The Bubble Sort Curve

A derivation of the curve that is approximated by a common visualization of the bubble sort diagram. Read the full proof on my site: https://linesthatconnect.github.io/blog/a-rigorous-derivation-of-the-bubble-sort-curve/ After uploading, I learned that this result appeared in the paper "Nonparametric testing via partial sorting" by Krzysztof Bisewski, H. M. Jansen, and Yoni Nazarathy. Read it here: https://arxiv.org/abs/2210.14546 The viral sorting algorithm video which first sparked my interest: https://www.youtube.com/watch?v=kPRA0W1kECg The animations in this video were created using Manim: https://www.manim.community/ Music credits: Fluidscape by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/ Night Music by Kevin Macleod river - Calm and Relaxing Piano Music by HarumachiMusic ... And a couple of my own songs: The Fog: https://soundcloud.com/lines-that-connect/the-fog Heavy Head, Light Rain: https://soundcloud.com/lines-that-connect/heavy-head-light-rain Thanks For Watching: https://soundcloud.com/lines-that-connect/thanks-for-watching Chapters: 00:00 Intro: 0:37 Laying the Background 3:20 How Bubble Sort Works 6:59 Mathematically Describing Diagrams 9:13 Stretching the Diagrams 11:52 Visual Derivation 14:38 Symbolic Derivation 16:48 Nice! 17:07 A Rigorous Solution