Abstract
It is reasonable to expect parallel machines to be faster than sequential ones. But exactly how much faster do we expect them to be? Various authors have observed that an exponential speedup is possible if sufficiently many processors are available. One such author has claimed (erroneously) that this is a counterexample to the parallel computation thesis. We show that even more startling speedups are possible. In fact, if enough processors are used, any recursive function can be computed in constant time. Although such machines clearly do not obey the parallel computation thesis, we argue that they still provide evidence in favour of it. In contrast, we show that an arbitrary polynomial speedup of sequential machines is possible on a model which satisfies the parallel computation thesis. If, as widely conjectured, P⊈POLYLOGSPACE, then there can be no exponential speedup on such a model.
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