Kinetic Compressed Quadtrees in the Black-Box Model with Applications to Collision Detection for Low-Density Scenes

Abstract

We present an efficient method for maintaining a compressed quadtree for a set of moving points in R d . Our method works in the blackbox KDS model, where we receive the locations of the points at regular time steps and we know a bound dmax on the maximum displacement of any point within one time step. When the number of points within any ball of radius dmax is at most k at any time, then our update algorithm runs in O(n log k) time. We generalize this result to constant-complexity moving objects in R d . The compressed quadtree we maintain has size O(n); under similar conditions as for the case of moving points it can be maintained in O(n log λ) time per time step, where λ is the density of the set of objects. The compressed quadtree can be used to perform broadphase collision detection for moving objects; it will report in O((λ + k)n) time a superset of all intersecting pairs of objects.

Amortized analysis of smooth quadtrees in all dimensions

“…In [dBRS12], de Berg et al study refinement of compressed quadtrees. They consider a refinement T 1 of a quadtree T 0 to be an extension of T 0 in which all boxes that were in T 0 have O(1) neighbors in T 1 .…”

A Practical Algorithm for Spatial Agglomerative Clustering

“…There are comparatively few papers which handle both dynamic and kinetic aspects si-multaneously. In particular, there are several results on kinetic kd-trees, quadtrees, and range spaces (see, e.g., [1,13,12]), and also on dynamic quadtrees (see [24] for a very recent example). We are not aware of dynamic and kinetic data structures for spatial partitioning problems.…”

On Modelling, Structuring and Capturing Geometric Information

“…A large number of papers in the literature explicitly or implicitly rely on the ability to efficiently navigate a quadtree as if it were a threaded quadtree. Our results readily imply improved bounds from amortized to worst-case [64,129,148,152,172]. Several dynamic applications are in graphics-related fields and are trivially made parallel, which enhances the need for worst-case bounds.…”

“…Recently, quadtrees have been applied in kinetic and uncertain settings that call for dynamic behaviour of the decomposition [64,129,148]. Bennett and Yap [22] show how to maintain a smooth and dynamic quadtree subject to amortized constant-time split and merge operations on the quadtree leaves.…”

“…Quadtrees and their higher-dimensional equivalents have been long studied in computational geometry [22,25,40,44,80,135,151,193]. Quadtrees are popular among practitioners because of their ease of implementation and good performance in many practical applications [12,21,24,64,90,152]. Given a planar point set P contained in a square, we can define the minimal quadtree that contains P to be the quadtree we obtain by recursively subdividing the root square until no leaf contains more than one (or a constant number of) point(s), refer to Figure 7.1(c).…”