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)|
|Number of pages||6|
|Journal||Proceedings of the National Academy of Sciences of the United States of America|
|State||Published - Sep 25 2018|
- Bipartite matching
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