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A Time Complexity Question
• Difficulty Level : Easy
• Last Updated : 29 Oct, 2020

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

## 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

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