Amortized Analysis of Smooth Quadtrees in All Dimensions

  • Huck Bennett
  • Chee Yap
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8503)


Quadtrees are a well-known data structure for representing geometric data in the plane, and naturally generalize to higher dimensions. A basic operation is to expand the tree by splitting a given leaf. A quadtree is smooth if adjacent leaf boxes differ by at most one in height.

In this paper, we analyze quadtrees that maintain smoothness with each split operation and also maintain neighbor pointers. Our main result shows that the smooth-split operation has an amortized cost of O(1) time for quadtrees of any fixed dimension D. This bound has exponential dependence on D which we show is unavoidable via a lower bound construction. We additionally give a lower bound construction showing an amortized cost of Ω(logn) for splits in a related quadtree model that does not maintain smoothness.


Maximal Chain Force Chain Neighbor Query Split Operation Private Correspondence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Huck Bennett
    • 1
  • Chee Yap
    • 1
  1. 1.Department of Computer Science, Courant InstituteNew York UniversityUSA

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