Lucky Cars and the Quicksort Algorithm

TL;DR

This paper reveals a surprising connection between the average comparisons in Quicksort and parking preferences of cars, showing that both counts follow the same recurrence relation and enumerate the same combinatorial structures.

Contribution

It establishes a novel link between Quicksort comparison counts and parking preference arrangements, providing new combinatorial insights.

Findings

Quicksort comparison counts match parking preference distributions.

Both counts satisfy the same recurrence relation.

The enumeration involves exactly one less lucky car in parking scenarios.

Abstract

Quicksort is a classical divide-and-conquer sorting algorithm. It is a comparison sort that makes an average of $2(n+1)H_n - 4n$ comparisons on an array of size $n$ ordered uniformly at random, where $H_n = \sum_{i=1}^n\frac{1}{i}$ is the $n$th harmonic number. Therefore, it makes $n!\left[2(n+1)H_n - 4n\right]$ comparisons to sort all possible orderings of the array. In this article, we prove that this count also enumerates the parking preference lists of $n$ cars parking on a one-way street with $n$ parking spots resulting in exactly $n-1$ lucky cars (i.e., cars that park in their preferred spot). For $n\geq 2$, both counts satisfy the second order recurrence relation $ f_n=2nf_{n-1}-n(n-1)f_{n-2}+2(n-1)! $ with $f_0=f_1=0$.