Editing Computational complexity (section)

{{Short description|Amount of resources to perform an algorithm}}
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In [[computer science]], the '''computational complexity''' or simply '''complexity''' of an [[algorithm]] is the amount of resources required to run it.<ref>{{citation
 | last = Vadhan | first = Salil | author-link = Salil Vadhan
 | editor1-last = van Tilborg | editor1-first = Henk C. A.
 | editor2-last = Jajodia | editor2-first = Sushil
 | contribution = Computational Complexity
 | contribution-url = https://people.seas.harvard.edu/~salil/research/ComputationalComplexity-2ndEd.pdf
 | doi = 10.1007/978-1-4419-5906-5_442
 | isbn = 9781441959065
 | pages = 235–240
 | publisher = Springer
 | title = Encyclopedia of Cryptography and Security
 | year = 2011}}</ref> Particular focus is given to [[time complexity|computation time]] (generally measured by the number of needed elementary operations) and [[space complexity|memory storage]] requirements. The complexity of a [[computational problem|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 {{math|''n'' → ''f''(''n'')}}, where {{math|''n''}} is the size of the input and {{math|''f''(''n'')}} is either the [[worst-case complexity]] (the maximum of the amount of resources that are needed over all inputs of size {{math|''n''}}) or the [[average-case complexity]] (the average of the amount of resources over all inputs of size {{math|''n''}}). [[Time complexity]] is generally expressed as the number of required elementary operations on an input of size {{math|''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 [[computer memory|memory]] required by an algorithm on an input of size {{math|''n''}}.