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PTIME
In computational complexity theory, P, also known as PTIME or DTIME(''n''O(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. Cobham's thesis holds that P is the class of computational problems that are "efficiently solvable" or " tractable". This is inexact: in practice, some problems not known to be in P have practical solutions, and some that are in P do not, but this is a useful rule of thumb. Definition A language ''L'' is in P if and only if there exists a deterministic Turing machine ''M'', such that * ''M'' runs for polynomial time on all inputs * For all ''x'' in ''L'', ''M'' outputs 1 * For all ''x'' not in ''L'', ''M'' outputs 0 P can also be viewed as a uniform family of boolean circuits. A language ''L'' is in P if and only if there exists a polynomial-time uniform family of boolean circuits \, such that * For all n \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class
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 general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. counting problems and function problems) and using other models of computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and quantum computers). The study of the relationships between complexity classes is a ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P Versus NP Problem
The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved. The informal term ''quickly'', used above, means the existence of an algorithm solving the task that runs in polynomial time, such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time). The general class of questions for which some algorithm can provide an answer in polynomial time is " P" or "class P". For some questions, there is no known way to find an answer quickly, but if one is provided with information showing what the answer is, it is possible to verify the answer quickly. The class of questions for which an answer can be ''verified'' in polynomial time is NP, which stands for "nondeterministic polynomial time".A nondeterministic Turing machine can move to a state that is not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computation Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expressed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
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. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tractable Problem
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. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Turing Machine
In computational complexity theory, an alternating Turing machine (ATM) is a non-deterministic Turing machine (NTM) with a rule for accepting computations that generalizes the rules used in the definition of the complexity classes NP and co-NP. The concept of an ATM was set forth by Chandra and Stockmeyer and independently by Kozen in 1976, with a joint journal publication in 1981. Definitions Informal description The definition of NP uses the ''existential mode'' of computation: if ''any'' choice leads to an accepting state, then the whole computation accepts. The definition of co-NP uses the ''universal mode'' of computation: only if ''all'' choices lead to an accepting state does the whole computation accept. An alternating Turing machine (or to be more precise, the definition of acceptance for such a machine) alternates between these modes. An alternating Turing machine is a non-deterministic Turing machine whose states are divided into two sets: existential states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSPACE
In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. Formal definition If we denote by SPACE(''t''(''n'')), the set of all problems that can be solved by Turing machines using ''O''(''t''(''n'')) space for some function ''t'' of the input size ''n'', then we can define PSPACE formally asArora & Barak (2009) p.81 :\mathsf = \bigcup_ \mathsf(n^k). PSPACE is a strict superset of the set of context-sensitive languages. It turns out that allowing the Turing machine to be nondeterministic does not add any extra power. Because of Savitch's theorem,Arora & Barak (2009) p.85 NPSPACE is equivalent to PSPACE, essentially because a deterministic Turing machine can simulate a non-deterministic Turing machine without needing much more space (even though it may use much more time).Arora & Barak (2009) p.86 Also, the complements of all problems in PSPACE are also in PSPACE, meaning tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DTIME
In computational complexity theory, DTIME (or TIME) is the computational resource of computation time for a deterministic Turing machine. It represents the amount of time (or number of computation steps) that a "normal" physical computer would take to solve a certain computational problem using a certain algorithm. It is one of the most well-studied complexity resources, because it corresponds so closely to an important real-world resource (the amount of time it takes a computer to solve a problem). The resource DTIME is used to define complexity classes, sets of all of the decision problems which can be solved using a certain amount of computation time. If a problem of input size ''n'' can be solved in , we have a complexity class (or ). There is no restriction on the amount of memory space used, but there may be restrictions on some other complexity resources (like alternation). Complexity classes in DTIME Many important complexity classes are defined in terms of DTIME, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP (complexity)
In computational complexity theory, NP (nondeterministic polynomial time) 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.''Polynomial time'' refers to how quickly the number of operations needed by an algorithm, relative to the size of the problem, grows. It is therefore a measure of efficiency of an algorithm. An equivalent definition of NP is the set of decision problems ''solvable'' in polynomial time by a nondeterministic Turing machine. This definition is the basis for the abbreviation NP; " nondeterministic, polynomial time". These two definitions are equivalent because the algorithm based on the Turing machine consists of two phases, the first of which consists of a guess abou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-complete
In computational complexity theory, a decision problem is P-complete (complete for the complexity class P) if it is in P and every problem in P can be reduced to it by an appropriate reduction. The notion of P-complete decision problems is useful in the analysis of: * which problems are difficult to parallelize effectively, * which problems are difficult to solve in limited space. specifically when stronger notions of reducibility than polytime-reducibility are considered. The specific type of reduction used varies and may affect the exact set of problems. Generically, reductions stronger than polynomial-time reductions are used, since all languages in P (except the empty language and the language of all strings) are P-complete under polynomial-time reductions. If we use NC reductions, that is, reductions which can operate in polylogarithmic time on a parallel computer with a polynomial number of processors, then all P-complete problems lie outside NC and so cannot be effecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circuit Complexity
In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of the Boolean circuits that compute them. A related notion is the circuit complexity of a recursive language that is decided by a uniform family of circuits C_,C_,\ldots (see below). Proving lower bounds on size of Boolean circuits computing explicit Boolean functions is a popular approach to separating complexity classes. For example, a prominent circuit class P/poly consists of Boolean functions computable by circuits of polynomial size. Proving that \mathsf\not\subseteq \mathsf would separate P and NP (see below). Complexity classes defined in terms of Boolean circuits include AC0, AC, TC0, NC1, NC, and P/poly. Size and depth A Boolean circuit with n input bits is a directed acyclic graph in which every node (usually called ''gates'' in this context) is either an input node of in-degree 0 labelle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
