The volume of the Union of Many Spheres And Point Inclusion Problems

Cover The volume of the Union of Many Spheres And Point Inclusion Problems
The volume of the Union of Many Spheres And Point Inclusion Problems
Paul G Spirakis
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(E. G. We can arbitrarily select a center, find the most distant of the centers of the C. 's to that center, add to the distance found the maximum radius and draw a sphere for S). If we don't use any preprocessing ideas, then B (n) = 0(n), leading to a time complexity of 0(n • t (n) • (_J i)). If we n U v(S) choose S to be the boundary of the union U C^ itself, then . — = 1 but then the task of selecting points from the interior of S uniformly randomly becomes a hard task, since a point belongi...ng to k > 1 spheres cannot be counted as a "whole" point. This leads to the following algorithm: Extended MC (1) Let S be the boundary of the union.
(2) Select N points (N >_ cn/e ^, c a constant) as follows: To select a random point in S, we select one C. At random and then we select one point P. , uniformly randomly within C. .
-1 (3) Compute M = E (cover(P^)) j=l -J (4) Output M.
Lemma 2 . Let c(n) be the time to compute the cover of a point. Let p(n) be the preprocessing time for this.


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