NC (complexity)

In computational complexity theory, the class NC (for Nick's class) is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem with input size n is NC if there exist constants c and k such that it can be solved in time O((log n)c) using O(nk) parallel processors.

Stephen Cook coined the name Nick's class after Nick Pippenger, who had done extensive research on circuits with polylogarithmic depth and polynomial size. As in the case of circuit complexity theory, usually the class has an extra constraint that the circuit family must be uniform (see below).

Equivalently, NC can be defined as those decision problems decidable by a uniform Boolean circuit (which can be calculated from the length of the input, for NC, we suppose we can compute the Boolean circuit of size n in logarithmic space in n) with polylogarithmic depth and a polynomial number of gates with a maximum fan-in of 2.

RNC is a class extending NC with access to randomness.

Problems in NC

Just as the class P can be thought of as the tractable problems (Cobham's thesis), so NC can be thought of as the problems that can be efficiently solved on a parallel computer. The parallel computer in the definition can be assumed to be a parallel, random-access machine (PRAM). That is a parallel computer with a central pool of memory, and any processor can access any bit of memory in constant time. The definition of NC is not affected by the choice of how the PRAM handles simultaneous access to a single bit by more than one processor.