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I know that the time complexity of quick sort in the worst case is $\Theta(n^2)$ and in the average case is $\Theta(n \log n)$. Can it be $\Theta(n\sqrt n)$ for certain inputs?

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    $\begingroup$ In the average case it's $\Theta(n\log n)$. It can probably be $\Theta(n\sqrt{n})$ for certain inputs of a certain structure. I suggest looking up the best and worst cases for quicksort, and trying to combine them. $\endgroup$ Feb 7, 2018 at 13:08

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Assume the array is sorted. If you always pick the element at n/2 as pivot then the time is O(n log n). If you always pick the first or last element it’s $O(n^2)$.

So which pivot gives you $O(n^{1.5})$?

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  • $\begingroup$ The third quarter? $\endgroup$
    – Ahmad
    Feb 8, 2018 at 5:54
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    $\begingroup$ Can you give a formal proof? $\endgroup$
    – Mr. Sigma.
    Feb 8, 2018 at 12:37
  • $\begingroup$ This answer is very much incomplete. And misleading too, for any constant $c$, if you always pick the element at $cn$ as your pivot the complexity is $\Theta(n \log n)$, and never $\Theta(n\sqrt n)$. $\endgroup$
    – orlp
    Sep 28, 2019 at 8:17
  • $\begingroup$ orlp, how is this misleading? Obviously picking the element at index cn for fixed 0 < c < n will not lead to Theta(n^(3/2)). So that is not the answer. But there is a simple answer. And finding that simple answers helps you understanding how likely / unlikely this case is. $\endgroup$
    – gnasher729
    Jun 16, 2022 at 15:18

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