  QuickXsort is a highly efficient in-place sequential sorting scheme that
mixes Hoare's Quicksort algorithm with X, where X can be chosen from a wider
range of other known sorting algorithms, like Heapsort, Insertionsort and
Mergesort. Its major advantage is that QuickXsort can be in-place even if X is
not. In this work we provide general transfer theorems expressing the number of
comparisons of QuickXsort in terms of the number of comparisons of X. More
specifically, if pivots are chosen as medians of (not too fast) growing size
samples, the average number of comparisons of QuickXsort and X differ only by
$o(n)$-terms. For median-of-$k$ pivot selection for some constant $k$, the
difference is a linear term whose coefficient we compute precisely. For
instance, median-of-three QuickMergesort uses at most $n \lg n - 0.8358n +
O(\log n)$ comparisons.
  Furthermore, we examine the possibility of sorting base cases with some other
algorithm using even less comparisons. By doing so the average-case number of
comparisons can be reduced down to $n \lg n- 1.4106n + o(n)$ for a remaining
gap of only $0.0321n$ comparisons to the known lower bound (while using only
$O(\log n)$ additional space and $O(n \log n)$ time overall).
  Implementations of these sorting strategies show that the algorithms
challenge well-established library implementations like Musser's Introsort.
