The probabilistic and deterministic parallel complexity of symmetric functions

Abstract

In this paper we prove lower bounds on the computational complexity of symmetric functions on parallel machines. The parallel models considered are PRIORITY and ARBITRARY, both widely used versions of the Parallel Random Access Machine (PRAM). We first consider PRIORITY. We prove that a PRIORITY PRAM with one shared memory cell and O(n) processors requires Ω(g(n)) time in order to decide if a binary vector of length n has at least g(n) 1's, for the case g(n)=o(n ^1/4). This is the decision problem for the threshold language L _g. Our lower bound is tight. For PRIORITY with m shared memory cells we prove an Ω(g(n)/m) lower bound. The limitation on the number of processors is essential, since with enough processors L _g can be decided by the same model in O(√g(n)) time. The limitation on g is important: for instance, for g(n)=n/2 L _g can be decided in time O(√g(n)), which is optimal [VW].

Our new technique for threshold languages enables us to show that for a large range of number of processors, PRIORITY and ARBITRARY with one shared memory cell require the same time complexity for computing large classes of symmetric functions. These results are of interest since PRIORITY is known to be more powerful than ARBITRARY (see [FMRW], for instance).

Next we show that a probabilistic PRIORITY with m≤n ^∈,(0≤∈<1) shared memory cells and any finite number of processors requires expected time more than 1−ε / 4 logn to compute PARITY of n bits. (Note that with enough processors and shared memory PRIORITY can compute PARITY in constant time even deterministically). For probabilistic PRIORITY without ROM we prove a tight Ω(n/m) lower bound. The probabilistic case is much more complicated than the deterministic case, hence we develop new techniques for probabilistic lower bounds.

This work was supported in part by the National Science Foundation under Grant No. DCR-8606366, and by the Office of Research and Graduate Studies of the Ohio State University. First Author's current address: Aiken Computation Lab., Harvard University, Cambridge, MA 02138.