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Analytic variations on quadtrees

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Abstract

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.

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Author information

Authors and Affiliations

  1. Algorithms Project, INRIA, Rocquencourt, F-78153, Le Chesnay, France

    Philippe Flajolet

  2. Informatik, E.T.H. Zentrum, CH-8092, Zurich, Switzerland

    Gaston Gonnet

  3. Ecole Normale Supérieure, LIENS, 45 rue d'Ulm, F-75005, Paris, France

    Claude Puech

  4. Department of Computer Science, Australian National University, 2601, Canberra, ACT, Australia

    J. M. Robson

Authors
  1. Philippe Flajolet
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  2. Gaston Gonnet
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  3. Claude Puech
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  4. J. M. Robson
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Additional information

Communicated by Bernard Chazelle.

The work of Philippe Flajolet was supported in part by the Basic Research Action of the E.C. under Contract No. 3075 (Project ALCOM). A preliminary version of this paper has been presented in the form of an extended abstract at the Second Annual Symposium on Discrete Algorithms [13].

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Flajolet, P., Gonnet, G., Puech, C. et al. Analytic variations on quadtrees. Algorithmica 10, 473–500 (1993). https://doi.org/10.1007/BF01891833

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  • DOI: https://doi.org/10.1007/BF01891833

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