05-1   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## 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 rectangular queries to higher-dimensional data. We give a declustering scheme with an additive error of $O_d(\log^{d-1} M)$ independent of the data size, where $d$ is the dimension, $M$ the number of storage devices and $d-1$ does not exceed the smallest prime power in the canonical decomposition of $M$ into prime powers. In particular, our schemes work for arbitrary $M$ in two and three dimensions, and arbitrary $M$ that is a power of two and at least $d-1$. For a lower bound, we show that a recent proof of a $\Omega_d(\log^{\frac{d-1}{2}} M)$ bound contains an error. We close the gap in the proof and thus establish the bound.

Mathematics Subject Classification (1991): 11K38

Keywords: Discrepancy, declustering, hypergraph