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, Bachmann-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. This distinction between upper and tight bounds is useful, and sometimes critical; most computer scientists would urge distinguishing the usage of O and Θ. In some other fields, however, the Θ notation is not commonly known.

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) was published 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 of the 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:

$g(x)\; =\; O(f(x)),.$

This use of the equals sign is an abuse of notation, 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:

$g\; in\; O(f),,$

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,

$g(x)\; =\; h(x)\; +\; O(f(x)),$

expresses the same as

$g(x)\; -\; h(x)\; in\; O(f(x)),.$

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:

$f(m)\; =\; O(m^n),,$

$g(n),,\; =\; O(m^n),.$

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

$g\; in\; O(f),,$

meaning the same as what is denoted by others as

$g(x)\; in\; O(f(x)),.$

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. Similar terminological and notational devices as for limits would resolve both this and the preceding anomaly, but are 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) = 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. 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) = 1,000,000n², 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 the big O notation captures what remains: we write any of

$T(n)=\; O(n^2)$

$T(n)in\; O(n^2)$

(read as "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,

$e^x=1+x+frac\{x^2\}\{2!\}+O(x^3)qquadhbox\{as\}\; xto\; 0$

expresses the fact that the error, the difference $e^x\; -\; left(1\; +\; x\; +frac\{x^2\}\{2!\}right)$, is smaller in absolute value than some constant times $left|x^3right|$ 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

$f(x)mbox\{\; is\; \}O(g(x))mbox\{\; as\; \}xtoinfty$

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

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

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:

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

if and only if

These 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) (as $xtoinfty$)

From the definition of order

$|f(x)|\; le\; ;\; M\; |g(x)|mbox\{\; for\; \}x>x\_0.$

Proof:

for all $x\; >\; 1$ (we take $x\_0\; =\; 1$):

$|6x^4\; -\; 2x^3\; +\; 5|\; le\; 6x^4\; +\; 2x^3\; +\; 5\; ,$

$|6x^4\; -\; 2x^3\; +\; 5|\; le\; 6x^4\; +\; 2x^4\; +\; 5x^4\; ,$

$|6x^4\; -\; 2x^3\; +\; 5|\; le\; 13x^4\; ,$

$|6x^4\; -\; 2x^3\; +\; 5|\; le\; 13\; ,|x^4\; |.\; ,$                       where M = 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:

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

There is also a second reason why that 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 $f\; in\; O(g)$, thinking of $O(g)$ 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 $ntoinfty$

$(n+1)^2\; =\; n^2\; +\; O(n)$

$(n+O(n^\{1/2\}))(n\; +\; O(log,n))^2\; =\; n^3\; +\; O(n^\{5/2\})$

$n^\{O(1)\}\; =\; O(e^n)$

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(eⁿ) 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.

Orders of common 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	Example
$Oleft(1right)$	constant	Determining if a number is even or odd
$Oleft(alpha(n)right)$	inverse Ackermann	Amortized time per operation when using a disjoint-set (union-find) data structure
$Oleft(log^*\; nright)$	iterated logarithmic	The find algorithm of Hopcroft and Ullman on a disjoint set
$Oleft(log\; log\; nright)$	log log n
$Oleft(log\; nright)$	logarithmic	Finding an item in a sorted array with the binary search algorithm
$Oleft(left(log\; nright)^cright)$	polylogarithmic	Deciding if n is prime with the AKS primality test
	fractional power	Searching in a kd-tree
$Oleft(nright)$	linear	Finding an item in an unsorted list, adding two n-digit numbers.
$Oleft(nlog\; nright)$	linearithmic, loglinear, or quasilinear	Sorting a list with heapsort, performing an FFT
$Oleft(\{n^2\}right)$	quadratic	Sorting a list with insertion sort, multiplying two n-digit numbers by simple algorithm, adding of two n×n matrices.
$Oleft(\{n^3\}right)$	cubic	Multiplying two n×n matrices by simple algorithm.
$Oleft(\{n^c\}right),\; c>1$	polynomial, sometimes called algebraic	Finding the shortest path on a weighted digraph with the Floyd-Warshall algorithm
$=e^\{(c+o(1))\; (ln\; n)^alpha\; (ln\; ln\; n)^\{1-alpha\}\}$	L-notation	Factoring a number using the special or general number field sieve
$Oleft(\{c^n\}right)$	exponential, sometimes called geometric	Determining if two logical statements are equivalent using brute force; finding the (exact) solution to the traveling salesman problem using dynamic programming.
$Oleft(n!right)$	factorial, sometimes called combinatorial	Determining if two logical statements are equivalent; solving the traveling salesman problem via brute-force search; finding the determinant of a matrix with expansion by minors.
$Oleft(\{n^n\}right)$	n to the n	Often used instead of $Oleft(n!right)$ to derive simpler formulas for asymptotic complexity.
$c\_1^\{O(c^n)\}$	double exponential	Finding a complete set of associative-commutative unifiers

Not as common, but even larger growth is possible, such as the single-valued version of the Ackermann function, A(n,n).

For any k>0 and c>0, O(n^c(log n)^k) is subset of O(n^(c+a)) for any a>0. So O(n^c(log n)^k) may be considered as polynomial with some bigger order.

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

$f(n)\; =\; 9\; log\; n\; +\; 5\; (log\; n)^3\; +\; 3n^2\; +\; 2n^3\; in\; O(n^3),!.$

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. 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 $c^2n^2\; in\; O(n^2)$. 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).

Changing of variable may affect the order of the resulting algorithm. For example, if an algorithm runs on the order of O(n) when n is the number of digits of the input number, then it has order O(log n) when n is the input number itself.

Product

$f\_1\; in\; O(g\_1)\; wedge$

 f_2in O(g_2), implies f_1  f_2in O(g_1  g_2),

$fcdot\; O(g)\; in\; O(f\; g)$

Sum

$f\_1\; in\; O(g\_1)\; wedge$

 f_2in O(g_2), implies f_1 + f_2in O(g_1 + g_2),

This implies $f\_1\; in\; O(g)\; land\; f\_2\; in\; O(g)\; implies\; f\_1+f\_2\; in\; O(g)$, which means that $O(g)$ is a convex cone.

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

Multiplication by a constant

Let $k\; in\; R\_\{>0\}$.

$O(k\; cdot\; g)\; =\; O(g)$

$fin\; O(g)\; Rightarrow\; kcdot\; fin\; O(g)$

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-o 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 algebraically defined functions $f(x)$ and $g(x)$, $lim\_\{x\; to\; infty\}f(x)/g(x)$ is generally found using L'Hôpital's rule.

For example,

• $2x\; in\; o(x^2)\; ,!$
• $2x^2\; not\; in\; o(x^2)$
• $1/x\; in\; o(1)$

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 related notations

Notation	Intuition	Definition
$f(n)\; in\; O(g(n))$	f is bounded above by $g$ (up to constant factor) asymptotically	exists (C>0), n_0 : forall(n>n_0) ; >f(n)| leq |Cg(n)| or $exists\; (C>0),\; n\_0\; :\; forall(n>n\_0)\; ;\; f(n)\; leq\; Cg(n)$
$f(n)\; in\; Omega(g(n))$	f is bounded below by $g$ (up to constant factor) asymptotically	exists (C>0), n_0 : forall (n>n_0) ; >Cg(n)| leq |f(n)|
$f(n)\; in\; Theta(g(n))$	f is bounded both above and below by $g$ asymptotically	exists (C,C'>0), n_0 : forall (n>n_0) ; >Cg(n)| < |f(n)| < |C'g(n)|
$f(n)\; in\; o(g(n))$	f is dominated by $g$ asymptotically	forall (C>0),exists n_0 : forall(n>n_0) ; >f(n)| < |Cg(n)|
$f(n)\; in\; omega(g(n))$	f dominates $g$ asymptotically	forall (C>0),exists n_0 : forall(n>n_0) ; >Cg(n)| < |f(n)|
$f(n)\; sim\; g(n)$	$f$ is equal to $g$ asymptotically	$lim\_\{n\; to\; infty\}\; frac\{f(n)\}\{g(n)\}\; =\; 1$

(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). $f(n)\; =\; tilde\{O\}\; (g(n))$ 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

$L\_n[alpha,c]=Oleft(e^\{(c+o(1))(ln\; n)^alpha(lnln\; n)^\{1-alpha\}\}right)$,

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.

To define Big O formally for multiple variables, suppose $f(vec\{x\})$ and $g(vec\{x\})$ are two functions defined on some subset of $mathbb\{R\}^n$. We say

$f(vec\{x\})mbox\{\; is\; \}O(g(vec\{x\}))mbox\{\; as\; \}vec\{x\}toinfty$

if and only if

$exists\; ;C,exists\; ;M>0mbox\{\; such\; that\; \}\; |f(vec\{x\})|\; le\; ;\; M\; |g(vec\{x\})|mbox\{\; for\; all\; \}vec\{x\}\; mbox\{\; with\; \}\; x\_i>C\; mbox\{\; for\; all\; \}i.$

For example, the statement

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

asserts that there exist constants C and M such that

$forall\; n,\; m>M:\; |g(n,m)|\; le\; C(n+m).$

where g(n,m) is defined by

$displaystyle\; f(n,m)\; =\; n^2\; +\; m^3\; +\; g(n,m).$

Note that this definition allows all of the coordinates of $vec\{x\}$ to increase to infinity. In particular, the statement

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

(i.e. $exists\; C\; exists\; M\; forall\; n\; forall\; m...$) is quite different from

$forall\; m:\; f(n,m)\; =\; hbox\{O\}(n^m)\; mbox\{\; as\; \}\; ntoinfty$

(i.e. $forall\; m\; exists\; C\; exists\; M\; forall\; n...$).

Graph theory

It is often useful to bound the running time of graph algorithms. Unlike most other computational problems, for a graph G = (V, E) there are two relevant parameters describing the size of the input: the number |V| of vertices in the graph and the number |E| 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. The symbols V and E are never used inside asymptotic notation with their literal meaning, so this abuse of notation does not risk ambiguity. For example $O(E\; +\; V\; log\; V)$ means $O((E,V)\; mapsto\; |E|\; +\; |V|cdotlog|V|)$ for a suitable metric of graphs. Another common convention—referring to the values |V| and |E| by the names n and m, respectively—sidesteps this ambiguity.

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,

$fsim\; g\; iff\; (f-g)\; in\; o(g)$

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, 2x is Θ(x), but 2x − 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

Further reading

• 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.
• G. H. Hardy. Orders of Infinity: The 'Infinitärcalcül' of Paul du Bois-Reymond, 1910.
• 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.
• 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
• 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.

External links

• Introduction to Asymptotic Notations