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:

Guess who's back? Back again?... for the rest, go listen to the frikkin' 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. 

Alright, folks! I know I keep this weblog very personal and there's art flowing all around this website, but let's talk some mathematics today! Specifically, we are here to discuss the notion of lower bounds for sorting algorithms. Now, when I say sorting algorithms, I am talking about comparison-based sorting algorithms. There are other sorting algorithms like counting sort, radix sort, bucket sort, etc. but they are a topic for another day. Now, buckle up for a long text-based post and a vomit load of mathematics, because, by the end of this article, we are going to show that any deterministic comparison-based sorting algorithm must take $\Omega(n \log n)$ time to sort an array of n elements in the worst case.

You started reading this article, and reached this point and wondered, "Wait, we are discussing the notion of lower bounds"? "What is a lower bound"? ... "To think that, what the fuck even is a bound"? Well, if you are pondering over that question, well ponder no more! I promised you a shit-ton of information and a butt-load of theory, so here we go!