Big O notation

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).

Uses

There are two formally close, but noticeably different usages of this notation: infinite asymptotics and infinitesimal asymptotics

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 T(n) = O(n²) and say that the algorithm has order of n² time complexity.

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

.

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(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.

Infinitesimal asymptotics

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

Properties

If a function may be bounded by a polynomial in n, then as n tends to zero, one may disregard higher-order terms of the polynomial. Notice the disinction with the case of infinite asymptotics. Notice also that this distinction is only of "pragmatic" or "mnemonic" value; the formal definition for the "big O" is the same for both cases.

Common orders of functions

Here is a list of classes of functions that are commonly encountered when analyzing algorithms. 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

Formal definition

The formal definition of big O uses limits. 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 computational complexity theory, only limits at ∞ are used; furthermore, only positive functions are considered, so the absolute value bars may be left out.

Multiple Variables

Big O can also be used with multiple variables and with other expressions on the right side of the equal sign. The statement

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.

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.

Related notation

Big O is the most commonly used of asymptotic notations for comparing functions. We will define them briefly by analogy with "big O", in terms of bounds.

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) 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).

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