An ${\mathcal{N} \mathcal{C}}$ Algorithm for Evaluating Monotone Planar Circuits

Abstract

Goldschlager first established that a special case of the monotone planar circuit problem can be solved by a Turing machine in $O(\log^{2} n)$ space. Subsequently, Dymond and Cook refined the argument and proved that the same class can be evaluated in $O(\log^{2} n)$ time with a polynomial number of processors. In this paper, we prove that the general monotone planar circuit value problem can be evaluated in $O(\log^{4} n)$ time with a polynomial number of processors, settling an open problem posed by Goldschlager and Parberry.