Computing Klee’s Measure of Grounded Boxes

Abstract

A well-known problem in computational geometry is Klee's measure problem, which asks for the volume of a union of axis-aligned boxes in d-space. In this paper, we consider Klee's measure problem for the special case where a 2-dimensional orthogonal projection of all the boxes has a common corner. We call such a set of boxes 2-grounded and, more generally, a set of boxes is k-grounded if in a k-dimensional orthogonal projection they share a common corner. Our main result is an O(n (d?1)/2log2 n) time algorithm for computing Klee's measure for a set of n 2-grounded boxes. This is an improvement of roughly $O(\sqrt{n})$ compared to the fastest solution of the general problem. The algorithm works for k-grounded boxes, for any k?2, and in the special case of k=d, also called the hypervolume indicator problem, the time bound can be improved further by a logn factor. The key idea of our technique is to reduce the d-dimensional problem to a semi-dynamic weighted volume problem in dimension d?2. The weighted volume problem requires solving a combinatorial problem of maintaining the sum of ordered products, which may be of independent interest.