New upper bounds in Klee's measure problem
SIAM Journal on Computing
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
Static and dynamic algorithms for k-point clustering problems
Journal of Algorithms
Computing the discrepancy with applications to supersampling patterns
ACM Transactions on Graphics (TOG)
On dynamic algorithms for algebraic problems
Journal of Algorithms
On the complexity of computing the measure of ∪[ai,bi]
Communications of the ACM
Internet packet filter management and rectangle geometry
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Lower bounds for dynamic algebraic problems
Information and Computation
Optimal Algorithms for List Indexing and Subset Rank
WADS '89 Proceedings of the Workshop on Algorithms and Data Structures
Semi-Online Maintenance of Geometric Optima and Measures
SIAM Journal on Computing
Efficient algorithms for shared camera control
Proceedings of the nineteenth annual symposium on Computational geometry
Lower bounds for algebraic computation trees
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Binary Space Partitions for Axis-Parallel Segments, Rectangles, and Hyperrectangles
Discrete & Computational Geometry
Computing the volume of the union of cubes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
More algorithms for all-pairs shortest paths in weighted graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Geometric clustering: fixed-parameter tractability and lower bounds with respect to the dimension
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Subquadratic algorithms for 3SUM
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
S-metric calculation by considering dominated hypervolume as klee's measure problem
Evolutionary Computation
The complexity of geometric problems in high dimension
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Rectangle-efficient aggregation in spatial data streams
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
Improved algorithms to network p-center location problems
Computational Geometry: Theory and Applications
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Given n axis-parallel boxes in a fixed dimension d ≥ 3, how efficiently can we compute the volume of the union? This standard problem in computational geometry, commonly referred to as Klee's measure problem, can be solved in time O(nd/2 log n) by an algorithm of Overmars and Yap (FOCS 1988). We give the first (albeit small) improvement: our new algorithm runs in time nd/22O(log*n), where log* denotes the iterated logarithm. For the related problem of computing the depth in an arrangement of n boxes, we further improve the time bound to near O(nd/2 logd/2-1 n), ignoring log\log n factors. Other applications and lower-bound possibilities are discussed. The ideas behind the improved algorithms are simple.