Remember the days when everyone and their pet iguana was raving about Arch Linux? You couldn't escape the ever-so-subtle "I use Arch BTW" remarks in every Linux forum. Well, move over, Arch, because NixOS is here to steal your thunder! Nowadays, it seems that you can't browse YouTube or read a blog without stumbling upon someone extolling the virtues of NixOS and how it is the epitome of computing perfection. But hey, who needs critical analysis when we can jump on the hype train and declare NixOS as the new Arch? Because that's exactly what's going on. NixOS has now become the self-proclaimed prodigy that's poised to dethrone Arch Linux as the holy grail of Linux distributions. The time is calling, my friends! It's time for you – the seasoned Linux enthusiast – to dust off your keyboard warrior capes and embark on a new crusade. So, grab your Tux plushie (or, your pitchforks if you belong to the world of devils) and let's embark on an adventure through the enigmatic world of NixOS (and let the memes commence)!

Guess who's back? Back again?... for the rest, go listen to the fucking song! I am not here to sing songs for you. Anyroad, what's up? Actually, no one cares... so, let's get started! Remember back in the day, when you visited your favourite forum and it would say something like "$N$ users online: $(n)$ members, $(x)$ guests" (of course $ {n, x} \in \mathbb{N} $)? This is called user presence. It's a simple way to show your users that they are not alone on your website. It's also a great way to show off your mad skills to your friends. So, let's get started!

Before we start, I would like to discuss your options to track user presence on your website. There are multiple ways to do this:

WebSockets: This is the most modern and probably the most efficient way to do this. WebSockets will provide you with a persistent, bidirectional communication channel between your server and the client. Talking in a broad perspective, you would set up a WebSocket server that listens for incoming connections and handles WebSocket events, and when a client establishes a WebSocket connection, register their presence by storing relevant data (e.g., user ID, session ID) on the server. Then you can implement mechanisms to track user activity, such as sending heartbeats or receiving client-initiated events. 

Are you back or are you new here? If you're back, why'd you even come back? If you're new, you are in for hell. People who are back, you know what's coming. People who are new, you don't know what's coming. But you will. You will. Last time, we discussed "Why Comparison-based Sorting Algorithms have $\Omega(n log n)$ Lower Bound" and this is a follow-up to that. Also, if you haven't read that, go read the previous article first, then come back here and read this. Alright, ready? Let's go.

Today, we will discuss average-case lower bounds for comparison-based sorting algorithms. Now, I don't expect your little brain to remember everything you were spoon-fed last time, so I'll give you a quick recap. As expected, our focus in this article, once again, is comparison-based sorting Algorithms. In our last article, we were able to define a comparison-based sorting algorithm.

Also, we were able to prove that any comparison-based sorting algorithm must take $\Omega(n log n)$ time in the worst case. And, we also defined a theorem for this proof. Do you remember what it was? Of course, you don't. Here's the theorem: