Phase changes in random point quadtrees

Hua Huai Chern, Michael Fuchs, Hsien Kuei Hwang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


We show that a wide class of linear cost measures (such as the number of leaves) in random d-dimensional point quadtrees undergo a change in limit laws: If the dimension d = 1, , 8, then the limit law is normal; if d 9 then there is no convergence to a fixed limit law. Stronger approximation results such as convergence rates and local limit theorems are also derived for the number of leaves, additional phase changes being unveiled. Our approach is new and very general, and also applicable to other classes of search trees. A brief discussion of Devroye's grid trees (covering m-ary search trees and quadtrees as special cases) is given. We also propose an efficient numerical procedure for computing the constants involved to high precision.

Original languageEnglish
Article number1240235
JournalACM Transactions on Algorithms
Issue number2
StatePublished - 1 May 2007


  • Analysis in distribution of algorithms
  • Asymptotic transfer
  • Central limit theorems
  • Depth
  • Differential equations
  • Grid trees
  • Local limit theorems
  • Mellin transforms
  • Page usage
  • Phase transitions
  • Quadtrees
  • Total path length

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