A limit field for orthogonal range searches in two-dimensional random point search trees

Nicolas Broutin, Henning Sulzbach

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Abstract

We consider the cost of general orthogonal range queries in random quadtrees. The cost of a given query is encoded into a (random) function of four variables which characterize the coordinates of two opposite corners of the query rectangle. We prove that, when suitably shifted and rescaled, the random cost function converges uniformly in probability towards a random field that is characterized as the unique solution to a distributional fixed-point equation. We also state similar results for 2-d trees. Our results imply for instance that the worst case query satisfies the same asymptotic estimates as a typical query, and thereby resolve an open question of Chanzy, Devroye and Zamora-Cura [Acta Inf., 37:355--383, 2001].
Original languageEnglish
JournalStochastic Processes and their Applications
Early online date8 Sep 2018
DOIs
Publication statusE-pub ahead of print - 8 Sep 2018

Keywords

  • quadtree
  • random partition
  • convergence in distribution
  • contraction method
  • range query
  • partial match
  • analysis of algorithms

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