eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-06-16
1
12
10.4230/DagSemProc.08081.2
article
Kinetic kd-Trees and Longest-Side kd-Trees
Abam, Mohammad
de Berg, Mark
Speckmann, Bettina
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08081/DagSemProc.08081.2/DagSemProc.08081.2.pdf
Kinetic data structures
kd-tree
longest-side kd-tree