# Fast and optimal simulations between CRCW PRAMs

## Abstract

We describe new simulations between different variants of the CRCW PRAM. On a Minimum PRAM, the minimum value is written in the event of a write conflict, whereas concurrent writing to the same cell is without effect on a Tolerant PRAM. Compared to other variants of the CRCW PRAM, the Minimum PRAM is very strong, and the Tolerant PRAM is very weak. In fact, the two variants are near opposite ends of the spectrum of CRCW PRAM variants commonly considered. We show that one step of a (randomized) Minimum PRAM with *n* processors can be simulated with high probability in *O*((log^{*}*n*)^{3}) time on a randomized Tolerant PRAM with *O(n*/(log^{*}*n*)^{3}) processors. The simulation is optimal, in the sense that the product of its slowdown and the number of simulating processors is within a constant factor of the number of simulated processors. It subsumes most previous work on randomized simulations between *n*-processor CRCW PRAMs with infinite memories.

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