Computational complexity theory and Time complexity

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Difference between Computational complexity theory and Time complexity

Computational complexity theory vs. Time complexity

Computational complexity theory is a branch of the theory of computation in theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm.

Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

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Big O notation

Big O notation is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity.

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Binary number

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one).

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Boolean satisfiability problem

In computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated as SATISFIABILITY or SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula.

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BPP (complexity)

In computational complexity theory, BPP, which stands for bounded-error probabilistic polynomial time is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded away from 1/2 for all instances.

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BQP

In computational complexity theory, BQP (bounded-error quantum polynomial time) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances.

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Cobham's thesis

Cobham's thesis, also known as Cobham–Edmonds thesis (named after Alan Cobham and Jack Edmonds), asserts that computational problems can be feasibly computed on some computational device only if they can be computed in polynomial time; that is, if they lie in the complexity class P (PTIME).

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Complexity class

In computational complexity theory, a complexity class is a set of problems of related resource-based complexity.

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Computational complexity

In computer science, the computational complexity, or simply complexity of an algorithm is the amount of resources required for running it.

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Decision problem

In computability theory and computational complexity theory, a decision problem is a problem that can be posed as a yes-no question of the input values.

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DSPACE

In computational complexity theory, DSPACE or SPACE is the computational resource describing the resource of memory space for a deterministic Turing machine.

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DTIME

In computational complexity theory, DTIME (or TIME) is the computational resource of computation time for a deterministic Turing machine.

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Euclidean algorithm

. EXAMPLES CAN BE FOUND BELOW, E.G., IN THE "Matrix method" SECTION.

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EXPTIME

In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems that have exponential runtime, i.e., that are solvable by a deterministic Turing machine in O(2p(n)) time, where p(n) is a polynomial function of n. In terms of DTIME, We know and also, by the time hierarchy theorem and the space hierarchy theorem, that so at least one of the first three inclusions and at least one of the last three inclusions must be proper, but it is not known which ones are.

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Formal language

In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols together with a set of rules that are specific to it.

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General number field sieve

In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than.

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Graph (discrete mathematics)

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".

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Graph isomorphism problem

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic.

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Integer factorization

In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers.

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Mathematical optimization

In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives.

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Non-deterministic Turing machine

In theoretical computer science, a Turing machine is a theoretical machine that is used in thought experiments to examine the abilities and limitations of computers.

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NP (complexity)

In computational complexity theory, NP (for nondeterministic polynomial time) is a complexity class used to describe certain types of decision problems.

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NP-completeness

In computational complexity theory, an NP-complete decision problem is one belonging to both the NP and the NP-hard complexity classes.

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NP-hardness

NP-hardness (''n''on-deterministic ''p''olynomial-time hardness), in computational complexity theory, is the defining property of a class of problems that are, informally, "at least as hard as the hardest problems in NP".

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P (complexity)

In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is a fundamental complexity class.

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Parallel computing

Parallel computing is a type of computation in which many calculations or the execution of processes are carried out concurrently.

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Parameterized complexity

In computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to multiple parameters of the input or output.

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Presburger arithmetic

Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929.

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Probabilistic Turing machine

In computability theory, a probabilistic Turing machine is a non-deterministic Turing machine which chooses between the available transitions at each point according to some probability distribution.

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Quantum algorithm

In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation.

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Quantum Turing machine

A quantum Turing machine (QTM), also a universal quantum computer, is an abstract machine used to model the effect of a quantum computer.

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Quicksort

Quicksort (sometimes called partition-exchange sort) is an efficient sorting algorithm, serving as a systematic method for placing the elements of an array in order.

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Randomized algorithm

A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic.

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RP (complexity)

In computational complexity theory, randomized polynomial time (RP) is the complexity class of problems for which a probabilistic Turing machine exists with these properties.

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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Time complexity

In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm.

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Travelling salesman problem

The travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science.

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Turing machine

A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules.

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ZPP (complexity)

In complexity theory, ZPP (zero-error probabilistic polynomial time) is the complexity class of problems for which a probabilistic Turing machine exists with these properties.

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