Maximum size of input according to the time complexity and maximum number of operations

This table gives the maximum input size $n$ you can use with an algorithm of a given complexity according to the maximum number of operations/units of time allowed.

This of course does not take into account any leading coefficients or more complicated expressions of the time complexity, but it gives a rough estimate.

For instance, if an $\mathcal{O}(n\log n)$ algorithm is used with a maximum number of operations of 1,000,000, the input should not be greater than 88,000.

For every computation, the chosen base of the logarithm is $e$.

Complexity	Example	$10^6$	$10^9$	$10^{12}$
$\mathcal{O}(\log\log n)$	Interpolation search	$10^{1.3\cdot 10^{4.3\cdot 10^5}}$	$10^{3.5\cdot 10^{4.3\cdot 10^{8}}}$	$10^{9.4\cdot 10^{4.3\cdot 10^{11}}}$
$\mathcal{O}(\log n)$	Binary search	$10^{4.3\cdot 10^5}$	$10^{4.3\cdot 10^{8}}$	$10^{4.3\cdot 10^{11}}$
$\mathcal{O}(n^{1/3})$		$10^{18}$	$10^{24}$	$10^{36}$
$\mathcal{O}(\sqrt n)$	Divisors of $n$	$10^{12}$	$10^{18}$	$10^{24}$
$\mathcal{O}(n)$	Array scan	$10^6$	$10^9$	$10^{12}$
$\mathcal{O}(n\log\log n)$	Sieve of Erastothenes	$3.9\cdot 10^5$	$3.4\cdot 10^8$	$3.1\cdot 10^{11}$
$\mathcal{O}(n\log n)$	Quick sort	$8.8\cdot 10^4$	$5.6\cdot 10^7$	$4.1\cdot 10^{10}$
$\mathcal{O}(n\log n \log\log n)$	Schönhage–Strassen's algorithm for multiplication	$4.0\cdot 10^4$	$2.1\cdot 10^7$	$1.4\cdot 10^{10}$
$\mathcal{O}(n(\log n)^2)$	Shell sort	$1.1\cdot 10^4$	$4.3\cdot 10^6$	$2.2\cdot 10^9$
$\mathcal{O}(n\sqrt n)$	Divisors $\forall k\leq n$	$10^4$	$10^6$	$10^9$
$\mathcal{O}(n^{\log_2 3})$	Karatsuba's algorithm for multiplication	$6103$	$4.8\cdot 10^5$	$3.7\cdot 10^7$
$\mathcal{O}(n^2)$	$n$ x $n$ grid scan	$1000$	$3.2\cdot 10^4$	$10^6$
$\mathcal{O}(n^2\log n)$		$407$	$10^4$	$2.8\cdot 10^5$
$\mathcal{O}(n^2\sqrt n)$		$251$	$3980$	$6.3\cdot 10^4$
$\mathcal{O}(n^{\log_{1.5} 3})$	Stooge sort	$164$	$2097$	$2.7\cdot 10^4$
$\mathcal{O}(n^{\log_2 7})$	Strassen's algorithm for matrix multiplication	$137$	$1600$	$1.9\cdot 10^4$
$\mathcal{O}(n^3)$	Naive matrix multiplication	$100$	$1000$	$10^4$
$\mathcal{O}(n^{\log_2 n})$	Slowsort	$22$	$44$	$80$
$\mathcal{O}(2^n)$	Power set	$20$	$30$	$40$
$\mathcal{O}(n\cdot 2^n)$		$16$	$25$	$35$
$\mathcal{O}(n^2\cdot 2^n)$	TSP via dynamic programming	$13$	$21$	$30$
$\mathcal{O}(e^n)$		$14$	$21$	$28$
$\mathcal{O}(10^n)$		$6$	$9$	$12$
$\mathcal{O}(n!)$	Naive TSP	$9$	$12$	$14$
$\mathcal{O}(n^n)$		$7$	$9$	$11$