NC (complexity)

In 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 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 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 (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 uniform Boolean circuits with polylogarithmic depth and a polynomial number of gates.

The NC Hierarchy

NCⁱ 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 \dots \subseteq \textbf{NC}^i \subseteq \dots \subseteq \textbf{NC}</math> which forms the NC-hierarchy. We can relate the NC classes to the space classes L and NL. From Papadimitriou 1994, Theorem 16.1: <math> \mathbf{NC^1 \subseteq L \subseteq NL \subseteq NC^2} </math> Similarly, we have that NCⁱ 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 complexity theory is whether or not every containment in the NC hierarchy is proper. It was observed by Papadimitriou that, if NCⁱ = NCⁱ⁺¹ for some i, then NCⁱ = NC^j for all <math>j \geq i</math>, and as a result, NCⁱ = 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 \dots</math> implies that the entire NC hierarchy "collapses" down to some level i. Thus, there are 2 possibilities:

1. <math>\textbf{NC}^1 \subset \dots \subset \textbf{NC}^i \subset \dots \subset \textbf{NC}</math>
2. <math>\textbf{NC}^1 \subset \dots \textbf{NC}^i = \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

• Greenlaw, Raymond, James Hoover, and Walter Ruzzo. Limits To Parallel computation; P-Completeness Theory. ISBN 0-19-508591-4
• Heribert Vollmer. Introduction to Circuit Complexity -- A Uniform Approach. ISBN 3-540-64310-9
• Christos Papadimitriou (1993). Computational Complexity, 1st edition, Addison Wesley. ISBN 0-201-53082-1.  Section 15.3: The class NC, pp.375–381.
• Dexter Kozen (2006). Theory of Computation. Springer. ISBN 1-84628-297-7.  Lecture 12: Relation of NC to Time-Space Classes