Cuttings for Disks and Axis-Aligned Rectangles in Three-Space

We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r ∈ N, an 1r -cutting is a covering of the space with simplices such that the interior of each simplex intersects at most n/r objects. For n pairwise disjoint disks in R and a parameter r ∈ N, we construct a 1r -cutting of size O(r). For n axis-aligned rectangles in R, we construct a 1r -cutting of size O(r ). As an application related to multi-point location in three-space, we present tight bounds on the cost of spanning trees across barriers. Given n points and a finite set of disjoint disk barriers in R, the points can be connected with a straight line spanning tree such that every disk is stabbed by at most O( √ n) edges of the tree. If the barriers are axis-aligned rectangles, then there is a straight line spanning tree such that every rectangle is stabbed by O(n) edges. Both bounds are best possible.