Little-o notation

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, Edmund Landau, 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.

In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. 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.

Big O notation characterizes functions according to their growth rates: 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 usually only provides an upper bound on the growth rate of the function.

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

Formal definition

Let $f,$ the function to be estimated, be a real or complex valued function, and let $g,$ the comparison function, be a real valued function. Let both functions be defined on some unbounded subset of the positive real numbers, and $g(x)$ be non-zero (often, but not necessarily, strictly positive) for all large enough values of $x.$ One writes
$f(x) = O\bigl( g(x) \bigr) \quad \text x \to \infty$
and it is read "$f(x)$ is big O of $g(x)$" or more often "$f(x)$ is of the order of $g(x)$" if the absolute value of $f(x)$ is at most a positive constant multiple of the absolute value of $g(x)$ for all sufficiently large values of $x.$ That is, $f(x) = O\bigl(g(x) \bigr)$ if there exists a positive real number $M$ and a real number $x_0$ such that
$, f(x), \le M\ , g(x), \quad \text x \ge x_0 ~.$
In many contexts, the assumption that we are interested in the growth rate as the variable $\ x\$ goes to infinity or to zero is left unstated, and one writes more simply that
$f(x) = O\bigl(g(x) \bigr).$
The notation can also be used to describe the behavior of $f$ near some real number $a$ (often, $a = 0$): we say
$f(x) = O\bigl( g(x) \bigr) \quad \text\ x \to a$
if there exist positive numbers $\delta$ and $M$ such that for all defined $x$ with $0 < , x-a, < \delta,$
$, f(x), \le M , g(x), .$
As $g(x)$ is non-zero for adequately large (or small) values of $x,$ both of these definitions can be unified using the limit superior:
$f(x) = O\bigl( g(x) \bigr) \quad \text\ x \to a$
if
$\limsup_ \frac < \infty.$
And in both of these definitions the limit point $a$ (whether $\infty$ or not) is a cluster point of the domains of $f$ and $g,$ i. e., in every neighbourhood of $a$ there have to be infinitely many points in common. Moreover, as pointed out in the article about the limit inferior and limit superior, the $\textstyle \limsup_$ (at least on the extended real number line) always exists.

In computer science, a slightly more restrictive definition is common: $f$ and $g$ are both required to be functions from some unbounded subset of the positive integers to the nonnegative real numbers; then $f(x) = O\bigl( g(x) \bigr)$ if there exist positive integer numbers $M$ and $n_0$ such that $, f(n), \le M , g(n),$ for all $n \ge n_0.$

Example

In typical usage the $O$ notation is asymptotical, that is, it refers to very large $x$. 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 $f(x)$ is a sum of several terms, if there is one with largest growth rate, it can be kept, and all others omitted.
*If $f(x)$ is a product of several factors, any constants (factors in the product that do not depend on $x$) can be omitted.
For example, let $f(x)=6x^4-2x^3+5$, and suppose we wish to simplify this function, using $O$ notation, to describe its growth rate as $x \rightarrow \infty$. This function is the sum of three terms: $6x^4$, $-2x^3$, and $5$. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of $x$, namely $6x^4$. Now one may apply the second rule: $6x^4$is a product of $6$ and $x^4$ in which the first factor does not depend on $x$. Omitting this factor results in the simplified form $x^4$. Thus, we say that $f(x)$ is a "big O" of $x^4$. Mathematically, we can write $f(x)=O(x^4)$. One may confirm this calculation using the formal definition: let $f(x)=6x^4-2x^3+5$ and $g(x)=x^4$. Applying the formal definition from above, the statement that $f(x)=O(x^4)$ is equivalent to its expansion,
$, f(x), \le M x^4$
for some suitable choice of a real number $x_0$ and a positive real number $M$ and for all $x>x_0$. To prove this, let $x_0=1$ and $M=13$. Then, for all $x>x_0$:
$\begin , 6x^4 - 2x^3 + 5, &\le 6x^4 + , 2x^3, + 5\\ &\le 6x^4 + 2x^4 + 5x^4\\ &= 13x^4 \end$
so
$, 6x^4 - 2x^3 + 5, \le 13 x^4 .$

Use

Big O notation has two main areas of application:
* In mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion.
* In computer science, it is useful in the analysis of algorithms.

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

There are two formally close, but noticeably different, usages of this notation:
* infinite asymptotics
* 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.

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 $T(n)=4n^2-2n+2$. 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 $4n^2$ is 1000 times as large as the $2n$ term. Ignoring the latter would have negligible effect on the expression's value for most purposes. Further, the coefficients become irrelevant if we compare to any other order of expression, such as an expression containing a term $n^3$ or $n^4$. Even if $T(n)=1000000n^2$, if $U(n)=n^3$, the latter will always exceed the former once grows larger than $1000000$, ''viz.'' $T(1000000)=1000000^3=U(1000000)$. Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm. So the big O notation captures what remains: we write either
:$T(n)= O(n^2)$
or
:$T(n) \in O(n^2)$
and say that the algorithm has ''order of '' time complexity. The sign "" is not meant to express "is equal to" in its normal mathematical sense, but rather a more colloquial "is", so the second expression is sometimes considered more accurate (see the "Equals sign" discussion below) while the first is considered by some as an abuse of notation.

Infinitesimal asymptotics

Big O can also be used to describe the error term in an approximation to a mathematical function. 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 is small:
\begin
e^x &=1+x+\frac+\frac+\frac+\dotsb &&\text x\\ pt &=1+x+\frac+O(x^3) &&\text x\to 0\\ pt &=1+x+O(x^2) &&\text x\to 0
\end
The middle expression (the one with $O(x^3)$) means the absolute-value of the error $e^x-(1+x+\frac)$ is at most some constant times $, x^3,$ when $x$ is close enough to $0$.

Properties

If the function can be written as a finite sum of other functions, then the fastest growing one determines the order of . For example,
:$f(n) = 9 \log n + 5 (\log n)^4 + 3n^2 + 2n^3 = O(n^3) \qquad\text n\to\infty .$
In particular, if a function may be bounded by a polynomial in , then as tends to ''infinity'', one may disregard ''lower-order'' terms of the polynomial. The sets and are very different. If is greater than one, then the latter grows much faster. A function that grows faster than for any is called ''superpolynomial''. One that grows more slowly than any exponential function of the form 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 .

We may ignore any powers of inside of the logarithms. The set is exactly the same as . The logarithms differ only by a constant factor (since ) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. On the other hand, exponentials with different bases are not of the same order. For example, and 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 , replacing by means the algorithm runs in the order of , and the big O notation ignores the constant . This can be written as . If, however, an algorithm runs in the order of , replacing with gives . This is not equivalent to in general. Changing variables may also affect the order of the resulting algorithm. For example, if an algorithm's run time is when measured in terms of the number of ''digits'' of an input number , then its run time is when measured as a function of the input number itself, because .

Product

:$f_1 = O(g_1) \text f_2 = O(g_2) \Rightarrow f_1 f_2 = O(g_1 g_2)$
:$f\cdot O(g) = O(f g)$

Sum

If $f_1 = O(g_1)$ and $f_2= O(g_2)$ then $f_1 + f_2 = O(\max(, g_1, , , g_2, ))$. It follows that if $f_1 = O(g)$ and $f_2 = O(g)$ then $f_1+f_2 \in O(g)$.

Multiplication by a constant

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

Multiple variables

Big ''O'' (and little o, Ω, etc.) can also be used with multiple variables. To define big ''O'' formally for multiple variables, suppose $f$ and $g$ are two functions defined on some subset of $\R^n$. We say
:$f(\mathbf)\textO(g(\mathbf))\quad\text\mathbf\to\infty$
if and only if there exist constants $M$ and $C > 0$ such that $, f(\mathbf), \le C , g(\mathbf),$ for all $\mathbf$ with $x_i \geq M$ for some $i.$
Equivalently, the condition that $x_i \geq M$ for some $i$ can be written $\, \mathbf\, _ \ge M$, where $\, \mathbf\, _$ denotes the Chebyshev norm. For example, the statement
:$f(n,m) = n^2 + m^3 + O(n+m) \quad\text n,m\to\infty$
asserts that there exist constants ''C'' and ''M'' such that
:$, f(n,m) - (n^2 + m^3), \le C , n+m,$
whenever either $m \geq M$ or $n \geq M$ holds. This definition allows all of the coordinates of $\mathbf$ to increase to infinity. In particular, the statement
:$f(n,m) = O(n^m) \quad \text n,m\to\infty$
(i.e., $\exists C \,\exists M \,\forall n \,\forall m\,\cdots$) is quite different from
:$\forall m\colon~f(n,m) = O(n^m) \quad\text n\to\infty$
(i.e., $\forall m \, \exists C \, \exists M \, \forall n \, \cdots$).

Under this definition, the subset on which a function is defined is significant when generalizing statements from the univariate setting to the multivariate setting. For example, if $f(n,m)=1$ and $g(n,m)=n$, then $f(n,m) = O(g(n,m))$ if we restrict $f$ and $g$ to abuse of notation