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

[@BackToBackSWE]

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.

[@fpv_am]

I've really applause you when I understand something from your explanation)

[@BackToBackSWE]

ye

[@mostafaomar5309]

@@BackToBackSWE Hi, you're doing a really good job, thank you, but I couldn't find any code in the description? :)

[@hil449]

you really should think about becoming a university professor if you enjoy teaching man. Your teaching methods are superb

[@34521ful]

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

[@BackToBackSWE]

This video is way too long and discourages clicks from passive users but whatever. It needs to be in detail since brevity would confuse and glaze over critical building blocks to the larger understanding.

[@ramondelarosa4895]

@@BackToBackSWE You really are a true MVP! Your videos on time complexity algorithms are a godsend! Cheers again! Keep up the good work!!

[@Ghareonn]

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

[@BackToBackSWE]

hey

[@name-np4gr]

i dont get it. wtf u learning this in college for?? you shoulda learned this in high school

[@cowcabobizle]

Well, he is.

[@MohammadAbrar-tx3fc]

Phase of education varies across the globe

[@paukatable]

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

[@BackToBackSWE]

im just so average that I had to learn harder than other lol

[@tomevans2417]

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

[@BackToBackSWE]

ye

[@minus-infinity]

thats why our youtuber bros are here 😜😜 they must have gone through same problems

[@cristian-razvanpopa7101]

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.

[@Rhunyc]

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!

[@BackToBackSWE]

Nice, welcome to the 💩show

[@supastar25]

He explains it so clearly on every step..subscribed

[@anthonylee1309]

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.

[@BackToBackSWE]

hahahaha thanks, I am but a boy

[@Egrodo1]

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 :)

[@BackToBackSWE]

Thanks and nah...we are going all the way this year.

[@charlieminaya3534]

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.

[@dragoslavmilutinovic5731]

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

[@BackToBackSWE]

thanks

[@starbloods0013]

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

[@BackToBackSWE]

nah, a college education is valuable srs. Program and build in ur own time. School itself sucks sucks sucks, but the output you get molds your mind into a critical thinking machine sorta...hard to explain.

[@starbloods0013]

@@BackToBackSWE gotchu. i'll take your word for it. thanks for the help and i hope ya'll keep making videos

[@adrijasamanta7949]

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.

[@BackToBackSWE]

thx

[@heathledger7291]

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.

[@BackToBackSWE]

yeah and thanks

[@hegyilevente221]

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!

[@BackToBackSWE]

ye, may the internet flourish

[@dawn_connor]

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.

[@karthikrockrr]

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

[@BackToBackSWE]

creator thanks u

[@sapokee2830]

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

[@BackToBackSWE]

Thank you, appreciate it 😄 Also check out our Free 5 Day DSA Interview Prep Mini-Course - backtobackswe.com/ 🎉

[@hrithikjaysingpure9623]

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

[@domyi6953]

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

[@BackToBackSWE]

I'm getting tired to be honest. So much effort, 4-6 hours a day. But, I'll keep going.

[@thestudycoven8771]

^ 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.

[@BackToBackSWE]

ty

[@vanducnguyen346]

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

[@BackToBackSWE]

ye, let's go to 100k

[@nullbrain8552]

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!!!

[@BackToBackSWE]

great to hear.

[@JV-jc7ci]

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.

[@akshitarora470]

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!

[@BackToBackSWE]

great and thanks. and I'm spending most of my time building backtobackswe.com code-wise and building our content team. and sure - that's my goal

[@amypellegrini1732]

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.

[@bulioh]

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

[@yarashamali8061]

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

[@BackToBackSWE]

sure thx

[@LenCedeno]

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!!

[@BackToBackSWE]

Thanks buddy! do try the 5 day free mini course or subscribe to our courses with 30% discount for some exclusive content b2bswe.co/3HhvIlV

[@antongrau7963]

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.

[@nolanhellen4888]

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

[@tejshenanigans8070]

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

[@BackToBackSWE]

They are too long to go mainstream

[@tinnguyen199]

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

[@tsilikitrikis]

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!!

[@tomaustin9166]

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! :)

[@scodz2007]

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[@BackToBackSWE]

thanks

[@Owen7768]

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 !

[@wsaylazer]

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

[@JujutsuMan]

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)

[@PenStab]

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!

[@Ian-bb7vv]

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

[@langwen8685]

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!

[@BackToBackSWE]

Sure

[@יעלליפשיץ-ס6ש]

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!!!

[@BackToBackSWE]

great and thx

[@KuldeepYadav-jw7jn]

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

[@BackToBackSWE]

thanks

[@КрасавчикИкиса]

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

[@ellenahpage8914]

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

[@yannaleyva4554]

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 :)

[@BackToBackSWE]

Table of Contents is critical. Thanks.

[@BornYooper]

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!

[@BackToBackSWE]

haha thanks

[@ekejma]

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.

[@mia-so4ry]

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

[@phenipatel7089]

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.

[@BackToBackSWE]

thanks, nice

[@suyuan9158]

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

[@BackToBackSWE]

hey

[@GabrielPerez-ml2go]

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

[@BackToBackSWE]

welcome

[@alan0428a]

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

[@vishnupriya47659]

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

[@coolio_catio]

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

[@theappgirl6615]

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

[@BackToBackSWE]

nice

[@josephcohen734]

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

[@DivinityinLove]

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.

[@frankgarcia6673]

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

[@BackToBackSWE]

Create a server farm and do it

[@jigarsingh600]

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

[@BackToBackSWE]

thanks lol

[@ARATHI2000]

Great job. Appreciate the Big-O complexity details!

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

[@BackToBackSWE]

thanks and thanks

[@ryancodrai487]

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.

[@sameerarasanga8975]

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

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

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This channel is a goldmine 👌 Superb explanation

[@BackToBackSWE]

thanks

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

[@BackToBackSWE]

great

[@willianfn]

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

[@zelda8697]

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

[@BackToBackSWE]

thanks :)

[@Me0w_Meaw]

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?

[@linhlieu1765]

can you explain why you have 1(n-1) instead of n on the right at 22:23

[@BackToBackSWE]

So we do (n-1) comparisons at that level with 1 subproblem.

[@danrussell1057]

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

[@AhmedMostafa-so2jp]

this is the best explaination ever to anything!!

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Just wow! This is a brilliant piece of material!

[@BackToBackSWE]

ye

[@anmjubaer]

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

[@BackToBackSWE]

thx and yeah

[@Youssiflagrosse]

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

[@BackToBackSWE]

Glad it was helpful!

[@Onelegtrev]

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

[@BackToBackSWE]

sure

[@Same_energy_101]

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

[@Same_energy_101]

24:30 🔥🔥🔥🔥🔥

[@BackToBackSWE]

nice

[@nniazoff]

Very clear and detailed. Great job.

[@BackToBackSWE]

thx

[@arvindojha1101]

Good job man ! , very neat and nice explanation.

[@BackToBackSWE]

thx

[@truthseeker8767]

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

[@BackToBackSWE]

thanks ahah

[@haoruikun1191]

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.

[@BackToBackSWE]

haha thanks, means a lot. Would love some feedback on other content - backtobackswe.com/

[@janewu4660]

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

[@abysir4461]

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

[@BackToBackSWE]

great

[@gabrielesimoncini1121]

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?

[@rishabh55]

So much effort in a video for merge sort

[@BackToBackSWE]

Very much so

[@smikkelbeer7890]

I got so confused at 28:48 because I forgot that 'i' was above the first '2' and didn't see it due to the light there lol...

[@shelby1999]

Understanding this concept finally in my 1000th video.

[@NagarajaT]

Excellent explanation I've even seen in a long time

[@BackToBackSWE]

Thank you, glad you liked it 😀 Do check out backtobackswe.com/platform/content and please recommend us to your family and friends 😀

[@saikiran-xh5lt]

I really appreciate your efforts, thanks, bro.

[@skhan3530]

Thanks a lot,it was great .Really helped me in understanding the concept. Appreciated! :)

[@canernm]

Thanks for the video. At 28:00 you say that base cases contribute 0 comparisons, however when you have two 1-element lists (base case) you still compare these 2 elements to find which is smaller so you can merge them into a 2-element list. Am I wrong ? Thanks in advance.

[@BackToBackSWE]

The base case is just an if statement that returns. You are talking about the merge right above the base cases I think

[@canernm]

@@BackToBackSWE Thanks for reply. Yeah I figured it out, its just that in my mind I calculated the work done starting from one level below the one you did haha. But yeah total work level is the same, thank you.

[@alex-gz7ud]

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

[@shishirpai380]

Everything makes perfect sense now! Thank you

[@harshraj3255]

Thank you so much for your effort! Great explanation. Godspeed to you!

[@Nyquiiist]

Could someone please point me to a resource that explains how we went from step 2 to step 3 at 30:15 ? Ive been out of school for quite a while, and just cant recall the reasoning behind doing (top bound - lower bound + 1). Thanks in advance!

[@BackToBackSWE]

Is this for the math part? Imagine it in your head and it'll help - if we want total items between 0 and 7 (inclusive of both 0 and 7), 7 - 0 gives us 7. But, there are 8 numbers, 0 1 2 3 4 5 6 7. So we add 1 since we excluded a bound. Or maybe Im talking about something else? I did this way back

[@Nyquiiist]

@@BackToBackSWE Haha I figured it out right after I posted this. Thank you for getting back though, much appreciated!

[@fairestmink4023]

Glad I found the explanation! This comment was very helpful. Thanks a lot!

[@hiteshnagothu887]

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?

[@BackToBackSWE]

We did do an asymptotic analysis

[@theManTheGuy_inTheMiddle]

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

[@BackToBackSWE]

Glad it helped😄 Also check out our FREE DSA Interview Prep Mini-Course - backtobackswe.com/ 🎉

[@youraverageprogrammer7532]

Amazing video, just earned a new subscriber thank you!

[@BackToBackSWE]

sure