, Volume 10, Issue 6, pp 473–500 | Cite as

Analytic variations on quadtrees

  • Philippe Flajolet
  • Gaston Gonnet
  • Claude Puech
  • J. M. Robson


Quadtrees constitute a hierarchical data structure which permits fast access to multidimensional data. This paper presents the analysis of the expected cost of various types of searches in quadtrees — fully specified and partial-match queries. The data model assumes random points with independently drawn coordinate values.

The analysis leads to a class of “full-history” divide-and-conquer recurrences. These recurrences are solved using generating functions, either exactly for dimensiond=2, or asymptotically for higher dimensions. The exact solutions involve hypergeometric functions. The general asymptotic solutions rely on the classification of singularities of linear differential equations with analytic coefficients, and on singularity analysis techniques.

These methods are applicable to the asymptotic solution of a wide range of linear recurrences, as may occur in particular in the analysis of multidimensional searching problems.

Key words

Analysis of algorithms Multidimensional search Quadtrees 


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  1. [1]
    Abramowitz, M., and Stegun, I. A.Handbook of Mathematical Functions. Dover, New York, 1973. A reprint of the tenth National Bureau of Standards edition, 1964.Google Scholar
  2. [2]
    Bentley, J. L.: Multidimensional binary search trees used for associative searching.Communications of the ACM,18(9) (Sept. 1975), 509–517.CrossRefGoogle Scholar
  3. [3]
    Bentley, J. L., and Friedman, J. H.: Data structures for range searching.ACM Computing Surveys,11(4) (1979), 397–409.Google Scholar
  4. [4]
    Bentley, J. L., and Stanat, D. F.: Analysis of range searching in quad trees.Information Processing Letters,3(6) (July 1975), 170–173.Google Scholar
  5. [5]
    Bronstein, M. On solutions of linear ordinary differential equations in their coefficient field. Technical Report 152, Department Informatik, ETH, January 1991.Google Scholar
  6. [6]
    Bronstein, M. Rational solution to differential equations related to quadtrees. Private communication, February 1991.Google Scholar
  7. [7]
    Buchta, C.: On the average number of maxima in a set of vectors.Information Processing Letters,33 (Nov. 1989), 63–65.Google Scholar
  8. [8]
    Devroye, L.: Branching processes in the analysis of the heights of trees.Acta Informatica,24 (1987), 277–298.Google Scholar
  9. [9]
    Devroye, L., and Laforest, L.: An analysis of randomd-dimensional quad trees.SIAM Journal on Computing,19 (1990), 821–832.Google Scholar
  10. [10]
    Finkel, R. A., and Bentley, J. L.: Quad trees, a data structure for retrieval on composite keys.Acta Informatica,4 (1974), 1–9.CrossRefGoogle Scholar
  11. [11]
    Fishburn, P. C., Odlyzko, A. M., and Roberts, F. S. Two-sided generalized Fibonacci sequences. Preprint, 1989.Google Scholar
  12. [12]
    Flajolet, P., and Lafforgue, T. Search costs in quadtrees and singularity perturbation asymptotics. 1993, submitted.Google Scholar
  13. [13]
    Flajolet, P., Gonnet, G., Puech, C., and Robson, J. M. The analysis of multidimensional searching in quad-trees.Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, Philadelphia, 1991, SIAM, Philadelphia, PA, pp. 100–109.Google Scholar
  14. [14]
    Flajolet, P., and Odlyzko, A. M.: Singularity analysis of generating functions.SIAM Journal on Discrete Mathematics,3(2) (1990), 216–240.Google Scholar
  15. [15]
    Flajolet, P., and Puech, C.: Partial match retrieval of multidimensional data.Journal of the ACM,33(2) (1986), 371–407.Google Scholar
  16. [16]
    Flajolet, P., and Soria, M.: Gaussian limiting distributions for the number of components in combinatorial structures.Journal of Combinatorial Theory, Series A,53 (1990), 165–182.Google Scholar
  17. [17]
    Gonnet, G. H., and Baeza-Yates, R.Handbook of Algorithms and Data Structures: in Pascal and C, second edn. Addison-Wesley, Reading, MA, 1991.Google Scholar
  18. [18]
    Guibas, L.: Problems.Journal of Algorithms,3(4) (1982), 362–380. 〈Problem 80-6〉, from the Stanford 1979 Algorithms Qualifying Examination. Solution by Eric S. Rosenthal, pp. 368–371.Google Scholar
  19. [19]
    Henrici, P.Applied and Computational Complex Analysis. Wiley, New York, 1977. Three volumes.Google Scholar
  20. [20]
    Hoshi, M., and Flajolet, P.: Page usage in a quadtree index.BIT,32 (1992), 384–402.Google Scholar
  21. [21]
    Iyengar, S. S., Rao, N. S. V., Kashyap, R. L., and Vaishnavi, V. K.: Multidimensional data structures: review and outlook.Advances in Computers,27 (1988), 69–119.Google Scholar
  22. [22]
    Knuth, D. E.The Art of Computer Programming, vol. 3: Sorting and Searching. Addison-Wesley, Reading, MA, 1973.Google Scholar
  23. [23]
    Labelle, G., and Laforest, L. Variations combinatoires autour des arborescences hyperquaternaires. Technical Report, LACIM, UQAM, Montreal, Nov. 1991.Google Scholar
  24. [24]
    Laforest, L. Étude des arbres hyperquaternaires. Technical Report 3, LACIM, UQAM, Montreal, Nov. 1990. (Author's Ph.D. thesis at McGill University.)Google Scholar
  25. [25]
    Louchard, G.: Exact and asymptotic distributions in digital and binary search trees.RAIRO Theoretical Informatics and Applications,21(4) (1987), 479–495.Google Scholar
  26. [26]
    Mathieu, C., Puech, C., and Yahia, H.: Average efficiency of data structures for binary image processing.Information Processing Letters,26 (Oct. 1987), 89–93.Google Scholar
  27. [27]
    Odlyzko, A. M., and Richmond, L. B.: Asymptotic expansions for the coefficients of analytic generating functions.Aequationes Mathematicae,28 (1985), 50–63.Google Scholar
  28. [28]
    Preparata, F. P., and Shamos, M. I.Computational Geometry, An Introduction. Springer-Verlag, New York, 1985.Google Scholar
  29. [29]
    Puech, C.Méthodes d'analyse de structures de données dynamiques. Doctorat és sciences, Université de Paris Sud, Orsay, 1984.Google Scholar
  30. [30]
    Puech, C., and Yahia, H. Quadtrees, octrees, hyperoctrees: a unified approach to tree data structures used in graphics, geometric modeling and image processing.Proceedings of the First ACM Symposium on Computational Geometry, Baltimore, 1985, pp. 272–280.Google Scholar
  31. [31]
    Régnier, M.: Analysis of grid file algorithms.BIT,25 (1985), 335–357.Google Scholar
  32. [32]
    Salvy, B. Asymptotique automatique et fonctions génératrices. Ph.D. thesis, École Polytechnique, 1991.Google Scholar
  33. [33]
    Samet, H.The Design and Analysis of Spatial Data Structures. Addison-Wesley, Reading, MA, 1990.Google Scholar
  34. [34]
    Sedgewick, R.Algorithms, second edn. Addison-Wesley, Reading, MA, 1988.Google Scholar
  35. [35]
    Wasow, W.Asymptotic Expansions for Ordinary Differential Equations. Dover, New York, 1987. A reprint of the Wiley edition, 1965.Google Scholar
  36. [36]
    Whittaker, E. T., and Watson, G. N.A Course of Modern Analysis, fourth edn. Cambridge University Press, Cambridge, 1927. Reprinted 1973.Google Scholar
  37. [37]
    Wimp, J., and Zeilberger, D.: Resurrecting the asymptotics of linear recurrences.Journal of Mathematical Analysis and Applications,111 (1985), 162–176.Google Scholar
  38. [38]
    Yahia, H. Analyse des structures de données arborescentes représentant des images. Doctorat de troisiéme cycle, Université de Paris Sud, Orsay, Dec. 1986.Google Scholar
  39. [39]
    Zeilberger, D.: A holonomic approach to special functions identities.Journal of Computational and Applied Mathematics,32 (1990), 321–368.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Philippe Flajolet
    • 1
  • Gaston Gonnet
    • 2
  • Claude Puech
    • 3
  • J. M. Robson
    • 4
  1. 1.Algorithms ProjectINRIALe ChesnayFrance
  2. 2.InformatikE.T.H. ZentrumZurichSwitzerland
  3. 3.Ecole Normale SupérieureLIENSParisFrance
  4. 4.Department of Computer ScienceAustralian National UniversityCanberraAustralia

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