No abstract

No abstract

No abstract

We study the depth (~-parallel time) complexity of widely used shared memory CRCW PRAM models. We first consider COMMON(m), ARBITRARY(m) and PRIORITY(m), for m<-n c (the communication width), assuming that the inputs are given in a read only memory (ROM) which can be read by all processors ([CO], [VW]).(1) With inputs in ROM, we prove that COMMON(n ~) is weaker than ARBITRARY(n'), ARBITRARY(n ~) is weaker than PRIORITY(n'), for e~l. The pioneering papers [FRW] and [FMRW] showed that if each processor can read only one input, then COMMON(n ~) is weaker than PRIORITY(n~). We remove this restriction. The case with ROM is important [VW] and requires new techniques.(2) We answer a conjecture by [VW] on computing PARITY on nondeterministic PRIORITY(l). Without ROM (each processor has one input), we prove a tight I~/(X/~-) bound.With ROM, we prove ~(loglogn) lower bound and O( n 1/3) upper bound.(3) We obtain optimal deterministic simulations of nondeterministie PRAM's, providing another new technique for obtaining tight lower bounds for and separation between nondeterministic models.(4) For models without ROM, we prove that symmetric functions with inputs from a finite domain cannot separate between ARBITRARY(l) and PRIORITY(I).In the second part, we for the first time successfully apply Kolmogorov-complexity to obtain lower bounds for PRIOR-ITY(oo) (the most powerful concurrent write PRAM with shared memory). We develop two new and general lower bound techniques which enable us to give very clean proofs for(1) The first depth-processor-input_size tradeoff,