Why Comparison Based Sorting Algorithms Are Ω(n*lg(n)) 

Table of Contents: (and clarifications) Introducing Today's Topic 0:00 - 0:38 Ad 0:38 - 1:11 The Approach We Are Going To Take 1:11 - 2:12 Step #1: Crafting Our Decision Tree Example 2:12 - 7:20 Making Observations About Our Tree 7:20 - 8:58 Step #2: Drawing Our Generalizations 8:58 - 11:50 Step #3: Solving The Relationship 11:50 - 16:48 The Final Solution/Result We Can Bound 16:48 - 17:51 Closing Thoughts 17:51 - 18:36 Clarifications: 10:58 -> I wasn't clear about the n!. We will have n! leaves, at least, because there are n! permutations of the final array. We have to have at least that many terminal states in the decision tree since those are all the possible "final destinations" we can reach through our comparisons. 14:49 -> Correction to what I said: "And...if we add 0 that does nothing for us."

I've never thought that i would watch such videos for their entertainment value. But here I am, enjoying the neatly prepeared math.

Your attitude makes reviewing algorithms fun, thanks for the good vibes.

bruh.. you just cleared up a semesters worth of confusion

Hey man, I really enjoyed the video and your energy. You kept the video so interesting and I really appreciate what you're doing. Subscribed!

Thank you so much for your videos ❤️ I got an offer from Amazon! 😀

Hey dude, I appreciate the tremendous work that must have gone to make this! You're too awesome man! Thanks a lot, never thought I'll be able to learn this concept as succinctly as you've done.

Love the ending. Thanks for explaining, nowhere i looked explained it with enough detail and simplicity as you. Thanks!

Thank you, it is very good that you make checkpoints during the explanation, 'if you get this part you are good'. (Something like this you said if I recall correct).

this helped me out man. Thanks for the tremendous work you did there. each part was beautifully explained.

Your videos are so good. I have studied algorithms for a while now, and I finally found your channel! I have used so much time on bad videos and explanations, but this is great. I think the reason for why I havent found any of your videos before is because of the titles. When Im looking for something I usually just type e.g. "Dijkstra", and yours are so long. Little tip maybe.

We learned this exact thing in class the other day... I was so confused and now I get it! Thanks so much! :) - UBC Comp Sci

This was cool! I’d love to see more of these.

Thank you! Video was amazing and you explained the concept very well!

Thanks a lot man. i was struggling a lot to understand it. its clear now

I just decided to like every video you have :) Great content

Love the way you simplify things 💕

always loved the way you teach !! Starting from beginning

think of the amount of effort and research he had done to put all this together ...... thanks a lot ,subscribed :)

finally a neat explanation for this point, thanks a lot 💕

That was simple and understandable. Thank you!

The best way of teaching complexities.Thank you 😊😊

Thank you so much, I finally understand so many fuzzy concepts because of you

i just about to finish my first year at computer science thanks to you, your videos are the best

The best explanation ever heard... great job.

Thank you for the great video! But can you make a video on similar approach/proof for Comparison based searching using decision trees? I think that would be great to understand too!

This is amazing, you are much better than our professer:) please keep doing it!

Hey! Videos like these help a LOT. Please make some videos on Masters theorem. I think I know the basic intuition behind it, but I'd love to know more. Thanks!

Excellent explanation ! Thank you so much sir ....

Hey i just stumbled upon your channel "really lucky" and i gotta say your style is very unique and clear any tips bro to learn data structures online too?

teacher dif my teacher rambling while you make connections and connect the dot now my note they shine !!

thanks a lot，great lesson!😃

Your videos are awesome! Also did a recent video about a silicon Valley salary analysis get taken down or something? Or am I seeing things

Excellent explanation, thank you

"...for entertainment reasons" - me: laughing, then realizing that it is way past my study time and that i am actually watching this for entertainment. Great Videos!

outro had me staring off into the night sky thinking about my place in the universe

amazing explaination , love , love , love

Simple Mathematical Explanation for a complex case

This is what I am trying to do n also the log n of big O notation relationship

Thanks so much!

Can you explain why it is valid to transform a summation of a function calculated with INTEGER indices into an integral (that takes the area under the curve, so it's not only the integer coordinates)?

Perfect explanation

oh my god I love you! thank you very much. I just found a new internet tutor :)

Perfect 👌🏽 explanation

Why can't my teachers show enthusiasm like this, I mean their lectures are fine and the explanations are OK, but they do it in a very mechanical way that I start day dreaming 5 minutes after the lecture starts. Anyways, great job brother.

just wow. thanks man

14:48 I think you mean "if we add 0, that does nothing for us"?

this is so good. not good it's great. thank you for such amazing video

Very Nice !!!!!!!!!!!!!!! Thanks

Cool explanation

I am curious if we can create "quaternary search" that would be using the && operator or "or" operator..in terms of big O it is log base 4 to the power of n. And then "tertiary search",up until "10th search"

Superb, but i have a question why taking the integral from 2 to n gives us the minimum bound?

You are amazing man ❤‍🔥❤❤

the website is out now right ?

Hey I really appreciate you efforts but what I know the maximum leaves would be the number of permutations of the array, which is 3! (3 factorial) for three elements. This is because there are six possible ways to arrange three unique elements on the other hand the minimum leaves will be 1 please correct me if I am wrong

Great job

thank you so much! just like a magic，i need to watch once again.

thanks bro.

would please cover segment tree algorithm implentation too??

Thank u

Are you going to take Design and Analysis of Algorithms?

Dude, You are just awesome. I've been following you for a while now and I'm really motivated from you. One question though: how did you learn maths? And I would be really happy if you can make a video on your journey to becoming a pro at algos and programming.

Hey. I'm not a Comp Sci student; I've only looked up sorting algorithms on RU-vid vids(especially MITOpenCourseware) so am not an expert but think I can weigh in here. Please correct me if I'm wrong. I thought the lower bound omega [Ω()] is the time it takes for the algo to finish sorting the best case input values i.e. the time to sort if you're really lucky with the array you're given. That would be the shortest possible depth of a leaf (e.g. on the leftmost branch of your diagram that compares A[1] to A[2], A[2] to A[3], A[3] to A[4], etc) which is n (in the best case of having an already sorted list) I say MINIMUM DEPTH of leaf/node as opposed to minimum number of leaves because only 'depth' number of comparisons are done one that branch. Some sources I checked were Wikipedia & GeeksforGeeks for best-case time where asymptotically sub-optimal algorithms like insertionsort & bubblesort actually have Ω(n) en.wikipedia.org/wiki/Sorting_algorithm www.geeksforgeeks.org/time-complexities-of-all-sorting-algorithms/ Please let me know what you think, I'm doing some coursework on the subject and am preparing for interviews so look forward to seeing more of your videos. Thanks

I cant get the minimum no. of leaves. How is that n!

I am watching this video to better understand the logic behind omega(n log n), then he said "IT'S FOR ENTERTAINMENT PURPOSES", I have never felt any smarter in my life 😂😂😂

this channel doesnt have a potential but is a potential seriously quality content

Thanks a lot for class bunkers like me! Anyways, university doesn't teach stuff! They want $$$. Thanks for providing good quality knowledgeable videos!

I have two doubts: 1. # of comparisons is O(2^h) but you just showed h >= O(n log n), but h is not the final answer. 2. You change the discrete summation into an integral, that doesn't seem to make sense. Integral is the area under curve and it is the limit of sum when the step goes to 0, but there is no concept of limit whatsoever in your equation

A,B,C,D,doit

Gold

if interview > my_math_skill : return jokes

It’s “number”, not “ amount”

Cool

支持一下

finally, updated again. ha

Im kind of shocked that people consider this level of math complicated......

Well, as a mathematician I cannot ignore a few mistakes that you made. Firstly, you cannot just say that integral is equal to the sum. In this case, it is true, but in general, it is false. I remember watched interesting proof for this integral. If I will find it, I will attach it in the comment. Secondly, distribution on the 15th minute is incorrect. The base of the log should be 2, not e. In general, it was an interesting video, but you should not break your explanation, because it breaks the focus.

Why are you so afraid of showing the Math ! You don't need to tell "dont worry about the math" Everytime you do some Math!