New upper bounds in Klee's measure problem
SIAM Journal on Computing
Voronoi diagrams in higher dimensions under certain polyhedral distance functions
Proceedings of the eleventh annual symposium on Computational geometry
On the complexity of computing the measure of ∪[ai,bi]
Communications of the ACM
Semi-Online Maintenance of Geometric Optima and Measures
SIAM Journal on Computing
Computing the volume of the union of cubes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Approximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects
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
An improved algorithm for computing the volume of the union of cubes
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
The union of probabilistic boxes: maintaining the volume
ESA'11 Proceedings of the 19th European conference on Algorithms
On Klee's measure problem for grounded boxes
Proceedings of the twenty-eighth annual symposium on Computational geometry
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The measure problem of Klee asks for the volume of the union of n axis-parallel boxes in a fixed dimension d. We give an ∂(n(d+2)/3) time algorithm for the special case of all boxes being cubes or, more generally, fat boxes. Previously, the fastest run-time was nd/2 2∂(log*n), achieved by the general case algorithm of Chan [SoCG 2008]. For the general problem our run-time would imply a complexity theoretic breakthrough for the k-clique problem and can thus be considered extremely unlikely.