Proximity problems on moving points
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Kinetic data structures: a state of the art report
WAFR '98 Proceedings of the third workshop on the algorithmic foundations of robotics on Robotics : the algorithmic perspective: the algorithmic perspective
Data structures for mobile data
Journal of Algorithms
Multidimensional binary search trees used for associative searching
Communications of the ACM
A segment-tree based kinetic BSP
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
K-D Trees Are Better when Cut on the Longest Side
ESA '00 Proceedings of the 8th Annual European Symposium on Algorithms
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Journal of Computer and System Sciences - Special issue on PODS 2000
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We propose a simple variant of kd-trees, called rank-based kd-trees, for sets of $n$ points in $\mathbb{R}^d$. We show that a rank-based kd-tree, like an ordinary kd-tree, supports orthogonal range queries 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 kinetic data structure (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, for sets of points in $\mathbb{R}^2$. Rank-based longest-side kd-trees can be kinetized efficiently as well, and like longest-side kd-trees, they support $\varepsilon$-approximate nearest-neighbor, $\varepsilon$-approximate farthest-neighbor, and $\varepsilon$-approximate range queries with convex ranges in $O((1/\epsilon)\log^2n)$ time. The KDS processes $O(n^3\log n)$ events in the worst case, assuming that the points follow constant-degree algebraic trajectories; each event can be handled in $O(\log^2n)$ time, and each point is involved in $O(\log n)$ certificates.