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:

    “The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    “He was thoughtful and grave—but the orders he gave
    Were enough to bewilder a crew.
    When he cried “Steer to starboard, but keep her head larboard!”
    What on earth was the helmsman to do?”
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    “That we can come here today and in the presence of thousands and tens of thousands of the survivors of the gallant army of Northern Virginia and their descendants, establish such an enduring monument by their hospitable welcome and acclaim, is conclusive proof of the uniting of the sections, and a universal confession that all that was done was well done, that the battle had to be fought, that the sections had to be tried, but that in the end, the result has inured to the common benefit of all.”
    —William Howard Taft (1857–1930)

    “Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
    —Friedrich Nietzsche (1844–1900)