# A local limit theorem for Quicksort key comparisons via multi-round smoothing

## TL;DR

This paper proves a full local limit theorem for the distribution of key comparisons in Quicksort using a novel multi-round smoothing technique, building on prior global and convergence results.

## Contribution

It introduces a multi-round smoothing method to establish the local limit theorem for Quicksort key comparisons, advancing the understanding of its distributional properties.

## Key findings

- Proves the full local limit theorem for Quicksort comparisons
- Develops a new multi-round smoothing technique
- Complements previous global distribution results

## Abstract

As proved by R\'egnier and R\"osler, the number of key comparisons required by the randomized sorting algorithm QuickSort to sort a list of $n$ distinct items (keys) satisfies a global distributional limit theorem. Fill and Janson proved results about the limiting distribution and the rate of convergence, and used these to prove a result part way towards a corresponding local limit theorem. In this paper we use a multi-round smoothing technique to prove the full local limit theorem.