Klee´s Measure Problem Made Easy

We present a new algorithm for a classic problem in computational geometry, Klee´s measure problem: given a set of n axis-parallel boxes in d-dimensional space, compute the volume of the union of the boxes. The algorithm runs in O(n^d/2) time for any constant d ≥ 3. Although it improves the previous best algorithm by “just” an iterated logarithmic factor, the real surprise lies in the simplicity of the new algorithm. We also show that it is theoretically possible to beat the O(n^d/2) time bound by logarithmic factors for integer input in the word RAM model, and for other variants of the problem. With additional work, we obtain an O(n^d/3 polylog n)-time algorithm for the important special case of orthants or unit hypercubes (which include the so-called “hypervolume indicator problem”), and an O(n^(d+1)/3 polylog n)-time algorithm for the case of arbitrary hypercubes or fat boxes, improving a previous O(n(^d+2)/3)-time algorithm by Bringmann.