The Union of Probabilistic Boxes: Maintaining the Volume

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2011-01-01
Yıldız, Hakan
Hershberger, John
Suri, Subhash
Suppose we have a set of n axis-aligned rectangular boxes in d-space, {B-1, B-2,..., B-n}, where each box B-i 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.

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Citation Formats
H. Yıldız, J. Hershberger, and S. Suri, “The Union of Probabilistic Boxes: Maintaining the Volume,” Saarbrücken, Almanya, 2011, vol. 6942, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/93998.