Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension
Journal of Combinatorial Theory Series A
Covering numbers, vapnik-červonenkis classes and bounds for the star-discrepancy
Journal of Complexity
On the tractability of multivariate integration and approximation by neural networks
Journal of Complexity
Bounds and constructions for the star-discrepancy via δ-covers
Journal of Complexity
Generating randomized roundings with cardinality constraints and derandomizations
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
On probabilistic results for the discrepancy of a hybrid-Monte Carlo sequence
Journal of Complexity
Algorithmic construction of low-discrepancy point sets via dependent randomized rounding
Journal of Complexity
Covering numbers, dyadic chaining and discrepancy
Journal of Complexity
Hardness of discrepancy computation and ε-net verification in high dimension
Journal of Complexity
A New Randomized Algorithm to Approximate the Star Discrepancy Based on Threshold Accepting
SIAM Journal on Numerical Analysis
On the inverse of the discrepancy for infinite dimensional infinite sequences
Journal of Complexity
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In the first part of this paper we derive lower bounds and constructive upper bounds for the bracketing numbers of anchored and unanchored axis-parallel boxes in the d-dimensional unit cube. In the second part we apply these results to geometric discrepancy. We derive upper bounds for the inverse of the star and the extreme discrepancy with explicitly given small constants and an optimal dependence on the dimension d, and provide corresponding bounds for the star and the extreme discrepancy itself. These bounds improve known results from [B. Doerr, M. Gnewuch, A. Srivastav, Bounds and constructions for the star-discrepancy via @d-covers, J. Complexity 21 (2005) 691-709], [M. Gnewuch, Bounds for the average L^p-extreme and the L^~-extreme discrepancy, Electron. J. Combin. 12 (2005) Research Paper 54] and [H. N. Mhaskar, On the tractability of multivariate integration and approximation by neural networks, J. Complexity 20 (2004) 561-590]. We also discuss an algorithm from [E. Thiemard, An algorithm to compute bounds for the star discrepancy, J. Complexity 17 (2001) 850-880] to approximate the star-discrepancy of a given n-point set. Our lower bound on the bracketing number of anchored boxes, e.g., leads directly to a lower bound of the running time of Thiemard's algorithm. Furthermore, we show how one can use our results to modify the algorithm to approximate the extreme discrepancy of a given set.