New upper bounds in Klee's measure problem
SIAM Journal on Computing
Semi-online maintenance of geometric optima and measures
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Computing the volume of the union of cubes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
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Computational Geometry: Algorithms and Applications
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ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
A (slightly) faster algorithm for Klee's measure problem
Computational Geometry: Theory and Applications
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EMO'03 Proceedings of the 2nd international conference on Evolutionary multi-criterion optimization
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
Parameterized average-case complexity of the hypervolume indicator
Proceedings of the 15th annual conference on Genetic and evolutionary computation
GECCO 2013 tutorial on evolutionary multiobjective optimization
Proceedings of the 15th annual conference companion on Genetic and evolutionary computation
Speeding up many-objective optimization by Monte Carlo approximations
Artificial Intelligence
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A well-known problem in computational geometry is Klee's measure problem, which asks for the volume of a union of axis-aligned boxes in d-space. In this paper, we consider Klee's measure problem for the special case where a 2-dimensional orthogonal projection of all the boxes has a common corner. We call such a set of boxes 2-grounded and, more generally, a set of boxes is k-grounded if in a k-dimensional orthogonal projection they share a common corner. Our main result is an O(n(d-1)/2log2n) time algorithm for computing Klee's measure for a set of n 2-grounded boxes. This is an improvement of roughly O(√n) compared to the fastest solution of the general problem. The algorithm works for k-grounded boxes, for any k ≥ 2, and in the special case of k=d, also called the hypervolume indicator problem, the time bound can be improved further by a log n factor. The key idea of our technique is to reduce the d-dimensional problem to a semi-dynamic weighted volume problem in dimension d-2. The weighted volume problem requires solving a combinatorial problem of maintaining the sum of ordered products, which may be of independent interest.