Why Is Merge Sort O(n * log(n))? The Really Really Long Answer. 

Table of Contents (If you watch this all, you will know why Merge Sort is upper bounded by O(n * log(n))) What This Video Will Cover 0:00 - 0:40 The Fundamental Subroutines 0:40 - 1:32 The Split Subroutine 1:32 - 8:35 The Split Subroutine In Code 8:35 - 11:05 [ Old Clip ] Merge Sort Execution Trace 11:05 - 14:38 The Merge Subroutine 14:38 - 16:54 The Recurrence Relation 16:54 - 21:45 Investigation: What Work Is Done Per Level? 21:45 - 27:10 Getting Exact: Solving For Total Work Across All Levels 27:10 - 30:50 My Whole Life Has Been A Lie 30:50 - 34:51 Checking Our Work 34:51 - 35:48 Wrap Up 35:48 - 36:30 The code for Merge Sort is in the description, fully commented for understanding.

The patience you had to go through every literal step, god damn, much appreciated (once again) man.

Bro, you should be the one teaching me algorithms instead of my college professor, good job!

In my 4 years of doing a Computer Science degree I never had a lecturer explain it this well

How COME a person can be this talented at teaching? Thank you a million times!

Hey man, just wanted to add a +1 to the supportive comments. I absolutely love how you do not jump over steps assuming people know what's happening. I have a brain that needs to know why every step is included even if some might find it common sense, so this is wonderful for me. I subscribed to you because I think you deserve it. :) Have a wonderful day!

For those getting stuck at [29:09], there is a 2^i before the parenthesis, but it might not be visible due to board reflections. Thank you for the video as it helped a lot to understand how its complexity is calculated for the first time.

Also found your videos through Leetcode. As someone without a traditional CS background, you have a talent for explaining things in a very clear way. I imagine making these videos must be exhausting, but I do hope you continue. Even if that means one per month :)

Coming from a non-computer science major, real understanding sorting part was always a real pain. (Even with my experience in working as a sw developer). Just after watching this video for about 5 mins, lightening struck my head. I've paid more than a few hundred bucks on algorithm + data structure courses, and was never satisfied once. Your explanation, though it is free, is by far the best, and the only one I was completely satisfied with. No, it's not just satisfaction, but the best I have ever encountered in my life. Thank you once again. Truly appreciate your great work.

^ THIS is how to teach The way most professors teach is as if they are giving a refresher of something the students have already learned rigorously. They should watch this video and take notes on how to teach.

literally the best explanation i've ever gotten. i finally get it. i go to a top university but i regret wasting my money when your videos teach me better than my professors

You are literally only youtuber who gives deep intuition behind this concept, you are giving us feeling like we could invent this by own, much respecttttt

You have just converted a process of time complexity of O(n ^ 1000000) to O(log n) in this video with your awesome teaching talent. Really like the way you teach with immense patience and dedication.

this is the stuff i was trying to find about mergesort, not just the how, but also the why, everyone else seems to think teaching splitting and merging functions is enough, but until u realize how the process is unfolding practically u never truely "realize" the essense of mergesort. The thing with recursive algorithms is that they seem very easy and obvious ,until u start thinking how the computer actually handles all the states,calls,order of calls,variables etc, thats probably why everyone tries to talk about recursion on a superficial level. This video is so great btw .Very in-depth.

The. Best. Explanation. Ever. I have months confused about what it actually meant for it to be nlogn, and it all makes so so much sense now. Thank you a million times. Your videos are THE best. Much appreciated.

This material is gold and it gives you an order of solving leetcode problems to understand merge sort in dept. Kudos to the creator !

This man is an absolute gold mine. Thank you so much, this is the clearest explanation of _anything_ I have ever seen.

You are a legend in teaching. Pure and simple explanation with a very patient attitude. It's clear that you don't do the videos to be done, you really want to transfer your knowledge, and you do it the best. I'm happy that i find you. Keep up the good work brother!

I found this channel through your comment in LeetCode, and I'm so glad I was able to find your channel before my tech interview. THANK YOU SO MUCH. KEEP IT GOING

This is the best explanation of the merge sort on youtube.

This makes me furious and frustrated that professors get paid too much to not want to teach or to teach poorly. You are a life saver and an example of how professors SHOULD BE teaching for what we are paying them for.

You have taken the amount of time for me to understand merge sort from O(n!) to O(n*log(n)), thank you

I find this to be incredibly valuable during a term where the only contact I get with my teachers is through zoom. I managed to get through my first computer science classes online with little to no issues but now that I am taking data structures I am finding that my assignments are taking 15+ hours and I think this is because of not getting to have those one on ones with my teacher. The way you have put together these videos has answered many questions that I didn't even know that I had. Thank you!!!

You need way more recognition man As a person who do programming, i always curios about the concrete math model behind what is commonly told I literally watch your video to find out how exactly the time complexity got deduced And man it was a damn good video Keep it up man We do need more curiosity people like you

I watched the whole video at night and did not even feel asleep! So interesting!Thanks for sharing the knowledge😄

why aren't your videos not mostly watched ? This explanation is strongly etched into my mind .Even if forget in future how did it get that time complexity ,you gave me confidence to derive the whole process .Thank you

First time I ever saw ANYBODY even attempt to explain how we arrive at log n or n log n or all these math conclusions. And I mean even in texts supposedly written to help you understand algorithms. Most times the lecturers/writers just rush to the final result. Way to go brother!!

I've seen (and read) lots of explanations with complexity analysis of merge sort, but this is the one that clicked for me! Thanks for this wonderful explanation.

i'm graduated from uni, but my data structures and algorithms professor sucked. this section of CS has been the bane of my existence for a long time now, and i've really appreciated this video. thanks so much. i needed a long explanation like this.

i mean WOW ! i was honestly struggling with my algorithm class this semester especially since its on zoom and the professor is all over the place and nothing made sense !!!!! but in just 36 minutes i understood what my professor tried to teach in 4 hours !!!!! so thank you so much ure a life saver ! would love to donate

I finally know why the time complexity of merge sort is nlog(n) now.....Thanks a lot!

This is the best analysis out there, you leave me in awe with your explanations time and again. Really waiting for new videos! This was the only one I had left by mistake, sad that there's no fresh content coming from you on youtube. Anyways, thanks a ton for all that you are providing to the community!

Thank you so much, I've never understood the analysis of Sorting Algorithms correctly but you gave a remarkable explanation on all my doubts on it.

After a couple of days and a lot of videos, finally I found a video that really help to understand the time complexity of merge sort. thank you !

Holy Molly, you rocks, this is the best explanation for a student, which I can find on the internet!

I love how you broke this down to the fundamentals. So helpful, I smashed the subscribe button so hard my clicker nearly exploded

Terrific work, hats off to your way of explanation. I follow tons of youtuber. Nobody isn't close to your explanation approach or skill. Keep up the awesome work!

I thought I'd skip it once I feel lost, but it never happened, great job. Wish u more subscribers

My god, this is fantastic. I have ADHD but you present this in such a thorough and interesting way, it's not problem paying attention. Thanks so much!

Thanks for you explanations. I am a student from germany. The content was also described in the script of my studies but your way to fill up this content with energy and a impressive way of didactics let me understand the topic. You have a sensitiv way to teach.

No one: Lecturers(not everyone): Me: This is how to put your effort in exemplifying down to the basics for the sake of learning!

That's the best explanation for any beginner to understand.

To the point well explained, couldn,,,'t imagine how 36 min goes so quickly

This forest explanation is the best recursion call stack explanation of an algorithm on internet!

This is literally by far the best explanation I have ever seen. Good going!

bro, yore easily better than all my college teachers at explaining this, hell yeah im subscribing

No other video on the internet can describe this any better, huge respect for you

If I could like this video 10000 times, I would.. Thank you so much for the clear explanation !

Really appreciate your patience and clear presentation. I understand why it's O(nlog n) after watching this video. In addition, I really love your clear pronunciation! Really really appreciate cus you do better than my college professor!

Hi guys, took me some time to understand at the end how the nlogn was extracted by the Σn. its very simple, because 'i' is not there in the sum(Σn) you Just adding n, logn times(logn=times you split the problem or levels). So here the serie has 3 terms but he start to count from 0 to logn - 1, so he has logn sums of n, or n + n +n =3*n=log(8)*n =n*logn im new to CS and try to Learn algorithms, great video man thank you very much!!

You have a great talent for explaining these concepts. I wish this content would have been available last year when I was taking my algorithms class!

My ivy-league college professor couldn't help me truly comprehend this time complexity, but you did it. I hope they fire that professor and hire you instead.

A true masterpiece of a video and a true masterpiece of a man. You are such a good teacher and I'm glad I can learn my University course of you! :)

thank you for this. what you are doing is really helping people. i think you know that, but still want to help confirm your work. thank you

I got to say that you are super AMAZING!!! My professor just skipped the total process.... like T(8)=T(n/2)+T(n/2)+(n-1) then bombed Worst case O(nlogn) You completely fill out all my gap about why that is O(nlogn)

Doing the math here was one of the high points of my review of computer science. One of the best videos because of the deeper understanding that is so effectively presented.

This is sooo informative. I finally understood Merge Sort and most importantly, why it has O(NlogN) time complexity. thank you so much ! :)

I feel that my brain has expanded 10 fold by watching your video. Thank you!

So much effort in a video for merge sort

you are amazing!! I'm planning to watch all your videos about sorts!!! Thank you so much for making this video. I can see the amount of effort that was put in to it!!!

Each sentence was a little incoherent individually, but as a whole this was a pretty clear explanation. Definitely cleared it up for me.

Awesome bro! I have never seen such a good explanation ever!

this was amazing, thank you so much. This video made me fall even deeply in love with CS and math!!

Great job. Appreciate the Big-O complexity details!

I think I love you.... Or at least your explanations and fantastic teaching skills xD Please keep up the fantastic work. Very high-quality and helpful, unlike most resources I have looked at previously! I also appreciate you including the table of comments with the times. Subscribed :)

Great explanation. I got lost towards the last 6 minutes once the distribution started and couldn't follow anything after we arrived at n log n. :/ But will return to watch again :) great break down. Thanks.

i would say you still need to watch few other videos to understand the concept of merge sort and the code as well, however he did great job explaining some concepts in this video, its amazingly helpful

I always had tough time understanding time complexities but now it's all clear after watching your videos❤️

You've taken something that twisted my head to epiphany. Thank you so much!

Thank you, it was so didactic. I loved your explanation and could really understand what was going on.

Thank you for your videos, they helped me a lot for my DSA exam

Very clear and detailed. Great job.

So clear, bro! Wonderful job, cannot wait to subscribe!

this is the best explaination ever to anything!!

This channel is a goldmine 👌 Superb explanation

I thought your explanation of the big o complexity was superb. I really liked you worked through the proof and then did the mathematical proof and showed that using the formula got the right answer. Definitely pushed my understanding further there.

I am watching this in LockDown...... Awesome Explanation.

Understanding this concept finally in my 1000th video.

Superb video. A huge effort. Thanks. Btw, can you do Master theorem? I find it a bit confusing.

Everything makes perfect sense now! Thank you

Just wow! This is a brilliant piece of material!

This was so so helpful, thank you so much for putting in all the work required to make this video

Thank you so much for your detailed explanation!! It really helps a lot!!!!

That is the hardcore explanation of O(logn).

Fantastic job !! Have we ignored the time or amount of work required to split each array until the base array arrives?? Let's say if a million items are given as input, does the amount of time to split until base has any significant effect on total running time asymptotically?

I always wanted to know where this came from. Awesome way to teach. Great job.

YOU ARE AMAZING!!!!! Thank you for making this video!!!!

Brah. I've owned text books that don't go into this much detail. This whole thing is SO appreciated

I understood why Merge Sort is O(n log n) in this way, and I want to ask if this reasoning is correct. You can think of the O(n log n ) in a different way. In fact, on each level (0,1,2) where the list is divided into 4s, 2s and 1s, when we do the merge step, we compare EACH number of the level, first in the left side and then (when we are done with the left side of all levels) on the right side of the level. So at the end of the day all numbers of all levels will be merged; so the logic is: For each level all the n numbers are merged The number of levels is obviously log(n), so we have to merge log(n) multiplied by n numbers. obviously, in the merge sort we merge first the left side and then the right side, but this is the same as merging all numbers of the same level log(n) times. Is it correct?

Hello guys , a little bit of help required, But first I would like to thank the genius for explaining this video at another level. I have came to learn a lot of things which I didn't even know. Doubt 1: I lost it at 22:10. one subproblem*(n-1)compsrisons. Why are we multiplying 1*(n-1) or 2(n/2-1) or so on... I am trying to match the sequence with this formula T(n) = 2T(n/2) + (n-1). But I can't find the pattern. Doubt 2: 30:10 The summation formula stands for (upper bound - lower bound) but he was adding extra +1 at the end. He did mention something about the bounds, but unfortunatly I did not get it which bonds he was talking about. Also 30:30 he multiplied n with the factor ( logn-1-0+1). Can anyone explain this to me ? please?

Excellent explanation I've even seen in a long time

you need to be at millions of subscribers. not hundereds of thousands

I'm very thankful past this video. Thank u. Hello from Brasil!!

I genuinely want to send over something to you on teacher's day

This is exactly what it should be!

Am I the only person that thought of Mr. Meeseeks from Rick and Morty - the recursive calls don't know how to solve the problem so they make a recursive call to solve the problem for them.

Amazing video, just earned a new subscriber thank you!

Good job man ! , very neat and nice explanation.

dude, I didn't understand shit when my professor taught this, but I get it now

I really appreciate your efforts, thanks, bro.