54,815 research outputs found
A New Randomized Algorithm to Approximate the Star Discrepancy Based on Threshold Accepting
We present a new algorithm for estimating the star discrepancy of arbitrary point sets. Similar to the algorithm for discrepancy approximation of Winker and Fang [SIAM J. Numer. Anal. 34 (1997), 2028{2042] it is based on the optimization algorithm threshold accepting. Our improvements include, amongst others, a non-uniform sampling strategy which is more suited for higher-dimensional inputs and additionally takes into account the topological characteristics of given point sets, and rounding steps which transform axis-parallel boxes, on which the discrepancy is to be tested, into critical test boxes. These critical test boxes provably yield higher discrepancy values, and contain the box that exhibits the maximum value of the local discrepancy. We provide comprehensive experiments to test the new algorithm. Our randomized algorithm computes the exact discrepancy frequently in all cases where this can be checked (i.e., where the exact discrepancy of the point set can be computed in feasible time). Most importantly, in higher dimension the new method behaves clearly better than all previously known methods
Improved Bounds and Schemes for the Declustering Problem
The declustering problem is to allocate given data on parallel working
storage devices in such a manner that typical requests find their data evenly
distributed on the devices. Using deep results from discrepancy theory, we
improve previous work of several authors concerning range queries to
higher-dimensional data. We give a declustering scheme with an additive error
of independent of the data size, where is the
dimension, the number of storage devices and does not exceed the
smallest prime power in the canonical decomposition of into prime powers.
In particular, our schemes work for arbitrary in dimensions two and three.
For general , they work for all that are powers of two.
Concerning lower bounds, we show that a recent proof of a
bound contains an error. We close the gap in
the proof and thus establish the bound.Comment: 19 pages, 1 figur
Deviations of ergodic sums for toral translations II. Boxes
We study the Kronecker sequence on the torus
when is uniformly distributed on We
show that the discrepancy of the number of visits of this sequence to a random
box, normalized by , converges as to a Cauchy
distribution. The key ingredient of the proof is a Poisson limit theorem for
the Cartan action on the space of dimensional lattices.Comment: 56 pages. This is a revised and expanded version of the prior
submission
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