In complexity theory, exponential time is the computation time of a problem where the time to complete the computation, m(n), is bounded by an exponential function of the problem size, n (i.e, as the size of the problem increases linearly, the time to solve the problem increases exponentially). In computer science, computational complexity theory is the branch of the theory of computation that studies the complexity, or efficiency, of solving computational problems. ... In computational complexity theory, computation time is a measure of how many steps are used by some abstract machine in a particular computation. ... The exponential function is one of the most important functions in mathematics. ... The word linear comes from the Latin word linearis, which means created by lines. ... The term exponential may refer to any of several topics in mathematics: Exponential distribution Exponential function Exponential growth, exponential decay Exponential time Matrix exponential Exponential map (in differential geometry) All relate in some fashion to exponents. ...

Written mathematically, there exists k > 1 such that m(n) = Θ(kⁿ) and there exists c such that m(n) = O(cⁿ). Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... It has been suggested that this article or section be merged into Asymptotic notation. ... It has been suggested that this article or section be merged into Asymptotic notation. ...

Computer scientists sometimes think of polynomial time as "fast", and anything slower than that as "slow". Exponential time would therefore be considered slow. There are algorithms which take time slower than polynomial time ("super-polynomial time") but faster than exponential time ("sub-exponential time"). These are also considered "slow". One example is the best known algorithm for integer factorization. In computational complexity theory, polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ... In number theory, the integer factorization problem is the problem of finding a non-trivial divisor of a composite number; for example, given a number like 91, the challenge is to find a number such as 7 which divides it. ...

See also

• Computational complexity theory
• Polynomial time
• Constant time
• Linear time
• Algorithm
• Big O notation