Abstract

Range searching is a well known problem in computational geometry. We consider this problem in the context of approximation, where an approximation parameter ε > 0 is provided. Most prior work on this problem has focused on the relative error model, where each range shape R is bounded, and points within distance ε · diam(R) of the range’s boundary may or may not be in-cluded. We introduce a different approximation model, called the absolute error model, in which points within distance ε of the range’s boundary may or may not be included, regardless of the diameter of the range. We consider sets of ranges consisting of general convex bodies, axis-aligned rectangles, halfspaces, Euclidean balls, and simplices. We examine a variety of problem formulations, including range searching under general commutative semigroups, idempotent semigroups, groups, range emptiness, and range re-porting. We apply our data structures to several related problems, including range sketching, approximate nearest neighbor searching, exact idempotent range searching, approximate range searching in the data stream model, and

Citations