resampling. For bounded-degree graphs, where each of the n nodes has degree at most d, and for an initial randomly-chosen set of s Voronoi nodes, we prove that, by extending the set of Voronoi nodes using an algorithm by Thorup and Zwick, each Voronoi region has size at most 4dn/s+1 nodes, and that the expected size of the extended set of Voronoi nodes is at most 2s log n.

Original language | English |
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Title of host publication | 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009 |

Pages | 183-191 |

Number of pages | 9 |

DOIs | |

Publication status | Published - 2009 Dec 1 |

Externally published | Yes |

Event | 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009 - Copenhagen, Denmark Duration: 2009 Jun 23 → 2009 Jun 26 |

### Other

Other | 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009 |
---|---|

Country | Denmark |

City | Copenhagen |

Period | 09/6/23 → 09/6/26 |

### Fingerprint

### Keywords

- Balancing
- Facility location
- Graph Voronoi diagram
- Territorial design

### ASJC Scopus subject areas

- Information Systems
- Biomedical Engineering
- Applied Mathematics

### Cite this

*6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009*(pp. 183-191). [5362361] https://doi.org/10.1109/ISVD.2009.26

**Balancing graph Voronoi diagrams.** / Honiden, Shinichi; Houle, Michael E.; Sommer, Christian.

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

*6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009.*, 5362361, pp. 183-191, 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009, Copenhagen, Denmark, 09/6/23. https://doi.org/10.1109/ISVD.2009.26

}

TY - GEN

T1 - Balancing graph Voronoi diagrams

AU - Honiden, Shinichi

AU - Houle, Michael E.

AU - Sommer, Christian

PY - 2009/12/1

Y1 - 2009/12/1

N2 - Many facility location problems are concerned with minimizing operation and transportation costs by partitioning territory into regions of similar size, each of which is served by a facility. For many optimization problems, the overall cost can be reduced by means of a partitioning into balanced subsets, especially in those cases where the cost associated with a subset is superlinear in its size. In this paper, we consider the problem of generating a Voronoi partition of a discrete graph so as to achieve balance conditions on the region sizes. Through experimentation, we first establish that the region sizes of randomly-generated graph Voronoi diagrams vary greatly in practice.We then show how to achieve a balanced partition of a graph via Voronoi site resampling. For bounded-degree graphs, where each of the n nodes has degree at most d, and for an initial randomly-chosen set of s Voronoi nodes, we prove that, by extending the set of Voronoi nodes using an algorithm by Thorup and Zwick, each Voronoi region has size at most 4dn/s+1 nodes, and that the expected size of the extended set of Voronoi nodes is at most 2s log n.

AB - Many facility location problems are concerned with minimizing operation and transportation costs by partitioning territory into regions of similar size, each of which is served by a facility. For many optimization problems, the overall cost can be reduced by means of a partitioning into balanced subsets, especially in those cases where the cost associated with a subset is superlinear in its size. In this paper, we consider the problem of generating a Voronoi partition of a discrete graph so as to achieve balance conditions on the region sizes. Through experimentation, we first establish that the region sizes of randomly-generated graph Voronoi diagrams vary greatly in practice.We then show how to achieve a balanced partition of a graph via Voronoi site resampling. For bounded-degree graphs, where each of the n nodes has degree at most d, and for an initial randomly-chosen set of s Voronoi nodes, we prove that, by extending the set of Voronoi nodes using an algorithm by Thorup and Zwick, each Voronoi region has size at most 4dn/s+1 nodes, and that the expected size of the extended set of Voronoi nodes is at most 2s log n.

KW - Balancing

KW - Facility location

KW - Graph Voronoi diagram

KW - Territorial design

UR - http://www.scopus.com/inward/record.url?scp=77951472302&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951472302&partnerID=8YFLogxK

U2 - 10.1109/ISVD.2009.26

DO - 10.1109/ISVD.2009.26

M3 - Conference contribution

AN - SCOPUS:77951472302

SN - 9780769537818

SP - 183

EP - 191

BT - 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009

ER -