Big O notation

Graphs of functions commonly used in the analysis of algorithms, showing the number of operations N versus input size n for each function

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 if we compare to any other order of expression, such as an expression containing a term n³ or n⁴. 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)). 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

${\displaystyle T(n)=O(n^{2})}$

or

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

and say that the algorithm has order of n² 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.[6]

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 x is small:

${\displaystyle {\begin{aligned}e^{x}&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dotsb &{\text{for all }}x\\[4pt]&=1+x+{\frac {x^{2}}{2}}+O(x^{3})&{\text{as }}x\to 0\\[4pt]&=1+x+O(x^{2})&{\text{as }}x\to 0\end{aligned}}}$

The second expression (the one with O(x³)) means the absolute-value of the error e^x − (1 + x + x²/2) is at most some constant times |x³| when x is close enough to 0.

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

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

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 ${\displaystyle f_{1}+f_{2}\in O(g)}$. In other words, this second statement says that ${\displaystyle O(g)}$ is a convex cone.

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

The statement "f(x) is O(g(x))" as defined above is usually written as f(x) = O(g(x)). Some consider this 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.[9] 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)."[10] In another letter, Knuth also pointed out that "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".[11]

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 constant C.[10] However, the use of the equals sign is customary.[9][10]

Big O notation can also be used in conjunction with other arithmetic operators in more complicated equations. 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,

${\displaystyle g(x)=h(x)+O(f(x))}$

expresses the same as

${\displaystyle g(x)-h(x)=O(f(x)).}$

Suppose an algorithm is being developed to operate on a set of n elements. Its developers are interested in finding a function T(n) that will express how long the algorithm will take to run (in some arbitrary measurement of time) in terms of the number of elements in the input set. The algorithm works by first calling a subroutine to sort the elements in the set and then perform its own operations. The sort has a known time complexity of O(n²), and after the subroutine runs the algorithm must take an additional 55n³ + 2n + 10 steps before it terminates. Thus the overall time complexity of the algorithm can be expressed as T(n) = 55n³ + O(n²). Here the terms 2n + 10 are subsumed within the faster-growing O(n²). Again, this usage disregards some of the formal meaning of the "=" symbol, but it does allow one to use the big O notation as a kind of convenient placeholder.

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 ${\displaystyle n\to \infty }$:

$${\displaystyle {\begin{aligned}(n+1)^{2}&=n^{2}+O(n),\\(n+O(n^{1/2}))\cdot (n+O(\log n))^{2}&=n^{3}+O(n^{5/2}),\\n^{O(1)}&=O(e^{n}).\end{aligned}}}$$

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 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. 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).

Big O is typeset as an italicized uppercase "O", as in the following example: ${\displaystyle O(n^{2})}$.[12][13] In TeX, it is produced by simply typing O inside math mode. Unlike Greek-named Bachmann–Landau notations, it needs no special symbol. Yet, some authors use the calligraphic variant ${\displaystyle {\mathcal {O}}}$ instead.[14][15]

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). 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 g(x) is strictly positive for all large enough values of x. One writes

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

if for every positive constant ε there exists a constant ${\displaystyle x_{0}}$ such that

${\displaystyle |f(x)|\leq \varepsilon g(x)\quad {\text{ for all }}x\geq x_{0}.}$[16]

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 has to be true for at least one constant M, the latter must hold for every positive constant ε, however small.[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 also little-o of g. For example, ${\displaystyle 2x^{2}=O(x^{2})}$ but ${\displaystyle 2x^{2}\neq o(x^{2})}$.

As g(x) is nonzero, or at least becomes nonzero beyond a certain point, the relation ${\displaystyle f(x)=o(g(x))}$ is equivalent to

${\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=0}$ (and this is in fact how Landau[16] originally defined the little-o notation).

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

It also satisfies a transitivity relation:

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

Another asymptotic notation is ${\displaystyle \Omega }$, read "big omega".[18] There are two widespread and incompatible definitions of the statement

${\displaystyle f(x)=\Omega (g(x))}$ as ${\displaystyle x\to a}$,

where a is some real number, ∞, or −∞, where f and g are real functions defined in a neighbourhood of a, and where g is positive in this neighbourhood.

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

In 1914 Godfrey Harold Hardy and John Edensor Littlewood introduced the new symbol ${\displaystyle \Omega }$,[19] which is defined as follows:

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

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

In 1916 the same authors introduced the two new symbols ${\displaystyle \Omega _{R}}$ and ${\displaystyle \Omega _{L}}$, defined as:[20]

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

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

These symbols were used by Edmund Landau, with the same meanings, in 1924.[21] After Landau, the notations were never used again exactly thus; ${\displaystyle \Omega _{R}}$ became ${\displaystyle \Omega _{+}}$ and ${\displaystyle \Omega _{L}}$ became ${\displaystyle \Omega _{-}}$.

These three symbols ${\displaystyle \Omega ,\Omega _{+},\Omega _{-}}$, as well as ${\displaystyle f(x)=\Omega _{\pm }(g(x))}$ (meaning that ${\displaystyle f(x)=\Omega _{+}(g(x))}$ and ${\displaystyle f(x)=\Omega _{-}(g(x))}$ are both satisfied), are now currently used in analytic number theory.[22][23]

We have

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

and more precisely

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

We have

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

and more precisely

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

however

${\displaystyle \sin x+1\not =\Omega _{-}(1)}$ as ${\displaystyle x\to \infty .}$

In 1976 Donald Knuth published a paper to justify his use of the ${\displaystyle \Omega }$-symbol to describe a stronger property.[24] Knuth wrote: "For all the applications I have seen so far in computer science, a stronger requirement ... is much more appropriate". He defined

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

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."[24]

Notation	Name[24]	Description	Formal definition	Limit definition[25][26][27][24][19]
${\displaystyle f(n)=o(g(n))}$	Small O; Small Oh	f is dominated by g asymptotically	${\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|<k\,g(n)}$	${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}$
${\displaystyle f(n)=O(g(n))}$	Big O; Big Oh; Big Omicron	${\displaystyle |f|}$ is bounded above by g (up to constant factor) asymptotically	${\displaystyle \exists k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|\leq k\,g(n)}$	${\displaystyle \limsup _{n\to \infty }{\frac {f(n)}{g(n)}}<\infty }$
${\displaystyle f(n)=\Theta (g(n))}$	Big Theta	f is bounded both above and below by g asymptotically	${\displaystyle \exists k_{1}>0\,\exists k_{2}>0\,\exists n_{0}\,\forall n>n_{0}\colon }$ ${\displaystyle k_{1}\,g(n)\leq f(n)\leq k_{2}\,g(n)}$	${\displaystyle f(n)=O(g(n))}$ and ${\displaystyle f(n)=\Omega (g(n))}$ (Knuth version)
${\displaystyle f(n)\sim g(n)}$	On the order of	f is equal to g asymptotically	${\displaystyle \forall \varepsilon >0\,\exists n_{0}\,\forall n>n_{0}\colon \left|{\frac {f(n)}{g(n)}}-1\right|<\varepsilon }$	${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1}$
${\displaystyle f(n)=\Omega (g(n))}$	Big Omega in complexity theory (Knuth)	f is bounded below by g asymptotically	${\displaystyle \exists k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)\geq k\,g(n)}$	${\displaystyle \liminf _{n\to \infty }{\frac {f(n)}{g(n)}}>0}$
${\displaystyle f(n)=\omega (g(n))}$	Small Omega	f dominates g asymptotically	${\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)>k\,g(n)}$	${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=\infty }$
${\displaystyle f(n)=\Omega (g(n))}$	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 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[27] (the Hardy–Littlewood version of ${\displaystyle \Omega }$, however, doesn't correspond to any such description).

Computer science uses the big ${\displaystyle O}$, big Theta ${\displaystyle \Theta }$, little ${\displaystyle o}$, little omega ${\displaystyle \omega }$ and Knuth's big Omega ${\displaystyle \Omega }$ notations.[28] Analytic number theory often uses the big ${\displaystyle O}$, small ${\displaystyle o}$, Hardy–Littlewood's big Omega ${\displaystyle \Omega }$ (with or without the +, − or ± subscripts) and ${\displaystyle \sim }$ notations.[22] The small omega ${\displaystyle \omega }$ notation is not used as often in analysis.[29]

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. 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.

In their book Introduction to Algorithms, Cormen, Leiserson, Rivest and Stein consider the set of functions f which satisfy

${\displaystyle f(n)=O(g(n))\quad (n\to \infty )~.}$

In a correct notation this set can, for instance, be called O(g), where

${\displaystyle O(g)=\{f:{\text{there exist positive constants}}~c~{\text{and}}~n_{0}~{\text{such that}}~0\leq f(n)\leq cg(n){\text{ for all }}n\geq n_{0}\}.}$[30]

The authors state that the use of equality operator (=) to denote set membership rather than the set membership operator (∈) is an abuse of notation, but that doing so has advantages.[6] Inside an equation or inequality, the use of asymptotic notation stands for an anonymous function in the set O(g), which eliminates lower-order terms, and helps to reduce inessential clutter in equations, for example:[31]

${\displaystyle 2n^{2}+3n+1=2n^{2}+O(n).}$

Another notation sometimes used in computer science is Õ (read soft-O): f(n) = Õ(g(n)) is shorthand for f(n) = O(g(n) log^k n) for some k.[32] Some authors write O^* for the same purpose.[33] 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 ${\displaystyle \ln n}$.