On the difficulty of range searching

  • Arne Andersson
  • Kurt Swanson
Invited Presentation
Part of the Lecture Notes in Computer Science book series (LNCS, volume 955)


The problem of range searching is fundamental and well studied, and a large number of solutions have been suggested in the literature. The only existing non-trivial lower bound that closely matches known upper bounds with respect to time/space tradeoff is given for the pointer machine model. However, the pointer machine prohibits a number of possible and natural operations, such as the use of arrays and bit manipulation. In particular, such operations have proven useful in some special cases such as one-dimensional and rectilinear queries.

In this article, we consider the general problem of (2-dimensional) range reporting allowing arbitrarily convex queries. We show that using a traditional approach, even when incorporating techniques like those used in fusion trees, a (poly-) logarithmic query time can not be achieved unless more than linear space is used. Our arguments are based on a new non-trivial lower bound in a model of computation which, in contrast to the pointer machine model, allows for the use of arrays and bit manipulation. The crucial property of our model, Layered Partitions, is that it can be used to describe all known algorithms for processing range queries, as well as many other data structures used to represent multi-dimensional data.

We show that Ω (log n/log T(n)) partitions must be used to allow queries in O(T(n)+ k) time, where k is the number of reported elements, for any growing function T(n). In some special cases, as for rectilinear queries, these partitions may be stored in compressed form, which has been exploited by the M-structure of Chazelle. However, so far there has been no indication that such compression would be feasible in the general case, in which case any algorithm based on our model, and supporting range searching in O(logcn+k) time requires Ω (n log n/log log n) space. (Note that it may be possible to obtain a better upper bound with an algorithm not adhering to the model of layered partitions.) Hence, we show that removing the restrictions of the pointer machine model does not help in obtaining a significantly improved time/space tradeoff — any solution based on traditional representations of point sets cannot combine linear space and polylogarithmic time.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. L. Bentley. Decomposable searching problems. Inform. Process. Lett., 8:244–251, 1979.CrossRefGoogle Scholar
  2. 2.
    J. L. Bentley and H. A. Maurer. Efficient worst-case data structures for range searching. Acta Inform., 13:155–168, 1980.CrossRefGoogle Scholar
  3. 3.
    B. Chazelle. Filtering search: a new approach to query-answering. In Proc. 24th Annu. IEEE Sympos. Found. Comput. Sci., pages 122–132, 1983.Google Scholar
  4. 4.
    B. Chazelle. Slimming down search structures: A functional approach to algorithm design. In Proc. 26th Annu. IEEE Sympos. Found. Comput. Sci., pages 165–174, 1985.Google Scholar
  5. 5.
    B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM J. Comput., 17:427–462, 1988.CrossRefGoogle Scholar
  6. 6.
    B. Chazelle. Lower bounds for orthogonal range searching, I: the reporting case. J. ACM, 37:200–212, 1990.CrossRefGoogle Scholar
  7. 7.
    B. Chazelle. Lower bounds for orthogonal range searching: II. the arithmetic model. Journal of the ACM, 37(3):439–463, July 1990.CrossRefGoogle Scholar
  8. 8.
    R. A. Finkel and J. L. Bentley. Quad trees: a data structure for retrieval on composite keys. Acta Inform., 4:1–9, 1974.CrossRefGoogle Scholar
  9. 9.
    M. L. Fredman and D. E. Willard. Blasting through the information theoretic barrier with fusion trees. In Proc. 22nd Annu. ACM Sympos. Theory Comput., pages 1–7, 1990.Google Scholar
  10. 10.
    G. S. Lueker. A data structure for orthogonal range queries. In Proc. 19th Annu. IEEE Sympos. Found. Comput. Sci., pages 28–34, 1978.Google Scholar
  11. 11.
    E. M. McCreight. Priority search trees. SIAM J. Comput., 14:257–276, 1985.CrossRefMathSciNetGoogle Scholar
  12. 12.
    P. B. Miltersen. Personal communication.Google Scholar
  13. 13.
    P. B. Miltersen. Lower bounds for union-split-find related problems on random access machines. In Proc. 26th Ann. ACM STOC, pages 625–634, 1994.Google Scholar
  14. 14.
    P. B. Miltersen, N. Nisan, S. Safra, and A. Wigderson. On data structures and asymmetric communication complexity. In Proc. 27th Annu. ACM Sympos. Theory Comput., 1995. To appear.Google Scholar
  15. 15.
    J. B. Saxe and J. L. Bentley. Transforming static data structures to dynamic structures. In Proc. 20th Annu. IEEE Sympos. Found. Comput. Sci., pages 148–168, 1979.Google Scholar
  16. 16.
    D. E. Willard. A new time complexity for orthogonal range queries. In Proc. 20th Allerton Conf. Commun. Control Comput., pages 462–471, 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Arne Andersson
    • 1
  • Kurt Swanson
    • 1
  1. 1.Dept. of Computer ScienceLund UniversityLundSweden

Personalised recommendations