Probabilistic log-space reductions and problems probabilistically hard for p
We present in this paper an interesting kind of reductions and its variations (the probabilistic NC reductions) which allow us to argue about the parallel complexity of problems not easily shown to be complete for P. We show that a problem complete under these reductions cannot be in NC unless P c RNC. Based on these reductions we also show the P-completeness of a probabilistic problem, namely, given a dag for some nodes of which exactly one of the incoming edges fail (all incoming edges to such a node have equal probability to fail) approximate, within an arbitrary given absolute performance ratio the longest distance we can travel, with certainty, along the edges of the dag.
In order to show the above result we show the P-completeness of the problem of approximating, with a given absolute performance ratio, the depth at which the ones arrive in a boolean monotone circuit.
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