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 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 ≥ d − 1 that are powers of two. Concerning lower bounds, we show that a recent proof of a Ω d (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 und

Geometric Characterization of Weyl’s Discrepancy Norm in Terms of Its n-Dimensional Unit Balls

“…Applications can be found in the field of numerical integration, especially for Monte Carlo methods in high dimensions, see, e.g., [28,36,40], or in computational geometry, see, e.g., [1,9,21]. For applications to data storage problems on parallel disks, see [10,13] and for halftoning images, see [31].This paper is motivated by [24] which applies Weyl's discrepancy concept in order to derive an ordering-dependent norm for measuring the (dis-)similarity between patterns. In this context the focus lies on evaluating the auto-misalignment that measures…”