Approximate Range Searching In The Absolute Error Model

Range searching is a well known problem in the area of geometric data structures. We consider this problem in the context of approximation, where an approximation parameter ε > 0 is provided, and points that are sufficiently close to the range’s boundary may or may not be included. Most prior work on approximating this problem has focused on the case of relative errors, in which it is assumed that each range shape R is bounded, and points that lie within distance ε · diam(R) of the range’s boundary may or may not be included. Here we consider a different approximation model, called the absolute error model, in which points that within distance ε of the range’s boundary may or may not be included, regardless of the diameter of the range. This model was first introduced by Chazelle, Liu, and Magen in the context of high-dimensional halfspace and spherical range searching. This absolute error model is more meaningful for many applications, and it leads to much more efficient and much simpler solutions than the relative error model. In this proposal, we present a number of results on range searching in the absolute error model under the assumption that the dimension d is a fixed constant. We consider range spaces consisting of orthogonal rectangles, halfspaces, Euclidean balls, simplices, and general convex bodies. We consider a variety of problem formulations, including range searching under semigroups and groups and range emptiness. We also consider the issue of how to construct complex range shapes from simpler ones. This issue of constructiveness does not arise in the case of exact range searching, and has not been considered in previous work on approximate range searching. We present a collection of simple techniques and constructions that can be applied to a wide variety of approximate range problems in this model.