Q:

Is my analysis for identifying the best sorting algorithm to solve this task correct?

This was an interview question and I am wondering if my analysis was correct:
A 'magic select' function basically generates the 'mth' smallest value in an array that has a size of n. The task was to sort the 'm' elements in ascending order using an efficient algorithm. My analysis was to first use the 'magic select' function to get the 'mth' smallest value. I then used a partition function to sort of create a pivot to get all smaller elements on the left. After that point, I felt that a bucket sort should accomplish the task of sorting the left half efficiently. 
I was just wondering if this was the best way to sort the 'm' smallest elements. I see the possibility of a quick sort being used here too. However, I thought that avoiding a comparison based sorting could lead to an O(n). Could radix sort or heap sort (O(nlogn)) be used for this too? If I didn't do it in the best possible way, which could be the best possible way to accomplish this? An array was the data structure I was allowed to use.
Many thanks!

A:

I'm pretty sure you can't do any better than any standard algorithm for selecting the k lowest elements out of an array in sorted order. The time complexity of your "magic machine" is O(n), which is the same time complexity you'd get from a standard selection algorithm like the median-of-medians algorithm or quickselect.
Consequently, your approaches seem very reasonable. I doubt you can do any better asymptotically.
Hope this helps!

