The volume of the union of many spheres and point inclusion problems

Abstract

We present here an O(n) probabilistic algorithm for computing the volume of the union of n spheres of possibly different radii. The method, which is an application of techniques developed by [Karp, Luby, 83], can be extended, in a straightforward manner, to compute the volume of the union of n objects (where each of them has an easy description e.g. boxes or spheres) in k dimensions. Its time complexity is then O(nk). We also examine the related problem of computing the number of spheres (or disks, in the plane) among a given set of spheres, containing a given point. For the case of n disks of the same radius r, we can answer such a query in time O(log²n) and O(n³) preprocessing space.

For the more general problem of n spheres of different radii, we can answer such queries in O(log²n) time and storage O(n log n), following a technique of [Chazelle, 83]. This leads to an O(n √n) expected time union estimation algorithm.

The probabilistic estimation of the union follows ideas developed by R. Karp and M. Luby (see [Karp, Luby, 83]). Some of our notation is heavily affected by their notation.

We also show how to use the above methods to test if n spheres have a (nonzero measure) intersection, in probabilistic time O(n).

This work was supported in part by the National Science Foundation Grant NSF-MCS83-00630.