# Problem Solving: Order of complexity

 ← Maths for big-O notation Order of complexity Limits of computation →

## Order of Complexity

Notation Name Example
$O(1)\,$  constant Determining if a number is even or odd; using a constant-size lookup table
$O(\log n)\,$  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.
$O(n)\,$  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.
$O(n\log n)=O(\log n!)\,$  linearithmic, loglinear, or quasilinear Performing a Fast Fourier transform; heapsort, quicksort (best and average case), or merge sort
$O(n^{2})\,$  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
$O(n^{c}),\;c>1$  polynomial or algebraic Tree-adjoining grammar parsing; maximum matching for bipartite graphs
$O(c^{n}),\;c>1$  exponential Finding the (exact) solution to the travelling salesman problem using dynamic programming; determining if two logical statements are equivalent using brute-force search
$O(n!)\,$  factorial Solving the travelling salesman problem via brute-force search; generating all unrestricted permutations of a poset; finding the determinant with expansion by minors.