Big O notation is a symbolism used in complexity theory, computer science, and mathematics to describe the asymptotic behavior of functions. More exactly, it is used to describe the asymptotic upper bound for the magnitude of a function in terms of another, usually simpler, function.
It was invented by the German number theorist Edmund Landau, hence it is also called Landau's symbol. The letter O was originally a capital omicron, and is never a digit zero.
In Wikipedia, the various notations described in this article 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 |
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2 Common orders of functions 3 Formal definition 4 Multiple Variables 5 Related notation |
There are two formally close, but noticeably different usages of his notation: infinite asymptotics an infinitesimal 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 T(n) = O(n2)
and say that the algorithm has order of n2 time complexity.
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(nc) 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(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.
Big O can also be used to describe the error term in an approximation
to a mathematical function. For instance,
Uses
Infinite asymptotics
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.Infinitesimal asymptotics
expresses the fact that the error is smaller in absolute value
than some constant times x3 if x is close enough to 0.
| 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 |
The formal definition of big O uses limitss.
Suppose f(x) and g(x) are two functions defined on
some subset of the real numbers.
In mathematics, both limits at ∞ and limits at a are used.
In computer science, only limits at ∞ are used; furthermore,
only positive functions are considered, so the absolute value bars may
be left out.
Big O can also be used with multiple variables and with other
expressions on the right side of the equal sign. The statement
Therefore, to be more formally correct, some people prefer to define O(g) as a
function that maps functions into sets of functions, with the
value O(g(x)) being the set of all functions that do not grow
faster then g(x). Under this convention, it is said, e.g.,
that f(x) belongs to class (or set) O(g(x)) and the
corresponding set membership notation is used.
Perhaps most commonly, one simply says "f(x) is
O(g(x))" without any formal notation for "is".
Another point of difficulty is that the parameter whose asymptotic
behavior is being examined is not always clear. A statement such as
f(x,y) = O(g(x,y)) requires some additional
explanation to make clear what is meant. Still, this problem is rare
in practice.Formal definition
where C is a constant. Intuitively, this means that f does not grow faster than g.
The notation can also be used to describe the behavior of f near
some real number a:
where C is a constant. Multiple Variables
asserts that there exist constants C and N such that
This is a mild abuse of the equality symbol, since it is neither
transitive nor symmetric. The notation
O(g(x)) = f(x) looks pretty strange.
| Notation | Analogy | |
| 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)) | asymptoticaly 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 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 esitmate 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) logkn) for some k. Essentially, it is Big-O, ignoring logarithmic factors. This notation is often used to describe a class of "nitpicking" estimates (since logkn is always o(n) for any constant k).
