Big O notation

In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithm's usage of computational resources (usually running time or memory). It is also called Big Oh notation, Landau notation, Bachmann-Landau notation, and asymptotic notation. Big O notation is also used in many other scientific and mathematical fields to provide similar estimations.

The symbol O is used to describe an asymptotic upper bound for the magnitude of a function in terms of another, usually simpler, function. There are also other symbols o, Ω, ω, and Θ for various other upper, lower, and tight bounds. Informally, the O notation is commonly employed to describe an asymptotic tight bound, but tight bounds are more formally and precisely denoted by the Θ (capital theta) symbol as described below. This distinction between upper and tight bounds is useful, and sometimes critical, and most computer scientists would urge distinguishing the usage of O and Θ, but in some other fields the Θ notation is not commonly known.

Usage

Big O notation has two main areas of application: in mathematics, it is usually used to characterize the residual term of a truncated infinite series, especially an asymptotic series; in computer science, it is useful in the analysis of the complexity of algorithms.

The notation was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book Analytische Zahlentheorie ("analytic number theory"), the first volume of which (not yet containing big O notation) was published in 1892. The notation was popularized in the work of another German number theorist Edmund Landau, hence it is sometimes called a Landau symbol. The big-O, standing for "order of", was originally a capital omicron; today the identical-looking Latin capital letter O is also used, but never the digit zero.

There are two formally close, but noticeably different, usages of this notation: infinite asymptotics and infinitesimal asymptotics. This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument.

Equals or member-of and other notational anomalies

In a way to be made precise below, O(f(x)) denotes the collection of functions g(x) – viewed as a function of variable x – that exhibit a growth that is limited to that of f(x) in some respect. The traditional notation for stating that g(x) belongs to this collection is:

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) = \mathcal{O}(f(x))\,.

This use of the equals sign is an abuse of notation, as the above statement is not an equation. It is improper to conclude from g(x) = O(f(x)) and h(x) = O(f(x)) that g(x) and h(x) are equal. One way to think of this is to consider "= O" one symbol here. To avoid the anomalous use, some authors prefer to write instead:

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) \in \mathcal{O}(f(x))\,,

without difference in meaning.

The common arithmetic operations are often extended to the class concept. For example, h(x) + O(f(x)) denotes the collection of functions having the growth of h(x) plus a part whose growth is limited to that of f(x). Thus,

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) = h(x) + \mathcal{O}(f(x))\,

expresses the same as

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) - h(x) \in \mathcal{O}(f(x))\,.

Another anomaly of the notation, although less exceptional, is that it does not make explicit which variable is the function argument, which may need to be inferred from the context if several variables are involved. The following two right-hand side big O notations have dramatically different meanings:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(m) = \mathcal{O}(m^n)\,,

Failed to parse (Missing texvc executable; please see math/README to configure.): g(n)\,\, = \mathcal{O}(m^n)\,.

The first case states that f(m) exhibits polynomial growth, while the second, assuming m > 1, states that g(n) exhibits exponential growth. So as to avoid all possible confusion, some authors use the notation

Failed to parse (Missing texvc executable; please see math/README to configure.): g \in \mathcal{O}(f)\,,

meaning the same as what is denoted by others as

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) \in \mathcal{O}(f(x))\,.

A final anomaly is that the notation does not make clear "where" the function growth is to be considered; infinitesimally near some point, or in the neighbourhood of infinity. This is in contrast with the usual notation for limits. Similar terminological and notational devices as for limits would resolve both this and the preceding anomaly, but are not in use.

Infinite asymptotics

Big O notation is useful when analyzing algorithms for efficiency. For example, the time (or the number of steps) it takes to complete a problem of size n might be found to be T(n) = 4n² - 2n + 2.

As n grows large, the n² term will come to dominate, so that all other terms can be neglected — for instance when n = 500, the term 4n² is 1000 times as large as the 2n term. Ignoring the latter would have negligible effect on the expression's value for most purposes.

Further, the coefficients become irrelevant as well if we compare to any other order of expression, such as an expression containing a term n³ or n². Even if T(n) = 1,000,000n², if U(n) = n³, the latter will always exceed the former once n grows larger than 1,000,000 (T(1,000,000) = 1,000,000³ = U(1,000,000)).

So the big O notation captures what remains: we write

Failed to parse (Missing texvc executable; please see math/README to configure.): T(n)\in \mathcal{O}(n^2)

(read as "big o of n squared") and say that the algorithm has order of n² time complexity.

Infinitesimal asymptotics

Big O can also be used to describe the error term in an approximation to a mathematical function. For example,

Failed to parse (Missing texvc executable; please see math/README to configure.): e^x=1+x+\frac{x^2}{2}+\mathcal{O}(x^3)\qquad\hbox{as}\ x\to 0

expresses the fact that the error, the difference Failed to parse (Missing texvc executable; please see math/README to configure.): e^x - \left(1 + x +\frac{x^2}{2}\right) , is smaller in absolute value than some constant times Failed to parse (Missing texvc executable; please see math/README to configure.): \left|x^3\right|

when Failed to parse (Missing texvc executable; please see math/README to configure.): x
is close enough to 0.

Formal definition

Suppose Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

and Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)
are two functions defined on

some subset of the real numbers. We say

such that for all x,Failed to parse (Missing texvc executable; please see math/README to configure.): |f(x)|\le  M\cdot g(x)

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)\mbox{ is }\mathcal{O}(g(x))\mbox{ as }x\to\infty

if and only if

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists \;x_0,\exists \;M>0\mbox{ such that } |f(x)| \le \; M |g(x)|\mbox{ for }x>x_0.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)\mbox{ is }\mathcal{O}(g(x))\mbox{ as }x\to a

if and only if

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists \;\delta >0,\exists \; M>0\mbox{ such that }|f(x)| \le \; M |g(x)|\mbox{ for }|x - a| < \delta.

If Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)

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

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)\mbox{ is }\mathcal{O}(g(x))\mbox{ as }x \to a

if and only if

Failed to parse (Missing texvc executable; please see math/README to configure.): \limsup_{x\to a} \left|\frac{f(x)}{g(x)}\right| < \infty.

In mathematics, both asymptotic behaviours near ∞ and near a are considered. In computational complexity theory, only asymptotics near ∞ are used; furthermore, only positive functions are considered, so the absolute value bars may be left out.

Example

Take the polynomials:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = 6x^4 -2x^3 +5 \,

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) = x^4. \,

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

Proof:

Failed to parse (Missing texvc executable; please see math/README to configure.): |6x^4 - 2x^3 + 5| \le 6x^4 + 2x^3 + 5 \,

        where x > 1

Failed to parse (Missing texvc executable; please see math/README to configure.): |6x^4 - 2x^3 + 5| \le 6x^4 + 2x^4 + 5x^4 \,

    because x3 < x4, and so on.

Failed to parse (Missing texvc executable; please see math/README to configure.): |6x^4 - 2x^3 + 5| \le 13x^4 \,

Failed to parse (Missing texvc executable; please see math/README to configure.): |6x^4 - 2x^3 + 5| \le 13 \,|x^4 |. \,

                       where C = 13 in this example

Matters of notation

The statement "Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

is Failed to parse (Missing texvc executable; please see math/README to configure.): O(g(x))

" as defined above is usually written as Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) =O(g(x)) . This is a slight abuse of notation; equality of two functions is not asserted, and it cannot be since the property of being Failed to parse (Missing texvc executable; please see math/README to configure.): O(g(x))

is not symmetric:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{O}(x)=\mathcal{O}(x^2)\mbox{ but }\mathcal{O}(x^2)\ne \mathcal{O}(x)

.

There is also a second reason why that notation is not precise. The symbol f(x) means the value of the function f for the argument x. Hence the symbol of the function is f and not f(x).

For these reasons, some authors prefer set notation and write Failed to parse (Missing texvc executable; please see math/README to configure.): f \in \mathcal{O}(g) , thinking of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{O}(g)

as the set of all functions dominated by g.

In more complex usage, O( ) can appear in different places in an equation, even several times on each side. For example, the following are true for Failed to parse (Missing texvc executable; please see math/README to configure.): n\to\infty

Failed to parse (Missing texvc executable; please see math/README to configure.): (n+1)^2 = n^2 + \mathcal{O}(n)

Failed to parse (Missing texvc executable; please see math/README to configure.): (n+\mathcal{O}(n^{1/2}))(n + \mathcal{O}(\log\,n))^2 = n^3 + \mathcal{O}(n^{5/2})

Failed to parse (Missing texvc executable; please see math/README to configure.): n^{\mathcal{O}(1)} = \mathcal{O}(e^n)

The meaning of such statements is as follows: for any functions which satisfy each O( ) on the left side, there are some functions satisfying each O( ) on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function f(n)=O(1), there is some function g(n)=O(eⁿ) such that n^f(n)=g(n)." In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side.

Orders of common functions

Here is a list of classes of functions that are commonly encountered when analyzing algorithms. All of these are as n increases to infinity. The slower-growing functions are listed first. c is an arbitrary constant.

Not as common, but even larger growth is possible, such as the single-valued version of the Ackermann function, A(n,n). Conversely, extremely slowly-growing functions such as the inverse of this function, often denoted α(n), are possible. Although unbounded, these functions are often regarded as being constant factors for all practical purposes.

Properties

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

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) = 9 \log n + 5 (\log n)^3 + 3n^2 + 2n^3 \in \mathcal{O}(n^3)\,\!.

In particular, if a function may be bounded by a polynomial in n, then as n tends to infinity, one may disregard lower-order terms of the polynomial.

Failed to parse (Missing texvc executable; please see math/README to configure.): O(n^c)

and Failed to parse (Missing texvc executable; please see math/README to configure.): O(c^n)
are very different.  The latter grows much, much faster, no matter how big the constant c is (as long as it is greater than one). A function that grows faster than any power of n is called superpolynomial.  One that grows more slowly than any exponential function of the form Failed to parse (Missing texvc executable; please see math/README to configure.): c^n
is called subexponential. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest known algorithms for integer factorization.

Failed to parse (Missing texvc executable; please see math/README to configure.): O(\log n)

is exactly the same as Failed to parse (Missing texvc executable; please see math/README to configure.): O(\log(n^c))

. The logarithms differ only by a constant factor, (since Failed to parse (Missing texvc executable; please see math/README to configure.): \log(n^c)=c \log n ) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. Exponentials with different bases, on the other hand, are not of the same order; assuming that they are is a common mistake. For example, Failed to parse (Missing texvc executable; please see math/README to configure.): 2^n

and Failed to parse (Missing texvc executable; please see math/README to configure.): 3^n
are not of the same order.

Changing units may or may not affect the order of the resulting algorithm. Changing units is equivalent to multiplying the appropriate variable by a constant wherever it appears. For example, if an algorithm runs in the order of n², replacing n by cn means the algorithm runs in the order of Failed to parse (Missing texvc executable; please see math/README to configure.): c^2n^2 , and the big O notation ignores the constant Failed to parse (Missing texvc executable; please see math/README to configure.): c^2 . This can be written as Failed to parse (Missing texvc executable; please see math/README to configure.): c^2n^2 \in \mathcal{O}(n^2) . If, however, an algorithm runs in the order of Failed to parse (Missing texvc executable; please see math/README to configure.): 2^n , replacing n with cn gives Failed to parse (Missing texvc executable; please see math/README to configure.): 2^{cn} = (2^c)^n . This is not equivalent to Failed to parse (Missing texvc executable; please see math/README to configure.): 2^n

(unless, of course, c=1).

Product

Failed to parse (Missing texvc executable; please see math/README to configure.): f_1 \in\mathcal{O}(g_1)\ \wedge\ f_2\in\mathcal{O}(g_2)\, \implies f_1 f_2\in\mathcal{O}(g_1 g_2)\,

Failed to parse (Missing texvc executable; please see math/README to configure.): f\cdot \mathcal{O}(g) \in \mathcal{O}(f g)

Sum

Failed to parse (Missing texvc executable; please see math/README to configure.): f_1 \in\mathcal{O}(g_1)\ \wedge\ f_2\in\mathcal{O}(g_2)\, \implies f_1 + f_2\in\mathcal{O}(g_1 + g_2)\,

This implies Failed to parse (Missing texvc executable; please see math/README to configure.): f_1 \in \mathcal{O}(g) \land f_2 \in \mathcal{O}(g) \implies f_1+f_2 \in \mathcal{O}(g)

, which means that Failed to parse (Missing texvc executable; please see math/README to configure.): O(g)

is a convex cone.

Failed to parse (Missing texvc executable; please see math/README to configure.): f + \mathcal{O}(g) \in \mathcal{O}(f + g)

Multiplication by a constant

Let Failed to parse (Missing texvc executable; please see math/README to configure.): k \in \R_{>0}

.

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{O}(k \cdot g) = \mathcal{O}(g)

Failed to parse (Missing texvc executable; please see math/README to configure.): f\in \mathcal{O}(g) \Rightarrow k\cdot f\in \mathcal{O}(g)

Related asymptotic notations

Big O is the most commonly used asymptotic notation for comparing functions, although in many cases Big O may be replaced with Θ for asymptotically tighter bounds (Theta, see below). Here, we define some related notations in terms of "big O":

Little-o notation

The relation Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) \in o(g(x))

is read as "Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)
is little-o of Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)

". Intuitively, it means that Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)

grows much faster than Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

. It assumes that f and g are both functions of one variable. Formally, it states that the limit of Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)/g(x)

is zero, as x approaches infinity. For algebraically defined functions Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)
and Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)

, Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{x \to \infty}f(x)/g(x)

is generally found using L'Hopital's Rule.

For example,

• Failed to parse (Missing texvc executable; please see math/README to configure.): 2x \in o(x^2) \,\!

• Failed to parse (Missing texvc executable; please see math/README to configure.): 2x^2 \not \in o(x^2)

• Failed to parse (Missing texvc executable; please see math/README to configure.): 1/x \in o(1)

Little-o notation is common in mathematics but rarer in computer science. In computer science the variable (and function value) is most often a natural number. In math, the variable and function values are often real numbers. The following properties can be useful:

• Failed to parse (Missing texvc executable; please see math/README to configure.): o(f) + o(f) \subseteq o(f)

• Failed to parse (Missing texvc executable; please see math/README to configure.): o(f)o(g) \subseteq o(fg)

• Failed to parse (Missing texvc executable; please see math/README to configure.): o(o(f)) \subseteq o(f)

• Failed to parse (Missing texvc executable; please see math/README to configure.): o(f) \subset O(f)

(and thus the above properties apply with most combinations of o and O).

As with big O notation, the statement "Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

is Failed to parse (Missing texvc executable; please see math/README to configure.): o(g(x))

" is usually written as Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = o(g(x)) , which is a slight abuse of notation.

Other related notations

is an asymptotic upper bound on Failed to parse (Missing texvc executable; please see math/README to configure.): f
(up to constant factor)

is an asymptotic lower bound on Failed to parse (Missing texvc executable; please see math/README to configure.): f
(up to constant factor)

is bounded tightly by Failed to parse (Missing texvc executable; please see math/README to configure.): g
asymptotically

is dominated by Failed to parse (Missing texvc executable; please see math/README to configure.): g
asymptotically

dominates Failed to parse (Missing texvc executable; please see math/README to configure.): g
asymptotically

(A mnemonic for these Greek letters is that "omicron" can be read "o-micron", i.e., "o-small", whereas "omega" can be read "o-mega" or "o-big".)

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

Another notation sometimes used in computer science is Õ (read soft-O). Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) = \tilde{O} (g(n))

is shorthand

for Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) = O(g(n) \log^kg(n))

for some k.  Essentially, it is Big-O, ignoring logarithmic factors. This notation is often used to describe a class of "nitpicking" estimates (since Failed to parse (Missing texvc executable; please see math/README to configure.): \log^kn
is always Failed to parse (Missing texvc executable; please see math/README to configure.): o(n^\epsilon)
for any constant Failed to parse (Missing texvc executable; please see math/README to configure.): k
and any Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon>0

).

The L notation, defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): L_n[\alpha, c] = O\left(e^{((c + o(1))(\ln n)^\alpha(\ln\ln n)^{1-\alpha})}\right)

,

is convenient for functions that are between polynomial and exponential.

Multiple variables

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

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n,m) = n^2 + m^3 + \hbox{O}(n+m) \mbox{ as } n,m\to\infty

asserts that there exist constants C and N such that

Failed to parse (Missing texvc executable; please see math/README to configure.): \forall n, m>N: |g(n,m)| \le C(n+m).

where g(n,m) is defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n,m) = n^2 + m^3 + g(n,m).

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

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n,m) = \hbox{O}(n^m) \mbox{ as } n,m\to\infty

is quite different from

Failed to parse (Missing texvc executable; please see math/README to configure.): \forall m: f(n,m) = \hbox{O}(n^m) \mbox{ as } n\to\infty.

Graph theory

It is often useful to bound the running time of graph algorithms. Unlike most other computational problems, for a graph G = (V, E) there are two relevant parameters describing the size of the input: the number |V| of vertices in the graph and the number |E| of edges in the graph. Inside asymptotic notation (and only there), it is common to use the symbols V and E, when someone really means |V| and |E|. We adopt this convention here to simplify asymptotic functions and make them easily readable. The symbols V and E are never used inside asymptotic notation with their literal meaning, so this abuse of notation does not risk ambiguity. For example Failed to parse (Missing texvc executable; please see math/README to configure.): O(E + V \log V)

means Failed to parse (Missing texvc executable; please see math/README to configure.): O((E,V) \mapsto |E| + |V|\cdot\log|V|)
for a suitable metric of graphs.  Another common convention—referring to the values |V| and |E| by the names n and m, respectively—sidesteps this ambiguity.

Generalizations and related usages

The generalization to functions taking values in any normed vector space is straightforward (replacing absolute values by norms), where f and g need not take their values in the same space. A generalization to functions g taking values in any topological group is also possible.

The "limiting process" x→x_o can also be generalized by introducing an arbitrary filter base, i.e. to directed nets f and g.

The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions,

Failed to parse (Missing texvc executable; please see math/README to configure.): f\sim g \iff (f-g) = o(g)

which is an equivalence relation and a more restrictive notion than the relationship "f is Θ(g)" from above. (It reduces to Failed to parse (Missing texvc executable; please see math/README to configure.): \lim f/g = 1

if f and g are positive real valued functions.)  For example, 2x is Θ(x), but 2x − x is not o(x).

See also

Wikibooks Data Structures has a page on the topic of

Big-O Notation

• Asymptotic expansion: Approximation of functions generalizing Taylor's formula
• Asymptotically optimal: A phrase frequently used to describe an algorithm that has an upper bound asymptotically within a constant of a lower bound for the problem
• Hardy notation: A different asymptotic notation
• Limit superior and limit inferior: An explanation of some of the limit notation used in this article
• Nachbin's theorem: A precise way of bounding complex analytic functions so that the domain of convergence of integral transforms can be stated

External links

• Introduction to Asymptotic Notations

References

Further reading

• Paul Bachmann. Die Analytische Zahlentheorie. Zahlentheorie. pt. 2 Leipzig: B. G. Teubner, 1894.
• Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen. 2 vols. Leipzig: B. G. Teubner, 1909.
• Marian Slodicka (Slodde vo de maten) & Sandy Van Wontergem. Mathematical Analysis I. University of Ghent, 2004.
• Donald Knuth. Big Omicron and big Omega and big Theta, ACM SIGACT News, Volume 8, Issue 2, 1976.
• Donald Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89683-4. Section 1.2.11: Asymptotic Representations, pp.107–123.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 3.1: Asymptotic notation, pp.41–50.
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X.  Pages 226–228 of section 7.1: Measuring complexity.
• Jeremy Avigad, Kevin Donnelly. Formalizing O notation in Isabelle/HOL
• Paul E. Black, "big-O notation", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 11 March 2005. Retrieved December 16, 2006.
• Paul E. Black, "little-o notation", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. Retrieved December 16, 2006.
• Paul E. Black, "Ω", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. Retrieved December 16, 2006.
• Paul E. Black, "ω", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 29 November 2004. Retrieved December 16, 2006.
• Paul E. Black, "Θ", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. Retrieved December 16, 2006.cs:Asymptotická složitost

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