Optimal bounds for decision problems on the CRCW PRAM

Abstract

Optimal Ω(log n /log log n ) lower bounds on the time for CRCW PRAMS with polynomially bounded numbers of processors or memory cells to compute parity and a number of related problems are proven. A strict time hierarchy of explicit Boolean functions of n bits on such machines that holds up to Ο(log n /log log n ) time is also exhibited. That is, for every time bound T within this range a function is exhibited that can be easily computed using polynomial resources in time T but requires more than polynomial resources to be computed in time T - 1. Finally, it is shown that almost all Boolean functions of n bits require log n - log log n + Ω(1) time when the number of processors is at most polynomial in n . The bounds do not place restrictions on the uniformity of the algorithms nor on the instruction sets of the machines.