Gravitational allocation on the sphere.

Abstract:

:Given a collection L of n points on a sphere [Formula: see text] 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 [Formula: see text] 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 [Formula: see text], which is optimal by a result of Ajtai, Komlós, and Tusnády.

authors

Holden N,Peres Y,Zhai A

doi

10.1073/pnas.1720804115

subject

Has Abstract

pub_date

2018-09-25 00:00:00

pages

9666-9671

issue

39

eissn

0027-8424

issn

1091-6490

pii

1720804115

journal_volume

115

pub_type

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