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

TLDR: A lower bound on the time required by the ROBUST PRAM to compute Boolean functions in terms of the number of different values each memory cell of the PRAM can contain and the degree of the function when expressed as a polynomial over a finite field is proved.

Abstract: 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 conflict. This paper addresses the question of whether a PRAM with such a weak form of write conflict 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 different values each memory cell of the PRAM can contain and the degree of the function when expressed as a polynomial over a finite field. 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. 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 O( log n) processors requires 0(log n) steps to compute OR. These results are obtained by defining a class of CRCW PRAMs, the fixed 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 fixed adversary PRAM can compute OR and, more generally, simulate a PRIORITY PRAM in constant time. Finally, we consider the effect of adding randomization to the ROBUST PRAM. For any algorithm that computes OR without error, its expected running time on its worst input is no better than the worst case deterministic time complexity of computing OR. However, allowing a small probability of error enables the ROBUST PRAM with single bit memory cells to compute OR in almost constant time. ] 1996 Academic Press, Inc.