Quicksort Is Optimal For Many Equal Keys

TL;DR

This paper proves that median-of-$k$ Quicksort with three-way partitioning is asymptotically optimal for sorting multisets with many duplicates, resolving a long-standing conjecture and advancing the analysis of Quicksort's efficiency with equal elements.

Contribution

It establishes the asymptotic optimality of median-of-$k$ Quicksort for multisets with many duplicates, confirming a conjecture by Sedgewick and Bentley.

Findings

Average comparisons are within a constant factor of the lower bound for many duplicates.

The constant factor approaches 1 as $k$ increases, indicating near-optimality.

First progress on Quicksort analysis with equal elements since 1977.

Abstract

I prove that the average number of comparisons for median-of-$k$ Quicksort (with fat-pivot a.k.a. three-way partitioning) is asymptotically only a constant $\alpha_k$ times worse than the lower bound for sorting random multisets with $\Omega(n^\varepsilon)$ duplicates of each value (for any $\varepsilon>0$). The constant is $\alpha_k = \ln(2) / \bigl(H_{k+1}-H_{(k+1)/2} \bigr)$, which converges to 1 as $k\to\infty$, so Quicksort is asymptotically optimal for inputs with many duplicates. This resolves a conjecture by Sedgewick and Bentley (1999, 2002) and constitutes the first progress on the analysis of Quicksort with equal elements since Sedgewick's 1977 article.