Our analysis of the bounding box heuristic is related to the idea of "realistic input models," which has become a topic of recent interest in computational geometry.
Our work is motivated less by the desire to show better combinatorial bounds for well behaved geometric structures, and more by the desire to validate an accepted belief that bounding boxes improve the performance of object intersection algorithms.
We start by considering a special case, where all objects have the same-size bounding box.
Since each object has bounding box size [[Alpha].sub.avg] and the average aspect ratio of [Laplace] is [[Alpha].sub.avg], we get (1/n) [[Sigma].sub.i] (1/[c.sub.i]) = 1, where [c.sub.i] = vol(c([P.sub.i])).
Consider a tiling of the plane by size [[Alpha].sub.avg] boxes that covers the portion of the plane occupied by the bounding boxes of the objects, namely, [union] b([P.sub.i]).
The assignment of objects to grid points is only for the proof, and we do not need to know the actual core geometry for the bounding box algorithm.)
We first consider the case where all objects have the same size bounding boxes.
Let [Laplace]' [is no subset or equal to] [Laplace] be a subset in which each object has the same size bounding box.