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In computer science (specifically computational complexity theory), the worst-case complexity measures the resources (e.g. running time, memory) that an algorithm requires given an input of arbitrary size (commonly denoted as n in asymptotic notation). It gives an upper bound on the resources required by the algorithm.
With respect to computational resources, asymptotic time complexity and asymptotic space complexity are commonly estimated. Other asymptotically estimated behavior include circuit complexity and various measures of parallel computation , such as the number of (parallel) processors.
[1]: 226 Since this function is generally difficult to compute exactly, and the running time for small inputs is usually not consequential, one commonly focuses on the behavior of the complexity when the input size increases—that is, the asymptotic behavior of the complexity. Therefore, the time complexity is commonly expressed using big O ...
Also, when implemented with the "shortest first" policy, the worst-case space complexity is instead bounded by O(log(n)). Heapsort has O(n) time when all elements are the same. Heapify takes O(n) time and then removing elements from the heap is O(1) time for each of the n elements. The run time grows to O(nlog(n)) if all elements must be distinct.
In the theoretical analysis of algorithms, the normal practice is to estimate their complexity in the asymptotic sense. The most commonly used notation to describe resource consumption or "complexity" is Donald Knuth's Big O notation, representing the complexity of an algorithm as a function of the size of the input .
Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann, [1] Edmund Landau, [2] and others, collectively called Bachmann–Landau notation or asymptotic notation.
Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics, however. Non-asymptotic bounds are provided by methods of approximation theory. Examples of applications are the following. In applied mathematics, asymptotic analysis is used to build numerical methods to approximate equation solutions.
In theoretical analysis of algorithms it is common to estimate their complexity in the asymptotic sense, i.e., to estimate the complexity function for arbitrarily large input. Big O notation, Big-omega notation and Big-theta notation are used to this end. [2]