License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.08081.2
URN: urn:nbn:de:0030-drops-15307
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2008/1530/
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Abam, Mohammad ; de Berg, Mark ; Speckmann, Bettina

Kinetic kd-Trees and Longest-Side kd-Trees

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08081.AbamMohammad.Pape.1530.pdf (0.2 MB)

Abstract

We propose a simple variant of kd-trees, called rank-based kd-trees, for sets of points in~$Reals^d$.
We show that a rank-based kd-tree, like an ordinary kd-tree, supports range search que-ries in~$O(n^{1-1/d}+k)$ time,
where~$k$ is the output size. The main advantage of rank-based kd-trees is that they can be efficiently kinetized:
the KDS processes~$O(n^2)$ events in the worst case, assuming that the points follow constant-degree algebraic trajectories,
each event can be handled in~$O(log n)$ time, and each point is involved in~$O(1)$ certificates.

We also propose a variant of longest-side kd-trees, called rank-based longest-side kd-trees (RBLS kd-trees, for short),
for sets of points in~$Reals^2$. RBLS kd-trees can be kinetized efficiently as well and like longest-side kd-trees,
RBLS kd-trees support nearest-neighbor, farthest-neighbor, and approximate range search queries in~$O((1/epsilon)log^2 n)$ time.
The KDS processes~$O(n^3log n)$ events in the worst case, assuming that the points follow constant-degree algebraic trajectories;
each event can be handled in~$O(log^2 n)$ time, and each point is involved in~$O(log n)$ certificates.

BibTeX - Entry

@InProceedings{abam_et_al:DagSemProc.08081.2,
  author =	{Abam, Mohammad and de Berg, Mark and Speckmann, Bettina},
  title =	{{Kinetic kd-Trees and Longest-Side kd-Trees}},
  booktitle =	{Data Structures},
  pages =	{1--12},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8081},
  editor =	{Lars Arge and Robert Sedgewick and Raimund Seidel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2008/1530},
  URN =		{urn:nbn:de:0030-drops-15307},
  doi =		{10.4230/DagSemProc.08081.2},
  annote =	{Keywords: Kinetic data structures, kd-tree, longest-side kd-tree}
}

Keywords: Kinetic data structures, kd-tree, longest-side kd-tree
Collection: 08081 - Data Structures
Issue Date: 2008
Date of publication: 16.06.2008

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