What is the Time Complexity for In-Place Quick sort?

There exists QuickSort implementations that runs on O(nlogn) worst-case, and as for your question, there is no better then O(nlogn) worst case comparison based sort, and as quick sort is 1, it is proved that O(nlogn) is cant be beaten anyway.

There is not a non-inplace version of quicksort. Since it is a recursive divide-et-impera algorithm is easy to show that as you say space complexity is at least O(log n). It can be less as noted if you use the iterative implementation, space compleity will be O(1).

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