Cuttings for disks and axis-aligned rectangles

Abstract

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 1/r-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 ℝ3 and a parameter r ∈ ℕ, we construct a 1/r-cutting of size O(r2). For n axis-aligned rectangles in ℝ3, we construct a 1/r-cutting of size O(r 3/2). 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 ℝ3, the points can be connected with a straight line spanning tree such that every disk cuts 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 cuts 0(n1/3) edges. Both bounds are the best possible. © Springer-Verlag Berlin Heidelberg 2007.