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Refined Asymptotics for the Number of Leaves of Random Point Quadtrees

Authors Michael Fuchs, Noela S. Müller, Henning Sulzbach

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Subject Classification

ACM Subject Classification
  • Theory of computation → Sorting and searching
Keywords
  • Quadtree
  • number of leaves
  • phase change
  • stochastic fixed-point equation
  • central limit theorem
  • positivity of variance
  • contraction method

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Abstract

In the early 2000s, several phase change results from distributional convergence to distributional non-convergence have been obtained for shape parameters of random discrete structures. Recently, for those random structures which admit a natural martingale process, these results have been considerably improved by obtaining refined asymptotics for the limit behavior. In this work, we propose a new approach which is also applicable to random discrete structures which do not admit a natural martingale process. As an example, we obtain refined asymptotics for the number of leaves in random point quadtrees. More applications, for example to shape parameters in generalized m-ary search trees and random gridtrees, will be discussed in the journal version of this extended abstract.

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Michael Fuchs, Noela S. Müller, and Henning Sulzbach. Refined Asymptotics for the Number of Leaves of Random Point Quadtrees. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.AofA.2018.23

Author Details

Michael Fuchs
  • Department of Applied Mathematics, National Chiao Tung University, 300 Hsinchu, Taiwan
Noela S. Müller
  • Institute for Mathematics, Goethe University, 60054 Frankfurt a.M., Germany
Henning Sulzbach
  • University of Birmingham, School of Mathematics, B15 2TT Birmingham, United Kingdom

Funding

  • Fuchs, Michael: Partially supported by MOST under the grants MOST-104-2923- M-009-006-MY3 and MOST-105-2115-M-009-010- MY2.

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