List of complexity classes
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This is a list of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics.
Many of these classes have a 'co' partner which consists of the complements of all languages in the original class. For example if a language L is in NP then the complement of L is in co-NP. (This does not mean that the complement of NP is co-NP—there are languages which are known to be in both, and other languages which are known to be in neither.)
"The hardest problems" of a class refer to problems which belong to the class such that every other problem of that class can be reduced to it. Furthermore, the reduction is also a problem of the given class, or its subset.
#P | Count solutions to an NP problem |
#P-complete | The hardest problems in #P |
2-EXPTIME | Solvable in doubly exponential time |
AC^{0} | A circuit complexity class of bounded depth |
ACC^{0} | A circuit complexity class of bounded depth and counting gates |
AC | A circuit complexity class |
AH | The arithmetic hierarchy |
AP | The class of problems alternating Turing machines can solve in polynomial time.^{[1]} |
APX | Optimization problems that have approximation algorithms with constant approximation ratio^{[1]} |
AM | Solvable in polynomial time by an Arthur-Merlin protocol^{[1]} |
BPP | Solvable in polynomial time by randomized algorithms (answer is probably right) |
BQP | Solvable in polynomial time on a quantum computer (answer is probably right) |
co-NP | "NO" answers checkable in polynomial time by a non-deterministic machine |
co-NP-complete | The hardest problems in co-NP |
DSPACE(f(n)) | Solvable by a deterministic machine with space O(f(n)). |
DTIME(f(n)) | Solvable by a deterministic machine in time O(f(n)). |
E | Solvable in exponential time with linear exponent |
ELEMENTARY | The union of the classes in the exponential hierarchy |
ESPACE | Solvable with exponential space with linear exponent |
EXP | Same as EXPTIME |
EXPSPACE | Solvable with exponential space |
EXPTIME | Solvable in exponential time |
FNP | The analogue of NP for function problems |
FP | The analogue of P for function problems |
FP^{NP} | The analogue of P^{NP} for function problems; the home of the traveling salesman problem |
FPT | Fixed-parameter tractable |
GapL | Logspace-reducible to computing the integer determinant of a matrix |
IP | Solvable in polynomial time by an interactive proof system |
L | Solvable with logarithmic (small) space |
LOGCFL | Logspace-reducible to a context-free language |
MA | Solvable in polynomial time by a Merlin-Arthur protocol |
NC | Solvable efficiently (in polylogarithmic time) on parallel computers |
NE | Solvable by a non-deterministic machine in exponential time with linear exponent |
NESPACE | Solvable by a non-deterministic machine with exponential space with linear exponent |
NEXP | Same as NEXPTIME |
NEXPSPACE | Solvable by a non-deterministic machine with exponential space |
NEXPTIME | Solvable by a non-deterministic machine in exponential time |
NL | "YES" answers checkable with logarithmic space |
NONELEMENTARY | Complement of ELEMENTARY. |
NP | "YES" answers checkable in polynomial time (see complexity classes P and NP) |
NP-complete | The hardest or most expressive problems in NP |
NP-easy | Analogue to P^{NP} for function problems; another name for FP^{NP} |
NP-equivalent | The hardest problems in FP^{NP} |
NP-hard | At least as hard as every problem in NP but not known to be in the same complexity class |
NSPACE(f(n)) | Solvable by a non-deterministic machine with space O(f(n)). |
NTIME(f(n)) | Solvable by a non-deterministic machine in time O(f(n)). |
P | Solvable in polynomial time |
P-complete | The hardest problems in P to solve on parallel computers |
P/poly | Solvable in polynomial time given an "advice string" depending only on the input size |
PCP | Probabilistically Checkable Proof |
PH | The union of the classes in the polynomial hierarchy |
P^{NP} | Solvable in polynomial time with an oracle for a problem in NP; also known as Δ_{2}P |
PP | Probabilistically Polynomial (answer is right with probability slightly more than ½) |
PPAD | Polynomial Parity Arguments on Directed graphs |
PR | Solvable by recursively building up arithmetic functions. |
PSPACE | Solvable with polynomial space. |
PSPACE-complete | The hardest problems in PSPACE. |
PTAS | Polynomial-time approximation scheme (a subclass of APX). |
R | Solvable in a finite amount of time. |
RE | Problems to which we can answer "YES" in a finite amount of time, but a "NO" answer might never come. |
RL | Solvable with logarithmic space by randomized algorithms (NO answer is probably right, YES is certainly right) |
RP | Solvable in polynomial time by randomized algorithms (NO answer is probably right, YES is certainly right) |
SL | Problems log-space reducible to determining if a path exist between given vertices in an undirected graph. In October 2004 it was discovered that this class is in fact equal to L. |
S_{2}P | one round games with simultaneous moves refereed deterministically in polynomial time^{[2]} |
TFNP | Total function problems solvable in non-deterministic polynomial time. A problem in this class has the property that every input has an output whose validity may be checked efficiently, and the computational challenge is to find a valid output. |
UP | Unambiguous Non-Deterministic Polytime functions. |
ZPL | Solvable by randomized algorithms (answer is always right, average space usage is logarithmic) |
ZPP | Solvable by randomized algorithms (answer is always right, average running time is polynomial) |
References[edit]
- ^ ^{a} ^{b} ^{c} Sanjeev Arora, Boaz Barak (2009), Computational Complexity: A Modern Approach, Cambridge University Press; 1 edition, ISBN 978-0-521-42426-4
- ^ "S_{2}P: Second Level of the Symmetric Hierarchy". Stanford University Complexity Zoo.
External links[edit]
- Complexity Zoo - list of over 500 complexity classes and their properties