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 disk blocks of the query stored by the scheme in any one of the disks. If vQv is the number of data blocks in Q and M is the number of disks then rt(Q) is at least vQv/M. One way to evaluate the performance of D with respect to a set of queries 𝑄 is to measure its additive error - the maximum difference between rt(Q) from vQv/M over all range queries Q e 𝑄.In this paper, 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 such declustering schemes will have an additive error of Ω(log M) when d = 2 and Ω(log d-1/2 M) when d > 2 with respect to range queries.Asymptotically optimal declustering schemes exist for 2-dimensional data. For data in larger dimensions, however, the best bound for additive errors is O(Md-1), which is extremely large. In this paper, we propose the two declustering schemes based on low discrepancy points in d-dimensions. When d is fixed, both schemes have an additive error of O(logd-1 M) with respect to range queries provided certain conditions are satisfied: the first scheme requires d ≥ 3 and M to be a power of a prime where the prime is at least d while the second scheme requires the size of the data to grow within some polynomial of M, with no restriction on M. These are the first known multidimensional declustering schemes with additive errors near optimal.