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Kinetic kd-Trees and Longest-Side kd-Trees

Authors Mohammad Abam, Mark de Berg, Bettina Speckmann



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Mohammad Abam
Mark de Berg
Bettina Speckmann

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Mohammad Abam, Mark de Berg, and Bettina Speckmann. Kinetic kd-Trees and Longest-Side kd-Trees. In Data Structures. Dagstuhl Seminar Proceedings, Volume 8081, pp. 1-12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)
https://doi.org/10.4230/DagSemProc.08081.2

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.
Keywords
  • Kinetic data structures
  • kd-tree
  • longest-side kd-tree

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