Kinetic kd-Trees and Longest-Side kd-Trees

Abstract

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.