Approximate range searching: The absolute model

Abstract

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. Most prior work on this problem has focused on the case of relative errors, where each range shape R is bounded, and points within distance ε ⋅ diam ( R ) of the range's boundary may or may not be included. We consider a different approximation model, called the absolute 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 range spaces consisting of halfspaces, Euclidean balls, simplices, axis-aligned rectangles, and general convex bodies. We consider a variety of problem formulations, including range searching under general commutative semigroups, idempotent semigroups, groups, and range emptiness. We show how idempotence can be used to improve not only approximate, but also exact halfspace range searching. Our data structures are much simpler than both their exact and relative model counterparts, and so are amenable to efficient implementation.