An Efficient Parallel Algorithm for the Layered Planar Monotone Circuit Value Problem
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A planar monotone circuit (PMC) is a Boolean circuit that can be embedded in the plane and that contains only AND and OR gates. A layered PMC is a PMC in which all input nodes are in the external face, and the gates can be assigned to layers in such a way that every wire goes between gates in successive layers. Goldschlager, Cook and Dymond, and others have developed NC 2 algorithms to evaluate a layered PMC when the output node is in the same face as the input nodes. These algorithms require a large number of processors (Ω(n 6 ), where n is the size of the input circuit).
In this paper we give an efficient parallel algorithm that evaluates a layered PMC of size n in \(O(\log^2 n)\) time using only a linear number of processors on an EREW PRAM. Our parallel algorithm is the best possible to within a polylog factor, and is a substantial improvement over the earlier algorithms for the problem.
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