Sorting Discrete i.i.d. Inputs: Quicksort is Optimal.

We prove the Sedgewick-Bentley conjecture on median-of-$k$ Quicksort on equal keys: The average number of comparisons for Quicksort with fat-pivot (a.k.a. three-way) partitioning is asymptotically only a constant times worse than the information-theoretic lower bound for sorting $n$ i.i.d. elements, and that constant converges 1 as $k to infty$. Hence, Quicksort with pivot sampling is an optimal distribution-sensitive algorithm for the i.i.d. sorting problem.