Computing Klee’s Measure of Grounded Boxes

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)/2 log 2 n ) time algorithm for computing Klee’s measure for a set of n 2-grounded boxes. This is an improvement of roughly 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 log n 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.