# A Time Complexity Question

What is the time complexity of following function fun()? Assume that log(x) returns log value in base 2.

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## C

`void` `fun()` `{` ` ` `int` `i, j;` ` ` `for` `(i = 1; i <= n; i++)` ` ` `for` `(j = 1; j <= ` `log` `(i); j++)` ` ` `printf` `(` `"GeeksforGeeks"` `);` `}` |

Time Complexity of the above function can be written as θ(log 1) + θ(log 2) + θ(log 3) + . . . . + θ(log n) which is θ(log n!)

Order of growth of ‘log n!’ and ‘n log n’ is same for large values of n, i.e., θ(log n!) = θ(n log n). So time complexity of fun() is θ(n log n).

The expression θ(log n!) = θ(n log n) can be easily derived from following Stirling’s approximation (or Stirling’s formula).

log n! = n*log n - n = O(n*log(n))

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

Sources:

http://en.wikipedia.org/wiki/Stirling%27s_approximation