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BigO notation

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Describes approximate behavior of a function

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Orders of approximation Scale analysis Big O notation Curve fitting False precision Significant figures
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BigO notation is amathematical notation that describes the approximate size of afunction on adomain. Big O is a member of afamily of notations invented by German mathematiciansPaul Bachmann^[1] andEdmund Landau^[2]and expanded by others, collectively calledBachmann–Landau notation. The letter O was chosen by Bachmann to stand forOrdnung, meaning theorder of approximation.

Incomputer science, big O notation is used toclassify algorithms according to how their run time or space requirements^[a] grow as the input size grows.^[3] Inanalytic number theory, big O notation is often used to express bounds on the growth of anarithmetical function; one well-known example is the remainder term in theprime number theorem.^[4]Inmathematical analysis, includingcalculus,Big O notation is used to bound the error when truncating apower series and to express the qualityof approximation of a real or complex valued functionby a simpler function.

Often, big O notation characterizes functions according to their growth rates as the variable becomes large: different functions with the sameasymptotic 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 theorder of the function. A description of a function in terms of big O notation only provides anupper bound on the growth rate of the function.

Associated with big O notation are several related notations, using the symbols $o {\displaystyle o}$ , $\sim$ , $\Omega$ , $\ll$ , $\gg$ , $\asymp$ , $\omega$ , and $\Theta$ to describe other kinds of bounds on growth rates.^[5]^[6]^[7]

Formal definition

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Let ${\textstyle f,}$ the function to be estimated, be either areal orcomplex valued function defined on adomain ${\textstyle D,}$ and let ${\textstyle g,}$ the comparison function, be a non-negative real valued function defined on the same set ${\textstyle D.}$ Common choices for the domain are intervals of real numbers, bounded or unbounded, the set of positive integers, the set ofcomplex numbers and tuples of real/complex numbers. With the domain written explicitly or understood implicitly, one writes

f(x)=O{\bigl (}g(x){\bigr )}\

which is read as " ${\textstyle f(x)}$ isbig ${\textstyle O}$ of ${\textstyle g(x)}$ " if there exists a positive real number ${\textstyle M}$ such that

\left|f(x)\right|\leq M\ g(x)\qquad ~{\mathsf {\ for\ all\ }}~\quad x\in D.

If $g(x)>0$ (i.e.g is also never zero) throughout the domain $D, {\displaystyle D,}$ an equivalent definition is that the ratio ${\textstyle {\frac {f(x)}{g(x)}}}$ isbounded, i.e. there is a positive real number $M {\displaystyle M}$ so that ${\textstyle {\Big |}{\frac {f(x)}{g(x)}}{\Big |}\leq M}$ for all $x\in D.$ These encompass all the uses of big ${\textstyle O}$ incomputer science and mathematics, including its use where the domain is finite, infinite, real, complex, single variate, or multivariate. In most applications, one chooses the function $g(x)$ appearing within theargument of ${\textstyle O{\bigl (}\cdot {\bigr )}}$ to be as simple a form as possible, omitting constant factors and lower order terms. The number ${\textstyle M}$ is called theimplied constant because it is normally not specified. When usingbig ${\textstyle O}$ notation, what matters is that some finite $M {\displaystyle M}$ exists, not its specific value. This simplifies the presentation of many analytic inequalities.

For functions defined on positive real numbers or positive integers, a more restrictive and somewhat conflicting definitionis still in common use,^[3]^[8] especially in computer science. When restricted to functions which areeventually positive, the notation

f(x)=O{\bigl (}g(x){\bigr )}\qquad ~{\mathsf {as}}\quad x\to \infty

means that for some real number ${\textstyle a,}$ ${\textstyle f(x)=O{\bigl (}g(x){\bigr )}}$ in the domain ${\textstyle \left[a,\infty \right).}$ Here, the expression ${\textstyle x\to \infty }$ doesn't indicate alimit, but the notion that the inequality holds forlarge enough ${\textstyle x.}$ The expression ${\textstyle x\to \infty }$ often is omitted.^[3]

Similarly, for a finite real number ${\textstyle a,}$ the notation

f(x)=O{\bigl (}g(x){\bigr )}\qquad ~{\text{ as }}\ x\to a

means that for some constant ${\textstyle c>0,}$ ${\textstyle f(x)=O{\bigl (}g(x){\bigr )}}$ on the interval $\left[a-c,a+c\right];$ that is, in a small neighborhood of $a . {\displaystyle a.}$ In addition, the notation $\ f(x)=h(x)+O{\bigl (}g(x){\bigr )}\$ means ${\textstyle f(x)-h(x)=O{\bigl (}g(x){\bigr )}.}$ More complicated expressions are also possible.

Despite the presence of the equal sign (=) as written, the expression ${\textstyle f(x)=O{\bigl (}g(x){\bigr )}}$ does not refer to anequality, but rather to an inequality relating ${\textstyle f}$ and ${\textstyle g.}$

In the 1930s,^[6] the Russian number theoristI.M. Vinogradov introduced the notation $\ll ,$ which has been increasingly used in number theory^[4]^[9]^[10] and other branches of mathematics, as an alternative to the ${\textstyle O}$ notation. We have

\ f\ll g\iff f=O{\bigl (}g{\bigr )}.

Frequently both notations are used in the same work.

Set version of big O

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In computer science^[3] it is common to definebig ${\textstyle O}$ as also defining aset of functions. With the positive (or non-negative) function $g(x)$ specified, one interprets ${\textstyle O{\bigl (}g(x){\bigr )}}$ as representing theset of all functions ${\textstyle {\tilde {f}}}$ that satisfy ${\textstyle {\tilde {f}}(x)=O{\bigl (}g(x){\bigr )}.}$ One can then equivalently write ${\textstyle f(x)\in O{\bigl (}g(x){\bigr )},}$ read as "the function ${\textstyle \ f(x)\ }$ is among the set of all functions oforder at most ${\textstyle g(x).}$ "

Examples with an infinite domain

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In typical usage the $O {\displaystyle O}$ notation is applied to an infinite interval of real numbers $[a,\infty )$ and captures the behavior of the function for very large $x {\displaystyle x}$ . 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 $f(x)$ 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 $x {\displaystyle x}$ ) can be omitted.

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

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

Example with a finite domain

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Big O can also be used to describe theerror 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, theexponential series and two expressions of it that are valid when $x {\displaystyle x}$ is small: ${\begin{aligned}e^{x}&=1+x+{\frac {\;x^{2}\ }{2!}}+{\frac {\;x^{3}\ }{3!}}+{\frac {\;x^{4}\ }{4!}}+\dotsb &&{\text{ for all finite }}x\\[4pt]&=1+x+{\frac {\;x^{2}\ }{2}}+O(|x|^{3})&&{\text{ for all }}|x|\leq 1\\[4pt]&=1+x+O(x^{2})&&{\text{ for all }}|x|\leq 1.\end{aligned}}$ The middle expression(the line with" $O(|x^{3}|)$ ") means the absolute-value of the error $\ e^{x}-(1+x+{\frac {\;x^{2}\ }{2}})\$ is at most some constant times $~|x^{3}|\$ when $\ x~$ is small.This is an example of the use ofTaylor's theorem.

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

Multivariate examples

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$x\sin y=O(x)\quad {\text{ for }}x\geq 1,y{\text{ any real number}}$

$3a^{2}+7ab+2b^{2}+a+3b+14\ll a^{2}+b^{2}\ll a^{2}\quad {\text{ for all }}a\geq b\geq 1$

${\frac {xy}{x^{2}+y^{2}}}=O(1)\quad {\text{ for all real }}x,y{\text{ that are not both }}0$

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

Here we have acomplex variable function of two variables.In general, any bounded function is $O(1)$ .

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

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

In all of these examples, the bound is uniformin both variables. Sometimes in a multivariate expression, one variable ismore important than others, and one may expressthat the implied constant $M {\displaystyle M}$ depends on oneor more of the variables using subscripts to the big O symbol or the $\ll$ symbol. For example, consider the expression

$(1+x)^{b}=1+O_{b}(x)\quad {\text{ for }}0\leq x\leq 1,b{\text{ any real number.}}$

This means that for each real number $b {\displaystyle b}$ , there is a constant $M_{b}$ ,which depends on $b {\displaystyle b}$ , so that for all $0\leq x\leq 1$ , $|(1+x)^{b}-1|\leq M_{b}\cdot x.$ This particular statement follows from thegeneral binomial theorem.

Another example, common in the theory ofTaylor series, is $e^{x}=1+x+O_{r}(x^{2})\quad {\text{ for all }}|x|\leq r,r{\text{ being any real number.}}$ Here the implied constant depends on the size of the domain.

The subscript convention applies to all of the othernotations in this page.

Properties

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Product

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

f\cdot O(g)=O(|f|g)

If the function $f {\displaystyle f}$ of a positive integer $n {\displaystyle n}$ 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)^{4}+3n^{2}+2n^{3}=O(n^{3})\qquad {\text{for }}n\geq 1.

Some general rules about growthtoward infinity; the 2nd and 3rd property belowcan be proved rigorously usingL'Hôpital's rule:

Large powers dominate small powers

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

Powers dominate logarithms

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For any positive $a, b, {\displaystyle a,b,}$ $(\log n)^{a}=O_{a,b}(n^{b}),$ no matter how large $a {\displaystyle a}$ is and how small $b {\displaystyle b}$ is. Here, the implied constant dependson both $a {\displaystyle a}$ and $b {\displaystyle b}$ .

Exponentials dominate powers

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

A function that grows faster than $n^{c}$ for any $c {\displaystyle c}$ is calledsuperpolynomial. One that grows more slowly than any exponential function of the form $c^{n}$ with $c>1$ is calledsubexponential. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest known algorithms forinteger factorization and the function $n^{\log n}$ .

We may ignore any powers of $n {\displaystyle n}$ inside of the logarithms. For any positive $c {\displaystyle c}$ , the notation $O(\log n)$ means exactly the same thing as $O(\log(n^{c}))$ , since $\log(n^{c})=c\log n$ . Similarly, logs with different constant bases are equivalent with respect to Big O notation. On the other hand, exponentials with different bases are not of the same order. For example, $2^{n}$ and $3^{n}$ are not of the same order.

More complicated expressions

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In more complicated usage, $O(\cdot )$ can appear in different places in an equation, even several times on each side. For example, the following are true for $n {\displaystyle n}$ a positive integer: ${\begin{aligned}(n+1)^{2}&=n^{2}+O(n),\\(n+O(n^{1/2}))\cdot (n+O(\log n))^{2}&=n^{3}+O(n^{5/2}),\\n^{O(1)}&=O(e^{n}).\end{aligned}}$ The meaning of such statements is as follows: forany functions which satisfy each $O(\cdot )$ on the left side, there aresome functions satisfying each $O(\cdot )$ 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 satisfying $f(n)=O(1)$ , there is some function $g(n)=O(e^{n})$ such that $n^{f(n)}=g(n)$ ". The implied constant in the statement " $g(n)=O(e^{n})$ " maydepend on the implied constant in the expression" $f(n)=O(1)$ ".

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

Vinogradov's ≫ and Knuth's big Ω

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

In 1976,Donald Knuth^[7]defined

f(x)=\Omega (g(x))\Longleftrightarrow g(x)=O(f(x))

which has the same meaning as Vinogradov's $f(x)\gg g(x)$ .

Much earlier, Hardy and Littlewood defined $\Omega$ differently, but this it seldom used anymore (Ivič's book^[9] being one exception).Justifying his use of the $\Omega$ -symbol to describe a stronger property,^[7] Knuth wrote: "For all the applications I have seen so far in computer science, a stronger requirement ... is much more appropriate". Knuth further wrote, "Although I have changed Hardy and Littlewood's definition of $\Omega$ , 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."^[7]

Indeed, Knuth's big $\Omega$ enjoys much more widespread use today than the Hardy–Littlewood big $\Omega$ , being a common featurein computer science and combinatorics.

Hardy's ≍ and Knuth's big Θ

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In analytic number theory,^[10] thenotation $f(x)\asymp g(x)$ means both $f(x)=O(g(x))$ and $g(x)=O(f(x))$ . This notation is originally due to Hardy.^[5] Knuth's notation for the same notion is $f(x)=\Theta (g(x))$ .^[7] Roughly speaking, these statements assert that $f(x)$ and $g(x)$ have thesame order. These notations mean that there are positive constants $M, N {\displaystyle M,N}$ so that $Ng(x)\leq f(x)\leq Mg(x)$ for all $x {\displaystyle x}$ in the common domain of $f, g {\displaystyle f,g}$ . When the functions are defined on the positive integers or positive real numbers, as with big O, writers oftentimes interpret statements $f(x)=\Omega (g(x))$ and $f(x)=\Theta (g(x))$ as holding for all sufficiently large $x {\displaystyle x}$ , that is, for all $x {\displaystyle x}$ beyond some point $x_{0}$ . Sometimes thisis indicated by appending $x\to \infty$ to the statement. For example, $2n^{2}-10n=\Theta (n^{2})$ is true for the domain $n\geq 6$ but false if thedomain is all positive integers, since the function is zero at $n=5$ .

Further examples

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$n^{3}+20n^{2}+n+12\asymp n^{3}\quad {\text{ for all }}n\geq 1.$

$(1+x)^{8}=x^{8}+\Theta (x^{7})\quad {\text{ for all }}x\geq 1.$

The notation

$f(n)=e^{\Omega (n)}\quad {\text{ for all }}n\geq 1,$ means that there is a positive constant $M {\displaystyle M}$ so that $f(n)\geq e^{Mn}$ for all $n\geq 1$ . By contrast, $f(n)=e^{-O(n)}\quad {\text{ for all }}n\geq 1,$ means that there is a positive constant $M {\displaystyle M}$ so that $f(n)\geq e^{-Mn}$ for all $n\geq 1$ and $f(n)=e^{\Theta (n)}\quad {\text{ for all }}n\geq 1,$ means that there are positive constants $M, N {\displaystyle M,N}$ so that $e^{Mn}\leq f(n)\leq e^{Nn}$ for all $n\geq 1$ .

For any domain $D {\displaystyle D}$ , $f(x)=g(x)+O(1)\Longleftrightarrow e^{f(x)}\asymp e^{g(x)},$ each statement being for all $x {\displaystyle x}$ in $D {\displaystyle D}$ .

Orders of common functions

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Further information:Time complexity § Table of common time complexities

"O(1)" redirects here. For the quasicoherent sheaf, seeProj 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 andn increases without bound. The slower-growing functions are generally listed first.

Notation	Name	Example
$O(1)$	constant	Finding the median value for a sorted array of numbers; Calculating $(-1)^{n}$ ; Using a constant-sizelookup table
$O(\alpha (n))$	inverse Ackermann function	Amortized complexity per operation for theDisjoint-set data structure
$O(\log \log n)$	double logarithmic	Average number of comparisons spent finding an item usinginterpolation search in a sorted array of uniformly distributed values
$O(\log n)$	logarithmic	Finding an item in a sorted array with abinary search or a balanced searchtree as well as all operations in abinomial heap
$O((\log n)^{c})$ ${\textstyle c>1}$	polylogarithmic	Matrix chain ordering can be solved in polylogarithmic time on aparallel random-access machine.
$O(n^{c})$ ${\textstyle 0<c<1}$	fractional power	Searching in ak-d tree
$O(n)$	linear	Finding an item in an unsorted list or in an unsorted array; adding twon-bit integers byripple carry
$O(n\log ^{*}n)$	nlog-starn	Performingtriangulation of a simple polygon using Seidel's algorithm,^[11] where $\log ^{}(n)={\begin{cases}0,&{\text{if }}n\leq 1\\1+\log ^{}(\log n),&{\text{if }}n>1\end{cases}}$
$O(n\log n)=O(\log n!)$	linearithmic, loglinear, quasilinear, or " $n\log n$ "	Performing afast Fourier transform; fastest possiblecomparison sort;heapsort andmerge sort
$O(n^{2})$	quadratic	Multiplying two $n {\displaystyle n}$ -digit numbers byschoolbook multiplication; simple sorting algorithms, such asbubble sort,selection sort andinsertion sort; (worst-case) bound on some usually faster sorting algorithms such asquicksort,Shellsort, andtree sort
$O(n^{c})$	polynomial or algebraic	Tree-adjoining grammar parsing; maximummatching forbipartite graphs; finding thedeterminant withLU decomposition
$L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}$ ${\textstyle 0<\alpha <1}$	L-notation orsub-exponential	Factoring a number using thequadratic sieve ornumber field sieve
$O(c^{n})$ ${\textstyle c>1}$	exponential	Finding the (exact) solution to thetravelling salesman problem usingdynamic programming; determining if two logical statements are equivalent usingbrute-force search
$O(n!)$	factorial	Solving thetravelling salesman problem via brute-force search; generating all unrestricted permutations of aposet; finding thedeterminant withLaplace expansion; enumeratingall partitions of a set

The statement $f(n)=O(n!)$ is sometimes weakened to $f(n)=O\left(n^{n}\right)$ to derive simpler formulas for asymptotic complexity.In many of these examples, the running time isactually $\Theta (g(n))$ , which conveys moreprecision.

Little-o notation

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"Little o" redirects here. For the baseball player, seeOmar Vizquel. For the Greek letter, seeOmicron.

For real or complex-valued functions of a real variable $x {\displaystyle x}$ with $g(x)>0$ forsufficiently large $x {\displaystyle x}$ , one writes^[2]

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

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

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

Intuitively, this means that $g(x)$ grows much faster than $f(x)$ , or equivalently $f(x)$ grows much slower than $g(x)$ .For example, one has

200x=o(x^{2})

and

1/x=o(1),

both as

x\to \infty .

When one is interested in the behavior of a function for large values of $x {\displaystyle x}$ , little-o notation makes astronger statement than the corresponding big-O notation: every function that is little-o of $g {\displaystyle g}$ is also big-O of $g {\displaystyle g}$ on some interval $[a,\infty )$ , but not every function that is big-O of $g {\displaystyle g}$ is little-o of $g {\displaystyle g}$ . For example, $2x^{2}=O(x^{2})$ but $2x^{2}\neq o(x^{2})$ for $x\geq 1$ .

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

c {\displaystyle c}

is a nonzero constant and

f=o(g)

then

c\cdot f=o(g)

, and

f=o(F)

and

g=o(G)

then

f\cdot g=o(F\cdot G).

f=o(F)

and

g=o(G)

then

f+g=o(F+G)

It also satisfies atransitivity relation:

f=o(g)

and

g=o(h)

then

f=o(h).

Little-o can also be generalized to the finite case:^[2] $f(x)=o(g(x))\quad {\text{ as }}x\to x_{0}$ if $\lim _{x\to x_{0}}{\frac {f(x)}{g(x)}}=0.$ In other words, $f(x)=\alpha (x)g(x)$ for some $\alpha (x)$ with $\lim _{x\to x_{0}}\alpha (x)=0$ .

This definition is especially useful in the computation oflimits usingTaylor series. For example:

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

Asymptotic notation

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A relation related to little-o is theasymptotic notation $\sim$ . For real valued functions $f, g {\displaystyle f,g}$ , the expression $f(x)\sim g(x)\quad {\text{ as }}x\to \infty$ means $\lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1.$ One can connect this to little-o by observing that $f(x)\sim g(x)$ is also equivalent to $f(x)=(1+o(1))g(x)$ . Here $o(1)$ refers to a function tending to zero as $x\to \infty$ . One reads this as" $f(x)$ isasymptotic to $g(x)$ ". For nonzero functions on the same (finite or infinite) domain, $\sim$ forms anequivalence relation.

One of the most famous theorems using the notation $\sim$ isStirling's formula $n!\sim {\bigg (}{\frac {n}{e}}{\bigg )}^{n}{\sqrt {2\pi n}}\quad {\text{ as }}n\to \infty .$ In number theory, the famousprime number theorem states that $\pi (x)\sim {\frac {x}{\log x}}\quad {\text{ as }}x\to \infty ,$ where $\pi (x)$ is the number of primes whichare at most $x {\displaystyle x}$ and $\log$ is thenatural logarithm of $x {\displaystyle x}$ .

As with little-o, there is a version with finite limits (two-sided orone-sided) as well, for example $\sin x\sim x\quad {\text{ as }}x\to 0.$

Further examples: $x^{a}=o_{a,b}(e^{bx})\quad {\text{ as }}x\to \infty ,{\text{ for any positive constants }}a,b,$ $f(x)=g(x)+o(1)\quad \Longleftrightarrow \quad e^{f(x)}\sim e^{g(x)}\quad (x\to \infty ).$ $\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\sim {\frac {1}{s-1}}\quad (x\to \infty ).$ The last asymptotic is a basic property of theRiemann zeta function.

Knuth's little 𝜔

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For eventually positive, real valued functions $f, g, {\displaystyle f,g,}$ the notation $f(x)=\omega (g(x))\quad {\text{ as }}x\to \infty$ means $\lim _{x\to \infty }{\frac {f(x)}{g(x)}}=\infty .$ In other words, $g(x)=o(f(x))$ .Roughly speaking, this means that $f(x)$ grows much faster than does $g(x)$ .

The Hardy–Littlewood Ω notation

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In 1914G.H. Hardy andJ.E. Littlewood introduced the new symbol $\ \Omega ,$ ^[12] which is defined as follows:

f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}\quad

\quad x\to \infty \quad

\quad \limsup _{x\to \infty }\ \left|{\frac {\ f(x)\ }{g(x)}}\right|>0~.

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

In 1916 the same authors introduced the two new symbols $\ \Omega _{R}\$ and $\ \Omega _{L}\ ,$ defined as:^[13]

f(x)=\Omega _{R}{\bigl (}\ g(x)\ {\bigr )}\quad

\quad x\to \infty \quad

\quad \limsup _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}>0\ ;

f(x)=\Omega _{L}{\bigl (}\ g(x)\ {\bigr )}\quad

\quad x\to \infty \quad

\quad ~\liminf _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}<0~.

These symbols were used byE. Landau, with the same meanings, in 1924.^[14] Authors that followed Landau, however, use a different notation for the same definitions:^[9] The symbol $\ \Omega _{R}\$ has been replaced by the current notation $\ \Omega _{+}\$ with the same definition, and $\ \Omega _{L}\$ became $\ \Omega _{-}~.$

These three symbols $\ \Omega \ ,\Omega _{+}\ ,\Omega _{-}\ ,$ as well as $\ f(x)=\Omega _{\pm }{\bigl (}\ g(x)\ {\bigr )}\$ (meaning that $\ f(x)=\Omega _{+}{\bigl (}\ g(x)\ {\bigr )}\$ and $\ f(x)=\Omega _{-}{\bigl (}\ g(x)\ {\bigr )}\$ are both satisfied), are now currently used inanalytic number theory.^[9]^[10]

Simple examples

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We have

\sin x=\Omega (1)\quad

\quad x\to \infty \ ,

and more precisely

\sin x=\Omega _{\pm }(1)\quad

\quad x\to \infty ,~

where $\Omega _{\pm }$ means that the left side is both $\Omega _{+}(1)$ and $\Omega _{-}(1)$ ,

We have

1+\sin x=\Omega (1)\quad

\quad x\to \infty \ ,

and more precisely

1+\sin x=\Omega _{+}(1)\quad

\quad x\to \infty \ ;

however

1+\sin x\neq \Omega _{-}(1)\quad

\quad x\to \infty ~.

Family of Bachmann–Landau notations

[edit]

For understanding the formal definitions, consult thelist of logic symbols used in mathematics.

Notation	Name^[7]	Description	Formal definition	Compact definition ^[4]^[5]^[7]^[12]^[15]^[16]
$f(n)=o(g(n))$	Small O; Small Oh; Little O; Little Oh	f is dominated byg asymptotically (for any constant factor $k {\displaystyle k}$ )	$\forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon \|f(n)\|\leq k\,g(n)$	$\lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0$
$f(n)=O(g(n))$ or $f(n)\ll g(n)$ (Vinogradov's notation)	Big O; Big Oh; Big Omicron	$\|f\|$ is bounded above byg (up to constant factor $k {\displaystyle k}$ )	$\exists k>0\,\forall n\in D\colon \|f(n)\|\leq k\,g(n)$	$\sup _{n\in D}{\frac {\left\|f(n)\right\|}{g(n)}}<\infty$
$f(n)\asymp g(n)$ (Hardy's notation) or $f(n)=\Theta (g(n))$ (Knuth notation)	Of the same order as (Hardy); Big Theta (Knuth)	f is bounded byg both above (with constant factor $k_{2}$ ) and below (with constant factor $k_{1}$ )	$\exists k_{1}>0\,\exists k_{2}>0\,\forall n\in D\colon$ $k_{1}\,g(n)\leq f(n)\leq k_{2}\,g(n)$	$f(n)=O(g(n))$ and $g(n)=O(f(n))$
$f(n)\sim g(n)$ as $n\to a$ , where $a {\displaystyle a}$ is finite, $\infty$ or $-\infty$	Asymptotic equivalence	f is equal togasymptotically	$\forall \varepsilon >0\,\exists n_{0}\,\forall n>n_{0}\colon \left\|{\frac {f(n)}{g(n)}}-1\right\|<\varepsilon$ (in the case $a=\infty$ )	$\lim _{n\to a}{\frac {f(n)}{g(n)}}=1$
$f(n)=\Omega (g(n))$ (Knuth's notation), or $f(n)\gg g(n)$ (Vinogradov's notation)	Big Omega in complexity theory (Knuth)	f is bounded below byg, up to a constant factor	$\exists k>0\,\forall n\in D\colon f(n)\geq k\,g(n)$	$\inf _{n\in D}{\frac {f(n)}{g(n)}}>0$
$f(n)=\omega (g(n))$ as $n\to a$ , where $a {\displaystyle a}$ can be finite, $\infty$ or $-\infty$	Small Omega; Little Omega	f dominatesg asymptotically	$\forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)>k\,g(n)$ (for $a=\infty$ )	$\lim _{n\to a}{\frac {f(n)}{g(n)}}=\infty$
$f(n)=\Omega (g(n))$	Big Omega in number theory (Hardy–Littlewood)	$\|f\|$ is not dominated byg asymptotically	$\exists k>0\,\forall n_{0}\,\exists n>n_{0}\colon \|f(n)\|\geq k\,g(n)$	$\limsup _{n\to \infty }{\frac {\left\|f(n)\right\|}{g(n)}}>0$

The limit definitions assume $g(n)>0$ for $n {\displaystyle n}$ in a neighborhood of the limit; when thelimit is $\infty$ , this means that $g(n)>0$ for sufficiently large $n {\displaystyle n}$ .

Computer science and combinatorics use the big $O {\displaystyle O}$ , big Theta $\Theta$ , little $o {\displaystyle o}$ , little omega $\omega$ and Knuth's big Omega $\Omega$ notations.^[3] Analytic number theory often uses the big $O {\displaystyle O}$ , small $o {\displaystyle o}$ , Hardy's $\asymp$ ,Hardy–Littlewood's big Omega $\Omega$ (with or without the +, − or ± subscripts), Vinogradov's $\ll$ and $\gg$ notations and $\sim$ notations.^[9]^[4]^[10] The small omega $\omega$ notation is not used as often in analysis or in number theory.^[17]

Quality of approximations using different notation

[edit]

Further information:Analysis of algorithms

Informally, especially in computer science, the big $O {\displaystyle O}$ notation often can be used somewhat differently to describe an asymptotictight bound where using big Theta $\Theta$ notation might be more factually appropriate in a given context.^[18]For example, when considering a function $T(n)=73n^{3}+22n^{2}+58$ , all of the following are generally acceptable, but tighter bounds (such as numbers 2,3 and 4 below) are usually strongly preferred over looser bounds (such as number 1 below).

$T(n)=O(n^{100})$
$T(n)=O(n^{3})$
$T(n)=\Theta (n^{3})$
$T(n)\sim 73n^{3}$ as $n\to \infty$ .

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 $n {\displaystyle n}$ , the inventors and users of the algorithm might be more inclined to put an upper bound on how long it will take to run without making an explicit statement about the lower bound or asymptotic behavior.

Extensions to the Bachmann–Landau notations

[edit]

Another notation sometimes used in computer science is ${\tilde {O}}$ (readsoft-O), which hides polylogarithmic factors. There are two definitions in use: some authors use $f(n)={\tilde {O}}(g(n))$ as shorthand for $f(n)=O(g(n)\log ^{k}n)$ for some $k {\displaystyle k}$ ^{[citation needed]}, while others use it as shorthand for $f(n)=O(g(n)\log ^{k}g(n))$ .^[19]When $g(n)$ is polynomial in $n {\displaystyle n}$ , there is no difference; however, the latter definition allows one to say, e.g. that $n2^{n}={\tilde {O}}(2^{n})$ while the former definition allows for $\log ^{k}n={\tilde {O}}(1)$ for any constant $k {\displaystyle k}$ . Some authors writeO^* for the same purpose as the latter definition.^[20] Essentially, it is bigO notation, ignoringlogarithmic factors because thegrowth-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 $\log ^{k}n=o(n^{\varepsilon })$ for any constant $k {\displaystyle k}$ and any $\varepsilon >0.$

Also, theL notation, defined as

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

is convenient for functions that are betweenpolynomial andexponential in terms of $\log n$ .

Generalizations and related usages

[edit]

The generalization to functions taking values in anynormed vector space is straightforward (replacing absolute values by norms), where $f {\displaystyle f}$ and $g {\displaystyle g}$ need not take their values in the same space. A generalization to functions $g {\displaystyle g}$ taking values in anytopological group is also possible^{[citation needed]}.The "limiting process" $x\to x_{0}$ can also be generalized by introducing an arbitraryfilter base, i.e. to directednets $f {\displaystyle f}$ and $g {\displaystyle g}$ . The $o {\displaystyle o}$ notation can be used to definederivatives anddifferentiability in quite general spaces, and also (asymptotical) equivalence of functions,

f\sim g\iff (f-g)\in o(g)

which is anequivalence relation and a more restrictive notion than the relationship " $f {\displaystyle f}$ is $\Theta (g)$ " from above. (It reduces to $\lim f/g=1$ if $f {\displaystyle f}$ and $g {\displaystyle g}$ are positive real valued functions.) For example, $2x=\Theta (x)$ is, but $2x-x\neq o(x)$ .

History

[edit]

We sketch the history of the Bois-Reymond, Bachmann–Landau, Hardy, Vinogradov and Knuth notations.

In 1870, Paul du Bois-Reymond^[21]defined $f(x)\succ \phi (x)$ , $f(x)\sim \phi (x)$ and $f(x)\prec \phi (x)$ to mean, respectively, $\lim _{x\to \infty }{\frac {f(x)}{\phi (x)}}=\infty ,\quad \lim _{x\to \infty }{\frac {f(x)}{\phi (x)}}>0,\quad \lim _{x\to \infty }{\frac {f(x)}{\phi (x)}}=0.$ These were not widely adopted and are not used today.The first and third enjoy a symmetry: $f(x)\prec \phi (x)$ means the same as $\phi (x)\succ f(x)$ . Later, Landau adopted $\sim$ in the narrowersense that the limit of $f(x)/\phi (x)$ equals 1. None of these notations is in use today.

The symbol O was first introduced by number theoristPaul Bachmann in 1894, in the second volume of his bookAnalytische Zahlentheorie ("analytic number theory").^[1] The number theoristEdmund 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.^[22]The symbol $\Omega$ (in the sense "is not ano of") was introduced in 1914 by Hardy and Littlewood.^[12] Hardy and Littlewood also introduced in 1916 the symbols $\Omega _{R}$ ("right") and $\Omega _{L}$ ("left"),.^[13] This notation $\Omega$ became somewhat commonly used in number theory at least since the 1950s.^[23]

The symbol $\sim$ , although it had been used before with different meanings,^[21] was given its modern definition by Landau in 1909^[2] and by Hardy in 1910.^[5] Just above on the same page of his tract Hardy defined the symbol $\asymp$ , where $f(x)\asymp g(x)$ means that both $f(x)=O(g(x))$ and $g(x)=O(f(x))$ are satisfied. The notation is still currently used in analytic number theory.^[24]^[10] In his tract Hardy also proposed the symbol $\mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-}$ , where $f\mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-} g$ means that $f\sim Kg$ for some constant $K\not =0$ (this corresponds to Bois-Reymond's notation $f\sim g$ ).

In the 1930s, Vinogradov^[6] popularized the notation $f(x)\ll g(x)$ and $g(x)\gg f(x)$ , both of which mean $f(x)=O(g(x))$ . This notation became standard in analytic number theory.^[4]

In the 1970s the big O was popularized in computer science byDonald Knuth, who proposed the different notation $f(x)=\Theta (g(x))$ for Hardy's $f(x)\asymp g(x)$ , and proposed a different definition for the Hardy and Littlewood Omega notation.^[7]

Hardy introduced the symbols $\preccurlyeq$ and advocated for Boid-Reymond's $\prec$ (as well as the already mentioned other symbols) in his 1910 tract "Orders of Infinity",^[5] but 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.^[25]Hardy's symbols $\preccurlyeq$ and $\mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-}$ are not used anymore.

Matters of notation

[edit]

Arrows

[edit]

In mathematics, an expression such as $x\to \infty$ indicates the presence of alimit. In big-O notation and related notations $\Omega ,\Theta ,\gg ,\ll ,\asymp$ , there is no implied limit, in contrast withlittle-o, $\sim$ and $\omega$ notations.Notation such as $f(x)=O(g(x))\;\;(x\to \infty )$ can be considered anabuse of notation.

Equals sign

[edit]

Some consider $f(x)=O(g(x))$ to also be anabuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. Asde Bruijn says, $O(x)=O(x^{2})$ is true but $O(x^{2})=O(x)$ is not.^[26]Knuth describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like $n=n^{2}$ from the identities $n=O(n^{2})$ and $n^{2}=O(n^{2})$ .^[27] In another letter, Knuth also pointed out that^[28]

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, some advocate for usingset notation and write $f(x)\in O(g(x))$ ,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 $h(x)=O(g(x))$ .^[27] However, the use of the equals sign is customary.^[26]^[27]and is more convenient in more complex expressions of the form $f(x)=g(x)+O(h(x))=O(k(x)).$

The Vinogradov notations $\ll$ and $\gg$ , which are widely used in number theory^[9]^[4]^[10]do not suffer from this defect, as they more clearly indicate that big-O indicates aninequality rather than anequality. They also enjoy a symmetry that big-O notation lacks: $f(x)\ll g(x)$ means the same as $g(x)\gg f(x)$ . In combinatorics and computer science, these notationsare rarely seen.^[3]

Typesetting

[edit]

Big O is typeset as an italicized uppercase "O", as in the following example: $O(n^{2})$ .^[29]^[30] InTeX, 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 ${\mathcal {O}}$ instead.^[31]^[32]

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 capitalomicron,^[7] probably in reference to his definition of the symbolOmega. The digitzero should not be used.

References and notes

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^^a ^bBachmann, Paul (1894).Analytische Zahlentheorie [Analytic Number Theory] (in German). Vol. 2. Leipzig: Teubner.
^^a ^b ^c ^d ^eLandau, Edmund (1909).Handbuch der Lehre von der Verteilung der Primzahlen [Handbook on the theory of the distribution of the primes] (in German). Leipzig: B.G. Teubner; reprinted as two volumes in one by Chelsea, 1974, with an appendix by Dr. Paul T. Bateman. pp. 59–63.
^^a ^b ^c ^d ^e ^fCormen, Thomas H.;Leiserson, Charles E.;Rivest, Ronald L.;Stein, Clifford (2022). "Characterizing running times".Introduction to Algorithms (4th ed.). MIT Press and McGraw-Hill.ISBN 978-0-262-53091-0.
^^a ^b ^c ^d ^e ^fIwaniec, Henryk; Kowalski, Emmanuel (2004).Analytic Number Theory. American Mathematical Society.
^^a ^b ^c ^d ^eHardy, G. H. (1910).Orders of Infinity: The 'Infinitärcalcül' of Paul du Bois-Reymond.Cambridge University Press. p. 2.
^^a ^b ^c ^dVinogradov, Matveevič (1934). "A new estimate forG(n) in Waring's problem".Doklady Akademii Nauk SSSR (in Russian).5 (5–6):249–253.
Translated in English in:
Vinogradov, Matveevič (1985).Selected works / Ivan Matveevič Vinogradov; prepared by the Steklov Mathematical Institute of the Academy of Sciences of the USSR on the occasion of his 90th birthday. Springer-Verlag.
^^a ^b ^c ^d ^e ^f ^g ^h ⁱKnuth, Donald (April–June 1976)."Big Omicron and big Omega and big Theta".SIGACT News.8 (2):18–24.doi:10.1145/1008328.1008329.S2CID 5230246.
^Sipser, Michael (2012).Introduction to the Theory of Computation (3 ed.). Boston, MA: PWS Publishin.
^^a ^b ^c ^d ^e ^fIvić, A. (1985).The Riemann Zeta-Function. John Wiley & Sons. chapter 9.
^^a ^b ^c ^d ^e ^fGérald Tenenbaum, Introduction to analytic and probabilistic number theory, « Notation », page xxiii. American Mathematical Society, Providence RI, 2015.
^Seidel, Raimund (1991), "A Simple and Fast Incremental Randomized Algorithm for Computing Trapezoidal Decompositions and for Triangulating Polygons",Computational Geometry,1:51–64,CiteSeerX 10.1.1.55.5877,doi:10.1016/0925-7721(91)90012-4
^^a ^b ^cHardy, G.H.;Littlewood, J.E. (1914)."Some problems of diophantine approximation:Part II. The trigonometrical series associated with the ellipticθ functions".Acta Mathematica.37: 225.doi:10.1007/BF02401834.Archived from the original on 2018-12-12. Retrieved2017-03-14.
^^a ^bHardy, G.H.;Littlewood, J.E. (1916). "Contribution to the theory of the Riemann zeta-function and the theory of the distribution of primes".Acta Mathematica.41:119–196.doi:10.1007/BF02422942.
^Landau, E. (1924). "Über die Anzahl der Gitterpunkte in gewissen Bereichen. IV" [On the number of grid points in known regions].Nachr. Gesell. Wiss. Gött. Math-phys. (in German):137–150.
^Balcázar, José L.; Gabarró, Joaquim."Nonuniform complexity classes specified by lower and upper bounds"(PDF).RAIRO – Theoretical Informatics and Applications – Informatique Théorique et Applications.23 (2): 180.ISSN 0988-3754.Archived(PDF) from the original on 14 March 2017. Retrieved14 March 2017 – via Numdam.
^Cucker, Felipe; Bürgisser, Peter (2013)."A.1 Big Oh, Little Oh, and Other Comparisons".Condition: The Geometry of Numerical Algorithms. Berlin, Heidelberg: Springer. pp. 467–468.doi:10.1007/978-3-642-38896-5.ISBN 978-3-642-38896-5.
^for example it is omitted in:Hildebrand, A.J."Asymptotic Notations"(PDF). Department of Mathematics.Asymptotic Methods in Analysis. Math 595, Fall 2009. Urbana, IL: University of Illinois.Archived(PDF) from the original on 14 March 2017. Retrieved14 March 2017.
^Cormen et al. 2022, p. 57.
^Cormen et al. 2022, p. 74–75.
^Andreas Björklund and Thore Husfeldt and Mikko Koivisto (2009)."Set partitioning via inclusion-exclusion"(PDF).SIAM Journal on Computing.39 (2):546–563.doi:10.1137/070683933.Archived(PDF) from the original on 2022-02-03. Retrieved2022-02-03. See sect.2.3, p.551.
^^a ^bBois-Reymond, Paul du (1870)."Sur la grandeur relative des infinis des fonctions".Annali di Matematica. Series 2.4:338–353.doi:10.1007/BF02420041.
^Erdelyi, A. (1956).Asymptotic Expansions. Courier Corporation.ISBN 978-0-486-60318-6.{{cite book}}:ISBN / Date incompatibility (help).
^E. C. Titchmarsh, The Theory of the Riemann Zeta-Function (Oxford; Clarendon Press, 1951)
^Hardy, G. H.;Wright, E. M. (2008) [1st ed. 1938]. "1.6. Some notations".An Introduction to the Theory of Numbers. Revised byD. R. Heath-Brown andJ. H. Silverman, with a foreword byAndrew Wiles (6th ed.). Oxford: Oxford University Press.ISBN 978-0-19-921985-8.
^Hardy, G. H. (1966–1979).Collected papers of G. H. Hardy (Including Joint papers with J. E. Littlewood and others), 7 vols. Clarendon Press, Oxford.
^^a ^bde Bruijn, N.G. (1958).Asymptotic Methods in Analysis. Amsterdam: North-Holland. pp. 5–7.ISBN 978-0-486-64221-5.Archived from the original on 2023-01-17. Retrieved2021-09-15.{{cite book}}:ISBN / Date incompatibility (help)
^^a ^b ^cGraham, Ronald;Knuth, Donald;Patashnik, Oren (1994).Concrete Mathematics (2 ed.). Reading, Massachusetts: Addison–Wesley. p. 446.ISBN 978-0-201-55802-9.Archived from the original on 2023-01-17. Retrieved2016-09-23.
^Donald Knuth (June–July 1998)."Teach Calculus with Big O"(PDF).Notices of the American Mathematical Society.45 (6): 687.Archived(PDF) from the original on 2021-10-14. Retrieved2021-09-05. (Unabridged version Archived 2008-05-13 at theWayback Machine)
^Donald E. Knuth, The art of computer programming. Vol. 1. Fundamental algorithms, third edition, Addison Wesley Longman, 1997. Section 1.2.11.1.
^Ronald L. Graham, Donald E. Knuth, and Oren Patashnik,Concrete Mathematics: A Foundation for Computer Science (2nd ed.), Addison-Wesley, 1994. Section 9.2, p. 443.
^Sivaram Ambikasaran and Eric Darve, An ${\mathcal {O}}(N\log N)$ Fast Direct Solver for Partial Hierarchically Semi-Separable Matrices,J. Scientific Computing57 (2013), no. 3, 477–501.
^Saket Saurabh and Meirav Zehavi, $(k,n-k)$ -Max-Cut: An ${\mathcal {O}}^{*}(2^{p})$ -Time Algorithm and a Polynomial Kernel,Algorithmica80 (2018), no. 12, 3844–3860.

Notes

[edit]

^Note that the "size" of the input is typically used as an indication of how challenging a giveninstance is, of the problem to be solved. The amount of [execution] time, and the amount of [memory] space required to compute the answer, (or to "solve' the problem), are seen as indicating the difficulty of thatinstance of the problem. For purposes ofComputational complexity theory, Big $O {\displaystyle O}$ notation is used for an upper bound on [the "order of magnitude" of] all 3 of those: the size of the input [data stream], the amount of [execution] time required, and the amount of [memory] space required.