Dynamic parallel complexity of computational circuits

Abstract

The dynamic parallel complexity of general computational circuits (defined in introduction) is discussed. We exhibit some relationships between parallel circuit evaluation and some uniform closure properties of a certain class of unary functions and present a systematic method for the design of processor efficient parallel algorithms for circuit evaluation. Using this method: (1) we improve the algorithm for parallel Boolean circuit evaluation; (2) we give a nontrivial upper bound for parallel min-max-plus circuit evaluation; (3) we partially answer the first open question raised in [MiRK85] by showing that all circuits over finite noncommutative semi-ring and circuits over infinite non-commutative semi-ring which has finite dimension over a commutative semi-ring can be evaluated in polylogarithmic time in its size and degree using M(n) processors. Moreover, we develop a theory for determining closure properties of certain classes of unary functions.