In [[computational complexity theory|complexity theory]], the class '''NC''' (for "Nick's Class") is the set of [[decision problem]]s 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 exist constants <math>c</math> and <math>k</math> such that it can be solved in time <math>O((\log n)^c)</math> using <math>O(n^k)</math> 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.
Just as the class '''[[P (complexity)|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 probably some tractable problems which 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'' ([[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 [[parallel random access machine|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.
== The NC Hierarchy ==
'''NC'''<sup>''i''</sup> is the class of decision problems decidable by uniform boolean circuits with a polynomial number of gates and depth <math>O((\log n)^i)</math>, or the class of decision problems solvable in time <math>O((\log n)^i)</math> on a parallel computer with a polynomial number of processors. Clearly, we have
<math>\textbf{NC}^1 \subseteq \textbf{NC}^2 \subseteq \cdots \subseteq \textbf{NC}^i \subseteq \cdots \subseteq \textbf{NC}</math>
which forms the '''NC'''-hierarchy.
We can relate the '''NC''' classes to the space classes '''[[L (complexity)|L]]''' and '''[[NL (complexity)|NL]]'''. From Papadimitriou 1994, Theorem 16.1:
<math> \mathbf{NC^1 \subseteq L \subseteq NL \subseteq NC^2 \subseteq P} </math><ref>[http://www.cs.mu.oz.au/677/notes/node9.html Nick's Class<!-- Bot generated title -->]</ref>
Similarly, we have that '''NC'''<sup>''i''</sup> is equivalent to the problems solvable on an [[alternating Turing machine]] with <math>O(\log n)</math> space and <math>(\log n)^{O(1)}</math> alternations.
=== Open problem: Is '''NC''' proper? ===
One major open question in [[computational complexity theory|complexity theory]] is whether or not every containment in the '''NC''' hierarchy is proper. It was observed by Papadimitriou that, if '''NC'''<sup>''i''</sup> = '''NC'''<sup>''i''+1</sup> for some ''i'', then '''NC'''<sup>''i''</sup> = '''NC'''<sup>''j''</sup> for all <math>j \geq i</math>, and as a result, '''NC'''<sup>''i''</sup> = '''NC'''. This observation is known as '''NC'''-hierarchy collapse because even a single equality in the chain of containments <math>\textbf{NC}^1 \subseteq \textbf{NC}^2 \subseteq \cdots</math> implies that the entire '''NC''' hierarchy "collapses" down to some level ''i''. Thus, there are 2 possibilities:
# <math>\textbf{NC}^1 \subset \cdots \subset \textbf{NC}^i \subset \cdots \subset \textbf{NC}</math>
# <math>\textbf{NC}^1 \subset \cdots \subset \textbf{NC}^i = \cdots = \textbf{NC}</math>
It is widely believed that (1) is the case, although no proof as to the truth of either statement has yet been discovered.
==References==
<references/>
* Greenlaw, Raymond, James Hoover, and Walter Ruzzo. ''Limits To Parallel computation; P-Completeness Theory''. ISBN 0-19-508591-4
* 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
* {{cite book|author = [[Christos Papadimitriou]] | year = 1993 | title = Computational Complexity | publisher = Addison Wesley | edition = 1st edition | id = ISBN 0-201-53082-1}} Section 15.3: The class '''NC''', pp.375–381.
* {{cite book|author = [[Dexter Kozen]] | year = 2006 | title = Theory of Computation | publisher = Springer | id = ISBN 1-84628-297-7}} Lecture 12: Relation of ''NC'' to Time-Space Classes
{{ComplexityClasses}}
[[Category:Complexity classes]] [[Category:Circuit complexity]]
[[de:NC (Komplexitätsklasse)]]
[[es:Clase de Nick]]
[[ko:NC (복잡도)]]
[[ja:NC (計算複雑性理論)]]
This site is not affiliated with or endorsed in any way by the Wikimedia Foundation or any of its affiliates. In fact, we fucking despise them.
|
