A (slightly) faster algorithm for Klee's measure problem

Given n axis-parallel boxes in a fixed dimension d≥3, how efficiently can we compute the volume of the union? This standard problem in computational geometry, commonly referred to as Klee's measure problem, can be solved in time O(nd/ ²logn) by an algorithm of Overmars and Yap (FOCS 1988). We give the first (albeit small) improvement: our new algorithm runs in time nd/ ²2 ^O( ^logn), where ^log denotes the iterated logarithm. For the related problem of computing the depth in an arrangement of n boxes, we further improve the time bound to near O(nd/ ²/logd/ ^2-1n), ignoring loglogn factors. Other applications and lower-bound possibilities are discussed. The ideas behind the improved algorithms are simple.