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In computational complexity theory, 'big O notation' is often used to describe how the size of the input data affects an algorithm's usage of computational resources (usually running time or memory). It is also called 'Big Oh notation', 'Landau notation', and 'asymptotic notation'. Big O notation is also used in many other scientific and mathematical fields to provide similar estimations.
The symbol ''O'' is used to describe an asymptotic upper bound for the magnitude of a function in terms of another, usually simpler, function. There are also other symbols ''o'', Ω, ω, and Θ for various other upper, lower, and tight bounds.
Informally, the ''O'' notation is commonly employed to describe an asymptotic tight bound, but tight bounds are more formally and precisely denoted by the Θ (capital theta) symbol as described below.

Contents

Usage

Equals or member-of and other notational anomalies

Infinite asymptotics

Infinitesimal asymptotics

'Formal definition'

Example

Matters of notation

Common orders of functions

Properties

Product

Sum

Multiplication by a constant

Related asymptotic notations

Little-o notation

Other standard computer science notation: Ω, ω, Θ, O, o, Õ, ∼

Multiple variables

Graph theory

Generalizations and related usages

See also

References

Usage

Big O notation has two main areas of application: in mathematics, it is usually used to characterize the residual term of a truncated infinite series, especially an asymptotic series; in computer science, it is useful in the analysis of the complexity of algorithms.
The notation was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book ''Analytische Zahlentheorie'' ("analytic number theory"), the first volume of which (not yet containing big O notation) came out in 1892. The notation was popularized in the work of another German number theorist Edmund Landau, hence it is sometimes called a Landau symbol. The big-O, standing for "order of", was originally a capital omicron; today the identical-looking Latin capital letter ''O'' is also used, but never the digit zero.
There are two formally close, but noticeably different, usages of this notation: infinite asymptotics and infinitesimal asymptotics. This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument.

Equals or member-of and other notational anomalies

In a way to be made precise below, ''O''(''f''(''x'')) denotes the collection of functions ''g''(''x'') – viewed as a function of variable ''x'' – that exhibit a growth that is limited to that of ''f''(''x'') in some respect. The traditional notation for stating that ''g''(''x'') belongs to this collection is:
:$g(x)\; =\; mathcal\{O\}(f(x)),.$
This is an anomalous and exceptional use of the equals sign in mathematics, as the above statement is not an equation. It is improper to conclude from ''g''(''x'') = ''O''(''f''(''x'')) and ''h''(''x'') = ''O''(''f''(''x'')) that ''g''(''x'') and ''h''(''x'') are equal. One way to think of this, is to consider "= ''O''" one symbol here. To avoid the anomalous use, some authors prefer to write instead:
:$g(x)\; in\; mathcal\{O\}(f(x)),,$
without difference in meaning.
The common arithmetic operations are often extended to the class concept. For example, ''h''(''x'') + ''O''(''f''(''x'')) denotes the collection of functions having the growth of ''h''(''x'') plus a part whose growth is limited to that of ''f''(''x''). Thus,
:$g(x)\; =\; h(x)\; +\; mathcal\{O\}(f(x)),$
expresses the same as
:$g(x)\; -\; h(x)\; in\; mathcal\{O\}(f(x)),.$
Another anomaly of the notation, although less exceptional, is that it does not make explicit which variable is the function argument, which may need to be inferred from the context if several variables are involved. The following two right-hand side big O notations have dramatically different meanings:
:$f(m)\; =\; mathcal\{O\}(m^n),,$
:$g(n),,\; =\; mathcal\{O\}(m^n),.$
The first case states that ''f''(''m'') exhibits polynomial growth, while the second, assuming ''m'' > 1, states that ''g''(''n'') exhibits exponential growth. So as to avoid all possible confusion, some authors use the notation
:$g\; in\; mathcal\{O\}(f),,$
meaning the same as what is denoted by others as
:$g(x)\; in\; mathcal\{O\}(f(x)),.$
A final anomaly is that the notation does not make clear "where" the function growth is to be considered; infinitesimally near some point, or in the neighbourhood of infinity. This is in contrast with the usual notation for limits. A similar notational device as for limits would resolve both this and the preceding anomaly, but is not in use.

Infinite asymptotics

Big O notation is useful when analyzing algorithms for efficiency. For example, the time (or the number of steps) it takes to complete a problem of size ''n'' might be found to be ''T''(''n'') = 4''n''² - 2''n'' + 2.
As ''n'' grows large, the ''n''² term will come to dominate, so that all other terms can be neglected—for instance when ''n'' = 500, the term 4''n''² is 1000 times as large as the 2''n'' term. Ignoring the latter would have negligible effect on the expression's value for most purposes.
Further, the coefficients become irrelevant as well if we compare to any other order of expression, such as an expression containing a term n³ or n². Even if ''T''(''n'') = 1,000,000''n''², if ''U''(''n'') = ''n''³, the latter will always exceed the former once ''n'' grows larger than 1,000,000 (''T''(1,000,000) = 1,000,000³ = ''U''(1,000,000)).
So the big O notation captures what remains: we write
:$T(n)in\; mathcal\{O\}(n^2)$
(pronounced "big o of n squared") and say that the algorithm has ''order of n²'' time complexity.

Infinitesimal asymptotics

Big O can also be used to describe the error term in an approximation
to a mathematical function. For example,
:
expresses the fact that the error, the difference
ight), is smaller in absolute value than some constant times $left|x^3$
ight| when $x$ is close enough to 0.

'Formal definition'

Suppose $f(x)$ and $g(x)$ are two functions defined on
some subset of the real numbers. We say

:$f(x)mbox\{\; is\; \}mathcal\{O\}(g(x))mbox\{\; as\; \}x\; oinfty$
if and only if
:$exists\; ;x\_0,exists\; ;M>0mbox\{\; such\; that\; \}\; |f(x)|\; le\; ;\; M\; |g(x)|mbox\{\; for\; \}x>x\_0.$
The notation can also be used to describe the behavior of ''f'' near
some real number ''a'': we say
:$f(x)mbox\{\; is\; \}mathcal\{O\}(g(x))mbox\{\; as\; \}x\; o\; a$
if and only if
:
If $g(x)$ is non-zero for values of ''x'' sufficiently close to ''a'', both of these definitions can be unified using the limit superior:
:$f(x)mbox\{\; is\; \}mathcal\{O\}(g(x))mbox\{\; as\; \}x\; o\; a$
if and only if
:
ight| < infty.
In mathematics, both asymptotic behaviours near ∞ and near ''a'' are considered. In computational complexity theory, only asymptotics near ∞ are used; furthermore, only positive functions are considered, so the absolute value bars may be left out.

Example

Take the polynomials:
:$f(x)\; =\; 6x^4\; -2x^3\; +5\; ,$
:$g(x)\; =\; x^4.\; ,$
We say ''f''(''x'') has order O(''g''(''x'')) or O(''x''⁴). From the definition of order, |''f(x)''| ≤ ''C'' |''g(x)''| for all ''x''>1, where ''C'' is a constant.
Proof:
:$|6x^4\; -\; 2x^3\; +\; 5|\; le\; 6x^4\; +\; 2x^3\; +\; 5\; ,$         where ''x'' > 1
:$|6x^4\; -\; 2x^3\; +\; 5|\; le\; 6x^4\; +\; 2x^4\; +\; 5x^4\; ,$     because ''x''³ < ''x''⁴, and so on.
:$|6x^4\; -\; 2x^3\; +\; 5|\; le\; 13x^4\; ,$
:$|6x^4\; -\; 2x^3\; +\; 5|\; le\; 13\; ,|x^4\; |.\; ,$                       where C = 13 in this example

Matters of notation

The statement "$f(x)$ is $O(g(x))$" as defined above is usually written as $f(x)\; =O(g(x))$. This is a slight abuse of notation; equality of two functions is not asserted, and it cannot be since the property of being $O(g(x))$ is not symmetric:
:$mathcal\{O\}(x)=mathcal\{O\}(x^2)mbox\{\; but\; \}mathcal\{O\}(x^2)$
e mathcal{O}(x).
There is also a second reason, while this notation is not precise. The symbol ''f''(''x'') means the value of the function ''f'' for the argument ''x''. Hence the symbol of the function is ''f'' and not ''f''(''x'').
For these reasons, some authors prefer set notation and write $f\; in\; mathcal\{O\}(g)$, thinking of $mathcal\{O\}(g)$ as the set of all functions dominated by ''g''.
In more complex usage, O( ) can appear in different places in an equation, even several times on each side. For example, the following are true for $n\; oinfty$
:$(n+1)^2\; =\; n^2\; +\; mathcal\{O\}(n)$
:$(n+mathcal\{O\}(n^\{1/2\}))(n\; +\; mathcal\{O\}(log,n))^2\; =\; n^3\; +\; mathcal\{O\}(n^\{5/2\})$
:$n^\{mathcal\{O\}(1)\}\; =\; mathcal\{O\}(e^n)$
The meaning of such statements is as follows: for ''any'' functions which satisfy each O( ) on the left side, there are ''some'' functions satisfying each O( ) on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function ''f''(''n'')=O(1), there is some function ''g''(''n'')=O(''e''^''n'') such that ''n''^''f''(''n'')=''g''(''n'')." In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side.

Common orders of functions

Here is a list of classes of functions that are commonly encountered when analyzing algorithms. All of these are as ''n'' increases to infinity. The slower-growing functions are listed first. ''c'' is an arbitrary constant.
{| class="wikitable"
!Notation !! Name !! Example
|-
|$mathcal\{O\}left(1$
ight) || constant || Determining if a number is even or odd
|-
|$mathcal\{O\}left(log^$
★ n
ight) || iterated logarithmic || The find algorithm of Hopcroft and Ullman on a disjoint set
|-
|$mathcal\{O\}left(log\; n$
ight) || logarithmic || Finding an item in a sorted list with the binary search algorithm
|-
|$mathcal\{O\}left(left(log\; n$
ight)^c
ight) || polylogarithmic || Deciding if ''n'' is prime with the AKS primality test
|-
|$mathcal\{O\}left(\{n^c\}$
ight), 0 || fractional power || searching in a kd-tree
|-
|$mathcal\{O\}left(n$
ight) || linear || Finding an item in an unsorted list
|-
|$mathcal\{O\}left(nlog\; n$
ight) || linearithmic, loglinear, or quasilinear || Sorting a list with heapsort, computing a FFT
|-
|$mathcal\{O\}left(\{n^2\}$
ight) || quadratic || Sorting a list with insertion sort, computing a DFT
|-
|$mathcal\{O\}left(\{n^c\}$
ight), c>1 || polynomial, sometimes called algebraic || Finding the shortest path on a weighted digraph with the Floyd-Warshall algorithm
|-
|$mathcal\{O\}left(\{c^n\}$
ight) || exponential, sometimes called geometric || Finding the (exact) solution to the traveling salesman problem (under the assumption that P ≠ NP)
|-
|$mathcal\{O\}left(n!$
ight) || factorial, sometimes called combinatorial || Determining if two logical statements are equivalent [1], traveling salesman problem, or any other NP-complete problem via brute-force search
|-
|$mathcal\{O\}left(c\_1^\{c\_2^n\}$
ight) || double exponential || Finding a complete set of associative-commutative unifiers [2]
|}
Not as common, but even larger growth is possible, such as the single-valued version of the Ackermann function, A(''n'',''n''). Conversely, extremely slowly-growing functions such as the inverse of this function, often denoted α(''n''), are possible. Although unbounded, these functions are often regarded as being constant factors for all practical purposes.

Properties

If a function ''f''(''n'') can be written as a finite sum of other functions, then the fastest growing one determines the order of
''f''(''n''). For example
:$f(n)\; =\; 9\; log\; n\; +\; 5\; (log\; n)^3\; +\; 3n^2\; +\; 2n^3\; in\; mathcal\{O\}(n^3),!.$
In particular, if a function may be bounded by a polynomial in ''n'', then as ''n'' tends to ''infinity'', one may disregard ''lower-order'' terms of the polynomial.
$O(n^c)$ and $O(c^n)$ are very different. The latter grows much, much faster, no matter how big the constant ''c'' is (as long as it is greater than one). A function that grows faster than any power of ''n'' is called ''superpolynomial''. One that grows more slowly than any exponential function of the form $c^n$ is called ''subexponential''. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest known algorithms for integer factorization.
$O(log\; n)$ is exactly the same as $O(log(n^c))$. The logarithms differ only by a constant factor, (since
$log(n^c)=c\; log\; n$) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. Exponentials with different bases, on the other hand, are not of the same order; assuming that they are is a common mistake. For example, $2^n$ and $3^n$ are 'not' of the same order.
Changing units may or may not affect the order of the resulting algorithm. Changing units is equivalent to multiplying the appropriate variable by a constant wherever it appears. For example, if an algorithm runs in the order of n², replacing n by cn means the algorithm runs in the order of $c^2n^2$, and the big O notation ignores the constant $c^2$. This can be written as $c^2n^2\; in\; mathcal\{O\}(n^2)$. If, however, an algorithm runs in the order of $2^n$, replacing n with cn gives $2^\{cn\}\; =\; (2^c)^n$. This is not equivalent to $2^n$ (unless, of course, c=1).

Product

:$f\_1(n)\; inmathcal\{O\}(g\_1(n))\; and$
f_2(n)inmathcal{O}(g_2(n)), Rightarrow f_1(n) f_2(n)inmathcal{O}(g_1(n) g_2(n)),
:$f(n)\; O(g(n))\; in\; O(f(n)\; g(n))$

Sum

:$f(n)+\; O(g(n))\; in\; O(f(n)+g(n))$

:$f\_\{1\}(n)\; in\; O(g\_1(n)),\; f\_\{2\}(n)\; in\; O(g\_2(n)),\; dots,\; f\_\{m\}(n)\; in\; O(g\_m(n))\; Rightarrow$
sumlimits_{i=1}^{m}f_{i}(n) in O(max, _{i=1}^m |g_i(n)| )
(see maxima and minima for the definition of 'max')

Multiplication by a constant

:$mathcal\{O\}(k\; g(n))\; =\; mathcal\{O\}(g(n)),quad\; k$
e 0
:$f(n)in\; O(g(n))\; Rightarrow\; kf(n)in\; O(g(n))$

Other useful relations are given in section below.

Related asymptotic notations

Big O is the most commonly used asymptotic notation for comparing functions, although in many cases Big O may be replaced with Θ for asymptotically tighter bounds (Theta, see below). Here, we define some related notations in terms of "big O":

Little-o notation

The relation $f(x)\; in\; o(g(x))$ is read as "$f(x)$ is little-oh of $g(x)$". Intuitively, it means that $g(x)$ grows much faster than $f(x)$. It assumes that ''f'' and ''g'' are both functions of one variable. Formally, it states that the limit of $f(x)/g(x)$ is zero, as ''x'' approaches infinity.
For example,

★ $2x\; in\; o(x^2)\; ,!$

★ $2x^2$
ot in o(x^2)
Little-o notation is common in mathematics but rarer in computer science. In computer science the variable (and function value) is most often a natural number. In math, the variable and function values are often real numbers. The following properties can be useful:

★ $o(f)\; +\; o(f)\; subseteq\; o(f)$

★ $o(f)o(g)\; subseteq\; o(fg)$

★ $o(o(f))\; subseteq\; o(f)$

★ $o(f)\; subset\; O(f)$ (and thus the above properties apply with most combinations of o and O).
As with big O notation, the statement "$f(x)$ is $o(g(x))$" is usually written as $f(x)\; =\; o(g(x))$, which is a slight abuse of notation.

Other standard computer science notation: Ω, ω, Θ, O, o, Õ, ∼

{|class="wikitable"
!Notation
!Definition
!Mathematical definition
!Alternative definition
|-
|$f(n)\; in\; mathcal\{O\}(g(n))$
|asymptotic upper bound
|
ight| < infty
|
|-
|$f(n)\; in\; Omega(g(n))$
|asymptotic lower bound
|
ight| > 0
|
|-
|$f(n)\; in\; Theta(g(n))$
|asymptotically tight bound
|
ight| leq limsup_{n o infty} left|rac{f(n)}{g(n)}
ight|< infty
|$f(n)\; in\; O(g(n))\; cap\; Omega(g(n))$
|-
|$f(n)\; in\; omega(g(n))$
|asymptotically dominant
|
ight| = infty
|
|-
|$f(n)\; in\; o(g(n))$
|asymptotically insignificant
|
ight| = 0
|
|-
|$f(n)\; sim\; g(n)$
|asymptotically equal
|
|$$
|}
(A mnemonic for these Greek letters is that "omicron" can be read "o-micron", i.e., "o-''small''", whereas "omega" can be read "o-mega" or "o-''big''".)
Aside from big-O, the notations Θ and Ω are the two most often used in computer science; The lower-case ω is rarely used.
Another notation sometimes used in computer science is Õ (read ''soft-O''). $f(n)\; =\; ilde\{O\}\; (g(n))$ is shorthand
for $f(n)\; =\; O(g(n)\; log^kg(n))$ for some ''k''. Essentially, it is Big-O, ignoring logarithmic factors. This notation is often used to describe a class of "nitpicking" estimates (since $log^kn$ is always $o(n^epsilon)$ for any constant $k$ and any $epsilon>0$).
The L notation, defined as
:
ight),
is convenient for functions that are between polynomial and exponential.

Multiple variables

Big O (and little o, and Ω...) can also be used with multiple variables. For example, the statement
:$f(n,m)\; =\; n^2\; +\; m^3\; +\; hbox\{O\}(n+m)\; mbox\{\; as\; \}\; n,m\; oinfty$
asserts that there exist constants ''C'' and ''N'' such that
:
where ''g''(''n'',''m'') is defined by
:$f(n,m)\; =\; n^2\; +\; m^3\; +\; g(n,m).$
To avoid ambiguity, the running variable should always be specified: the statement
:$f(n,m)\; =\; hbox\{O\}(n^m)\; mbox\{\; as\; \}\; n,m\; oinfty$
is quite different from
:

Graph theory

It is often useful to bound the running time of graph algorithms. Unlike most other computational problems, in graphs, there are two relevant parameters describing the size of the input, |V| and |E|; |V| is the number of vertices in the graph, while |E| is the number of edges in the graph. Inside asymptotic notation (and only there), it is common to use the symbols V and E, when someone really means |V| and |E|. We adopt this convention here to simplify asymptotic functions and make them easily readable. Keep in mind that the symbols V and E are never used inside asymptotic notation with their literal meaning, so there is no risk of ambiguity. For example $O(E\; +\; V\; log\; V)$ means $O((E,V)\; mapsto\; |E|\; +\; |V|cdotlog|V|)$ for a suitable metric of graphs.

Generalizations and related usages

The generalization to functions taking values in any normed vector space is straightforward (replacing absolute values by norms), where ''f'' and ''g'' need not take their values in the same space. A generalization to functions ''g'' taking values in any topological group is also possible.
The "limiting process" ''x→x_o'' can also be generalized by introducing an arbitrary filter base, i.e. to directed nets ''f'' and ''g''.
The ''o'' notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions,
:$fsim\; g\; iff\; (f-g)\; =\; o(g)$
which is an equivalence relation and a more restrictive notion than the relationship "''f'' is Θ(''g'')" from above. (It reduces to $lim\; f/g\; =\; 1$ if ''f'' and ''g'' are positive real valued functions.) For example, 2''x'' is Θ(''x''), but 2''x'' − ''x'' is not ''o''(''x'').

See also

★ Asymptotic expansion: Approximation of functions generalizing Taylor's formula.

★ Asymptotically optimal: A phrase frequently used to describe an algorithm that has an upper bound asymptotically within a constant of a lower bound for the problem

★ Hardy notation: A different asymptotic notation

★ Limit superior and limit inferior: An explanation of some of the limit notation used in this article

★ Nachbin's theorem: A precise way of bounding complex analytic functions so that the domain of convergence of integral transforms can be stated.

References

★ Paul Bachmann. ''Die Analytische Zahlentheorie. Zahlentheorie''. pt. 2 Leipzig: B. G. Teubner, 1894.

★ Edmund Landau. ''Handbuch der Lehre von der Verteilung der Primzahlen''. 2 vols. Leipzig: B. G. Teubner, 1909.

★ Marian Slodicka (Slodde vo de maten) & Sandy Van Wontergem. ''Mathematical Analysis I''. University of Ghent, 2004.

★ Donald Knuth. ''Big Omicron and big Omega and big Theta'', ACM SIGACT News, Volume 8, Issue 2, 1976.

★ Donald Knuth. ''The Art of Computer Programming'', Volume 1: ''Fundamental Algorithms'', Third Edition. Addison-Wesley, 1997. ISBN 0-201-89683-4. Section 1.2.11: Asymptotic Representations, pp.107–123.

★ Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. ''Introduction to Algorithms'', Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 3.1: Asymptotic notation, pp.41–50.

★ Introduction to the Theory of Computation, Michael Sipser, , , PWS Publishing, 1997, ISBN 0-534-94728-X Pages 226–228 of section 7.1: Measuring complexity.

★ Jeremy Avigad, Kevin Donnelly. ''Formalizing O notation in Isabelle/HOL''

★ Paul E. Black, "big-O notation", in ''Dictionary of Algorithms and Data Structures'' [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 11 March 2005. Retrieved December 16, 2006.

★ Paul E. Black, "little-o notation", in ''Dictionary of Algorithms and Data Structures'' [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. Retrieved December 16, 2006.

★ Paul E. Black, "Ω", in ''Dictionary of Algorithms and Data Structures'' [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. Retrieved December 16, 2006.

★ Paul E. Black, "ω", in ''Dictionary of Algorithms and Data Structures'' [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 29 November 2004. Retrieved December 16, 2006.

★ Paul E. Black, "Θ", in ''Dictionary of Algorithms and Data Structures'' [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. Retrieved December 16, 2006.