On the Klee’s Measure Problem in Small Dimensions

  • Bogdan S. Chlebus
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1521)


The Klee’s measure problem is to compute the volume of the union of a given set of n isothetic boxes in a d-dimensional space. The fastest currently known algorithm for this problem, developed by Overmars and Yap [6], runs in time O(nd/2 log n). We present an alternative simple approach with the same asymptotic performance. The exposition is restricted to dimensions three and four.


Measure Problem Segment Tree Spatial Data Structure Skeleton Tree Algebraic Computation Tree 
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. 1.
    J.L. Bentley, Algorithms for Klee’s rectangle problem, Unpublished notes, Dept. of Computer Science, CMU, 1977.Google Scholar
  2. 2.
    R.A. Finkel and J.L. Bentley, Quad-trees; a data structure for retrieval on composite keys, Acta Informatica 4 1974 1–9.zbMATHCrossRefGoogle Scholar
  3. 3.
    M.L. Fredman and B. Weide, The complexity of computing the measure of U[ai,bi], Comm. ACM 21 1978 540–544.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    S. S. Iyengar, N. S. V. Rao, R. L. Kashyap, and V. K. Vaishnavi, Multidimensional data structures: Review and outlook, Advances in Computers 27 1988 69–119.Google Scholar
  5. 5.
    V. Klee, Can the measure of S[ai, bi] be computed in less than O(n log n) steps?, Amer. Math. Monthly 84 1977 284–285.CrossRefMathSciNetGoogle Scholar
  6. 6.
    M.H. Overmars and C.K. Yap, New upper bounds in Klee’s measure problem, SIAM J. Comput. 20 1991 1034–1045; a preliminary version: New upper bounds in Klee’s measure problem, Proceedings, 29th Ann. Symposium on Foundations of Computer Science, 1988, pp. 550–556.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    F.P. Preparata and M.I. Shamos, Computational Geometry, Springer-Verlag, 1985.Google Scholar
  8. 8.
    H. Samet, The Design and Analysis of Spatial Data Structures, Addison-Wesley, 1990.Google Scholar
  9. 9.
    J. van Leeuwen and D. Wood, The measure problem for rectangular ranges in d-space, J. Algorithms 2 1981 282–300.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Bogdan S. Chlebus
    • 1
  1. 1.Instytut InformatykiUniwersytet WarszawskiWarszawaPoland

Personalised recommendations