Amortized analysis of smooth quadtrees in all dimensions

Huck Bennett, Chee Yap

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

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(log n) for splits in a related quadtree model that does not maintain smoothness.

Original languageEnglish (US)
Title of host publicationAlgorithm Theory, SWAT 2014 - 14th Scandinavian Symposium and Workshops, Proceedings
PublisherSpringer Verlag
Pages38-49
Number of pages12
ISBN (Print)9783319084039
DOIs
StatePublished - 2014
Event14th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2014 - Copenhagen, Denmark
Duration: Jul 2 2014Jul 4 2014

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8503 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other14th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2014
CountryDenmark
CityCopenhagen
Period7/2/147/4/14

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Bennett, H., & Yap, C. (2014). Amortized analysis of smooth quadtrees in all dimensions. In Algorithm Theory, SWAT 2014 - 14th Scandinavian Symposium and Workshops, Proceedings (pp. 38-49). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8503 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-08404-6_4