The Union of Probabilistic Boxes: Maintaining the Volume

Suppose we have a set of n axis-aligned rectangular boxes in d-space, {B1, B2, ..., Bn}, where each box Bi is active (or present) with an independent probability pi. We wish to compute the expected volume occupied by the union of all the active boxes. Our main result is a data structure for maintaining the expected volume over a dynamic family of such probabilistic boxes at an amortized cost of O(n(d-1)/2 log n) time per insert or delete. The core problem turns out to be one-dimensional: we present a new data structure called an anonymous segment tree, which allows us to compute the expected length covered by a set of probabilistic segments in logarithmic time per update. Building on this foundation, we then generalize the problem to d dimensions by combining it with the ideas of Overmars and Yap [13]. Surprisingly, while the expected value of the volume can be efficiently maintained, we show that the tail bounds, or the probability distribution, of the volume are intractable-- specifically, it is NP-hard to compute the probability that the volume of the union exceeds a given value V even when the dimension is d = 1.