Limits on the power of concurrent-write parallel machines

Abstract

The computation of specific functions using the most general form of concurrentread-concurrent-write parallel RAM is considered. It is shown that such a machine can compute any function of Boolean inputs in time log n -log log n + 0( 1) given a polynomial number of processors and memory cells and that this bound is tight for integer addition. Despite this evidence of the power of the model we show that a very simple function, namely parity, requires time Q (Jiogn) to compute given a polynomial bound on the number of processors, independent of the number of memory cells. 13"

Two lower bound arguments with "inaccessible" numbers

“…Using the same lemma, we also show that the lower bound of Q(log n) remains valid even if the operation set of the PRAM consists of arbitrary functions of bounded arity (which makes the machine as powerful as that in [2]). In the proof, Gallai's theorem from Ramsey theory (see [lo]) is applied in a similar way as in [S].…”

“…. x,), where the input numbers xi have polynomially in n many bits, on PRAMS with exponentially many processors and arbitrary instructions, see [2].) The first step of the proof is a lemma: with a computation time of t < log n parallel steps a PRAM can compute only functions that can be expressed by a definition of cases in terms of functions which depend on < 2' < n variables.…”