The Dynamic Parallel Complexity of Computational Circuits

We establish connections between parallel circuit evaluation and uniform algebraic closure properties of unary function classes. We use this connection in the development of time-efficient and processor-efficient parallel algorithms for the evaluation of algebraic circuits. Our algorithm provides a nontrivial upper bound on the parallel complexity of the circuit value problem over $\{{\Bbb R},\min,\max,+\}$ and $\{{\Bbb R}^{+},\min,\max,\times\}$. We partially answer an open question of Miller, Ramachandran, and Kaltofen by showing that circuits over a polynomial-bounded noncommutative semiring and circuits over infinite noncommutative semirings with a polynomial-bounded dimension over a commutative semiring can be evaluated in polylogarithmic time in their size and degree using a polynomial number of processors. We also present an improved parallel algorithm for Boolean circuits.