Gravitational allocation on the sphere
2018 ◽
Vol 115
(39)
◽
pp. 9666-9671
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Given a collection L of n points on a sphere Sn2 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(logn). 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(logn), which is optimal by a result of Ajtai, Komlós, and Tusnády.
1988 ◽
Vol 46
◽
pp. 946-947
1987 ◽
Vol 45
◽
pp. 588-589
Keyword(s):
2018 ◽
1996 ◽
Vol 76
(05)
◽
pp. 682-688
◽
Keyword(s):
Keyword(s):