Big O notation - Citizendia

In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithm's usage of computational resources (usually running time or memory). Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation In Computational complexity theory, a computational resource is a resource used by some Computational models in the solution of Computational problems It is also called Big Oh notation, Landau notation, Bachmann-Landau notation, and asymptotic notation. Big O notation is also used in many other scientific and mathematical fields to provide similar estimations.

The symbol O is used to describe an asymptotic upper bound for the magnitude of a function in terms of another, usually simpler, function. In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing In Mathematics, especially in Order theory, an upper bound of a Subset S of some Partially ordered set ( P, &le The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering There are also other symbols o, Ω, ω, and Θ for various other upper, lower, and tight bounds. Informally, the O notation is commonly employed to describe an asymptotic tight bound, but tight bounds are more formally and precisely denoted by the Θ (capital theta) symbol as described below. Theta (uppercase Θ, lowercase θ or ϑ; Θήτα is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth This distinction between upper and tight bounds is useful, and sometimes critical; most computer scientists would urge distinguishing the usage of O and Θ. In some other fields, however, the Θ notation is not commonly known.

Usage

Big O notation has two main areas of application: in mathematics, it is usually used to characterize the residual term of a truncated infinite series, especially an asymptotic series; in computer science, it is useful in the analysis of the complexity of algorithms. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their To analyze an Algorithm is to determine the amount of resources (such as time and storage necessary to execute it Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation

The notation was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book Analytische Zahlentheorie ("analytic number theory"), the first volume of which (not yet containing big O notation) was published in 1892. Paul Gustav Heinrich Bachmann ( June 22 1837 &ndash March 31 1920) was a German Mathematician. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes The notation was popularized in the work of another German number theorist Edmund Landau, hence it is sometimes called a Landau symbol. Edmund Georg Hermann (Yehezkel Landau ( February 14, 1877 – February 19, 1938) was a German Jewish Mathematician The big-O, standing for "order of", was originally a capital omicron; today the identical-looking Latin capital letter O is also used, but never the digit zero. Omicron or Omikron (uppercase Ο, lowercase ο, literally "small o": Όμικρον o mikron, micron meaning 'small' in contrast O is the fifteenth letter of the modern Latin Alphabet. Its name in English is spelled o (oʊ plural usually o's or os; sometimes

There are two formally close, but noticeably different, usages of this notation: infinite asymptotics and infinitesimal asymptotics. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have 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.

Equals or member-of and other notational anomalies

In a way to be made precise below, O(f(x)) denotes the collection of functions g(x) – viewed as a function of variable x – that exhibit a growth that is limited to that of f(x) in some respect. The traditional notation for stating that g(x) belongs to this collection is:

$g(x) = O(f(x))\,.$

This use of the equals sign is an abuse of notation, as the above statement is not an equation. History The "=" symbol that is now universally accepted by mathematics for equality was first recorded by Welsh mathematician Robert Recorde in The In Mathematics, abuse of notation occurs when an author uses a Mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent It is improper to conclude from g(x) = O(f(x)) and h(x) = O(f(x)) that g(x) and h(x) are equal. One way to think of this is to consider "= O" one symbol here. To avoid the anomalous use, some authors prefer to write instead:

$g(x) \in O(f(x))\,,$

without difference in meaning.

The common arithmetic operations are often extended to the class concept. For example, h(x) + O(f(x)) denotes the collection of functions having the growth of h(x) plus a part whose growth is limited to that of f(x). Thus,

$g(x) = h(x) + O(f(x))\,$

expresses the same as

$g(x) - h(x) \in O(f(x))\,.$

Another anomaly of the notation, although less exceptional, is that it does not make explicit which variable is the function argument, which may need to be inferred from the context if several variables are involved. The following two right-hand side big O notations have dramatically different meanings:

$f(m) = O(m^n)\,,$

$g(n)\,\, = O(m^n)\,.$

The first case states that f(m) exhibits polynomial growth, while the second, assuming m > 1, states that g(n) exhibits exponential growth. So as to avoid all possible confusion, some authors use the notation

$g \in O(f)\,,$

meaning the same as what is denoted by others as

$g(x) \in O(f(x))\,.$

A final anomaly is that the notation does not make clear "where" the function growth is to be considered; infinitesimally near some point, or in the neighbourhood of infinity. This is in contrast with the usual notation for limits. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Similar terminological and notational devices as for limits would resolve both this and the preceding anomaly, but are not in use.

Infinite asymptotics

Big O notation is useful when analyzing algorithms for efficiency. To analyze an Algorithm is to determine the amount of resources (such as time and storage necessary to execute it 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. The word term is from the Latin terminus "boundary line limit" from the Indo-European root ter- "peg post boundary" Ignoring the latter would have negligible effect on the expression's value for most purposes.

Further, the coefficients become irrelevant as well if we compare to any other order of expression, such as an expression containing a term n³ or n². In Mathematics, a coefficient is a Constant multiplicative factor of a certain object 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)).

So the big O notation captures what remains: we write any of

T (n) = O (n 2)

$T(n)\in O(n^2)$

(read as "big o of n squared") and say that the algorithm has order of n² time complexity.

Infinitesimal asymptotics

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

$e^x=1+x+\frac{x^2}{2}+O(x^3)\qquad\hbox{as}\ x\to 0$

expresses the fact that the error, the difference $e^x - \left(1 + x +\frac{x^2}{2}\right)$ , is smaller in absolute value than some constant times $\left|x^3\right|$ when $x$ is close enough to 0. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

Formal definition

Suppose $f (x)$ and $g (x)$ are two functions defined on some subset of the real numbers. In Mathematics, the real numbers may be described informally in several different ways We say

Theory of O-Notation: f is in the order of g (i. e.

f (x) = O (g (x))

) if and only if there exists a positive real number M and a real number

x 0

such that for all x, $|f(x)|\le M\cdot g(x)$ , wherever

x > x 0

$f(x)\mbox{ is }O(g(x))\mbox{ as }x\to\infty$

if and only if

$\exists \;x_0,\exists \;M>0\mbox{ such that } |f(x)| \le \; M |g(x)|\mbox{ for }x>x_0.$

The notation can also be used to describe the behavior of f near some real number a: we say

$f(x)\mbox{ is }O(g(x))\mbox{ as }x\to a$

if and only if

$\exists \;\delta >0,\exists \; M>0\mbox{ such that }|f(x)| \le \; M |g(x)|\mbox{ for }|x - a| < \delta.$

If $g (x)$ is non-zero for values of x sufficiently close to a, both of these definitions can be unified using the limit superior:

$f(x)\mbox{ is }O(g(x))\mbox{ as }x \to a$

if and only if

$\limsup_{x\to a} \left|\frac{f(x)}{g(x)}\right| < \infty.$

In mathematics, both asymptotic behaviours near ∞ and near a are considered. ↔ In Mathematics, the limit inferior and limit superior (also called infimum limit and supremum limit, or liminf and limsup In computational complexity theory, only asymptotics near ∞ are used; furthermore, only positive functions are considered, so the absolute value bars may be left out. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources

Example

Take the polynomials:

$f(x) = 6x^4 -2x^3 +5 \,$

$g(x) = x^4. \,$

We say f(x) has order O(g(x)) or O(x⁴) (as $x\to\infty$ )

From the definition of order

$|f(x)| \le \; M |g(x)|\mbox{ for }x>x_0.$

Proof:

$|6x^4 - 2x^3 + 5| \le 6x^4 + 2x^3 + 5 \,$ where x > 0

$|6x^4 - 2x^3 + 5| \le 6x^4 + 2x^4 + 5x^4 \,$ because x³ < x⁴, and so on. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

$|6x^4 - 2x^3 + 5| \le 13x^4 \,$

$|6x^4 - 2x^3 + 5| \le 13 \,|x^4 |. \,$ where M = 13 in this example

Matters of notation

The statement " $f (x)$ is $O (g (x))$ " as defined above is usually written as $f (x) = O (g (x))$ . This is a slight abuse of notation; equality of two functions is not asserted, and it cannot be since the property of being $O (g (x))$ is not symmetric:

$O(x)=O(x^2)\mbox{ but }O(x^2)\ne O(x)$ . In Mathematics, abuse of notation occurs when an author uses a Mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition

There is also a second reason why that notation is not precise. The symbol f(x) means the value of the function f for the argument x. Hence the symbol of the function is f and not f(x).

For these reasons, some authors prefer set notation and write $f \in O(g)$ , thinking of $O (g)$ as the set of all functions dominated by g. In Set theory and its applications to Logic, Mathematics, and Computer science, set-builder notation (sometimes simply "set notation"

In more complex usage, O( ) can appear in different places in an equation, even several times on each side. For example, the following are true for $n\to\infty$

(n + 1) 2 = n 2 + O (n)

$(n+O(n^{1/2}))(n + O(\log\,n))^2 = n^3 + O(n^{5/2})$

n O (1) = O (e n)

The meaning of such statements is as follows: for any functions which satisfy each O( ) on the left side, there are some functions satisfying each O( ) on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function f(n)=O(1), there is some function g(n)=O(eⁿ) such that n^f(n)=g(n). " In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side.

Orders of common 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	Example
$O\left(1\right)$	constant	Determining if a number is even or odd
$O\left(\alpha(n)\right)$	inverse Ackermann	Amortized time per operation when using a disjoint-set (union-find) data structure
$O\left(\log^* n\right)$	iterated logarithmic	The `find` algorithm of Hopcroft and Ullman on a disjoint set
$O\left(\log \log n\right)$	log log n
$O\left(\log n\right)$	logarithmic	Finding an item in a sorted array with the binary search algorithm
$O\left(\left(\log n\right)^c\right)$	polylogarithmic	Deciding if n is prime with the AKS primality test
$O\left({n^c}\right), 0<c<1$	fractional power	Searching in a kd-tree
$O\left(n\right)$	linear	Finding an item in an unsorted list, adding two n-digit numbers. In Recursion theory, the Ackermann function or Ackermann-Péter function is a simple example of a general recursive function that is not primitive In Computer science, especially Analysis of algorithms, amortized analysis refers to finding the average running time per operation over a Worst-case Given a set of elements it is often useful to break them up or partition them into a number of separate nonoverlapping sets. In Computer science, the iterated logarithm of n, written log* n (usually read " log star " is the number of times the John Edward Hopcroft (born October 7, 1939) is a renowned theoretical Computer scientist. Jeffrey D Ullman (born November 22, 1942) is a renowned Computer scientist. Given a set of elements it is often useful to break them up or partition them into a number of separate nonoverlapping sets. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce A binary search algorithm (or binary chop) is a technique for locating a particular value in a sorted list of values A polylogarithmic function in n is a Polynomial in the Logarithm of n, a_k \log^k(n + \cdots + a_1 \log(n + a_0 The AKS primality test (also known as Agrawal-Kayal-Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving In Computer science, a k d-tree (short for k-dimensional tree) is a space-partitioning Data structure for organizing The word linear comes from the Latin word linearis, which means created by lines.
$O\left(n\log n\right)$	linearithmic, loglinear, or quasilinear	Sorting a list with heapsort, performing an FFT
$O\left({n^2}\right)$	quadratic	Sorting a list with insertion sort, multiplying two n-digit numbers by simple algorithm, adding of two n×n matrices. In Computer science, a linearithmic function ( Portmanteau of linear and logarithmic) is a function of the form n · Heapsort (method is a comparison-based Sorting algorithm, and is part of the Selection sort family A quadratic function, in Mathematics, is a Polynomial function of the form f(x=ax^2+bx+c \\! where a \ne 0 \\! Insertion sort is a simple Sorting algorithm, a Comparison sort in which the sorted array (or list is built one entry at a time
$O\left({n^3}\right)$	cubic	Multiplying two n×n matrices by simple algorithm. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve.
$O\left({n^c}\right), c>1$	polynomial, sometimes called algebraic	Finding the shortest path on a weighted digraph with the Floyd-Warshall algorithm
$L n [α, c],0 < α < 1$ $=e^{(c+o(1)) (\ln n)^\alpha (\ln \ln n)^{1-\alpha}}$	L-notation	Factoring a number using the special or general number field sieve
$O\left({c^n}\right)$	exponential, sometimes called geometric	Determining if two logical statements are equivalent using brute force; finding the (exact) solution to the traveling salesman problem using dynamic programming. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations The L - notation is often used to express the Computational complexity of certain Algorithms for difficult Number theory problems eg The special number field sieve (SNFS is a special-purpose Integer factorization algorithm In Mathematics, the general number field sieve (GNFS is the most efficient classical Algorithm known for factoring integers larger than 100 The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found The Travelling salesman problem ( TSP) in Operations research is a problem in discrete or Combinatorial optimization. In Mathematics and Computer science, dynamic programming is a method of solving problems exhibiting the properties of Overlapping subproblems and
$O\left(n!\right)$	factorial, sometimes called combinatorial	Determining if two logical statements are equivalent^[1]; solving the traveling salesman problem via brute-force search; finding the determinant of a matrix with expansion by minors. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k The Travelling salesman problem ( TSP) in Operations research is a problem in discrete or Combinatorial optimization. In Computer science, brute-force search or exhaustive search, also known as generate and test, is a trivial but very general problem-solving technique In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Linear algebra, the Laplace expansion of the Determinant ofan n × n square matrix B expresses the determinant
$O\left({n^n}\right)$	n to the n	Often used instead of $O\left(n!\right)$ to derive simpler formulas for asymptotic complexity.
$c_1^{O(c^n)}$	double exponential	Finding a complete set of associative-commutative unifiers^[2]

Not as common, but even larger growth is possible, such as the single-valued version of the Ackermann function, A(n,n). A double exponential function is a Constant raised to the power of an Exponential function. In Recursion theory, the Ackermann function or Ackermann-Péter function is a simple example of a general recursive function that is not primitive

For k>0 and any c>1, O(n^c log(n^k)) is subset of O(log(n^(c+a))) for any a>0. So O(n^c log(n^k)) may be considered as polynomial with some bigger order.

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

$f(n) = 9 \log n + 5 (\log n)^3 + 3n^2 + 2n^3 \in O(n^3)\,\!.$

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 (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 $c n$ 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.

$O (log n)$ 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. Exponentials with different bases, on the other hand, are not of the same order; assuming that they are is a common mistake. For example, $2 n$ and $3 n$ 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 $c 2 n 2$ , and the big O notation ignores the constant $c 2$ . This can be written as $c^2n^2 \in O(n^2)$ . If, however, an algorithm runs in the order of $2 n$ , replacing n with cn gives $2 c n = (2 c) n$ . This is not equivalent to $2 n$ (unless, of course, c=1).

Changing of variable may affect the order of the resulting algorithm. For example, if an algorithm runs in the order of O(n) of number of digits of the input number, it has order O(log n) of the input number.

Product

$f_1 \in O(g_1)\ \wedge\ f_2\in O(g_2)\, \implies f_1 f_2\in O(g_1 g_2)\,$

$f\cdot O(g) \in O(f g)$

Sum

$f_1 \in O(g_1)\ \wedge\ f_2\in O(g_2)\, \implies f_1 + f_2\in O(g_1 + g_2)\,$

This implies $f_1 \in O(g) \land f_2 \in O(g) \implies f_1+f_2 \in O(g)$ , which means that

O (g)

is a convex cone. In Linear algebra, a convex cone is a Subset of a Vector space that is closed under Linear combinations with positive coefficients

$f + O(g) \in O(f + g)$

Multiplication by a constant

Let $k \in \R_{>0}$ .

$O(k \cdot g) = O(g)$

$f\in O(g) \Rightarrow k\cdot f\in O(g)$

Related asymptotic notations

Big O is the most commonly used asymptotic notation for comparing functions, although in many cases Big O may be replaced with Θ for asymptotically tighter bounds (Theta, see below). Theta (uppercase Θ, lowercase θ or ϑ; Θήτα is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth Here, we define some related notations in terms of "big O":

Little-o notation

The relation $f(x) \in o(g(x))$ is read as " $f (x)$ is little-o of $g (x)$ ". Intuitively, it means that $g (x)$ grows much faster than $f (x)$ . It assumes that f and g are both functions of one variable. Formally, it states that the limit of $f (x) / g (x)$ is zero, as x approaches infinity. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" For algebraically defined functions $f (x)$ and $g (x)$ , $\lim_{x \to \infty}f(x)/g(x)$ is generally found using L'Hôpital's rule. In Calculus, l'Hôpital's rule (also called Bernoulli 's rule; often misspelled l'Hospital) uses Derivatives to help compute limits

For example,

$2x \in o(x^2) \,\!$
$2x^2 \not \in o(x^2)$
$1/x \in o(1)$

Little-o notation is common in mathematics but rarer in computer science. In computer science the variable (and function value) is most often a natural number. In math, the variable and function values are often real numbers. The following properties can be useful:

$o(f) + o(f) \subseteq o(f)$
$o(f)o(g) \subseteq o(fg)$
$o(o(f)) \subseteq o(f)$
$o(f) \subset O(f)$ (and thus the above properties apply with most combinations of o and O).

As with big O notation, the statement " $f (x)$ is $o (g (x))$ " is usually written as $f (x) = o (g (x))$ , which is a slight abuse of notation.

Other related notations

Notation	Intuition	Definition
$f(n) \in O(g(n))$	$f$ is bounded above by $g$ (up to constant factor) asymptotically	$\exists (C>0), n_0 : \forall(n>n_0) \; \|f(n)\| \leq \|Cg(n)\|$
$f(n) \in \Omega(g(n))$	$f$ is bounded below by $g$ (up to constant factor) asymptotically	$\exists (C>0), n_0 : \forall (n>n_0) \; \|Cg(n)\| \leq \|f(n)\|$
$f(n) \in \Theta(g(n))$	$f$ is bounded both above and below by $g$ asymptotically	$\exists (C,C'>0), n_0 : \forall (n>n_0) \; \|Cg(n)\| < \|f(n)\| < \|C'g(n)\|$
$f(n) \in o(g(n))$	$f$ is dominated by $g$ asymptotically	$\forall (C>0),\exists n_0 : \forall(n>n_0) \; \|f(n)\| < \|Cg(n)\|$
$f(n) \in \omega(g(n))$	$f$ dominates $g$ asymptotically	$\forall (C>0),\exists n_0 : \forall(n>n_0) \; \|Cg(n)\| < \|f(n)\|$
$f(n) \sim g(n)$	$f$ is equal to $g$ asymptotically	$\lim_{n \to \infty} \frac{f(n)}{g(n)} = 1$

(A mnemonic for these Greek letters is that "omicron" can be read "o-micron", i. e. , "o-small", whereas "omega" can be read "o-mega" or "o-big". )

Aside from big-O, the notations Θ and Ω are the two most often used in computer science; The lower-case ω is rarely used.

Another notation sometimes used in computer science is Õ (read soft-O). $f(n) = \tilde{O} (g(n))$ is shorthand for $f (n) = O (g (n)log k g (n))$ 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 $o (n ε)$ for any constant $k$ and any $ε > 0$ ).

The L notation, defined as

$L_n[\alpha, c] = O\left(e^{((c + o(1))(\ln n)^\alpha(\ln\ln n)^{1-\alpha})}\right)$ ,

is convenient for functions that are between polynomial and exponential. The L - notation is often used to express the Computational complexity of certain Algorithms for difficult Number theory problems eg

Multiple variables

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

To define Big O formally for multiple variables, suppose $f(\vec{x})$ and $g(\vec{x})$ are two functions defined on some subset of $\mathbb{R}^n$ . We say

$f(\vec{x})\mbox{ is }O(g(\vec{x}))\mbox{ as }\vec{x}\to\infty$

if and only if

$\exists \;C,\exists \;M>0\mbox{ such that } |f(\vec{x})| \le \; M |g(\vec{x})|\mbox{ for all }\vec{x} \mbox{ with } x_i>C \mbox{ for all }i.$

For example, the statement

$f(n,m) = n^2 + m^3 + \hbox{O}(n+m) \mbox{ as } n,m\to\infty$

asserts that there exist constants C and M such that

$\forall n, m>M: |g(n,m)| \le C(n+m).$

where g(n,m) is defined by

$\displaystyle f(n,m) = n^2 + m^3 + g(n,m).$

Note that this definition allows all of the coordinates of $\vec{x}$ to increase to infinity. ↔ In particular, the statement

$f(n,m) = \hbox{O}(n^m) \mbox{ as } n,m\to\infty$

(i. e. $\exists C \exists M \forall n \forall m...$ ) is quite different from

$\forall m: f(n,m) = \hbox{O}(n^m) \mbox{ as } n\to\infty$

(i. e. $\forall m \exists C \exists M \forall n...$ ).

Graph theory

It is often useful to bound the running time of graph algorithms. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. Unlike most other computational problems, for a graph G = (V, E) there are two relevant parameters describing the size of the input: the number |V| of vertices in the graph and the number |E| of edges in the graph. Inside asymptotic notation (and only there), it is common to use the symbols V and E, when someone really means |V| and |E|. An asymptote of a real-valued function y=f(x is a curve which describes the behavior of f as either x or y goes to infinity We adopt this convention here to simplify asymptotic functions and make them easily readable. The symbols V and E are never used inside asymptotic notation with their literal meaning, so this abuse of notation does not risk ambiguity. For example $O (E + V log V)$ means $O((E,V) \mapsto |E| + |V|\cdot\log|V|)$ for a suitable metric of graphs. Another common convention—referring to the values |V| and |E| by the names n and m, respectively—sidesteps this ambiguity.

Generalizations and related usages

The generalization to functions taking values in any normed vector space is straightforward (replacing absolute values by norms), where f and g need not take their values in the same space. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to A generalization to functions g taking values in any topological group is also possible. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the

The "limiting process" x→x_o can also be generalized by introducing an arbitrary filter base, i. In Mathematics, a filter is a special Subset of a Partially ordered set. e. to directed nets f and g. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics

The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions,

$f\sim g \iff (f-g) \in o(g)$

which is an equivalence relation and a more restrictive notion than the relationship "f is Θ(g)" from above. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" (It reduces to $\lim f/g = 1$ if f and g are positive real valued functions. ) For example, 2x is Θ(x), but 2x − x is not o(x).

External links

Introduction to Asymptotic Notations

Contents

Usage

Equals or member-of and other notational anomalies

Infinite asymptotics

Infinitesimal asymptotics

Formal definition

Example

Matters of notation

Orders of common functions

Properties

Product

Sum

Multiplication by a constant

Related asymptotic notations

Little-o notation

Other related notations

Multiple variables

Graph theory

Generalizations and related usages

See also

External links

References

Further reading