Analysis of a bounding box heuristic for object intersection

Bounding boxes are commonly used in computer graphics and other fields to improve the performance of algorithms that should process only the intersecting objects. A bounding-box-based heuristic avoids unnecessary intersection processing by eliminating the pairs whose bounding boxes are disjoint. Empirical evidence suggests that the heuristic works well in many practical applications, although its worst-case performance can be bad for certain pathological inputs. What is a pathological input, however, is not well understood, and consequently there is no guarantee that the heuristic will always work well in a specific application. In this paper, we analyze the performance of bounding box heuristic in terms of two natural shape parameters, aspect ratio and scale factor. These parameters can be used to realistically measure the degree to which the objects are pathologically shaped. We derive tight worst-case bounds on the performance for bounding box heuristic. One of the significant contributions of our paper is that we only require that objects be well shaped on average. Somewhat surprisingly, the bounds are significantly different from the case when all objects are well shaped.