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Big-O Notation

Big O notation

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Example of Big O notation: f(x) ∈ O(g(x)) as there exists c > 0 (e.g., c = 1) and x0 (e.g., x0 = 5) such that f(x) ≤ cg(x) whenever x ≥ x0.

Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul BachmannEdmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation.

In computer science, big O notation is used to classify algorithms according to how their running time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem.

Big O notation characterizes functions according to their growth rates: different functions with the same growth rate may be represented using the same O notation.

The letter O is used because the growth rate of a function is also referred to as the order of the function. A description of a function in terms of big O notation usually only provides an upper bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates.

Big O notation is also used in many other fields to provide similar estimates.

Formal definition

Let f be a real or complex valued function and g a real valued function, both defined on some unbounded subset of the real positive numbers, such that g(x) is strictly positive for all large enough values of x.[4] One writes

{\displaystyle f(x)=O(g(x)){\text{ as }}x\to \infty }{\displaystyle f(x)=O(g(x)){\text{ as }}x\to \infty }

if and only if for all sufficiently large values of x, the absolute value of f(x) is at most a positive constant multiple of g(x). That is, f(x) = O(g(x)) if and only if there exists a positive real number M and a real number x0 such that

{\displaystyle |f(x)|\leq \;Mg(x){\text{ for all }}x\geq x_{0}.}{\displaystyle |f(x)|\leq \;Mg(x){\text{ for all }}x\geq x_{0}.}

In many contexts, the assumption that we are interested in the growth rate as the variable x goes to infinity is left unstated, and one writes more simply that

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

The notation can also be used to describe the behavior of f near some real number a (often, a = 0): we say

{\displaystyle f(x)=O(g(x)){\text{ as }}x\to a}{\displaystyle f(x)=O(g(x)){\text{ as }}x\to a}

if and only if there exist positive numbers δ and M such that

{\displaystyle |f(x)|\leq \;Mg(x){\text{ when }}0<|x-a|<\delta .}{\displaystyle |f(x)|\leq \;Mg(x){\text{ when }}0<|x-a|<\delta .}

As g(x) is chosen to be non-zero for values of x sufficiently close to a, both of these definitions can be unified using the limit superior:

{\displaystyle f(x)=O(g(x)){\text{ as }}x\to a}{\displaystyle f(x)=O(g(x)){\text{ as }}x\to a}

if and only if

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

Example

In typical usage, the formal definition of O notation is not used directly; rather, the O notation for a function f is derived by the following simplification rules:

  • 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 (terms in the product that do not depend on x) can be omitted.

For example, let f(x) = 6x4 − 2x3 + 5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms: 6x4, −2x3, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the second rule: 6x4 is a product of 6 and x4 in which the first factor does not depend on x. Omitting this factor results in the simplified form x4. Thus, we say that f(x) is a "big-oh" of (x4). Mathematically, we can write f(x) = O(x4). One may confirm this calculation using the formal definition: let f(x) = 6x4 − 2x3 + 5 and g(x) = x4. Applying the formal definition from above, the statement that f(x) = O(x4) is equivalent to its expansion,

{\displaystyle |f(x)|\leq \;Mx^{4}}{\displaystyle |f(x)|\leq \;Mx^{4}}

for some suitable choice of x0 and M and for all x > x0. To prove this, let x0 = 1 and M = 13. Then, for all x > x0:

{\displaystyle {\begin{aligned}|6x^{4}-2x^{3}+5|&\leq 6x^{4}+|2x^{3}|+5\\&\leq 6x^{4}+2x^{4}+5x^{4}\\&=13x^{4}\end{aligned}}}{\displaystyle {\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

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

Usage

Big O notation has two main areas of application:

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

There are two formally close, but noticeably different, usages of this notation:

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

Graphs of functions commonly used in the analysis of algorithms, showing the number of operations N versus input size nfor 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) = 1,000,0003U(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})}{\displaystyle \ T(n)=O(n^{2})}

or

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

and say that the algorithm has order of n2 time complexity. Note that "=" 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.[5]

Infinitesimal asymptotics

Big O can also be used to describe the error term in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term. Consider, for example, the exponential series and two expressions of it that are valid when 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\\&=1+x+{\frac {x^{2}}{2}}+O(x^{3})&{\text{as }}x\to 0\\&=1+x+O(x^{2})&{\text{as }}x\to 0\\\end{aligned}}}{\displaystyle {\begin{aligned}e^{x}&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dotsb &{\text{for all }}x\\&=1+x+{\frac {x^{2}}{2}}+O(x^{3})&{\text{as }}x\to 0\\&=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 ex − (1 + x + x2/2) is at most some constant times |x3| when x is close enough to 0.

Properties

If the function f can be written as a finite sum of other functions, then the fastest growing one determines the order of f(n). For example,

{\displaystyle f(n)=9\log n+5(\log n)^{3}+3n^{2}+2n^{3}=O(n^{3})\,,\qquad {\text{as }}n\to \infty \,\!.}f(n)=9\log n+5(\log n)^{3}+3n^{2}+2n^{3}=O(n^{3})\,,\qquad {\text{as }}n\to \infty \,\!.

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. Another thing to notice is the sets O(nc) and O(cn) are very different. If c is greater than one, then the latter grows much faster. A function that grows faster than nc for any c is called superpolynomial. One that grows more slowly 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 and the function nlog n.

We may ignore any powers of n inside of the logarithms. The set O(log n) is exactly the same as O(log(nc)). The logarithms differ only by a constant factor (since log(nc) = c log n) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. On the other hand, exponentials with different bases are not of the same order. For example, 2nand 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 cnmeans the algorithm runs in the order of c2n2, and the big O notation ignores the constant c2. This can be written as c2n2 = O(n2). If, however, an algorithm runs in the order of 2n, replacing n with cn gives 2cn = (2c)n. This is not equivalent to 2n in general. Changing variables may also affect the order of the resulting algorithm. For example, if an algorithm's run time is O(n) when measured in terms of the number n of digits of an input number x, then its run time is O(log x) when measured as a function of the input number x itself, because n = O(log x).

Product

{\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_{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)}f\cdot O(g)=O(fg)

Sum

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

This implies {\displaystyle f_{1}=O(g){\text{ and }}f_{2}=O(g)\Rightarrow f_{1}+f_{2}\in O(g)}f_{1}=O(g){\text{ and }}f_{2}=O(g)\Rightarrow f_{1}+f_{2}\in O(g), which means that {\displaystyle O(g)}O(g) is a convex cone.

Multiplication by a constant

Let k be constant. Then:
{\displaystyle \ O(|k|g)=O(g)}{\displaystyle \ O(|k|g)=O(g)} if k is nonzero.
{\displaystyle f=O(g)\Rightarrow kf=O(g).}f=O(g)\Rightarrow kf=O(g).

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 {\displaystyle f}fand {\displaystyle g}g are two functions defined on some subset of {\displaystyle \mathbb {R} ^{n}}\mathbb {R} ^{n}. We say

{\displaystyle f({\vec {x}}){\text{ is }}O(g({\vec {x}})){\text{ as }}{\vec {x}}\to \infty }f({\vec {x}}){\text{ is }}O(g({\vec {x}})){\text{ as }}{\vec {x}}\to \infty

if and only if

{\displaystyle \exists M\,\exists C>0{\text{ such that for all }}{\vec {x}}{\text{ with }}x_{i}\geq M{\text{ for some }}i,|f({\vec {x}})|\leq C|g({\vec {x}})|.}\exists M\,\exists C>0{\text{ such that for all }}{\vec {x}}{\text{ with }}x_{i}\geq M{\text{ for some }}i,|f({\vec {x}})|\leq C|g({\vec {x}})|.

Equivalently, the condition that {\displaystyle x_{i}\geq M}x_{i}\geq M for some {\displaystyle i}i can be replaced with the condition that {\displaystyle \|{\vec {x}}\|_{\infty }\geq M}{\displaystyle \|{\vec {x}}\|_{\infty }\geq M}, where {\displaystyle \|{\vec {x}}\|_{\infty }}{\displaystyle \|{\vec {x}}\|_{\infty }} denotes the Chebyshev norm. For example, the statement

{\displaystyle f(n,m)=n^{2}+m^{3}+O(n+m){\text{ as }}n,m\to \infty }{\displaystyle f(n,m)=n^{2}+m^{3}+O(n+m){\text{ as }}n,m\to \infty }

asserts that there exist constants C and M such that

{\displaystyle \forall \|(n,m)\|_{\infty }\geq M:|g(n,m)|\leq C|n+m|,}{\displaystyle \forall \|(n,m)\|_{\infty }\geq M:|g(n,m)|\leq C|n+m|,}

where g(n,m) is defined by

{\displaystyle f(n,m)=n^{2}+m^{3}+g(n,m).}{\displaystyle f(n,m)=n^{2}+m^{3}+g(n,m).}

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

{\displaystyle f(n,m)=O(n^{m}){\text{ as }}n,m\to \infty }{\displaystyle f(n,m)=O(n^{m}){\text{ as }}n,m\to \infty }

(i.e., {\displaystyle \exists C\,\exists M\,\forall n\,\forall m\dots }\exists C\,\exists M\,\forall n\,\forall m\dots ) is quite different from

{\displaystyle \forall m\colon f(n,m)=O(n^{m}){\text{ as }}n\to \infty }\forall m\colon f(n,m)=O(n^{m}){\text{ as }}n\to \infty

(i.e., {\displaystyle \forall m\,\exists C\,\exists M\,\forall n\dots }\forall m\,\exists C\,\exists M\,\forall n\dots ).

Note that under this definition, the subset on which a function is defined is significant when generalizing statements from the univariate setting to the multivariate setting. For example, if {\displaystyle f(n,m)=1}{\displaystyle f(n,m)=1} and {\displaystyle g(n,m)=n}{\displaystyle g(n,m)=n}, then {\displaystyle f(n,m)=O(g(n,m))}{\displaystyle f(n,m)=O(g(n,m))} if we restrict {\displaystyle f}f and {\displaystyle g}g to {\displaystyle [1,\infty )^{2}}{\displaystyle [1,\infty )^{2}}, but not if they are defined on {\displaystyle [0,\infty )^{2}}{\displaystyle [0,\infty )^{2}}.

This is not the only generalization of big O to multivariate functions, and in practice, there is some inconsistency in the choice of definition.

Matters of notation

Equals sign

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. 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)."

For these reasons, it would be more precise to use set notation and write f(x) ∈ 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. However, the use of the equals sign is customary. Knuth pointed out that "mathematicians customarily use the = sign as they use the word 'is' in English: Aristotle is a man, but a man isn't necessarily Aristotle."

Other arithmetic operators

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

expresses the same as

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

Example 

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.

Multiple usages

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 }n\to \infty

{\displaystyle (n+1)^{2}=n^{2}+O(n)}{\displaystyle (n+1)^{2}=n^{2}+O(n)}
{\displaystyle (n+O(n^{1/2}))(n+O(\log n))^{2}=n^{3}+O(n^{5/2})}{\displaystyle (n+O(n^{1/2}))(n+O(\log n))^{2}=n^{3}+O(n^{5/2})}
{\displaystyle n^{O(1)}=O(e^{n}).}{\displaystyle 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 somefunctions 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 nf(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 nO(1) = O(en) does not imply the false statement O(en) = nO(1)

Typesetting

Big O consists of just an uppercase "O". Unlike Greek-named Bachmann–Landau notations, it needs no special symbol. Yet, commonly used calligraphic variants, like {\displaystyle {\mathcal {O}}}{\mathcal {O}}, are available, in LaTeX and derived typesetting systems.

Orders of common functions

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 and n increases without bound. The slower-growing functions are generally listed first.

NotationNameExample
{\displaystyle O(1)}O(1)constantDetermining if a binary number is even or odd; Calculating {\displaystyle (-1)^{n}}(-1)^{n}; Using a constant-size lookup table
{\displaystyle O(\log \log n)}O(\log \log n)double logarithmicNumber of comparisons spent finding an item using interpolation search in a sorted array of uniformly distributed values
{\displaystyle O(\log n)}O(\log n)logarithmicFinding an item in a sorted array with a binary search or a balanced search tree as well as all operations in a Binomial heap
{\displaystyle O((\log n)^{c})}{\displaystyle O((\log n)^{c})}
{\displaystyle \scriptstyle c>1}{\displaystyle \scriptstyle c>1}
polylogarithmicMatrix chain ordering can be solved in polylogarithmic time on a parallel random-access machine.
{\displaystyle O(n^{c})}O(n^{c})
{\displaystyle \scriptstyle 0<c<1}{\displaystyle \scriptstyle 0<c<1}
fractional powerSearching in a k-d tree
{\displaystyle O(n)}O(n)linearFinding an item in an unsorted list or in an unsorted array; adding two n-bit integers by ripple carry
{\displaystyle O(n\log ^{*}n)}{\displaystyle O(n\log ^{*}n)}log-star nPerforming triangulation of a simple polygon using Seidel's algorithm, or the union–find algorithm. Note that {\displaystyle \log ^{*}(n)={\begin{cases}0,&{\text{if }}n\leq 1\\1+\log ^{*}(\log n),&{\text{if }}n>1\end{cases}}}\log ^{*}(n)={\begin{cases}0,&{\text{if }}n\leq 1\\1+\log ^{*}(\log n),&{\text{if }}n>1\end{cases}}
{\displaystyle O(n\log n)=O(\log n!)}{\displaystyle O(n\log n)=O(\log n!)}linearithmic, loglinear, or quasilinearPerforming a fast Fourier transform; Fastest possible comparison sortheapsort and merge sort
{\displaystyle O(n^{2})}O(n^{2})quadraticMultiplying two n-digit numbers by a simple algorithm; simple sorting algorithms, such as bubble sortselection sort and insertion sort; (worst case) bound on some usually faster sorting algorithms such as quicksortShellsort, and tree sort
{\displaystyle O(n^{c})}O(n^{c})polynomial or algebraicTree-adjoining grammar parsing; maximum matching for bipartite graphs; finding the determinant with LU decomposition
{\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}}{\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}}
{\displaystyle \scriptstyle 0<\alpha <1}{\displaystyle \scriptstyle 0<\alpha <1}
L-notation or sub-exponentialFactoring a number using the quadratic sieve or number field sieve
{\displaystyle O(c^{n})}O(c^{n})
{\displaystyle \scriptstyle c>1}{\displaystyle \scriptstyle c>1}
exponentialFinding the (exact) solution to the travelling salesman problem using dynamic programming; determining if two logical statements are equivalent using brute-force search
{\displaystyle O(n!)}O(n!)factorialSolving the travelling salesman problem via brute-force search; generating all unrestricted permutations of a poset; finding the determinant with Laplace expansion; enumerating all partitions of a set

The statement {\displaystyle f(n)=O(n!)}{\displaystyle f(n)=O(n!)} is sometimes weakened to {\displaystyle f(n)=O\left(n^{n}\right)}f(n)=O\left(n^{n}\right) to derive simpler formulas for asymptotic complexity. For any {\displaystyle k>0}k>0 and {\displaystyle c>0}c>0{\displaystyle O(n^{c}(\log n)^{k})}O(n^{c}(\log n)^{k}) is a subset of {\displaystyle O(n^{c+\varepsilon })}O(n^{c+\varepsilon }) for any {\displaystyle \varepsilon >0}\varepsilon >0, so may be considered as a polynomial with some bigger order.

Related asymptotic notations

Big O is the most commonly used asymptotic notation for comparing functions. Together with some other related notations it forms the family of Bachmann–Landau notations.

Little-o notation

"Little o" redirects here. For the baseball player, see Omar Vizquel.

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). Let as before f be a real or complex valued function and g a real valued function, both defined on some unbounded subset of the real positive numbers, such that g(x) is strictly positive for all large enough values of x. One writes

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

If for every positive constant ε there exists a constant N such that

{\displaystyle |f(x)|\leq \varepsilon g(x)\qquad {\text{for all }}x\geq N~.}{\displaystyle |f(x)|\leq \varepsilon g(x)\qquad {\text{for all }}x\geq N~.} For example, one has
{\displaystyle 2x=o(x^{2})}{\displaystyle 2x=o(x^{2})} and {\displaystyle 1/x=o(1).}{\displaystyle 1/x=o(1).}

The difference between the earlier definition for the big-O notation and the present 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. 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})}{\displaystyle 2x^{2}=O(x^{2})} but {\displaystyle 2x^{2}\neq o(x^{2}).}{\displaystyle 2x^{2}\neq o(x^{2}).}

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

{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=0}{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=0} (and this is in fact how Landau 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)}{\displaystyle f=o(g)} then {\displaystyle c\cdot f=o(g)}{\displaystyle c\cdot f=o(g)}, and
if {\displaystyle f=o(F)}{\displaystyle f=o(F)} and {\displaystyle g=o(G)}{\displaystyle g=o(G)} then {\displaystyle f\cdot g=o(F\cdot G).}{\displaystyle f\cdot g=o(F\cdot G).}

It also satisfies a transitivity relation:

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

Big Omega notation

There are two very widespread and incompatible definitions of the statement

{\displaystyle f(x)=\Omega (g(x))\ (x\rightarrow a),}f(x)=\Omega (g(x))\ (x\rightarrow 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 first one (chronologically) is used in analytic number theory, and the other one in computational complexity theory. When the two subjects meet, this situation is bound to generate confusion.

The Hardy–Littlewood definition

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

{\displaystyle f(x)=\Omega (g(x))\ (x\rightarrow \infty )\;\Leftrightarrow \;\limsup _{x\to \infty }\left|{\frac {f(x)}{g(x)}}\right|>0.}{\displaystyle f(x)=\Omega (g(x))\ (x\rightarrow \infty )\;\Leftrightarrow \;\limsup _{x\to \infty }\left|{\frac {f(x)}{g(x)}}\right|>0.}

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

In 1916 the same authors introduced the two new symbols {\displaystyle \Omega _{R}}\Omega _{R} and {\displaystyle \Omega _{L}}\Omega _{L},[14] thus defined:

{\displaystyle f(x)=\Omega _{R}(g(x))\ (x\rightarrow \infty )\;\Leftrightarrow \;\limsup _{x\to \infty }{\frac {f(x)}{g(x)}}>0}f(x)=\Omega _{R}(g(x))\ (x\rightarrow \infty )\;\Leftrightarrow \;\limsup _{x\to \infty }{\frac {f(x)}{g(x)}}>0;
{\displaystyle f(x)=\Omega _{L}(g(x))\ (x\rightarrow \infty )\;\Leftrightarrow \;\liminf _{x\to \infty }{\frac {f(x)}{g(x)}}<0.}{\displaystyle f(x)=\Omega _{L}(g(x))\ (x\rightarrow \infty )\;\Leftrightarrow \;\liminf _{x\to \infty }{\frac {f(x)}{g(x)}}<0.}

Although Donald Knuth wrote otherwise, Edmund Landau did use these three symbols, with the same meanings, in 1924.

These Hardy–Littlewood symbols are prototypes, which after Landau were never used again exactly thus.

{\displaystyle \Omega _{R}}\Omega _{R} became {\displaystyle \Omega _{+}}\Omega _{+}, and {\displaystyle \Omega _{L}}\Omega _{L} became {\displaystyle \Omega _{-}.}{\displaystyle \Omega _{-}.}

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

Simple examples

We have

{\displaystyle \sin x=\Omega (1)\ (x\rightarrow \infty ),}\sin x=\Omega (1)\ (x\rightarrow \infty ),

and more precisely

{\displaystyle \sin x=\Omega _{\pm }(1)\ (x\rightarrow \infty ).}\sin x=\Omega _{\pm }(1)\ (x\rightarrow \infty ).

We have

{\displaystyle \sin x+1=\Omega (1)\ (x\rightarrow \infty ),}\sin x+1=\Omega (1)\ (x\rightarrow \infty ),

and more precisely

{\displaystyle \sin x+1=\Omega _{+}(1)\ (x\rightarrow \infty );}\sin x+1=\Omega _{+}(1)\ (x\rightarrow \infty );

however

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

The Knuth definition

In 1976 Donald Knuth published a paper to justify his use of the {\displaystyle \Omega }\Omega -symbol to describe a stronger property. 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))}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 }\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".

Family of Bachmann–Landau notations

NotationName[15]DescriptionFormal DefinitionLimit Definition[18][19][20][15][13]
{\displaystyle f(n)=o(g(n))}f(n)=o(g(n))Small O; Small Ohf is dominated by gasymptotically{\displaystyle \forall k>0\;\exists n_{0}\;\forall n>n_{0}\;|f(n)|<k\cdot g(n)}{\displaystyle \forall k>0\;\exists n_{0}\;\forall n>n_{0}\;|f(n)|<k\cdot g(n)}{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}
{\displaystyle f(n)=O(g(n))}f(n)=O(g(n))Big O; Big Oh; Big Omicron{\displaystyle |f|}|f| is bounded above by g (up to constant factor) asymptotically{\displaystyle \exists k>0\;\exists n_{0}\;\forall n>n_{0}\;|f(n)|\leq k\cdot g(n)}{\displaystyle \exists k>0\;\exists n_{0}\;\forall n>n_{0}\;|f(n)|\leq k\cdot g(n)}{\displaystyle \limsup _{n\to \infty }{\frac {\left|f(n)\right|}{g(n)}}<\infty }{\displaystyle \limsup _{n\to \infty }{\frac {\left|f(n)\right|}{g(n)}}<\infty }
{\displaystyle f(n)=\Theta (g(n))}f(n)=\Theta (g(n))Big Thetaf is bounded both above and below by gasymptotically{\displaystyle \exists k_{1}>0\;\exists k_{2}>0\;\exists n_{0}\;\forall n>n_{0}}\exists k_{1}>0\;\exists k_{2}>0\;\exists n_{0}\;\forall n>n_{0}{\displaystyle k_{1}\cdot g(n)\leq f(n)\leq k_{2}\cdot g(n)}k_{1}\cdot g(n)\leq f(n)\leq k_{2}\cdot g(n){\displaystyle f(n)=O(g(n))}f(n)=O(g(n)) and {\displaystyle f(n)=\Omega (g(n))}f(n)=\Omega (g(n)) (Knuth version)
{\displaystyle f(n)\sim g(n)}{\displaystyle f(n)\sim g(n)}On the order off is equal to gasymptotically{\displaystyle \forall \varepsilon >0\;\exists n_{0}\;\forall n>n_{0}\;\left|{f(n) \over g(n)}-1\right|<\varepsilon }\forall \varepsilon >0\;\exists n_{0}\;\forall n>n_{0}\;\left|{f(n) \over g(n)}-1\right|<\varepsilon {\displaystyle \lim _{n\to \infty }{f(n) \over g(n)}=1}{\displaystyle \lim _{n\to \infty }{f(n) \over g(n)}=1}
{\displaystyle f(n)=\Omega (g(n))}f(n)=\Omega (g(n))Big Omega in number theory (Hardy–Littlewood){\displaystyle |f|}|f| is not dominated by gasymptotically{\displaystyle \exists k>0\;\forall n_{0}\;\exists n>n_{0}\;|f(n)|\geq k\cdot g(n)}{\displaystyle \exists k>0\;\forall n_{0}\;\exists n>n_{0}\;|f(n)|\geq k\cdot g(n)}{\displaystyle \limsup _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|>0}{\displaystyle \limsup _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|>0}
{\displaystyle f(n)=\Omega (g(n))}f(n)=\Omega (g(n))Big Omega in complexity theory (Knuth)f is bounded below by gasymptotically{\displaystyle \exists k>0\;\exists n_{0}\;\forall n>n_{0}\;f(n)\geq k\cdot g(n)}\exists k>0\;\exists n_{0}\;\forall n>n_{0}\;f(n)\geq k\cdot g(n){\displaystyle \liminf _{n\to \infty }{\frac {f(n)}{g(n)}}>0}{\displaystyle \liminf _{n\to \infty }{\frac {f(n)}{g(n)}}>0}
{\displaystyle f(n)=\omega (g(n))}f(n)=\omega (g(n))Small Omegaf dominates gasymptotically{\displaystyle \forall k>0\;\exists n_{0}\;\forall n>n_{0}\ |f(n)|>k\cdot |g(n)|}\forall k>0 \; \exists n_0 \; \forall n>n_0 \  |f(n)| > k\cdot |g(n)|{\displaystyle \lim _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|=\infty }{\displaystyle \lim _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|=\infty }

The limit definitions assume {\displaystyle g(n)\neq 0}{\displaystyle g(n)\neq 0} for sufficiently large n. The table is (partly) sorted from smallest to largest, in the sense that o, O, Θ, ∼, (Knuth's version of) Ω, ω on functions correspond to <, ≤, ≈, =, ≥, > on the real line (the Hardy-Littlewood version of Ω, however, doesn't correspond to any such description).

Computer science uses the big O, Big Theta Θ, little o, little omega ω and Knuth's big Omega Ω notations. Analytic number theory often uses the big O, small o, Hardy–Littlewood's big Omega Ω (with or without the +, - or ± subscripts) and {\displaystyle \sim }\sim notations.[17] The small omega ω notation is not used as often in analysis.

Use in computer science

Further information: Analysis of algorithms

Informally, especially in computer science, the Big O notation often is permitted to be somewhat abused 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 (i.e., numbers 2 and 3 below) are usually strongly preferred over looser bounds (i.e., 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 (bullets number 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.

Other notation

In their book Introduction to AlgorithmsCormenLeisersonRivest and Stein consider the set of functions g to which some function f belongs when it satisfies

{\displaystyle f(n)=O(g(n))\ (n\rightarrow \infty ).}{\displaystyle f(n)=O(g(n))\ (n\rightarrow \infty ).}

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

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

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. 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:

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

Extensions to the Bachmann–Landau notations

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) logk g(n)) for some k. 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 logk n is always o(nε) 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 }}}{\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}\ln n.

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[citation needed]. The "limiting process" x → xo 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,

{\displaystyle f\sim g\iff (f-g)\in o(g)}f\sim 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).

History (Bachmann–Landau, Hardy, and Vinogradov notations)

The symbol O 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 number theorist Edmund Landau adopted it, and was thus inspired to introduce in 1909 the notation o; hence both are now called Landau symbols. These notations were used in applied mathematics during the 1950s for asymptotic analysis.The symbol {\displaystyle \Omega }\Omega  (in the sense "is not an o of") was introduced in 1914 by Hardy and Littlewood. Hardy and Littlewood also introduced in 1918 the symbols {\displaystyle \Omega _{R}}\Omega _{R} ("right") and {\displaystyle \Omega _{L}}\Omega _{L} ("left"), precursors of the modern symbols {\displaystyle \Omega _{+}}\Omega _{+} ("is not smaller than a small o of") and {\displaystyle \Omega _{-}}\Omega _{-} ("is not larger than a small o of"). Thus the Omega symbols (with their original meanings) are sometimes also referred to as "Landau symbols". This notation {\displaystyle \Omega }\Omega  became commonly used in number theory at least since the 1950s. In the 1970s the big O was popularized in computer science by Donald Knuth, who introduced the related Theta notation, and proposed a different definition for the Omega notation.

Landau never used the Big Theta and small omega symbols.

Hardy's symbols were (in terms of the modern O notation)

{\displaystyle f\preccurlyeq g\iff f\in O(g)}{\displaystyle f\preccurlyeq g\iff f\in O(g)}   and   {\displaystyle f\prec g\iff f\in o(g);}f\prec g\iff f\in o(g);

(Hardy however never defined or used the notation {\displaystyle \prec \!\!\prec }\prec \!\!\prec , nor {\displaystyle \ll }\ll , as it has been sometimes reported). It should also be noted that Hardy introduces the symbols {\displaystyle \preccurlyeq }\preccurlyeq  and {\displaystyle \prec }\prec  (as well as some other symbols) in his 1910 tract "Orders of Infinity", and makes use of it only in three papers (1910–1913). In his nearly 400 remaining papers and books he consistently uses the Landau symbols O and o.

Hardy's notation is not used anymore. On the other hand, in the 1930s, the Russian number theorist Ivan Matveyevich Vinogradov introduced his notation {\displaystyle \ll }\ll , which has been increasingly used in number theory instead of the {\displaystyle O}O notation. We have

{\displaystyle f\ll g\iff f\in O(g),}f\ll g\iff f\in O(g),

and frequently both notations are used in the same paper.

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 capital omicron, probably in reference to his definition of the symbol Omega. The digit zero should not be used.

See also

Further reading

  • Hardy, G. H. (1910). Orders of Infinity: The 'Infinitärcalcül' of Paul du Bois-ReymondCambridge University Press.
  • Knuth, Donald (1997). "1.2.11: Asymptotic Representations". Fundamental Algorithms. The Art of Computer Programming. 1(3rd ed.). Addison–Wesley. ISBN 0-201-89683-4.
  • Cormen, Thomas H.Leiserson, Charles E.Rivest, Ronald L.Stein, Clifford (2001). "3.1: Asymptotic notation". Introduction to Algorithms (2nd ed.). MIT Press and McGraw–Hill. ISBN 0-262-03293-7.
  • Sipser, Michael (1997). Introduction to the Theory of Computation. PWS Publishing. pp. 226–228. ISBN 0-534-94728-X.
  • Avigad, Jeremy; Donnelly, Kevin (2004). Formalizing O notation in Isabelle/HOL (PDF). International Joint Conference on Automated Reasoning. doi:10.1007/978-3-540-25984-8_27.
  • Black, Paul E. (11 March 2005). Black, Paul E., ed. "big-O notation". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.
  • Black, Paul E. (17 December 2004). Black, Paul E., ed. "little-o notation". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.
  • Black, Paul E. (17 December 2004). Black, Paul E., ed. "Ω". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.
  • Black, Paul E. (17 December 2004). Black, Paul E., ed. "ω". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.
  • Black, Paul E. (17 December 2004). Black, Paul E., ed. "Θ". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.

References and notes

  1. Jump up to:a b Bachmann, Paul (1894). Analytische Zahlentheorie[Analytic Number Theory] (in German). 2. Leipzig: Teubner.
  2. Jump up to:a b Landau, 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. p. 883.
  3. Jump up^ Mohr, Austin. "Quantum Computing in Complexity Theory and Theory of Computation" (PDF). p. 2. Retrieved 7 June2014.
  4. Jump up^ Landau, 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. p. 31.
  5. Jump up to:a b Thomas H. Cormen et al., 2001, Introduction to Algorithms, Second Edition[page needed]
  6. Jump up^ Cormen, Thomas; Leiserson, Charles; Rivest, Ronald; Stein, Clifford (2009). Introduction to Algorithms (Third ed.). MIT. p. 53.
  7. Jump up^ Howell, Rodney. "On Asymptotic Notation with Multiple Variables" (PDF). Retrieved 2015-04-23.
  8. Jump up^ N. G. de Bruijn (1958). Asymptotic Methods in Analysis. Amsterdam: North-Holland. pp. 5–7. ISBN 978-0-486-64221-5.
  9. Jump up to:a b Graham, RonaldKnuth, DonaldPatashnik, Oren (1994). Concrete Mathematics (2 ed.). Reading, Massachusetts: Addison–Wesley. p. 446. ISBN 978-0-201-55802-9.
  10. Jump up^ Donald Knuth (June–July 1998). "Teach Calculus with Big O" (PDF)Notices of the American Mathematical Society45 (6): 687. (Unabridged version)
  11. Jump up^ Tom (24 June 2014). "Big O and related notations in LaTeX". texblog.
  12. Jump up to:a b Landau, 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. p. 61.
  13. Jump up to:a b c Hardy, G. H.; Littlewood, J. E. (1914). "Some problems of diophantine approximation: Part II. The trigonometrical series associated with the elliptic ϑ-functions". Acta Mathematica37: 225. doi:10.1007/BF02401834.
  14. Jump up to:a b G. H. Hardy and J. E. Littlewood, « Contribution to the theory of the Riemann zeta-function and the theory of the distribution of primes », Acta Mathematica, vol. 41, 1916.
  15. Jump up to:a b c d e f Donald Knuth. "Big Omicron and big Omega and big Theta", SIGACT News, Apr.-June 1976, 18–24.
  16. Jump up^ E. Landau, "Über die Anzahl der Gitterpunkte in gewissen Bereichen. IV." Nachr. Gesell. Wiss. Gött. Math-phys. Kl. 1924, 137–150.
  17. Jump up to:a b Aleksandar Ivić. The Riemann zeta-function, chapter 9. John Wiley & Sons 1985.
  18. Jump up^ Balcázar, José L.; Gabarró, Joaquim. "Nonuniform complexity classes specified by lower and upper bounds"(PDF)RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications23 (2): 180. ISSN 0988-3754. Retrieved 14 March 2017.
  19. Jump up^ 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. ISBN 978-3-642-38896-5. Retrieved 14 March2017.
  20. Jump up to:a b Vitányi, PaulMeertens, Lambert (April 1985). "Big Omega versus the wild functions" (PDF)ACM SIGACT News16 (4): 56–59. doi:10.1145/382242.382835.
  21. Jump up^ Cormen, Thomas H.Leiserson, Charles E.Rivest, Ronald L.Stein, Clifford (2001) [1990]. Introduction to Algorithms(2nd ed.). MIT Press and McGraw-Hill. pp. 41–50. ISBN 0-262-03293-7.
  22. Jump up^ for example it is omitted in: Hildebrand, A.J. "Asymptotic Notations" (PDF)Asymptotic Methods in Analysis. Math 595, Fall 2009. Retrieved 14 March 2017.
  23. Jump up^ Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest (2009). Introduction to Algorithms (3rd ed.). Cambridge/MA: MIT Press. p. 47. ISBN 978-0262533058When we have only an asymptotic upper bound, we use O-notation. For a given function g(n), we denote by O(g(n)) (pronounced “big-oh of gof n” or sometimes just “oh of g of n”) the set of functions O(g(n)) = { f(n) : there exist positive constants c and n0 such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0}
  24. Jump up^ Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest (2009). Introduction to Algorithms (3rd ed.). Cambridge/MA: MIT Press. p. 45. ISBN 978-0262533058Because θ(g(n)) is a set, we could write "f(n) ∈ θ(g(n))" to indicate that f(n) is a member of θ(g(n)). Instead, we will usually write f(n) = θ(g(n)) to express the same notion. You might be confused because we abuse equality in this way, but we shall see later in this section that doing so has its advantages.
  25. Jump up^ Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest (2009). Introduction to Algorithms (3rd ed.). Cambridge/MA: MIT Press. p. 49. ISBN 978-0262533058When the asymptotic notation stands alone (that is, not within a larger formula) on the right-hand side of an equation (or inequality), as in n = O(n²), we have already defined the equal sign to mean set membership: n ∈ O(n²). In general, however, when asymptotic notation appears in a formula, we interpret it as standing for some anonymous function that we do not care to name. For example, the formula 2n2 + 3n + 1 = 2n2 + θ(n) means that 2n2 + 3n + 1 = 2n2 + f(n), where f(n) is some function in the set θ(n). In this case, we let f(n) = 3n + 1, which is indeed in θ(n). Using asymptotic notation in this manner can help eliminate inessential detail and clutter in an equation.
  26. Jump up^ Erdelyi, A. (1956). Asymptotic ExpansionsISBN 978-0486603186.
  27. Jump up^ E. C. Titchmarsh, The Theory of the Riemann Zeta-Function (Oxford; Clarendon Press, 1951)
  28. Jump up^ See for instance "A new estimate for G(n) in Waring's problem" (Russian). Doklady Akademii Nauk SSSR 5, No 5-6 (1934), 249–253. Translated in English in: 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, 1985.
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