Table of common time complexities
Further information: Computational complexity of mathematical operations
The following table summarises some classes of commonly encountered time complexities. In
the table, poly(x) = xO(1), i.e., polynomial in x.
Name
Complexity
class
constant time
Running
Examples of
time (T(n)) running times
O(1)
inverse
Ackermann time
iterated
logarithmic time
10
O(α(n))
O(log* n)
log-logarithmic
logarithmic time DLOGTIME O(log n)
log n, log(n2)
polylogarithmic
poly(log n)
(log n)2
time
O(nc) where 0 1/2 2/3
n ,n
fractional power
<c<1
cubic time
polynomial time
P
quasi-polynomial
QP
time
sub-exponential
time
SUBEXP
Amortized time per operation
using a bounded priority
queue[1]
Binary search
Searching in a kd-tree
Finding the smallest item in an
unsorted array
Seidel's polygon triangulation
O(n log* n)
algorithm.
O(n log n)
n log n, log n! Fastest possible comparison sort
2
O(n )
n2
Bubble sort; Insertion sort
Naive multiplication of two n×n
3
3
O(n )
n
matrices. Calculating partial
correlation.
Karmarkar's algorithm for
2O(log n) =
n, n log n, n10 linear programming; AKS
poly(n)
primality test
Best-known O(log2 n)poly(log n)
log log n
log n
2
n
,n
approximation algorithm for the
directed Steiner tree problem.
Assuming complexity theoretic
O(2nε) for all
O(2log nlog log n)
conjectures, BPP is contained in
ε>0
O(n)
"n log star n"
time
linearithmic time
quadratic time
Determining if an integer
(represented in binary) is even
or odd
Amortized time per operation
using a disjoint set
Distributed coloring of cycles
O(log log n)
linear time
Example algorithms
n
1 http://en.wikipedia.org/wiki/Time_complexity#Table_of_common_time_complexities (first definition)
sub-exponential
time
(second
definition)
SUBEXP.[2]
o(n)
2
O(n)
2
Best-known algorithm for
integer factorization and graph
isomorphism
n1/3
n
exponential time E
2
1.1 , 10
factorial time
O(n!)
n!
exponential time EXPTIME
2poly(n)
2n, 2n2
n
double
2-EXPTIME 22poly(n)
exponential time
23n
Solving the traveling salesman
problem using dynamic
programming
Solving the traveling salesman
problem via brute-force search
Deciding the truth of a given
statement in Presburger
arithmetic
2 http://en.wikipedia.org/wiki/Time_complexity#Table_of_common_time_complexities