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:

    “There are nine orders of angels, to wit, angels, archangels, virtues, powers, principalities, dominations, thrones, cherubim, and seraphim.”
    —Gregory the Great, Pope (c. 540–604)

    “One cannot be a good historian of the outward, visible world without giving some thought to the hidden, private life of ordinary people; and on the other hand one cannot be a good historian of this inner life without taking into account outward events where these are relevant. They are two orders of fact which reflect each other, which are always linked and which sometimes provoke each other.”
    —Victor Hugo (1802–1885)

    “If we use common words on a great occasion, they are the more striking, because they are felt at once to have a particular meaning, like old banners, or everyday clothes, hung up in a sacred place.”
    —George Eliot [Mary Ann (or Marian)

    “Let us stop being afraid. Of our own thoughts, our own minds. Of madness, our own or others’. Stop being afraid of the mind itself, its astonishing functions and fandangos, its complications and simplifications, the wonderful operation of its machinery—more wonderful because it is not machinery at all or predictable.”
    —Kate Millett (b. 1934)