Big O notation

It was first introduced by German number theorist Paul Bachmann in his 1892 book Analytische Zahlentheorie. The notation was popularized in the work of another German number theorist Edmund Landau, hence it is sometimes called a Landau symbol. The O was originally a capital omicron; today the capital letter O is used, but never the digit zero.

In Wikipedia, the various notations described in this article, including Omega notation and Theta notation are used for approximating formulas (e.g. those in the sum article), for analysis of algorithms (e.g. those in the heapsort article), and for the definitions of terms in complexity theory (e.g. polynomial time).

Table of contents

1 Uses 1.1 Infinite asymptotics 1.2 Infinitesimal asymptotics 2 Formal definition 3 Matters of notation 4 Common orders of functions 5 Properties 6 Related asymptotic notations: O, o, Ω, ω, Θ, Õ 7 Multiple variables

Uses

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.

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. Further, the constants will depend on the precise details of the implementation and the hardware it runs on, so they should also be neglected. Big O notation captures what remains: we write

Infinitesimal asymptotics

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

Formal definition

Suppose f(x) and g(x) are two functions defined on some subset of the real numbers. We say

f(x) is O(g(x)) as x → ∞

there exists numbers x₀ and M such that |f(x)| ≤ M |g(x)| for x > x₀.

f(x) is O(g(x)) as x → a

there exists numbers δ>0 and M such that |f(x)| ≤ M |g(x)| for |x - x₀| < δ.

f(x) is O(g(x)) as x → a

In mathematics, both asymptotic behaviors 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.

Matters of notation

The statement "f(x) is O(g(x))" as defined above is often written as f(x) = O(g(x)). This is a slight abuse of notation: we are not really asserting the equality of two functions. Here is a bad example:

Common orders of functions

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

notation	name
O(1)	constant
O(log n)	logarithmic
O([log n]c)	polylogarithmic
O(n)	linear
O(n log n)	sometimes called "linearithmic"
O(n2)	quadratic
O(nc)	polynomial, sometimes "geometric"
O(cn)	exponential
O(n!)	factorial
O(nn)	?

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

.

O(n^c) and O(cⁿ) are very different. The latter grows much, much faster, no matter how big the constant c is. A function that grows faster than any power of n is called superpolynomial. One that grows slower than an 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 algorithms known for integer factorization.

O(log n) is exactly the same as O(log(n^c)). The logarithms differ only by a constant factor, (since log(n^c)=c log n) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent.

If a function f(x) may be bounded by a polynomial in x, then as x tends to zero, one may disregard higher-order terms of the polynomial. Notice the distinction with the case of infinite asymptotics.

Related asymptotic notations: O, o, Ω, ω, Θ, Õ

Big O is the most commonly used asymptotic notation for comparing functions. We will define some others briefly in terms of "big O":

Notation	Definition
f(n) = O(g(n))	asymptotic upper bound
f(n) = o(g(n))	asymptotically negligible (M = 0)
f(n) = Ω(g(n))	asymptotic lower bound (iff g(n) = O(f(n)))
f(n) = ω(g(n))	asymptotically dominant (iff g(n) = o(f(n)))
f(n) = Θ(g(n))	asymptotically tight bound (iff both f(n) = O(g(n)) and g(n) = O(f(n)))

Here is a hint (and mnemonics) why Landau selected these Greek letters: "omicron" is "o-micron", i.e., "o-small", whereas "omega" is "o-BIG".

The relation f(n) = o(g(n)) is read as "f(n) is little-oh of g(n)". Intuitively, it means that g(n) grows much faster than f(n). Formally, it states that the limit of f(n)/g(n) is zero.

The notations Θ and Ω are often used in computer science; the lower-case o is common in mathematics but rare in computer science. The lower-case ω is rarely used.

In casual use, O is commonly used where Θ is meant, i.e., a tight estimate is implied. For example, one might say "heapsort is O(n log n) in average case" when the intended meaning was "heapsort is Θ(n log n) in average case". Both statements are true, but the latter is a stronger claim.

Another notation sometimes used in computer science is Õ (read Soft-O). f(n) = Õ(g(n)) is shorthand for f(n) = O(g(n) log^kn) for some k. Essentially, it is Big-O, ignoring logarithmic factors. This notation is often used to describe a class of "nitpicking" estimates (since log^kn is always o(n) for any constant k).

Multiple variables

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