Kinetic KD-trees and longest-side KD-trees
Mohammad Ali Abam, Mark de Berg, Bettina Speckmann · 2007
We propose a simple variant of kd-trees, called rank-based kd-trees, for sets of points in Rd. We show that a rank-based kd-tree, like an ordinary kd-tree, supports range search queries in O(n1−1/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(n2) events in the worst case, assuming that the points follow constant-degree algebraic trajectories, each event can be handled in O(logn) 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 R2. 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/ε) log2 n) time. The KDS processes O(n3 logn) events in the worst case, assuming that the points follow constant-degree algebraic trajectories; each event can be handled in O(log2 n) time, and each point is involved in O(logn) certificates. Background. Due to the increased availability of GPS systems and to other technological advances, motion data is becoming more and more available in a variety of application areas: air-traffic control,