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In [[computational complexity theory|complexity theory]], the class '''NC''' ("Nick's Class") is the set of [[decision problem|decision problems]] decidable in polylogarithmic time on a [[parallel computing|parallel computer]] with a polynomial number of processors.  In other words, a problem is in '''NC''' if there are constants ''c'' and ''k'' such that it can be solved in time O((log ''n'')<sup>''c''</sup>) using O(''n''<sup>''k''</sup>) parallel processors.  

Just as the class [[Complexity classes P and NP|P]] can be thought of as the tractable problems, so NC can be thought of as the problems that can be efficiently solved on a parallel computer.  NC is a subset of P because parallel computers can be simulated by sequential ones. It is unknown whether NC = P, but most researchers suspect this to be false, meaning that there are some tractable problems which are probably "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]] 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'' ([[pParallel 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 [[pParallel random access mMachine|PRAM]] for descriptions of those models.  

Equivalently, NC can be defined as those decision problems decidable by [[Boolean circuit|uniform Boolean circuits]] with polylogarithmic depth and a polynomial number of gates.

'''NC'''<sup>''i''</sup> is the class of decision problems decidable by uniform boolean circuits with a polynomial number of gates and depth O((log ''n'')<sup>''i''</sup>), or the class of decision problems solvable in time O((log ''n'')<sup>''i''</sup>) on a parallel computer with a polynomial number of processors.

[[Stephen Cook]] coined the name NC ("Nick's class") after [[Nick Pippenger]], who had done extensive research on circuits with polylogarithmic depth and polynomial size.

==Reference==
* Greenlaw, Raymond, James Hoover, and Walter Ruzzo. ''Limits To Parallel computation; P-Completeness Theory''. ISBN 0195085914
* Heribert Vollmer. ''[http://www.thi.uni-hannover.de/forschung/publikationen/cc/index.en.php Introduction to Circuit Complexity -- A Uniform Approach]''. ISBN 3-540-64310-9

{{ComplexityClasses}}[[Category:Complexity classes]]

[[de:NC (Komplexitätsklasse)]]