BIG O NOTATION
In mathematics, computer science, and related fields, big O notation (also known as Big Oh notation, Landau notation, Bachmann–Landau notation, and asymptotic notation) describes the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. Big O notation allows its users to simplify functions in order to concentrate on their growth rates: different functions with the same growth rate may be represented using the same O notation.
Although developed as a part of pure mathematics, this notation is now frequently also used in the analysis of algorithms to describe an algorithm's usage of computational resources: the worst case or average case running time or memory usage of an algorithm is often expressed as a function of the length of its input using big O notation. This allows algorithm designers to predict the behavior of their algorithms and to determine which of multiple algorithms to use, in a way that is independent of computer architecture or clock rate. Because Big O notation discards multiplicative constants on the running time, and ignores efficiency for low input sizes, it does not always reveal the fastest algorithm in practice or for practically-sized data sets. But the approach is still very effective for comparing the scalability of various algorithms as input sizes become large.
A description of a function in terms of big O notation usually only provides an upper bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates. Big O notation is also used in many other fields to provide similar estimates.
Formal definitionLet f(x) and g(x) be two functions defined on some subset of the real numbers. One writes
The notation can also be used to describe the behavior of f near some real number a (often, a = 0): we say
ExampleIn typical usage, the formal definition of O notation is not used directly; rather, the O notation for a function f(x) is derived by the following simplification rules:
- If f(x) is a sum of several terms, the one with the largest growth rate is kept, and all others omitted.
- If f(x) is a product of several factors, any constants (terms in the product that do not depend on x) are omitted.
One may confirm this calculation using the formal definition: let f(x) = 6x4 − 2x3 + 5 and g(x) = x4. Applying the formal definition from above, the statement that f(x) = O(x4) is equivalent to its expansion,
UsageBig O notation has two main areas of application. In mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated
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 asymptoticsBig 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 if we compare to any other order of expression, such as an expression containing a term n3 or n4. 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)). Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm.
So the big O notation captures what remains: we write either
Note that "=" is not meant to express "is equal to" in its normal mathematical sense, but rather a more colloquial "is", so the second expression is technically accurate (see the "Equals sign" discussion below) while the first is a common abuse of notation.
Infinitesimal asymptoticsBig O can also be used to describe the error term in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term. For example,
PropertiesIf 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 (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 cn 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.
O(logn) is exactly the same as O(log(nc)). The logarithms differ only by a constant factor (since log(nc) = clogn) 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. For example, 2n and 3n 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 c2n2, and the big O notation ignores the constant c2. This can be written as . If, however, an algorithm runs in the order of 2n, replacing n with cn gives 2cn = (2c)n. This is not equivalent to 2n in general.
Changing of variable may affect the order of the resulting algorithm. For example, if an algorithm's running time is O(n) when measured in terms of the number n of digits of an input number x, then its running time is O(log x) when measured as a function of the input number x itself, because n = Θ(log x).
Multiplication by a constant
Multiple variablesBig O (and little o, and Ω…) can also be used with multiple variables.
To define Big O formally for multiple variables, suppose and are two functions defined on some subset of . We say
Matters of notation
Equals signThe statement "f(x) is O(g(x))" as defined above is usually written as f(x) = O(g(x)). Some consider this to be an abuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As it is said, O(x) = O(x2) is true but O(x2) = O(x) is not.Knuth describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like n = n2 from the identities n = O(n2) and n2 = O(n2)."
For these reasons, it would be more precise to use set notation and write f(x) ∈ O(g(x)), thinking of O(g(x)) as the class of all functions h(x) such that |h(x)| ≤ C|g(n)| for some constant C. However, the use of the equals sign is customary. Knuth pointed out that "mathematicians customarily use the = sign as they use the word 'is' in English: Aristotle is a man, but a man isn’t necessarily Aristotle."