Abstract
Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically ditributed as a normal variable. The limit distribution property extends to quadtrees of all dimensions a result only known so far to hold for binary search trees.
The analysis is based on a technique of singularity perturbation that appears to be of some generality. For quadtrees, this technique is applied to linear differential equations satisfied by interventing bivariate generating functions
Article PDF
Similar content being viewed by others
References
M. Abramowitz and I. A. Stegun.Handbook of Mathematical Functions. Dover, 1973. A reprint of the tenth National Bureau of Standards edition, 1964.
E. A. Bender. Central and local limit theorems applied to asymptotic enumeration.Journal of Combinatorial Theory, 15:91–111, 1973.
F. Bergeron, P. Flajolet, and B. Salvy. Varieties of increasing trees. In J.-C. Raoult, editor,CAAP ’92, pages 24–48 (Proceedings of the 17th Colloquium on Trees in Algebra and Programming, Rennes, France, February 1992). Lecture Notes in Computer Science, Volume 581. Springer-Verlag, Berlin, 1992.
P. Billingsley.Probability and Measure, 2nd edition, Wiley, New York, 1986.
G. G. Brown and B. O. Shubert. On random binary trees.Mathematics of Operations Research, 9(1):43–65, 1984.
E. A. Coddington and M. Levinson,Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.
L. Devroye and L. Laforest, An analysis of randomd-dimensional quad trees.SIAM Journal on Computing, 19:821–832, 1990.
M. Drmota. Asymptotic distributions and a multivariate Darboux method in enumeration problems. Manuscript, 1990.
R. A. Finkel and J. L. Bentley. Quad trees, a data structure for retrieval on composite keys.Acta Informatica, 4:1–9, 1974.
P. Flajolet, G. Gonnet, C. Puech, and J. M. Robson. Analytic variation on quadtrees.Algorithmica, 10:473–500, 1993.
P. Flajolet and A. M. Odlyzko. Singularity analysis of generating functions.SIAM Journal on Discrete Mathematics, 3(2):216–240, 1990.
P. Flajolet and C. Puech. Partial match retrieval of multidimensional data.,Journal of the ACM, 22(2):371–407, 1986.
P. Flajolet and M. Soria. Gaussian limiting distributions for the number of components in combinatorial structures.Journal of Combinatorial Theory, Series A, 53:165–182, 1990.
P. Flajolet and M. Soria. General combinatorial schemas: Gaussian limit distributions and exponential tails.Discrete Mathematics, 114:159–180, 1993.
J. Françon. Arbres binaires de recherche: Propriétés combinatoires et applications.RAIRO Informatique Théorique, 10(12):35–50, 1976.
J. Françon. On the analysis of algorithms for trees.Theoretical Computer Science, 4:155–169, 1977.
Z. Gao and L. B. Richmond. Central and local limit theorems applied to asymptotic enumerations, IV: Multivariate generating functions.Journal of Computational and Applied Mathematics, 41:177–186, 1992.
G. H. Gonnet and R. Baeza-Yates,Handbook of Algorithms and Data Structures: in Pascal and C, 2nd edition. Addison-Wesley. Reading, MA, 1991.
P. Henrici.Applied and Computational Complex Analysis, 3 volumes. Wiley, New York, 1977.
M. Hoshi and P. Flajolet. Page usage in a quadtree index.BIT, 32:384–402, 1992.
P. Jacquet and M. Régnier. Trie partitioning process: Limiting distributions. In P. Franchi-Zanetacchi, editor,CAAP ’86 pages 196–210 (Proceedings of the 11th Colloquium on Trees in Algebra and Programming, Nice France, March 1986). Lecture Notes in Computer Science, Volume 214. Springer-Verlag, Berlin, 1986.
D. E. Knuth,The Art of Computer Programming, Volume 3. Addison-Wesley, Reading, MA, 1973.
L. Laforest, Étude des arbres hyperquaternaires. Technical Report 3, LACIM, UQAM, Montreal, Nov. 1990. (Author’s Ph.D. Thesis at McGill University.)
G. Louchard. Exact and asymptotic distributions in digital and binary search trees,RAIRO Theoretical Informatics and Applications, 21(4):479–495, 1987.
E. Lukacs.Characteristic Functions. Griffin, London, 1970.
W. L. Lynch. More combinatorial problems on certain trees.Computer Journal, 7:299–302, 1965.
H. Mahmoud.Evolution of Random Search Trees. Wiley, New York, 1992.
H. M. Mahmoud and B. Pittel. Analysis of the space of search trees under the random insertion algorithm.Journal of Algorithms, 10:52–75, 1989.
H. Samet.Applications of Spatial Data Structures. Addition-Wesley, Reading, MA, 1990.
H. Samet.The Design and Analysis of Spatial Data Structures. Addison-Wesley, MA, 1990.
R. Sedgewick.Algorithms, 2nd edition. Addison-Wesley, Reading, MA, 1988.
W. Wasow.Asymptotic Expansions for Ordinary Differential Equations. Dover, 1987. A reprint of the Wiley edition, 1965.
E. T. Whittaker and G. N. Watson.A Course of Modern Analysis, 4th edition. Cambridge University Press, Cambridge, 1927. Reprint 1973.
Author information
Authors and Affiliations
Additional information
This work was partly supported by the ESPRIT Basic Research Action No. 7141 (ALCOM II).
Rights and permissions
About this article
Cite this article
Flajolet, P., Lafforgue, T. Search costs in quadtrees and singularity perturbation asymptotics. Discrete Comput Geom 12, 151–175 (1994). https://doi.org/10.1007/BF02574372
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02574372