Big O Notation - Orders of Common Functions

Orders of Common Functions

Further information: Time complexity#Table of common time complexities

Here is a list of classes of functions that are commonly encountered when analyzing the running time of an algorithm. In each case, c is a constant and n increases without bound. The slower-growing functions are generally listed first.

Notation Name Example
constant Determining if a number is even or odd; using a constant-size lookup table
double logarithmic Finding an item using interpolation search in a sorted array of uniformly distributed values.
logarithmic Finding an item in a sorted array with a binary search or a balanced search tree as well as all operations in a Binomial heap.
fractional power Searching in a kd-tree
linear Finding an item in an unsorted list or a malformed tree (worst case) or in an unsorted array; Adding two n-bit integers by ripple carry.
n log-star n Performing triangulation of a simple polygon using Seidel's algorithm. (Note log^*(n) = \begin{cases} 0, & \text{if }n \leq 1 \\ 1 + \log^*(\log n), & \text{if }n>1 \end{cases}
linearithmic, loglinear, or quasilinear Performing a Fast Fourier transform; heapsort, quicksort (best and average case), or merge sort
quadratic Multiplying two n-digit numbers by a simple algorithm; bubble sort (worst case or naive implementation), Shell sort, quicksort (worst case), selection sort or insertion sort
polynomial or algebraic Tree-adjoining grammar parsing; maximum matching for bipartite graphs

L-notation or sub-exponential Factoring a number using the quadratic sieve or number field sieve
exponential Finding the (exact) solution to the travelling salesman problem using dynamic programming; determining if two logical statements are equivalent using brute-force search
factorial Solving the traveling salesman problem via brute-force search; generating all unrestricted permutations of a poset; finding the determinant with expansion by minors.

The statement is sometimes weakened to to derive simpler formulas for asymptotic complexity. For any and, is a subset of for any, so may be considered as a polynomial with some bigger order.

Read more about this topic:  Big O Notation

Famous quotes containing the words orders of, orders, common and/or functions:

    “Your money’s no good here. Orders of the house.”
    —Stanley Kubrick (b. 1928)

    “God is a foreman with certain definite views
    Who orders life in shifts of work and leisure.”
    —Seamus Heaney (b. 1939)

    “Whenever there are in any country uncultivated lands and unemployed poor, it is clear that the laws of property have been so far extended as to violate natural right. The earth is given as a common stock for man to labor and live on.... The small landowners are the most precious part of a state.”
    —Thomas Jefferson (1743–1826)

    “If photography is allowed to stand in for art in some of its functions it will soon supplant or corrupt it completely thanks to the natural support it will find in the stupidity of the multitude. It must return to its real task, which is to be the servant of the sciences and the arts, but the very humble servant, like printing and shorthand which have neither created nor supplanted literature.”
    —Charles Baudelaire (1821–1867)