Limitations of the QRQW and EREW PRAM models

Abstract

We consider parallel random access machines (PRAMs) with restricted access to the shared memory resulting from handling congestion of memory requests. We study the (SIMD) QRQW PRAM model where multiple requests are queued and serviced one at a time. We also consider exclusive read exclusive write (EREW) PRAM and its modification obtained by adding a single bus.

For the QRQW PRAMs we investigate the case when the machine can measure the duration of a single step. Even for such a (powerful) QRQW PRAM PARITY of n bits (PARITY_n) requires Ω(log n) time while OR of n bits can be computed deterministically in a constant time. On randomized QRQW PRAM the function PARITY_n is still difficult. We prove a lower time bound Ω(√ log n/log log n) for algorithms that succeed with probability 0.5+∃ (∃>0). These bounds show that implementing concurrent writes may degradate runtime of a CRCW PRAM algorithm.

The simple 2-compaction problem is known to be hard for EREW PRAM. The same time bound Ω(√log n) for this problem has been proved for both deterministic and randomized EREW PRAM. We show that this is not a coincidence since the time complexity of this problem is the same for deterministic and randomized case. The technique which we apply is quite general and may be used to obtain similar results for any problem where the number of input configurations is small.

We also show that improving time bound Ω(√ log n) for 2-compaction on EREW PRAM requires novel and more sophisticated techniques.

Partially supported by KBN grants 8 S503 002 07, 2 P301 034 07, DFG-Sonderforschungsbereich 376 “Massive ParallelitÄt” and EU ESPRIT Long Term Research Project 20244 (ALCOM-IT)