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

  • Paul G. Spirakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 182)


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).


Voronoi Diagram Query Time Monte Carlo Technique Probabilistic Algorithm Current Union 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Bentley, Maurer, 80]
    "Efficient worst-case data structures for range searching," by J.L. Bentley and H.A. Maurer, Acta Inform. Vol. 13, pp. 155–168, 1980.Google Scholar
  2. [Chazelle, 83]
    "Filtering Search: A New Approach to Query-Answering," by B. Chazelle, 24th Annual Symposium on Foundations of Computer Science, Nov. 1983.Google Scholar
  3. [Cole,Yap,83]
    "Geometric Retrieal Problems," by R. Cole and C. Yap, 24th Annual Symposium on Foundations of Computer Science, Nov. 1983.Google Scholar
  4. [Feller,57]
    An introduction to probability theory and its applications, Vol. 1, Wiley, 1957.Google Scholar
  5. [Karp,Luby,83]
    "Monte Carlo Algorithms for Enumeration and Reliability Problems" by R. Karp and M. Luby, 24th Annual Symposium on Foundations of Computer Science, Nov. 1983.Google Scholar
  6. [Lee,82]
    "On k-Nearest Neighbor Voronoi Diagrams in the Plane" by D. Lee, IEEE Trans. Computing, No. 6, 1982.Google Scholar
  7. [Schwartz, Sharir, 83]
    "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.Google Scholar
  8. [Sharir, 83]
    "Intersection and closest-pair problems for a set of planar objects" by M. Sharir, Courant Inst. Tech. Report No. 56, Feb. 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Paul G. Spirakis
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNYUSA

Personalised recommendations