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Scandinavian Workshop on Algorithm Theory

SWAT 2014: Algorithm Theory – SWAT 2014 pp 38–49Cite as

Amortized Analysis of Smooth Quadtrees in All Dimensions

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8503))

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

This work was supported by NSF Grant CCF-0917093.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Courant Institute, New York University, USA

    Huck Bennett & Chee Yap

Authors
  1. Huck Bennett
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  2. Chee Yap
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Editor information

Editors and Affiliations

  1. Tepper School of Business, Carnegie Mellon University, 15213, Pittsburgh, PA, USA

    R. Ravi

  2. DTU Informatics, 2800, Kongens Lyngby, Denmark

    Inge Li Gørtz

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© 2014 Springer International Publishing Switzerland

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Bennett, H., Yap, C. (2014). Amortized Analysis of Smooth Quadtrees in All Dimensions. In: Ravi, R., Gørtz, I.L. (eds) Algorithm Theory – SWAT 2014. SWAT 2014. Lecture Notes in Computer Science, vol 8503. Springer, Cham. https://doi.org/10.1007/978-3-319-08404-6_4

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  • DOI: https://doi.org/10.1007/978-3-319-08404-6_4

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08403-9

  • Online ISBN: 978-3-319-08404-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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