New upper bounds in Klee's measure problem
SIAM Journal on Computing
Efficient maintenance of the union intervals on a line, with applications
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computing the volume of the union of cubes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
A (slightly) faster algorithm for Klee's measure problem
Computational Geometry: Theory and Applications
Klee's measure problem on fat boxes in time ∂(n(d+2)/3)
Proceedings of the twenty-sixth annual symposium on Computational geometry
An improved algorithm for computing the volume of the union of cubes
Proceedings of the twenty-sixth annual symposium on Computational geometry
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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.