Fast algorithms for collision and proximity problems involving moving geometric objects

Abstract

Consider a set of geometric objects, such as points or axes-parallel hyperrectangles in ℝ^d, that move with constant but possibly different velocities along linear trajectories. Efficient algorithms are presented for several problems defined on such objects, such as determining whether any two objects ever collide and computing the minimum inter-point separation or minimum diameter that ever occurs. In particular, two open problems from the literature are solved: Deciding in o(n ²) time if there is a collision in a set of n moving points in ℝ², where the points move at constant but possibly different velocities, and the analogous problem for detecting a red-blue collision between sets of red and blue moving points. The strategy used involves reducing the given problem on moving objects to a different problem on a set of static objects, and then solving the latter problem using techniques based on sweeping, orthogonal range searching, simplex composition, and parametric search.

The research of these authors was supported in part by NSF grant CCR-92-00270.

This author was supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM II).