Variance of the number of Comparisons of Randomized Quicksort

TL;DR

This paper provides a clear, self-contained proof of the variance formula for the number of comparisons in randomized Quicksort, aiming to aid understanding despite the result being previously known.

Contribution

It offers a detailed, self-contained proof of the variance of comparisons in randomized Quicksort, filling a gap in accessible explanations.

Findings

Explicit variance formula for randomized Quicksort comparisons

Detailed proof accessible to students and researchers

Clarification of known theoretical result

Abstract

This paper gives a straightforward self-contained proof of the formula for the variance of the number of comparisons used by the Quicksort sorting algorithm when pivots are chosen uniformly at random. The result has been known for some time but we had not found a single source giving all details of the argument in a self-contained and explicit way. We hope these details may be of interest to students and others wishing to understand this calculation. No originality is claimed for the result.