### Abstract

We consider the problem of balancing the load among several service-providing facilities, while keeping the total cost low. Let D be the underlying demand region, and let p
_{1},... ,p
_{m} be m points representing m facilities. We consider the following problem: Subdivide D into m equal-area regions R
_{1}.,..., R
_{m}, so that region R
_{i} is served by facility p
_{i}, and the average distance between a point q in D and the facility that serves q is minimal. We present constant-factor approximation algorithms for this problem, with the additional requirement that the resulting regions must be convex. As an intermediate result we show how to partition a convex polygon into m = 2
^{k} equal-area convex subregions so that the fatness of the resulting regions is within a constant factor of the fatness of the original polygon. We also prove that our partition is, up to a constant factor, the best one can get if one's goal is to maximize the fatness of the least fat subregion. We also discuss the structure of the optimal partition for the aforementioned load balancing problem: indeed, we argue that it is always induced by an additive-weighted Voronoi diagram for an appropriate choice of weights.

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

Title of host publication | Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06 |

Pages | 301-308 |

Number of pages | 8 |

Volume | 2006 |

State | Published - 2006 |

Event | 22nd Annual Symposium on Computational Geometry 2006, SCG'06 - Sedona, AZ, United States Duration: Jun 5 2006 → Jun 7 2006 |

### Other

Other | 22nd Annual Symposium on Computational Geometry 2006, SCG'06 |
---|---|

Country | United States |

City | Sedona, AZ |

Period | 6/5/06 → 6/7/06 |

### Fingerprint

### Keywords

- Additive-weighted Voronoi diagrams
- Approximation algorithms
- Fat partitions
- Fatness
- Geometric optimization
- Load balancing

### ASJC Scopus subject areas

- Software
- Geometry and Topology
- Safety, Risk, Reliability and Quality
- Chemical Health and Safety

### Cite this

*Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06*(Vol. 2006, pp. 301-308)

**Minimum-cost load-balancing partitions.** / Aronov, Boris; Carmi, Paz; Katz, Matthew J.

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

*Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06.*vol. 2006, pp. 301-308, 22nd Annual Symposium on Computational Geometry 2006, SCG'06, Sedona, AZ, United States, 6/5/06.

}

TY - GEN

T1 - Minimum-cost load-balancing partitions

AU - Aronov, Boris

AU - Carmi, Paz

AU - Katz, Matthew J.

PY - 2006

Y1 - 2006

N2 - We consider the problem of balancing the load among several service-providing facilities, while keeping the total cost low. Let D be the underlying demand region, and let p 1,... ,p m be m points representing m facilities. We consider the following problem: Subdivide D into m equal-area regions R 1.,..., R m, so that region R i is served by facility p i, and the average distance between a point q in D and the facility that serves q is minimal. We present constant-factor approximation algorithms for this problem, with the additional requirement that the resulting regions must be convex. As an intermediate result we show how to partition a convex polygon into m = 2 k equal-area convex subregions so that the fatness of the resulting regions is within a constant factor of the fatness of the original polygon. We also prove that our partition is, up to a constant factor, the best one can get if one's goal is to maximize the fatness of the least fat subregion. We also discuss the structure of the optimal partition for the aforementioned load balancing problem: indeed, we argue that it is always induced by an additive-weighted Voronoi diagram for an appropriate choice of weights.

AB - We consider the problem of balancing the load among several service-providing facilities, while keeping the total cost low. Let D be the underlying demand region, and let p 1,... ,p m be m points representing m facilities. We consider the following problem: Subdivide D into m equal-area regions R 1.,..., R m, so that region R i is served by facility p i, and the average distance between a point q in D and the facility that serves q is minimal. We present constant-factor approximation algorithms for this problem, with the additional requirement that the resulting regions must be convex. As an intermediate result we show how to partition a convex polygon into m = 2 k equal-area convex subregions so that the fatness of the resulting regions is within a constant factor of the fatness of the original polygon. We also prove that our partition is, up to a constant factor, the best one can get if one's goal is to maximize the fatness of the least fat subregion. We also discuss the structure of the optimal partition for the aforementioned load balancing problem: indeed, we argue that it is always induced by an additive-weighted Voronoi diagram for an appropriate choice of weights.

KW - Additive-weighted Voronoi diagrams

KW - Approximation algorithms

KW - Fat partitions

KW - Fatness

KW - Geometric optimization

KW - Load balancing

UR - http://www.scopus.com/inward/record.url?scp=33748051075&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748051075&partnerID=8YFLogxK

M3 - Conference contribution

SN - 1595933409

SN - 9781595933409

VL - 2006

SP - 301

EP - 308

BT - Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06

ER -