Exponential Time

Related Concepts

Computational Complexity; O-Notation; Polynomial Time; Subexponential Time

Definition

An exponential-time algorithm is one whose running time grows as an exponential function of the size of its input.

Theory

Let x denote the length of the input to an algorithm (typically in bits, but other measures are sometimes used), and let T(x) denote the running time of the algorithm on inputs of length x. Then the algorithm is exponential-time if the running time satisfies

$$T(x) \leq c{b}^{x}$$

for all sufficiently large x, where the coefficient and the base b;>;1 are constants. The running time will also satisfy the bound

$$T(x) \leq {(b^\prime)}^{c^\prime x}$$

for another base b ^′;>;1, and an appropriate constant c ^′. The term “exponential” comes from the fact that the size x is in the exponent. In O-notation, this would be written T(x);=;O(b ^x), or equivalently $T(x) = {({b}^{{\prime}})}^{O(x)}$. The notations 2^O(x) and e ^O(x) are common.

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