Algorithms for parallel memory,II: Hierarchical multilevel memories

In this paper we introduce parallel versions of two hierarchical memory models and give optimal algorithms in these models for sorting, FFT, and matrix multiplication. In our parallel models, there areP memory hierarchies operating simultaneously; communication among the hierarchies takes place at a base memory level. Our optimal sorting algorithm is randomized and is based upon the probabilistic partitioning technique developed in the companion paper for optimal disk sorting in a two-level memory with parallel block transfer. The probability of using/times the optimal running time is exponentially small in (log ) logP.A summarized version of this research was presented at the 22nd Annual ACM Symposium on Theory of Computing, Baltimore, MD, May 1990. This work was done while the first author was at Brown University. Support was provided in part by a National Science Foundation Presidential Young Investigator Award with matching funds from IBM, by NSF Research Grants DCR-8403613 and CCR-9007851, by Army Research Office Grant DAAL03-91-G-0035, and by the Office of Naval Research and the Defense Advanced Research Projects Agency under Contract N00014-91-J-4052 ARPA Order 8225. This work was done in part while the second author was at Brown University supported by a Bellcore graduate fellowship and at Bellcore.