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.
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References
"Efficient worst-case data structures for range searching," by J.L. Bentley and H.A. Maurer, Acta Inform. Vol. 13, pp. 155–168, 1980.
"Filtering Search: A New Approach to Query-Answering," by B. Chazelle, 24th Annual Symposium on Foundations of Computer Science, Nov. 1983.
"Geometric Retrieal Problems," by R. Cole and C. Yap, 24th Annual Symposium on Foundations of Computer Science, Nov. 1983.
An introduction to probability theory and its applications, Vol. 1, Wiley, 1957.
"Monte Carlo Algorithms for Enumeration and Reliability Problems" by R. Karp and M. Luby, 24th Annual Symposium on Foundations of Computer Science, Nov. 1983.
"On k-Nearest Neighbor Voronoi Diagrams in the Plane" by D. Lee, IEEE Trans. Computing, No. 6, 1982.
"On the Piano Mover's Problem: I. The Special Case of a Rigid Polygonal Body Moving Amidst Polygonal Barriers" by J.T. Schartz and M. Sharir in Comm. Pure Appl. Math., 1983.
"Intersection and closest-pair problems for a set of planar objects" by M. Sharir, Courant Inst. Tech. Report No. 56, Feb. 1983.
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© 1984 Springer-Verlag Berlin Heidelberg
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Spirakis, P.G. (1984). The volume of the union of many spheres and point inclusion problems. In: Mehlhorn, K. (eds) STACS 85. STACS 1985. Lecture Notes in Computer Science, vol 182. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0024021
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DOI: https://doi.org/10.1007/BFb0024021
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