Big O notation
Big O notation is a type of symbolism used in complexity theory, computer science, and mathematics to describe the asymptotic behavior of functionss. More exactly, it is used to describe an asymptotic upper bound for the magnitude of a function in terms of another, usually simpler, function.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 |
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) = 4n^{2} - 2n + 2.
As n grows large, the n^{2} 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
expresses the fact that the error is smaller in absolute value than some constant times x^{3} if x is close enough to 0.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 x_{0} and M such that |f(x)| ≤ M |g(x)| for x > x_{0}.
- 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 - x_{0}| < 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
By this reason, some authors prefer a set notation and write f ∈ O(g), thinking of O(g) as the set of all functions dominated by g.Furthermore, an "equation" of the form
- f(x) = h(x) + O(g(x))
Common orders of functions
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(n^{2}) | quadratic |
O(n^{c}) | polynomial, sometimes "geometric" |
O(c^{n}) | exponential |
O(n!) | factorial |
O(n^{n}) | ? |
Properties
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.O(n^{c}) and O(c^{n}) 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 c^{n} 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
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))) |
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 later 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^{k}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 log^{k}n 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
asserts that there exist constants C and N such that