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 in 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[1][2] coined the name "Nick's class" after Nick Pippenger, who had done extensive research[3] on circuits with polylogarithmic depth and polynomial size.[4]
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.[5] NC is a subset of P because polylogarithmic parallel computations can be simulated by polynomial-time sequential ones. It is unknown whether NC = P, but most researchers suspect this to be false, meaning that there are probably some tractable problems that are "inherently sequential" and cannot significantly be sped up by using parallelism. Just as the class NP-complete can be thought of as "probably intractable", so the class P-complete, when using NC reductions, can be thought of as "probably not parallelizable" or "probably inherently sequential".
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. It can be CRCW, CREW, or EREW. See PRAM for descriptions of those models.
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.
- ^ Cook, S.A. (1981). "Towards a complexity theory of synchronous parallel computation". L'Enseignement Mathématique. 27: 99–124. Archived from the original on 2022-03-10.
- ^ Cook, Stephen A. (1985-01-01). "A taxonomy of problems with fast parallel algorithms". Information and Control. International Conference on Foundations of Computation Theory. 64 (1): 2–22. doi:10.1016/S0019-9958(85)80041-3. ISSN 0019-9958.
- ^ Pippenger, Nicholas (1979). "On simultaneous resource bounds". 20th Annual Symposium on Foundations of Computer Science (SFCS 1979): 307–311. doi:10.1109/SFCS.1979.29. ISSN 0272-5428. S2CID 7029313.
- ^ Arora & Barak (2009) p.120
- ^ Arora & Barak (2009) p.118