Big O notation

Big O notation is a mathematical notation that describes the approximate size of a function on a domain. 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. The letter O was chosen by Bachmann to stand for Ordnung, meaning the order of approximation.

Example of Big O notation: ${\displaystyle {\color {red}f(x)}=O({\color {blue}g(x)})}$ as ${\displaystyle x\to \infty }$ since there exists ${\displaystyle M>0}$ (e.g., ${\displaystyle M=1}$) and ${\displaystyle x_{0}}$ (e.g.,${\displaystyle x_{0}=5}$) such that ${\displaystyle 0\leq {\color {red}f(x)}\leq M{\color {blue}g(x)}}$ whenever ${\displaystyle x\geq x_{0}}$.

In computer science, big O notation is used to classify algorithms according to how their run time or space requirements^[a] grow as the input size grows.^[3] In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; one well-known example is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates.

Often, big O notation characterizes functions according to their growth rates as the variable becomes large: different functions with the same asymptotic growth rate may be represented using the same O notation. The letter O is used because the growth rate of a function is also referred to as the order of the function. A description of a function in terms of big O notation only provides an upper bound on the growth rate of the function.

Associated with big O notation are several related notations, using the symbols ${\displaystyle o}$, ${\displaystyle \Omega }$, ${\displaystyle \omega }$, and ${\displaystyle \Theta }$ to describe other kinds of bounds on asymptotic growth rates.^[3]

Applications

• In mathematics, it is commonly used in analysis to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion, and for approximating complicated functions by simpler ones.
• In number theory and combinatorics it is used to describe the approximate growth of various functions which count objects, such as prime numbers and graphs.
• In computer science, it is useful in the analysis of algorithms.^[3]

In most applications, the function ${\displaystyle g(x)}$ appearing within the ${\displaystyle O(\cdot )}$ is typically chosen to be as simple as possible, omitting constant factors and lower order terms.

Formal definition

Let ${\displaystyle f,}$ the function to be estimated, be a real or complex valued function defined on a domain ${\displaystyle D}$, and let ${\displaystyle g,}$ the comparison function, be a non-negative real valued function defined on the same set ${\displaystyle D}$. Common choices for the domain are intervals of real numbers, bounded or unbounded, and the set of positive integers^[4]. With the domain written explicitly or understood implicitly, one writes $${\displaystyle f(x)=O{\bigl (}g(x){\bigr )}}$$ and it is read "${\displaystyle f(x)}$ is big O of ${\displaystyle g(x)}$" if ${\displaystyle f(x)/g(x)}$ is a bounded function. In other words, ${\displaystyle f(x)=O{\bigl (}g(x){\bigr )}}$ if there exists a positive real number ${\displaystyle M}$ such that $${\displaystyle |f(x)|\leq M\ |g(x)|\quad {\text{ for all }}x\in D~.}$$ When ${\displaystyle D}$ is an infinite interval of real numbers, there is an older, alternate notation which is still in common use ^[5]. The notation $${\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}x\to \infty }$$ means that for some real number ${\displaystyle a}$, ${\displaystyle f(x)=O(g(x))}$ in the domain ${\displaystyle [a,\infty )}$, and $${\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a}$$ means that for some constant ${\displaystyle c>0}$, ${\displaystyle f(x)=O(g(x))}$ on the interval ${\displaystyle [c-a,c+a]}$, that is, in a small neighborhood of ${\displaystyle a}$. In addition, the notation $${\displaystyle f(x)=h(x)+O(g(x))}$$ means ${\displaystyle f(x)-h(x)=O(g(x))}$.

Despite the use of the equal sign in the notation, ${\displaystyle f(x)=O(g(x))}$ does not refer to an equation but to an inequality relating ${\displaystyle f}$ and ${\displaystyle g}$.

In the 1930s,^[6] the Russian number theorist Ivan Matveyevich Vinogradov introduced his notation ${\displaystyle \ll }$, which has been increasingly used in number theory instead of the ${\displaystyle O}$ notation. We have

${\displaystyle f\ll g\iff f=O(g),}$

and frequently both notations are used in the same paper. ^[7]

Examples with an infinite domain

In typical usage the ${\displaystyle O}$ notation is applied to an infinite interval of real numbers ${\displaystyle [a,\infty )}$ and captures the behavior of the function for very large ${\displaystyle x}$. In this setting, the contribution of the terms that grow "most quickly" will eventually make the other ones irrelevant. As a result, the following simplification rules can be applied:

• If ${\displaystyle f(x)}$ is a sum of several terms, if there is one with largest growth rate, it can be kept, and all others omitted.
• If ${\displaystyle f(x)}$ is a product of several factors, any constants (factors in the product that do not depend on ${\displaystyle x}$) can be omitted.

For example, let ${\displaystyle f(x)=6x^{4}-2x^{3}+5}$, and suppose we wish to simplify this function, using ${\displaystyle O}$ notation, to describe its growth rate for large ${\displaystyle x}$. This function is the sum of three terms: ${\displaystyle 6x^{4}}$, ${\displaystyle -2x^{3}}$, and ${\displaystyle 5}$. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of ${\displaystyle x}$, namely ${\displaystyle 6x^{4}}$. Now one may apply the second rule: ${\displaystyle 6x^{4}}$is a product of ${\displaystyle 6}$ and ${\displaystyle x^{4}}$ in which the first factor does not depend on ${\displaystyle x}$. Omitting this factor results in the simplified form ${\displaystyle x^{4}}$. Thus, we say that ${\displaystyle f(x)}$ is a "big O" of ${\displaystyle x^{4}}$. Mathematically, we can write ${\displaystyle f(x)=O(x^{4})}$ for all ${\displaystyle x\geq 1}$. One may confirm this calculation using the formal definition: let ${\displaystyle f(x)=6x^{4}-2x^{3}+5}$ and ${\displaystyle g(x)=x^{4}}$. Applying the formal definition from above, the statement that ${\displaystyle f(x)=O(x^{4})}$ is equivalent to its expansion, $${\displaystyle |f(x)|\leq Mx^{4}}$$ for some suitable choice of a positive real number ${\displaystyle M}$ and for all ${\displaystyle x\geq 1}$. To prove this, let ${\displaystyle M=13}$. Then, for all ${\displaystyle x\geq 1}$: $${\displaystyle {\begin{aligned}|6x^{4}-2x^{3}+5|&\leq 6x^{4}+|-2x^{3}|+5\\&\leq 6x^{4}+2x^{4}+5x^{4}\\&=13x^{4}\end{aligned}}}$$ so $${\displaystyle |6x^{4}-2x^{3}+5|\leq 13x^{4}.}$$ While it is also true, by the same argument, that ${\displaystyle f(x)=O(x^{10})}$, this is a less precise approximation of the function ${\displaystyle f}$. On the other hand, the statement ${\displaystyle f(x)=O(x^{3})}$ is false, because the term ${\displaystyle 6x^{4}}$ causes ${\displaystyle f(x)/x^{3}}$ to be unbounded.

When a function ${\displaystyle T(n)}$ describes the number of steps required in an algorithm with input ${\displaystyle n}$, an expression such as $${\displaystyle T(n)=O(n^{2})}$$ with the implied domain being the set of positive integers, can be interpreted as saying that the algorithm has order of ${\displaystyle n^{2}}$ time complexity.

Example with a finite domain

Big O can also be used to describe the error term in an approximation to a mathematical function on a finite interval. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term. Consider, for example, the exponential series and two expressions of it that are valid when x is small: $${\displaystyle {\begin{aligned}e^{x}&=1+x+{\frac {\;x^{2}\ }{2!}}+{\frac {\;x^{3}\ }{3!}}+{\frac {\;x^{4}\ }{4!}}+\dotsb &&~{\mathsf {\ for\ all\ finite\ }}~x\\[4pt]&=1+x+{\frac {\;x^{2}\ }{2}}+O(|x|^{3})&&~{\mathsf {\ for\ all\ }}~|x|\leq 1\\[4pt]&=1+x+O(x^{2})&&~{\mathsf {\ for\ all\ }}~|x|\leq 1.\end{aligned}}}$$ The middle expression (the line with "${\displaystyle O(x^{3})}$") means the absolute-value of the error ${\displaystyle \ e^{x}-(1+x+{\frac {\;x^{2}\ }{2}})\ }$ is at most some constant times ${\displaystyle ~|x^{3}|\ }$ when ${\displaystyle \ x~}$ is small. This is an example of the use of Taylor's theorem.

The behavior of a given function may be very different on finite domains than on infinite domains, for example, $${\displaystyle (x+1)^{8}=x^{8}+O(x^{7})\quad {\text{ for }}x\geq 1}$$ while $${\displaystyle (x+1)^{8}=1+8x+O(x^{2})\quad {\text{ for }}|x|\leq 1.}$$

Multivariate examples

$${\displaystyle m^{2}+5mn+3n^{2}+m-4n+12=O((m+n)^{2})\quad {\text{ for }}m\geq 1,n\geq 1.}$$

$${\displaystyle x\sin y=O(x)\quad {\text{ for }}x\geq 1,y\in \mathbb {R} .}$$

$${\displaystyle x^{it}=O(1)\quad {\text{ for }}x\neq 0,t\in \mathbb {R} .}$$

Here we have a complex variable function of two variables. In general, any bounded function is ${\displaystyle O(1)}$.

$${\displaystyle (x+y)^{10}=O(x^{10})\quad {\text{ for }}x\geq 1,-2\leq y\leq 2.}$$

The last example illustrates a mixing of finite and infinite domains on the different variables.

Properties

Product

${\displaystyle f_{1}=O(g_{1}){\text{ and }}f_{2}=O(g_{2})\Rightarrow f_{1}f_{2}=O(g_{1}g_{2})}$

${\displaystyle f\cdot O(g)=O(|f|g)}$

Sum

If ${\displaystyle f_{1}=O(g_{1})}$ and ${\displaystyle f_{2}=O(g_{2})}$ then ${\displaystyle f_{1}+f_{2}=O(\max(g_{1},g_{2}))}$. It follows that if ${\displaystyle f_{1}=O(g)}$ and ${\displaystyle f_{2}=O(g)}$ then ${\displaystyle f_{1}+f_{2}=O(g)}$.

Multiplication by a constant

Let k be a nonzero constant. Then ${\displaystyle O(|k|\cdot g)=O(g)}$. In other words, if ${\displaystyle f=O(g)}$, then ${\displaystyle k\cdot f=O(g).}$

Transitive property

If ${\displaystyle f=O(g)}$ and ${\displaystyle g=O(h)}$ then ${\displaystyle f=O(h)}$.

If the function ${\displaystyle f}$ of a positive integer ${\displaystyle n}$ can be written as a finite sum of other functions, then the fastest growing one determines the order of ${\displaystyle f(n)}$. For example,

${\displaystyle f(n)=9\log n+5(\log n)^{4}+3n^{2}+2n^{3}=O(n^{3})\qquad {\text{for }}n\geq 1.}$

Some general rules about growth toward infinity; these can be proved rigorously using L'Hôpital's rule:

Large powers dominate small powers

For ${\displaystyle b>a\geq 0}$, then $${\displaystyle n^{a}=O(n^{b}).}$$

Powers dominate logarithms

For any positive ${\displaystyle a,b,}$ $${\displaystyle (\log n)^{a}=O(n^{b}),}$$ no matter how large ${\displaystyle a}$ is and how small ${\displaystyle b}$ is.

Exponentials dominate powers

For any positive ${\displaystyle a,b,}$ $${\displaystyle n^{a}=O(e^{bn}),}$$ no matter how large ${\displaystyle a}$ is and how small ${\displaystyle b}$ is.

A function that grows faster than ${\displaystyle n^{c}}$ for any c is called superpolynomial. One that grows more slowly than any exponential function of the form cⁿ 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 and the function n^{log n}.

We may ignore any powers of n inside of the logarithms. The set 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, that is, ${\displaystyle \log _{a}n=O(\log _{b}n)}$ for any ${\displaystyle a,b>1}$. On the other hand, exponentials with different bases are not of the same order. For example, 2ⁿ and 3ⁿ are not of the same order.

Uses of Big-O in more complicated expressions

In more complicated usage, O(·) can appear in different places in an equation, even several times on each side. For example, the following are true for ${\displaystyle n}$ a positive integer: $${\displaystyle {\begin{aligned}(n+1)^{2}&=n^{2}+O(n),\\(n+O(n^{1/2}))\cdot (n+O(\log n))^{2}&=n^{3}+O(n^{5/2}),\\n^{O(1)}&=O(e^{n}).\end{aligned}}}$$ 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(eⁿ) such that n^f(n) = g(n)". In this use the "=" is a formal symbol that unlike the usual use of "=" is not a symmetric relation. Thus for example n^O(1) = O(eⁿ) does not imply the false statement O(eⁿ) = n^O(1).

Some further examples: $${\displaystyle {\begin{aligned}f=O(g)\;&\Rightarrow \;\int _{a}^{b}f=O{\bigg (}\int _{a}^{b}g{\bigg )}\\f(x)=g(x)+O(1)\;&\Rightarrow \;e^{f(x)}=O(e^{g(x)})\\(1+O(1/x))^{O(x)}&=O(1)\quad {\text{ for }}x>0\\\sin x&=O(|x|)\quad {\text{ for all real }}x.\end{aligned}}}$$

Matters of notation

Arrows

In mathematics, an expression such as ${\displaystyle x\to \infty }$ indicates the presence of a limit. In big-O notation, there is no implied limit, in contrast with little-o notation.

Equals sign

Some consider ${\displaystyle f(x)=O(g(x))}$ to be an abuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As de Bruijn says, O[x] = O[x²] is true but O[x²] = O[x] is not.^[8] Knuth describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like n = n² from the identities n = O[n²] and n² = O[n²]".^[9] In another letter, Knuth also pointed out that^[10]

the equality sign is not symmetric with respect to such notations [as, in this notation,] mathematicians customarily use the '=' sign as they use the word 'is' in English: Aristotle is a man, but a man isn't necessarily Aristotle.

For these reasons, it would be more precise to use set notation and write f(x) ∈ O[g(x)] – read as: "f(x) is an element of O[g(x)]", or "f(x) is in the set O[g(x)]" – thinking of O[g(x)] as the class of all functions h(x) such that |h(x)| ≤ C |g(x)| for some positive real number C.^[9] However, the use of the equals sign is customary.^[8][9]

Typesetting

Big O is typeset as an italicized uppercase "O", as in the following example: ${\displaystyle O(n^{2})}$.^[11][12] In TeX, it is produced by simply typing 'O' inside math mode. Unlike Greek-named Bachmann–Landau notations, it needs no special symbol. However, some authors use the calligraphic variant ${\displaystyle {\mathcal {O}}}$ instead.^[13][14]

Orders of common functions

Further information: Time complexity § Table of common time complexities

"O(1)" redirects here. For the quasicoherent sheaf, see Proj construction § The twisting sheaf of Serre.

Here is a list of classes of functions that are commonly encountered when analyzing the running time of an algorithm. In each case, c is a positive constant and n increases without bound. The slower-growing functions are generally listed first.

Notation	Name	Example
${\displaystyle O(1)}$	constant	Finding the median value for a sorted array of numbers; Calculating ${\displaystyle (-1)^{n}}$; Using a constant-size lookup table
${\displaystyle O(\alpha (n))}$	inverse Ackermann function	Amortized complexity per operation for the Disjoint-set data structure
${\displaystyle O(\log \log n)}$	double logarithmic	Average number of comparisons spent finding an item using interpolation search in a sorted array of uniformly distributed values
${\displaystyle O(\log n)}$	logarithmic	Finding an item in a sorted array with a binary search or a balanced search tree as well as all operations in a binomial heap
${\displaystyle O((\log n)^{c})}$ ${\textstyle c>1}$	polylogarithmic	Matrix chain ordering can be solved in polylogarithmic time on a parallel random-access machine.
${\displaystyle O(n^{c})}$ ${\textstyle 0<c<1}$	fractional power	Searching in a k-d tree
${\displaystyle O(n)}$	linear	Finding an item in an unsorted list or in an unsorted array; adding two n-bit integers by ripple carry
${\displaystyle O(n\log ^{*}n)}$	n log-star n	Performing triangulation of a simple polygon using Seidel's algorithm,[15] where ${\displaystyle \log ^{*}(n)={\begin{cases}0,&{\text{if }}n\leq 1\\1+\log ^{*}(\log n),&{\text{if }}n>1\end{cases}}}$
${\displaystyle O(n\log n)=O(\log n!)}$	linearithmic, loglinear, quasilinear, or "n log n"	Performing a fast Fourier transform; fastest possible comparison sort; heapsort and merge sort
${\displaystyle O(n^{2})}$	quadratic	Multiplying two n-digit numbers by schoolbook multiplication; simple sorting algorithms, such as bubble sort, selection sort and insertion sort; (worst-case) bound on some usually faster sorting algorithms such as quicksort, Shellsort, and tree sort
${\displaystyle O(n^{c})}$	polynomial or algebraic	Tree-adjoining grammar parsing; maximum matching for bipartite graphs; finding the determinant with LU decomposition
${\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}}$ ${\textstyle 0<\alpha <1}$	L-notation or sub-exponential	Factoring a number using the quadratic sieve or number field sieve
${\displaystyle O(c^{n})}$ ${\textstyle c>1}$	exponential	Finding the (exact) solution to the travelling salesman problem using dynamic programming; determining if two logical statements are equivalent using brute-force search
${\displaystyle O(n!)}$	factorial	Solving the travelling salesman problem via brute-force search; generating all unrestricted permutations of a poset; finding the determinant with Laplace expansion; enumerating all partitions of a set

The statement ${\displaystyle f(n)=O(n!)}$ is sometimes weakened to ${\displaystyle f(n)=O\left(n^{n}\right)}$ to derive simpler formulas for asymptotic complexity.

Little-o notation

"Little o" redirects here. For the baseball player, see Omar Vizquel. For the Greek letter, see Omicron.

Intuitively, the assertion "f(x) is o(g(x))" (read "f(x) is little-o of g(x)") means that g(x) grows much faster than f(x), or equivalently f(x) grows much slower than g(x). As before, let f be a real or complex valued function and g a real valued function, both defined on some unbounded subset of the positive real numbers, such that ${\displaystyle g(x)}$ is strictly positive for all large enough values of x. One writes ^[16]

${\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to \infty }$

if $${\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=0.}$$ That is, for every positive constant ε there exists a constant ${\displaystyle x_{0}}$ such that

${\displaystyle |f(x)|\leq \varepsilon g(x)\quad {\text{ for all }}x\geq x_{0}.}$

For example, one has

${\displaystyle 2x=o(x^{2})}$ and ${\displaystyle 1/x=o(1),}$     both as ${\displaystyle x\to \infty .}$

The difference between the definition of the big-O notation and the definition of little-o is that while the former expresses an inequality for all elements of a domain, the latter expresses an inequality that holds no matter how small the implied constant is, as long as ${\displaystyle x}$ is sufficiently large.^[17] In this way, little-o notation makes a stronger statement than the corresponding big-O notation: every function that is little-o of g is also big-O of g, but not every function that is big-O of g is little-o of g. For example, ${\displaystyle 2x^{2}=O(x^{2})}$ but ${\displaystyle 2x^{2}\neq o(x^{2})}$.

Little-o respects a number of arithmetic operations. For example,

if c is a nonzero constant and ${\displaystyle f=o(g)}$ then ${\displaystyle c\cdot f=o(g)}$, and

if ${\displaystyle f=o(F)}$ and ${\displaystyle g=o(G)}$ then ${\displaystyle f\cdot g=o(F\cdot G).}$

if ${\displaystyle f=o(F)}$ and ${\displaystyle g=o(G)}$ then ${\displaystyle f+g=o(F+G)}$

It also satisfies a transitivity relation:

if ${\displaystyle f=o(g)}$ and ${\displaystyle g=o(h)}$ then ${\displaystyle f=o(h).}$

Little-o can also be generalized to the finite case:^[18]

${\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to x_{0}}$ if $${\displaystyle \lim _{x\to x_{0}}{\frac {f(x)}{g(x)}}=0.}$$ In other words, ${\displaystyle f(x)=\alpha (x)g(x)}$ for some ${\displaystyle \alpha (x)}$ with ${\displaystyle \lim _{x\to x_{0}}\alpha (x)=0}$.

This definition is especially useful in the computation of limits using Taylor series. For example:

${\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+\ldots =x+o(x^{2}){\text{ as }}x\to 0}$, so ${\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=\lim _{x\to 0}{\frac {x+o(x^{2})}{x}}=\lim _{x\to 0}1+o(x)=1}$

Big Omega notation

Another asymptotic notation is ${\displaystyle \Omega }$, read "big omega".^[19] There are two widespread and incompatible definitions of the statement

${\displaystyle f(x)=\Omega (g(x))}$

where f and g are real functions.

The Hardy–Littlewood definition is used mainly in analytic number theory, and the Knuth definition mainly in computational complexity theory and in combinatorics; the definitions are not equivalent.

The Hardy–Littlewood definition

In 1914 G.H. Hardy and J.E. Littlewood introduced the new symbol ${\displaystyle \ \Omega \ ,}$^[20] which is defined as follows:

${\displaystyle f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}\quad }$ as ${\displaystyle \quad x\to \infty \quad }$ if ${\displaystyle \quad \limsup _{x\to \infty }\ \left|{\frac {\ f(x)\ }{g(x)}}\right|>0~.}$

Thus ${\displaystyle ~f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}~}$ is the negation of ${\displaystyle ~f(x)=o{\bigl (}\ g(x)\ {\bigr )}~.}$

In 1916 the same authors introduced the two new symbols ${\displaystyle \ \Omega _{R}\ }$ and ${\displaystyle \ \Omega _{L}\ ,}$ defined as:^[21]

${\displaystyle f(x)=\Omega _{R}{\bigl (}\ g(x)\ {\bigr )}\quad }$ as ${\displaystyle \quad x\to \infty \quad }$ if ${\displaystyle \quad \limsup _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}>0\$ ;}

${\displaystyle f(x)=\Omega _{L}{\bigl (}\ g(x)\ {\bigr )}\quad }$ as ${\displaystyle \quad x\to \infty \quad }$ if ${\displaystyle \quad ~\liminf _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}<0~.}$

These symbols were used by E. Landau, with the same meanings, in 1924.^[22] Authors that followed Landau, however, use a different notation for the same definitions:^{[citation needed]} The symbol ${\displaystyle \ \Omega _{R}\ }$ has been replaced by the current notation ${\displaystyle \ \Omega _{+}\ }$ with the same definition, and ${\displaystyle \ \Omega _{L}\ }$ became ${\displaystyle \ \Omega _{-}~.}$

These three symbols ${\displaystyle \ \Omega \ ,\Omega _{+}\ ,\Omega _{-}\ ,}$ as well as ${\displaystyle \ f(x)=\Omega _{\pm }{\bigl (}\ g(x)\ {\bigr )}\ }$ (meaning that ${\displaystyle \ f(x)=\Omega _{+}{\bigl (}\ g(x)\ {\bigr )}\ }$ and ${\displaystyle \ f(x)=\Omega _{-}{\bigl (}\ g(x)\ {\bigr )}\ }$ are both satisfied), are now currently used in analytic number theory.^[23][24]

Simple examples

We have

${\displaystyle \sin x=\Omega (1)\quad }$ as ${\displaystyle \quad x\to \infty \ ,}$

and more precisely

${\displaystyle \sin x=\Omega _{\pm }(1)\quad }$ as ${\displaystyle \quad x\to \infty ~.}$

We have

${\displaystyle 1+\sin x=\Omega (1)\quad }$ as ${\displaystyle \quad x\to \infty \ ,}$

and more precisely

${\displaystyle 1+\sin x=\Omega _{+}(1)\quad }$ as ${\displaystyle \quad x\to \infty \$ ;}

however

${\displaystyle 1+\sin x\neq \Omega _{-}(1)\quad }$ as ${\displaystyle \quad x\to \infty ~.}$

The Knuth definition

In 1976 Donald Knuth published a paper to justify his use of the ${\displaystyle \Omega }$-symbol to describe a stronger property.^[25] Knuth wrote: "For all the applications I have seen so far in computer science, a stronger requirement ... is much more appropriate". He defined

${\displaystyle f(x)=\Omega (g(x))\Longleftrightarrow g(x)=O(f(x))}$

with the comment: "Although I have changed Hardy and Littlewood's definition of ${\displaystyle \Omega }$, I feel justified in doing so because their definition is by no means in wide use, and because there are other ways to say what they want to say in the comparatively rare cases when their definition applies."^[25]

Vinogradov^[6] introduced the notation ${\displaystyle f(x)\gg g(x)}$, which means the same as ${\displaystyle g(x)=O(f(x))}$. Vinogradov's two notations enjoy visual symmetry, as for positive functions ${\displaystyle f,g}$, we have $${\displaystyle f(x)\ll g(x)\Longleftrightarrow g(x)\gg f(x).}$$

Family of Bachmann–Landau notations

Notation	Name[25]	Description	Formal definition	Limit definition[26][27][28][25][20]
${\displaystyle f(n)=o(g(n))}$	Small O; Small Oh; Little O; Little Oh	f is dominated by g asymptotically (for any constant factor ${\displaystyle k}$)	${\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|\leq k\,g(n)}$	${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}$
${\displaystyle f(n)=O(g(n))}$ (Landau's notation) or ${\displaystyle f(n)\ll g(n)}$ (Vinogradov's notation)	Big O; Big Oh; Big Omicron	${\displaystyle |f|}$ is bounded above by g (up to constant factor ${\displaystyle k}$)	${\displaystyle \exists k>0\,\forall n\in D\colon |f(n)|\leq k\,g(n)}$	${\displaystyle \sup _{n\in D}{\frac {\left|f(n)\right|}{g(n)}}<\infty }$
${\displaystyle f(n)\asymp g(n)}$ (Hardy's notation) or ${\displaystyle f(n)=\Theta (g(n))}$ (Knuth notation)	Of the same order as (Hardy); Big Theta (Knuth)	f is bounded by g both above (with constant factor ${\displaystyle k_{2}}$) and below (with constant factor ${\displaystyle k_{1}}$)	${\displaystyle \exists k_{1}>0\,\exists k_{2}>0\,\forall n\in D\colon }$ ${\displaystyle k_{1}\,g(n)\leq f(n)\leq k_{2}\,g(n)}$	${\displaystyle f(n)=O(g(n))}$ and ${\displaystyle g(n)=O(f(n))}$
${\displaystyle f(n)\sim g(n)}$ as ${\displaystyle n\to a}$, where ${\displaystyle a}$ is finite, ${\displaystyle \infty }$ or ${\displaystyle -\infty }$	Asymptotic equivalence	f is equal to g asymptotically	${\displaystyle \forall \varepsilon >0\,\exists n_{0}\,\forall n>n_{0}\colon \left|{\frac {f(n)}{g(n)}}-1\right|<\varepsilon }$ (in the case ${\displaystyle a=\infty }$)	${\displaystyle \lim _{n\to a}{\frac {f(n)}{g(n)}}=1}$
${\displaystyle f(n)=\Omega (g(n))}$ (Knuth's notation), or ${\displaystyle f(n)\gg g(n)}$ (Vinogradov's notation)	Big Omega in complexity theory (Knuth)	f is bounded below by g, up to a constant factor	${\displaystyle \exists k>0\,\forall n\in D\colon f(n)\geq k\,g(n)}$	${\displaystyle \inf _{n\in D}{\frac {f(n)}{g(n)}}>0}$
${\displaystyle f(n)=\omega (g(n))}$ as ${\displaystyle n\to a}$, where ${\displaystyle a}$ can be finite, ${\displaystyle \infty }$ or ${\displaystyle -\infty }$	Small Omega; Little Omega	f dominates g asymptotically	${\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)>k\,g(n)}$ (for ${\displaystyle a=\infty }$)	${\displaystyle \lim _{n\to a}{\frac {f(n)}{g(n)}}=\infty }$
${\displaystyle f(n)=\Omega (g(n))}$	Big Omega in number theory (Hardy–Littlewood)	${\displaystyle |f|}$ is not dominated by g asymptotically	${\displaystyle \exists k>0\,\forall n_{0}\,\exists n>n_{0}\colon |f(n)|\geq k\,g(n)}$	${\displaystyle \limsup _{n\to \infty }{\frac {\left|f(n)\right|}{g(n)}}>0}$

The limit definitions assume ${\displaystyle g(n)>0}$ for ${\displaystyle n}$ in a neighborhood of the limit; when the limit is ${\displaystyle \infty }$, this means that ${\displaystyle g(n)>0}$ for sufficiently large ${\displaystyle n}$. The table is (partly) sorted from smallest to largest, in the sense that ${\displaystyle o,O,\Theta ,\sim ,}$ (Knuth's version of) ${\displaystyle \Omega ,\omega }$ on functions correspond to ${\displaystyle <,\leq ,\approx ,=,}$${\displaystyle \geq ,>}$ on the real line^[28] (the Hardy–Littlewood version of ${\displaystyle \Omega }$, however, doesn't correspond to any such description).

Computer science and combinatorics use the big ${\displaystyle O}$, big Theta ${\displaystyle \Theta }$, little ${\displaystyle o}$, little omega ${\displaystyle \omega }$ and Knuth's big Omega ${\displaystyle \Omega }$ notations.^[29] Analytic number theory often uses the big ${\displaystyle O}$, small ${\displaystyle o}$, Hardy's ${\displaystyle \asymp }$,^[30] Hardy–Littlewood's big Omega ${\displaystyle \Omega }$ (with or without the +, − or ± subscripts), Vinogradov's ${\displaystyle \ll }$ and ${\displaystyle \gg }$ notations and ${\displaystyle \sim }$ notations.^[23] The small omega ${\displaystyle \omega }$ notation is not used as often in analysis or in number theory.^[31]

Use in computer science

Further information: Analysis of algorithms

Informally, especially in computer science, the big O notation often can be used somewhat differently to describe an asymptotic tight bound where using big Theta Θ notation might be more factually appropriate in a given context.^[32] For example, when considering a function T(n) = 73n³ + 22n² + 58, all of the following are generally acceptable, but tighter bounds (such as numbers 2 and 3 below) are usually strongly preferred over looser bounds (such as number 1 below).

1. T(n) = O(n¹⁰⁰)
2. T(n) = O(n³)
3. T(n) = Θ(n³)

The equivalent English statements are respectively:

1. T(n) grows asymptotically no faster than n¹⁰⁰
2. T(n) grows asymptotically no faster than n³
3. T(n) grows asymptotically as fast as n³.

So while all three statements are true, progressively more information is contained in each. In some fields, however, the big O notation (number 2 in the lists above) would be used more commonly than the big Theta notation (items numbered 3 in the lists above). For example, if T(n) represents the running time of a newly developed algorithm for input size n, the inventors and users of the algorithm might be more inclined to put an upper asymptotic bound on how long it will take to run without making an explicit statement about the lower asymptotic bound.

Extensions to the Bachmann–Landau notations

Another notation sometimes used in computer science is Õ (read soft-O), which hides polylogarithmic factors. There are two definitions in use: some authors use f(n) = Õ(g(n)) as shorthand for f(n) = O(g(n) log^k n) for some k, while others use it as shorthand for f(n) = O(g(n) log^k g(n)).^[33] When g(n) is polynomial in n, there is no difference; however, the latter definition allows one to say, e.g. that ${\displaystyle n2^{n}={\tilde {O}}(2^{n})}$ while the former definition allows for ${\displaystyle \log ^{k}n={\tilde {O}}(1)}$ for any constant k. Some authors write O^* for the same purpose as the latter definition.^[34] Essentially, it is big O notation, ignoring logarithmic factors because the growth-rate effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s). This notation is often used to obviate the "nitpicking" within growth-rates that are stated as too tightly bounded for the matters at hand (since log^k n is always o(n^ε) for any constant k and any ε > 0).

Also, the L notation, defined as

${\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }},}$

is convenient for functions that are between polynomial and exponential in terms of ${\displaystyle \ln n}$.

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^{[citation needed]}. 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,

${\displaystyle f\sim g\iff (f-g)\in o(g)}$

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, ${\displaystyle 2x}$ is Θ(x), but 2x − x is not o(x).

History (Bachmann–Landau, Hardy, and Vinogradov notations)

The symbol O was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book Analytische Zahlentheorie ("analytic number theory").^[1] The number theorist Edmund Landau adopted it, and was thus inspired to introduce in 1909 the notation o;^[2] hence both are now called Landau symbols. These notations were used in applied mathematics during the 1950s for asymptotic analysis.^[35] The symbol ${\displaystyle \Omega }$ (in the sense "is not an o of") was introduced in 1914 by Hardy and Littlewood.^[20] Hardy and Littlewood also introduced in 1916 the symbols ${\displaystyle \Omega _{R}}$ ("right") and ${\displaystyle \Omega _{L}}$ ("left"),^[21] precursors of the modern symbols ${\displaystyle \Omega _{+}}$ ("is not smaller than a small o of") and ${\displaystyle \Omega _{-}}$ ("is not larger than a small o of"). Thus the Omega symbols (with their original meanings) are sometimes also referred to as "Landau symbols". This notation ${\displaystyle \Omega }$ became commonly used in number theory at least since the 1950s.^[36]

The symbol ${\displaystyle \sim }$, although it had been used before with different meanings,^[28] was given its modern definition by Landau in 1909^[37] and by Hardy in 1910.^[38] Just above on the same page of his tract Hardy defined the symbol ${\displaystyle \asymp }$, where ${\displaystyle f(x)\asymp g(x)}$ means that both ${\displaystyle f(x)=O(g(x))}$ and ${\displaystyle g(x)=O(f(x))}$ are satisfied. The notation is still currently used in analytic number theory.^[39][30] In his tract Hardy also proposed the symbol ${\displaystyle \mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-} }$, where ${\displaystyle f\mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-} g}$ means that ${\displaystyle f\sim Kg}$ for some constant ${\displaystyle K\not =0}$.

In the 1930s, Vinogradov^[6] popularized the notation ${\displaystyle f(x)\ll g(x)}$ and ${\displaystyle g(x)\gg f(x)}$, both of which mean ${\displaystyle f(x)=O(g(x))}$. This notation became standard in analytic number theory.^[7]

In the 1970s the big O was popularized in computer science by Donald Knuth, who proposed the different notation ${\displaystyle f(x)=\Theta (g(x))}$ for Hardy's ${\displaystyle f(x)\asymp g(x)}$, and proposed a different definition for the Hardy and Littlewood Omega notation.^[25]

Two other symbols coined by Hardy were (in terms of the modern O notation)

${\displaystyle f\preccurlyeq g\iff f=O(g)}$   and   ${\displaystyle f\prec g\iff f=o(g);}$

(Hardy however never defined or used the notation ${\displaystyle \prec \!\!\prec }$, nor ${\displaystyle \ll }$, as it has been sometimes reported). Hardy introduced the symbols ${\displaystyle \preccurlyeq }$ and ${\displaystyle \prec }$ (as well as the already mentioned other symbols) in his 1910 tract "Orders of Infinity", and made use of them only in three papers (1910–1913). In his nearly 400 remaining papers and books he consistently used the Landau symbols O and o. Hardy's symbols ${\displaystyle \preccurlyeq }$ and ${\displaystyle \prec }$ (as well as ${\displaystyle \mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-} }$) are not used anymore.

The big-O originally stands for "order of" ("Ordnung", Bachmann 1894), and is thus a Latin letter. Neither Bachmann nor Landau ever call it "Omicron". The symbol was much later on (1976) viewed by Knuth as a capital omicron,^[25] probably in reference to his definition of the symbol Omega. The digit zero should not be used.

See also

• Asymptotic computational complexity
• Asymptotic expansion: Approximation of functions generalizing Taylor's formula
• Asymptotically optimal algorithm: A phrase frequently used to describe an algorithm that has an upper bound asymptotically within a constant of a lower bound for the problem
• Big O in probability notation: O_p, o_p
• Limit inferior and limit superior: An explanation of some of the limit notation used in this article
• Master theorem (analysis of algorithms): For analyzing divide-and-conquer recursive algorithms using big O notation
• Nachbin's theorem: A precise method of bounding complex analytic functions so that the domain of convergence of integral transforms can be stated
• Order of approximation
• Order of accuracy
• Computational complexity of mathematical operations