On Klee's Measure Problem for Grounded Boxes

摘要

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 O(√(1/2)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 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.