From discrepancy to declustering

Abstract

Declustering schemes allocate data blocks among multiple disks to enable parallel retrieval. Given a declustering scheme D , its response time with respect to a query Q , rt ( Q ), is defined to be the maximum number of data blocks of the query stored by the scheme in any one of the disks. If | Q | is the number of data blocks in Q and M is the number of disks, then rt ( Q ) is at least ⌈| Q |/ M ⌉. One way to evaluate the performance of D with respect to a set of range queries Q is to measure its additive error ---the maximum difference of rt ( Q ) from ⌈| Q |/ M ⌉ over all range queries Q ∈ Q.In this article, we consider the problem of designing declustering schemes for uniform multidimensional data arranged in a d -dimensional grid so that their additive errors with respect to range queries are as small as possible. It has been shown that for a fixed dimension d ≥ 2, any declustering scheme on an M d grid, a grid with length M on each dimension, will always incur an additive error with respect to range queries of Ω(log M ) when d = 2 and Ω(log d −1/2 M ) when d > 2.Asymptotically optimal declustering schemes exist for 2-dimensional data. However, the best general upper bound known so far for the worst-case additive errors of d -dimensional declustering schemes, d ≥ 3, is O ( M d −1 ), which is large when compared to the lower bound. In this article, we propose two declustering schemes based on low-discrepancy points in d -dimensions. When d is fixed, both schemes have an additive error of O (log d −1 M ) with respect to range queries, provided that certain conditions are satisfied: the first scheme requires that the side lengths of the grid grow at a rate polynomial in M , while the second scheme requires d ≥ 2 and M = p t where d ≤ p ≤ C , C a constant, and t is a positive integer such that t ( d − 1) ≥ 2. These are the first multidimensional declustering schemes with additive errors proven to be near optimal.