I'm studying a variation of Quicksort in which the algorithm samples a subarray of size $f(n)< n$ ($n$ is the size of the input array) and then chooses the pivot from this subarray. The pivot is accepted if it lies in the middle third of the sampled array, otherwise it is rejected and the subarray is sampled again until the pivot is accepted. Is there any reference in which this variation has been studied?
Sebastian Wyld from the University of Liverpool has been extensively working on the analysis of various versions of QuickSort: His 2016 thesis, "Dual-Pivot Quicksort and Beyond: Analysis of Multiway Partitioning and Its Practical Potential", explored the analysis of the complexity of "Multiway Quicksort, i.e., partitioning the input in one step around several pivots" and includes
- an extensive survey of such variants,
- an average case analysis of the generic one in terms of both
- the number of comparisons and
- the number of accesses to external memory, and
- some intricate results about the case where keys can be repeated.
Sebastian is very kind, and if you have questions about his work (after studying it seriously), I do believe that he would answer any question you may have.
I hope it helps!