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

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Question: Analyze the total work that Merge Sort performs as an exact function of n, the length of the input list.
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[@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.

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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.

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[@BackToBackSWE]

hey

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

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Well, he is.

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Phase of education varies across the globe

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[@BackToBackSWE]

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

[@Rhunyc]

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[@BackToBackSWE]

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He explains it so clearly on every step..subscribed

[@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

[@Egrodo1]

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[@BackToBackSWE]

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

[@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.

[@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

[@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

[@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

[@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

[@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]

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[@BackToBackSWE]

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

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[@BackToBackSWE]

ye, may the internet flourish

[@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

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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]

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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.

[@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

[@tinnguyen199]

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

[@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/ 🎉

[@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

[@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

[@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

[@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.

[@bulioh]

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

[@hrithikjaysingpure9623]

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

[@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]

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[@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.

[@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.

[@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

[@ellenahpage8914]

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

[@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.

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

[@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)

[@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

[@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

[@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

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[@BackToBackSWE]

thanks

[@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!!

[@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

[@alan0428a]

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

[@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 !

[@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!

[@КрасавчикИкиса]

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

[@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.

[@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

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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.

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

[@BackToBackSWE]

thanks

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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! :)

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

[@BackToBackSWE]

welcome

[@anmjubaer]

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

[@BackToBackSWE]

thx and yeah

[@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

[@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?

[@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

[@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/

[@wsaylazer]

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

[@sameerarasanga8975]

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

[@vrashankraom]

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

[@BackToBackSWE]

thanks and 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

[@vishnupriya47659]

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

[@ARATHI2000]

Great job. Appreciate the Big-O complexity details!

[@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 :)

[@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.

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

[@BackToBackSWE]

great

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

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

[@BackToBackSWE]

thanks lol

[@rishabh55]

So much effort in a video for merge sort

[@BackToBackSWE]

Very much so

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

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

[@BackToBackSWE]

hey

[@serogozin]

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

[@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?

[@Onelegtrev]

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

[@BackToBackSWE]

sure

[@willianfn]

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

[@hiimgosu3174]

Hi there, I have a question. I totally understand the complete mathematical way of getting the nlogn-n+1. However, for the T(n)=2T(n/2)+(n-1), this (n-1) means n-1 "comparison", but T(n) means total time used at this level. these 2 items have different unit, should we multiply (n-1) with a constant which stands for time cost of each comparison? Kindly help clarify. Thank you sir

[@nniazoff]

Very clear and detailed. Great job.

[@BackToBackSWE]

thx

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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.

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this is the best explaination ever to anything!!

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

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I didn't see this directly linked in your description, but this explains the derivation of the geometric series formula: en.wikipedia.org/wiki/Geometric_progression

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

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

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Is there an optimal sequence using which I can watch all the videos you've put up?

[@BackToBackSWE]

Just watch videos you are weak on We have a more category-based ordering on backtobackswe.com (paywall though)

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This channel is a goldmine 👌 Superb explanation

[@BackToBackSWE]

thanks

[@TripedalTroductions]

This is nitpicking but according to the ISO 31-11 standard, log base 2 denoted as lb(x) for binary logarithm, where as log base 10 is lg(x), but log(x) is not defined in the standard. Personally I hate it. Why wouldn't log(x) be log base 10 and lg(x) be the binary logarithm? This is madness. Great video by the way. This is the videos I go to by default now cause the quick and dirty pretty visuals from other channels just don't cut it for me.

[@BackToBackSWE]

fascinating

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Just wow! This is a brilliant piece of material!

[@BackToBackSWE]

ye

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Understanding this concept finally in my 1000th video.

[@abysir4461]

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

[@BackToBackSWE]

great

[@shishirpai380]

Everything makes perfect sense now! Thank you

[@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.

[@craxl2287]

Hi, I am working on my Math IA (research project for IB HL Mathematics) for my High School Math course. For my project, I am trying to understand why Merge Sort has a big O notation of O(nlogn). Your video makes a lot of sense and really helped me understand most of the complicated concepts. Even still I am just a tad confused at time 28:10. You said Log(8)=3 but does that not assume that the base for the log is 2? Did I miss something in the video where you specified this? Also should the big O notation actually by O(nlog(base 2)(n)) and is instead simplified to just O(nlogn)?

[@BackToBackSWE]

"You said Log(8)=3 but does that not assume that the base for the log is 2?" Yes. "Also should the big O notation actually by O(nlog(base 2)(n)) and is instead simplified to just O(nlogn)?" The base of the log does not matter since it is asymptotic. You can intellectually understand this, but you can take the actual limit to see this is the case (I forgot the math behind it, been 1 year since we did this in algos).

[@alex-gz7ud]

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

[@zavierjack4709]

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

[@zavierjack4709]

24:30 🔥🔥🔥🔥🔥

[@BackToBackSWE]

nice

[@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 😀

[@tibzblaze1708]

So the number of comparisons to get from level 1 to level 2 would be 2(N/2 - 1) right?

[@Youssiflagrosse]

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

[@BackToBackSWE]

Glad it was helpful!

[@insanebhuvi3726]

Give an example input list that requires merge- sort and heap-sort to take O(nlog n) time to sort ,but insertion -sort runs in O(n) time. What if you reverse this list I want the python code for this question can u please try ???

[@saikiran-xh5lt]

I really appreciate your efforts, thanks, bro.

[@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/ 🎉

[@blatogh1277]

That is the hardcore explanation of O(logn).