It is notable, however, that the parallel algorithms for the problems just mentioned only require a number of processors which is polynomial in size of their input \(n\) and also only require time polylogarithmic in \(n\) (i.e. \(O(\log^k(n))\) for fixed \(k\)). This motivates the definition of \(\textbf{NC}\) as the class of problems which can be solved by a PRAM machine in time \(O(\log^c(n))\) using \(O(n^k)\) processors (for fixed \(c,k \in \mathbb{N}\)).[31] In particular, the amount of work (defined as the sum of the number of steps required by each processor) performed by a machine satisfying this definition will be polynomial in the size of its input.