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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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
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