Kinetic data structures: a state of the art report

Suppose we are simulating a collection of continuously moving bodies, rigid or deformable, whose instantaneous motion follows known laws. As the simulation proceeds, we are interested in maintaining certain quantities of interest (for example, the separation of the closest pair of objects), or detecting certain discrete events (for example, collisions) which may, in turn, alter the motion laws of the objects. In this paper we present a general framework for addressing such problems and the tools for designing and analyzing relevant algorithms, which we call kinetic data structures. We discuss kinetic data structures for a variety of fundamental geometric problems, such as the maintenance of convex hulls, Voronoi and Delaunay diagrams, closest pairs, and intersection and visibility problems. We also brie y address the issues that arise in implementing such structures robustly and efciently. The resulting techniques satisfy three desirable properties: (1) they exploit the continuity of the motion of the objects to gain e ciency, (2) the number of events processed by the algorithms is close to the minimum necessary in the worst case, and (3) any object may change its ` ight plan' at any moment with a low cost update to the simulation data structures. For computer applications dealing with motion in the physical world, kinetic data structures lead to simulation performance unattainable by other means. In addition, they raise fundamentally new combinatorial and algorithmic questions whose study may prove fruitful for other disciplines as well.