Big O notation

Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. Its purpose is to characterize a function's behavior for large inputs in a simple but rigorous way that enables comparison to other functions. More precisely, it is used to describe an asymptotic upper bound for the magnitude of a function in terms of another, usually simpler, function. It 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, and in computer science, it is useful in the analysis of the complexity of algorithms.

In informal usage, the O notation is commonly employed to describe an asymptotic tight bound, but tight bounds are more formally and properly denoted by the Θ (big Theta) symbol as described below.

It was first introduced by German 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 capital letter O is also used, but never the digit zero.

Uses

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.

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) = 4n² - 2n + 2.

As n grows large, the n² term will come to dominate, so that all other terms can be neglected—for instance when n = 500, the term 4n² is 1000 times as large as the 2n term, and so 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 2ⁿ. Even if T(n) = 1,000,000n², if U(n) = n³, 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 Big O notation captures what remains: we write

$T(n)\in O(n^{2})$

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,

$e^{x}=1+x+{\frac {x^{2}}{2}}+{\hbox{O}}(x^{3})\qquad {\hbox{as}}\ x\to 0$

expresses the fact that the error (the difference $e^{x}-(1+x+{\frac {x^{2}}{2}})$) is smaller in absolute value than some constant times x³. This error is negligible if 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

$f(x){\mbox{is }}O(g(x)){\mbox{ as }}x\to \infty$

if and only if

$\exists \;x_{0},\exists \;M>0{\mbox{ such that }}|f(x)|\leq \;M|g(x)|{\mbox{ for }}x>x_{0}$.

The notation can also be used to describe the behavior of f near some real number a: we say

$f(x){\mbox{ is }}O(g(x)){\mbox{ as }}x\to a$

if and only if

$\exists \;\delta >0,\exists \;M>0{\mbox{ such that }}|f(x)|\leq \;M|g(x)|{\mbox{ for }}|x-a|<\delta$.

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:

$f(x){\mbox{ is }}O(g(x)){\mbox{ as }}x\to a$

if and only if

$\limsup _{x\to a}\left|{\frac {f(x)}{g(x)}}\right|<\infty .$

Theory of O-Notation: f is in the order of g (f(x) = O(g(x))) if and only if there exists a positive real number M and a real number x₀ such that for all X,
|f(x)| <= M . |g(x)|, wherever x > x₀

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(x⁴). From the definition of order, |f(x)| ≤ C |g(x)| for all x>1, where C is a constant.

Proof:

$|6x^{4}-2x^{3}+5|\leq 6x^{4}+2x^{3}+5\,$         where x > 1

$|6x^{4}-2x^{3}+5|\leq 6x^{4}+2x^{4}+5x^{4}\,$     because x³ < x⁴, and so on.

$|6x^{4}-2x^{3}+5|\leq 13x^{4}\,$

$|6x^{4}-2x^{3}+5|\leq 13\,|x^{4}|.\,$                       where C = 13 in this example

Matters of notation

The statement "$f(x)$ is $O(g(x))$" as defined above is often written as $f(x)=O(g(x))$. This is a slight abuse of notation: we are not really asserting the equality of two functions. The property of being $O(g(x))$ is not symmetric:

$O(x)=O(x^{2}){\mbox{ but }}O(x^{2})\neq O(x)$.

For this reason, some authors prefer a set notation and write $f\in O(g)$, thinking of $O(g)$ as the set of all functions dominated by g.

Furthermore, an "equation" of the form

$f(x)=h(x)+O(g(x))$

should be understood as "the difference of $f(x){\mbox{ and }}h(x){\mbox{ is }}O(g(x))$".

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.

Notation	Name
O(1)	constant
O(log* n)	iterated logarithmic
O(log n)	logarithmic
O((log n)c)	polylogarithmic
O(n)	sublinear
O(n)	linear
O(n log n)	linearithmic, loglinear, quasilinear or supralinear
O(n2)	quadratic
O(nc), c > 1	polynomial, sometimes called algebraic
O(cn)	exponential, sometimes called geometric
O(n!)	factorial, sometimes called combinatorial
O(nn)	-

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

$f(n)=10\log n+5(\log n)^{3}+7n+3n^{2}+6n^{3}\in {\hbox{O}}(n^{3})\,\!$.

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(cⁿ) 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 slower 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.

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, 2ⁿ and 3ⁿ 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 c²n², and the big O notation ignores the constant c². This can be written as $c^{2}n^{2}\in O(n^{2})$. If, however, an algorithm runs in the order of 2ⁿ, replacing n with cn gives 2^cn = (2^c)ⁿ. This is not equivalent to 2ⁿ (unless, of course, c=1).

Product

$O(f(n))O(g(n))=O(f(n)g(n))\,$

Sum

$O(f(n))+O(g(n))=O(\max \lbrace f(n),g(n)\rbrace )\,$

Multiplication by a constant

$O(kg(n))=O(g(n)),\quad k\neq 0$

Addition of a constant

$O(k\ +g(n))=O(g(n))$ unless g(n) ∈ o(1), in which case it is O(1).

Other useful relations are given in section Big O and little o below.

Related asymptotic notations: O, o, Ω, ω, Θ, Õ

Big O is the most commonly used asymptotic notation for comparing functions, although it is often actually an informal substitute for Θ (Theta, see below). Here, we define some related notations in terms of "big O":

Notation	Definition	Mathematical definition
$f(n)\in O(g(n))$	asymptotic upper bound	$\limsup _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|<\infty$
$f(n)\in o(g(n))$	asymptotically negligible	$\lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0$
$f(n)\in \Omega (g(n))$	asymptotic lower bound	$\liminf _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|>0$
$f(n)\in \omega (g(n))$	asymptotically dominant	$\lim _{n\to \infty }{\frac {f(n)}{g(n)}}=\infty$
$f(n)\in \Theta (g(n))$	asymptotically tight bound	$0<\liminf _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|\leq \limsup _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|<\infty$

(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".)

The relation $f(n)=o(g(n))$ is read as "$f(n)$ is little-oh of $g(n)$". Intuitively, it means that $g(n)$ grows much faster than $f(n)$. Formally, it states that the limit of $f(n)/g(n)$ is zero.

Aside from big-O, the notations Θ and Ω are the two most often used in computer science; the lower-case o is common in mathematics but rarer in computer science. The lower-case ω is rarely used.

In casual use, O is commonly used where Θ is meant, i.e., when a tight estimate is implied. For example, one might say "heapsort is ($O(n\log n)$) in the average case" when the intended meaning was "heapsort is $\Theta (n\log n)$ in the average case". Both statements are true, but the latter is a stronger claim.

Another notation sometimes used in computer science is Õ (read Soft-O). $f(n)={\tilde {O}}(g(n))$ is shorthand for $f(n)=O(g(n)\log ^{k}g(n))$ for some k. Essentially, it is Big-O, ignoring logarithmic factors. This notation is often used to describe a class of "nitpicking" estimates (since $\log _{k}n$ is always $o(n)$ for any constant $k$).

Big O and little o

The following properties can be useful:

• $o(f)+o(f)\in o(f)$
• $o(f)o(g)\in o(fg)$
• $o(o(f))\in o(f)$
• $o(f)\in O(f)$ (and thus the above properties apply with most combinations of o and O).

Multiple variables

Big O (and little o, and Ω...) can also be used with multiple variables. For example, the statement

$f(n,m)=n^{2}+m^{3}+{\hbox{O}}(n+m){\mbox{ as }}n,m\to \infty$

asserts that there exist constants C and N such that

$\forall n,m>N:f(n,m)\leq n^{2}+m^{3}+(n+m)C.$

To avoid ambiguity, the running variable should always be specified: the statement

$f(n,m)={\hbox{O}}(n^{m}){\mbox{ as }}n,m\to \infty$

is quite different from

$\forall m:f(n,m)={\hbox{O}}(n^{m}){\mbox{ as }}n\to \infty .$

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 $O(E+VlgV)$ means $O((E,V)\mapsto |E|+|V|\cdot \log |V|)$ for a suitable metric of graphs.

See also

• Asymptotic notation: similar relations, used in computational complexity
• Asymptotic expansion: approximation of functions generalizing Taylor's formula.
• Nachbin's theorem: a precise way of bounding complex analytic functions so that the domain of convergence of integral transforms can be stated.

References

• Marian Slodička. 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 0262032937. Section 3.1: Asymptotic notation, pp.41–50.
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Pages 226–228 of section 7.1: Measuring complexity.
• Jeremy Avigad, Kevin Donnelly. Formalizing O notation in Isabelle/HOL