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 $O_d(\log^{d-1} M)$ 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 dimensions two and three. For general $d$, they work for all $M\geq d-1$ that are powers of two. Concerning lower bounds, 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.Comment: 19 pages, 1 figur

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 Od(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 dimensions two and three. For general d, they work for all M ≥ d − 1 that are powers of two. Concerning lower bounds, we show that a recent proof of a Ωd(log d−1 2 M) bound contains an error. We close the gap in the proof and thus establish the bound. ∗supported by the DFG-Graduiertenkolleg 357 “Effiziente Algorithmen un

Improved Bounds and Schemes for the Declustering Problem ⋆

Abstract. 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 among the devices. Using deep results from discrepancy theory, we improve previous work of several authors concerning rectangular queries of higher-dimensional data. For this problem, we give a declustering scheme with an additive error of Od(log d−1 M) independent of the data size, where d is the dimension, M the number of storage devices and d−1 not larger than the smallest prime power in the canonical decomposition of M. Thus, in particular, our schemes work for arbitrary M in two and three dimensions, and arbitrary M ≥ d−1 that is a power of two. These cases seem to be the most relevant in applications. For a lower bound, we show that a recent proof of a Ωd(log d−1 2 M) bound contains a critical error. Using an alternative approach, we establish this bound.