## Abstract

Given a collection L of n points on a sphere S2 n of surface area n, a fair allocation is a partition of the sphere into n parts each of area 1, and each is associated with a distinct point of L. We show that, if the n points are chosen uniformly at random and if the partition is defined by a certain "gravitational" potential, then the expected distance between a point on the sphere and the associated point of L is O( √ log n).We use our result to define a matching between two collections of n independent and uniform points on the sphere and prove that the expected distance between a pair of matched points is O( p log n), which is optimal by a result of Ajtai, Komlós, and Tusnády.

Original language | English (US) |
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Pages (from-to) | 9666-9671 |

Number of pages | 6 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 115 |

Issue number | 39 |

DOIs | |

State | Published - Sep 25 2018 |

## Keywords

- Allocation
- Bipartite matching
- Gravity
- Transportation

## ASJC Scopus subject areas

- General