Limits on the Power of Parallel Random Access Machines with Weak Forms of Write Conflict Resolution

The ROBUST PRAM is a concurrent-read concurrent-write (CRCW) parallel random access machine in which any value might appear in a memory cell as a result of a write connict. This paper addresses the question of whether a PRAM with such a weak form of write connict resolution can compute functions faster than the concurrent-read exclusive-write (CREW) PRAM. We prove a lower bound on the time required by the ROBUST PRAM to compute Boolean functions in terms of the number of diierent values each memory cell of the PRAM can contain and the degree of the function when expressed as a polynomial over a nite eld. In the case of 1-bit memory cells, our lower bound for the problem of computing the OR of n Boolean variables exactly matches Cook, Dwork, and Reischuk's upper bound on the CREW PRAM 2]. We extend our result to obtain a lower bound depending on the number of processors, for computing Boolean functions on the ROBUST PRAM, even with memory cells of unbounded size. A particular consequence is that the ROBUST PRAM with 2 2 O(p log n) processors requires (p log n) steps to compute OR. These results are obtained by deening a class of CRCW PRAMs, the xed adversary PRAMs, all of which are at least as powerful as the ROBUST PRAM. We prove our lower bounds using carefully chosen PRAMs from this class. We also show the limitations of this technique by describing how, with n-bit memory cells, any PRAM in this class can compute OR and, more generally, simulate a PRIORITY PRAM in constant time. Finally, we show that the ROBUST PRAM with 1-bit memory cells can compute OR in O((log n) 2) steps using O(n=(log n) 2) processors with o(1) probability of error. Thus, adding randomization increases the computational power of the ROBUST PRAM. This is in contrast to the situation for the CREW PRAM, where adding randomization can at best decrease time by a constant factor 3].