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In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.
In computational complexity theory, NP is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.
In computational complexity theory, a decision problem is P-complete if it is in P and every problem in P can be reduced to it by an appropriate reduction.
In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems that are solvable by a deterministic Turing machine in exponential time, i.e., in O(2p(n)) time, where p(n) is a polynomial function of n.
The space complexity of an algorithm or a data structure is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is the memory required by an algorithm until it executes completely. This includes the memory space used by its inputs, called input space, and any other (auxiliary) memory it uses during execution, which is called auxiliary space.
In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly analyzed resources are time and memory.
In computational complexity theory, Savitch's theorem, proved by Walter Savitch in 1970, gives a relationship between deterministic and non-deterministic space complexity. It states that for any function ,
In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.
In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem.
In computational complexity theory, the complexity class NEXPTIME is the set of decision problems that can be solved by a non-deterministic Turing machine using time .
In computational complexity theory, NL is the complexity class containing decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space.
In computational complexity theory, L is the complexity class containing decision problems that can be solved by a deterministic Turing machine using a logarithmic amount of writable memory space. Formally, the Turing machine has two tapes, one of which encodes the input and can only be read, whereas the other tape has logarithmic size but can be read as well as written. Logarithmic space is sufficient to hold a constant number of pointers into the input and a logarithmic number of boolean flags, and many basic logspace algorithms use the memory in this way.
In computational complexity theory, SL is the complexity class of problems log-space reducible to USTCON, which is the problem of determining whether there exists a path between two vertices in an undirected graph, otherwise described as the problem of determining whether two vertices are in the same connected component. This problem is also called the undirected reachability problem. It does not matter whether many-one reducibility or Turing reducibility is used. Although originally described in terms of symmetric Turing machines, that equivalent formulation is very complex, and the reducibility definition is what is used in practice.
Randomized Logarithmic-space (RL), sometimes called RLP, is the complexity class of computational complexity theory problems solvable in logarithmic space and polynomial time with probabilistic Turing machines with one-sided error. It is named in analogy with RP, which is similar but has no logarithmic space restriction.
In computational complexity theory, the Immerman–Szelepcsényi theorem states that nondeterministic space complexity classes are closed under complementation. It was proven independently by Neil Immerman and Róbert Szelepcsényi in 1987, for which they shared the 1995 Gödel Prize. In its general form the theorem states that NSPACE(s(n)) = co-NSPACE(s(n)) for any function s(n) ≥ log n. The result is equivalently stated as NL = co-NL; although this is the special case when s(n) = log n, it implies the general theorem by a standard padding argument. The result solved the second LBA problem.
In computer science, st-connectivity or STCON is a decision problem asking, for vertices s and t in a directed graph, if t is reachable from s.
A symmetric Turing machine is a Turing machine which has a configuration graph that is undirected.
In computational complexity theory of computer science, the structural complexity theory or simply structural complexity is the study of complexity classes, rather than computational complexity of individual problems and algorithms. It involves the research of both internal structures of various complexity classes and the relations between different complexity classes.
In computational complexity theory, a problem is NP-complete when:
- It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no".
- When the answer is "yes", this can be demonstrated through the existence of a short solution.
- The correctness of each solution can be verified quickly and a brute-force search algorithm can find a solution by trying all possible solutions.
- The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified.
References
- Papadimitriou, Christos H. (1994). Computational Complexity. Reading, Massachusetts: Addison-Wesley. ISBN 0-201-53082-1.
- Rytter, Wojciech (1986), "The space complexity of the unique decipherability problem", Information Processing Letters, 23 (1): 1–3, doi:10.1016/0020-0190(86)90121-3, MR 0853618 .
- Jones, Neil D.; Lien, Y. Edmund; Laaser, William T (1976). "New problems complete for nondeterministic log space". Mathematical Systems Theory. 10 (1): 1–17. doi:10.1007/BF01683259. S2CID 11530713.