## The computational complexity of a problem as a function of the size of the input

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**Computational complexity**

In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to time and memory requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem.

The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory. Both areas are highly related, as the complexity of an algorithm is always an upper bound on the complexity of the problem solved by this algorithm. Moreover, for designing efficient algorithms, it is often fundamental to compare the complexity of a specific algorithm to the complexity of the problem to be solved. Also, in most cases, the only thing that is known about the complexity of a problem is that it is lower than the complexity of the most efficient known algorithms. Therefore, there is a large overlap between analysis of algorithms and complexity theory.

As the amount of resources required to run an algorithm generally varies with the size of the input, the complexity is typically expressed as a function n → f(n), where n is the size of the input and f(n) is either the worst-case complexity (the maximum of the amount of resources that are needed over all inputs of size n) or the average-case complexity (the average of the amount of resources over all inputs of size n). Time complexity is generally expressed as the number of required elementary operations on an input of size n, where elementary operations are assumed to take a constant amount of time on a given computer and change only by a constant factor when run on a different computer. Space complexity is generally expressed as the amount of memory required by an algorithm on an input of size n.

**Complexity as a function of input size**

For clarity, only time complexity is considered in this section, but everything applies (with slight modifications) to the complexity with respect to other resources.

It is impossible to count the number of steps of an algorithm on all possible inputs. As the complexity generally increases with the size of the input, the complexity is typically expressed as a function of the size n (in bits) of the input, and therefore, the complexity is a function of n. However, the complexity of an algorithm may vary dramatically for different inputs of the same size. Therefore, several complexity functions are commonly used.

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