Asymptotically optimal declustering schemes for range queries

Declustering techniques have been widely adopted in parallel storage systems (e.g. disk arrays) to speed up bulk retrieval of multidimensional data. A declustering scheme distributes data items among multiple devices, thus enabling parallel I/O access and reducing query response time. We measure the performance of any declustering scheme as its worst case additive deviation from the ideal scheme. The goal thus is to design declustering schemes with as small an additive error as possible. We describe a number of declustering schemes with additive error O(log M) for 2-dimensional range queries, where M is the number of disks. These are the first results giving such a strong bound for any value of M. Our second result is a lower bound on the additive error. In 1997, Abdel-Ghaffar and Abbadi showed that except for a few stringent cases, additive error of any 2-dim declustering scheme is at least one. We strengthen this lower bound to Omega((log M)(d-1/2)) for d-dim schemes and to Omega (log M) for 2-dim schemes, thus proving that the 2-dim schemes described in this paper are (asymptotically) optimal. These results are obtained by establishing a connection to geometric discrepancy, a widely studied area of mathematics. We also present simulation results to evaluate the performance of these schemes in practice.