Asymptotic Notations 101: Big O, Big Omega, & Theta (Asymptotic Analysis Bootcamp)

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Today we will initiate a discussion on something that I have lied to you about for a very long time. This will be as simple as possible.
We will not only consider the informal definition but rather also look at the mathematical understandings behind why we call these asymptotic “bounds”.
Again, we care about this because the true colors of an algorithm can only be seen in the asymptotic nature of runtime and space.
So imagine this, we have these components:
A function T(n) which is the actual number of comparisons, swaps...just...resources an algorithm needs in terms of time or space. It is a function of n. When n changes, T(n) changes.
Our job is to classify behaviour.
A bound O( f(n) ) is the function that we choose to apply for the specific bounding.
The definitions, an example:
"T(n) is O(f(n))" iff for some constants c and n0, T(n) less than or equal to c * f(n) for all n greater than or equal to n0
In English...this means...we can say that f(n) is a fundamental function that can upper bound T(n)'s value for all n going on forever.
We have an infinite choice for what c is.
Our constant does not change behavior, it changes "steepness" of the graph.
We are saying that...if I declare f(n) as an upper bound, then I can find a constant c to multiply against f(n) to ALWAYS always always keep T(n) beneath my c * f(n)...T(n) will never beat c * f(n) for infinite n values...hence asymptotic bounding.
If we can't find this c then f(n) fails as an upper bound because it does not satisfy the asymptotic requirement.
So why are constants dropped?
Well...think about what we just did. The injection of the arbitrary c as a multiple onto a base function removes the need for a constant. It adds no meaning to a bound because it is conceptually already a part of the definition of what a bound is.
Big Bounds
Big O: Upper bound on an algorithm's runtime
Theta (Θ): This is a "tight" or "exact" bound. It is a combination of Big
For example:
An algorithm taking Ω(n log n) takes at least n log n time, but has no upper limit.
An algorithm taking Θ(n log n) is far preferential since it takes at least n log n (Ω(n log n)) and no more than n log n (O(n log n)).
Big Omega (Ω): Lower bound on an algorithm's runtime.
Little Bounds
Little o: Upper bound on an algorithm's runtime but the asymptotic runtime cannot equal the upper bound.
There is no little theta (θ).
Little Omega (ω): Lower bound on an algorithm's runtime but the asymptotic runtime cannot equal the lower bound.
If you can't get an exact upper bound, try lower bounding (although it is less useful to be honest).
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#asymptoticnotations

[@LegacyCS2]

I really appreciate how you said at the beginning "Do you understand this? Because I don't, this is too thick for me, but this is the typical introduction you'd get in actual classes". This whole sentence resonated so much with me. Sometimes I cannot grasp those abstract ideas in math, and I just need to have the concepts explained to me like im 5 years old, and your video was able to make me understand this in just 20 minutes, when I've been struggling with this for 3 years. Thank you man, keep it up!

[@eusebiupopescu-i2b]

wanted to comment the exact same thing, HUGE thanks!!!!!

[@DavidVCastro]

Your comment resonates with me so much!

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[@BackToBackSWE]

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[@momofro1819]

I agree, I haven't seen videos so detailed and easy to understand. Most of the videos on KZbin don't go this far into the small details that really help beginners build a foundation for their understanding.

[@crisjos6167]

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This is Ben from 5 months in the future. Cool video man.

[@BackToBackSWE]

This is Ben 1 month after this comment

[@IceyBoy47]

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[@yashjaiswal2335]

This is yash a student of IT. And here is a massive thanks for explaining the concepts.

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[@BackToBackSWE]

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[@diegocastro7434]

Hey man just wanted to thank you. Reviewing big O for my algorithms class and I am happy to say this is hands down the best explanation for this out there!

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[@alkendimacale8047]

omg wtf. I actually understood this holy shit. You're amazing dude, wish you were my prof hahahahha

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[@af_125]

I have paused at 15 minutes to say - thank you! This video helped me understand this topic very nicely. I thought I was good at algorithms and data structures and time complexity stuff last year until they introduced this from a mathematical standpoint this year. This really helped.

[@BackToBackSWE]

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[@rohitkarnati8604]

This man literally called me out haha, I was scrolling to look at the comments and got off task and he said "you still with me" 4:57 and I was like yes sir.

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[@BackToBackSWE]

Have you tried the platform yet? Check out the free DSA Mini-Course 👉backtobackswe.com/five-day Table of Contents: Introducing Asymptotic Bounding 0:00 - 0:46 Big O: The Asymptotic Upper Bound 0:46 - 1:05 Introducing Our Way of Thinking 1:05 - 1:40 What Does Asymptotic Mean? 1:40 - 2:39 Formally Defining Big O 2:39 - 5:35 This Is What I Mean By "Bounding" 5:35 - 8:05 This Bound Is Not Tight 8:05 - 8:55 Trying A Different Bound 8:55 - 12:06 Trying An Aggressive Bound 12:06 - 14:59 WHY DO WE DROP CONSTANTS? 14:59 - 16:11 Big Omega (Ω): The Lower Bound 16:11 - 18:40 Theta (Θ): The Exact Bound 18:40 - 21:37 Little o & Little Omega (ω) 21:37 - 22:08 Wrap Up 22:08 - 22:56 Errors: I misspoke sometimes and reversed sentences...oh well, the point got across at least I hope. There is no code for this video. Hey.

[@rataozinhoster]

tks man, didn't understand fckn anything, but im seeing this video more times through this semstr.

[@solomonelisha3543]

Instablaster

[@saloni8818]

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[@brianperez5000]

This was legitimately the most helpful video I've ever seen. When my professor tried explaining this concept, it sounded like gibberish for 45 minutes. After watching this video, I completely understand the concept. Thank you so much for your help!

[@TheZombyKillr]

Thank you so much for explaining this. As a graduate student in my 5th year of CS, I have only had bad luck with professors. No one would ever use examples or simplistic breakdowns to explain the concept, as you did in this video. You state the problem, simplify it, solve it, and work back up to the complex level we started at, in a way that makes it easy to understand, and very easy to follow. My assignments finally seem clear to me now. You've earned a sub :) Have a good week!

[@BackToBackSWE]

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[@suparnaprasad8187]

Absolutely amazing. Learnt more in 23 minutes than 1.5 hours of class. Looking forward to more such content!!

[@ExarchiasGhost]

Finally someone that explains the theory behind big O. I am in a phase that I had to master the algorithmic analysis, (to some extent of course), and I spent an awful time of searching to find this part of theory. Everyone just assumes that we know this part and just throwing fancy words instead of explaining. I am grateful for your explanation.

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[@ExarchiasGhost]

@@BackToBackSWE it did a lot :)

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[@samanthasavoy5303]

I don't normally comment on youtube, but I had to say that you are a phenomenal teacher. Cannot thank you enough for how you explained this concept!!!

[@collengura]

Apart from Abdul Bari's asymptotic series, I have not come across any such well explained asymptotic analysis. You do it so well that it sticks in the mind forever. All my grey areas concerning Big O concept have been cleared. No need to claim anything. Well done; and keep it up!

[@nononono-lm8id]

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[@BackToBackSWE]

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[@nehagawhane6869]

Pausing at 8:33 where it magically starts making sense to me, you are amazing! I dont comment often but you deserve a wide reach and endless growth, thank you for doing this!

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[@mustafa_masoud]

man!, i have been taking an online course in Algorithms, blv me lots of sources and videos that i did watch trying to get the meaning of different bounds, only your video which made me totally get it clearly and in a very smart easy way,, i can't thank you enough, keep up the good work.

[@BackToBackSWE]

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[@ev3rything491]

Amazing! Im a teacher trying to get his head around analytic number theory and felt like he needed to refresh his memory on Big-O notation. Gotta say, this one is great. It made it all much more intuitive. Now, whenever I see O(...) Im just thinking "O(...) = behaviour of the function under consideration". Thanks!

[@slnaisld2121]

had to pause the video to say that the way you explain this is so refreshing. I thought it was going to be hard to understand but your explanation really helped. Thank you!!!

[@johnmcsudden3176]

Like others before me, I've also paused the video to just say thank you. Your explanation is absolutely wonderful and you being able to take on the point of view of a new learner and are able to successfully work from there, including the delivery and the understanding is nothing short of astonishing. I don't normally comment on tutorials because they rarely impress me, but really - great work. Wish my professors teached like this. Liked and subscribed.

[@BackToBackSWE]

sure! all the best friend

[@Kirmeins]

NOW We're talking! All those notations simply confused the heck out of me! Thanks for clearing that up! Now I know what they're all about! So helpful

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[@matthewrenze5922]

You're a hero! This is much more understandable than my algorithms class!

[@BackToBackSWE]

thx

[@ShaliniNegi24]

Finallyyyyyyyy understood the topic! Thank you!

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[@jaihohoho]

This deserves more views! I have literally watched dozens of videos about Big O and other asymptotic notations and I still couldnt grasp them until I’ve seen this! Great work! Also, your’e very attractive #daddy

[@BackToBackSWE]

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[@intellagent7622]

your videos have improved. Your older vids I had to go at .75 speed, but now you talk clearer slower and its easier to understand. good job for improving

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[@flarepaints5516]

Absolutely love your videos! I've been studying this for a week and struggling, but after watching your video it all makes sense 🙌🏿

[@BackToBackSWE]

great to hear

[@lisztk1928]

Dude we have homeworks that ask us to prove mathematically if a function is Big O, Big Omega or Theta of another function. This video gave me a great intuition. Keep up.

[@BackToBackSWE]

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[@TrendRain]

bruh, you are f***** genius. I spent roughly about 7-8 hours truly understanding asymptotic bounding. You made it all clear. I see there are not many views in your channel, but keep it real, keep it simple you will eventually gonna make this the biggest and the best channel for CS student. Thanks for making things clear.

[@BackToBackSWE]

thnaks

[@blue-go4gr]

wow man just wow the fact the you started out with big oh unlike all other teachers and even the book it self then went to omega all while building up for the main star 'theta' .... man ????!!!!!! after understanding the concept it only make sense to start this way what the hell is the book doing !!! you're soo good at teaching thank you so much 🤍🤍🤍 the thing that i like the most about your vids is that you explain 'the thing behind things' so it all make sense not just make it understandable and that is actully all it takes to make things understandable for lots of us

[@zuzuz18]

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Wow, this is one of the best videos I have seen on this topic, you really break it down and speak in clear terms. Thank you so much! Keep up the great work.

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[@assaultszn3557]

i didn't understand Big O properly from being in university for 1 year, however after watching this video, i understand perfectly. You're Awesome!

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great

[@cranjismcbasketball8113]

I was so confused reading this topic in my textbook. You have made it so clear and simple. I will surely be watching more of your videos. Thanks so much

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[@alandangg]

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[@danielbarajas4625]

Bless up man. I have an exam tomorrow and I definitely feel 100% confident about this topic.

[@BackToBackSWE]

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[@BackToBackSWE]

great, thank you.

[@BRUCE59468]

Fantastic video! You explained something clearly that my professor could not.

[@BackToBackSWE]

thanks

[@DavidVCastro]

OMG, thank you!! I am in an Advanced Algorithms class that has a prereq of Mathematical Systems for Computer Science that I missed, and am apparently exempt from (no idea why!), and I've been struggling to keep up. You broke things down so well, and didn't assume nearly as much knowledge as my textbook assumes!

[@dragofand]

That part where you talk about the function you get describing a behavior was very, very useful! Thank you! I was really missing that from the video lectures i was watching.

[@BackToBackSWE]

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[@ihavenoname6771]

Never comment or like anyone KZbin but this dude KNOWS what he's talking. and thank to him, I got every single point he explained super clear!

[@crazyotaku5754]

i usually dont comment in youtube . but i have to say this. this video has the best explanation of asymptotic bounding in the whole youtube !!! .... You are a remarkable and very knowledgeable teacher. thanks a lot !

[@kayveemo]

You've taught me something in 20 minutes that my professor couldn't in 2 hours. Thank you so much, your videos are incredible

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[@juanong4709]

Amazingly clear explanation. Literally felt the need to write a comment to thank you for making this!

[@BackToBackSWE]

thank you for watching, you are loved

[@tedprice1]

Thanks for breaking this down. You flipped the way I was thinking about big-O and now it makes sense with crystal clarity. Thank you very much.

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[@BackToBackSWE]

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[@seanfly5554]

Thank you so much! I've been struggling with understanding this for the past 2 days, now I clearly get it thanks to this one video.

[@BackToBackSWE]

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Bro you're saving my ass for these first few lectures in Data Structures. Our teacher is god awful at explaining things and I can't believe she and you were both talking about asymptotic bounding. Huge props

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Thank you so much for this video. I have been pulling my hair out trying to learn this thing, i almost accepted the fate that i might fail this subject again, but you have actually given me some hope that i can finish this task! thank you.. i actually understand it now im glad i invested the 24 minutes to watch this. It makes so much sense

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[@JeffCaldwell-nemo_omen]

It took this video to make me realize that my understanding of what something like `f(n) = O(g(n))` means was completely backwards. I thought it meant that f(n) must be larger than g(n), but nonono. Whatever g(n) is needs to be larger than f(n). Wish I'd seen this before my last assignment. Glad I saw it before the first exam. Great video!

[@darkteam5515]

WOW! I spent 6+ hours watching different videos trying to figure out Big O and your video is the only one that makes sense to me. Thank you sooooo much

[@BackToBackSWE]

Happy to help

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[@BackToBackSWE]

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[@eusebiurizescu5441]

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[@MrTrevor0567]

First video of yours I’ve watched. This explanation made SO MUCH MORE SENSE than my professor. You’re awesome man!

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thx

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[@detroitpistonsplayoffs]

I can’t wrap my head around how such a young guy can have such a better grasp of these concepts than the professors at my university and the textbook they gave us to use, and all the other people who are made of KZbin videos on the subject. Oh, and my useless tutoring center. all of these people and resources don’t understand the concept well enough to actually test it. You are amazing!

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[@linuxtubers7313]

@@BackToBackSWE Welcome

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You have no idea how this is helping me as a college student. I wish i can thank you in person but this is the best that I can do. Thank you so much for the video. I really appreciate it my man.

[@uriel4829]

AMAZING VIDEO! I could not for the life of me understand big o from reading textbooks and from my lectures, but It finally clicked here. Cheers mate

[@BackToBackSWE]

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[@merzoukismahane8420]

i just needed to open my youtube acount and stop the video for a while to tell you that you made it easy and great ,and you're just the best one who explained that very well .thank you

[@DanielRamirez-di4po]

Bruhhhh.....all the Computer Science classes I've taken thus far and have PAID for and this dude just taught me Big O, Big Omega, and Theta way better than any of them did.......in 23 minutes......on KZbin.....for free.

[@LorneGiles]

I don't normally leave comments, but I wanted to take the time to say thank you. I have to say this is far clearer, more illustrative, and more understanding of the difficulty faced in digesting this topic than either my lecturer or the prescribed textbook (Shaffer) could muster. Keep up the good work as it was excellent and now favourited.

[@cesarmtz1992]

I’m so glad I decided to watch some coding videos before bed. This is going to be tomorrow’s lecture and you are explaining it so well!

[@tysonfontanez]

my professor took 4 classes to explain this in a way that didn't make sense and then I found this video and understood the concepts in 23 minutes, you sir are a legend

[@MrCodigoFuente]

3 mins into the video was enough to make me subscribe. This guy knows his stuff!

[@matteosalvatore9826]

Man i'm studying algorithms and data structure in my university and i gotta say that iunderstood this argument better with you than with my prof. slides. YOU ARE SUCH AND AMAZING EXPLAINER THX

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[@garrettw9756]

Really really REALLY thankful that I stumbled onto this video. Your explanations are crystal clear and I can honestly say that I've never understood this subject matter as well as I do now having watched this video. I cannot thank you enough!

[@blaiswilliam]

English isn't even my first language and I understood perfectly, very well explained, thanks so much!

[@BackToBackSWE]

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[@hugocast]

10:34 The whole "not a specific function, but A BEHAVIOR" part was what made it click for me. Thank you for the Aha moment :)

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[@thevgancheetah]

Dude, this is as clear as one can explain!! I feel like I understand everything now! Wow!! Great great great job!!!

[@BackToBackSWE]

thanks

[@harshpujara7774]

Dudeee.. i was trying to learn this topic since yesterday, but nothing worked and then i got your video in suggestion and boom, everything's clear. Thanks a lot

[@Irthiza]

This is the best explanation I have ever got on asymptotic notations. I will never forget this concept.

[@BackToBackSWE]

glad it helped

[@MJRezai]

I usually don't comment for videos, but this video, which is the first one I've seen from this channel, shows how clearly you can make complicated topics be understood. and I think I owe you a thank you. 👍

[@BackToBackSWE]

Hey, thanks

[@tomh6010]

You just paraphrased 5 weeks of my university course and I understand this waaay more than I did previously. Every credit.

[@rnyah_]

Omg i dont know how to express my feelings now You are such a great teacher and your way of explaining is the best i saw a lot of videos but no one gave me like you did I really don't know what to say Thank you so much

[@diamond9296]

Man out here teaching better than the top universities in the world, for free and in less time. Free vs few thousand dollars per class, 23mins vs 3 hour lecture unnecessarily complicating what is already unintuitive. Absolute champion!! If you were an algorithm, you'd be O(1)

[@BeALeaderAndAnInspir]

Hey man, thanks a lot. Finally, I got this after like 3 hours when I found your vid here. Haha... Keep doing the great work. You are a great inspiration.

[@BackToBackSWE]

nice

[@pashazzubuntu]

You're great and I subscribed, as your ability to tell/teach is something we rarely find on KZbin. While I'm here, I have a really subtle question. So there's an algorithm for choosing a pivot for qselect, that is the median of medians. I tried to prove that 5 is an optimal number for groups of medians, so I got this: T(n) = T(n/5) + T(0.7n) + O(n) where T(n/5) is Qselect 3rd element in each group of five T(0.7n) is derived from the fact that a median of medians is not more than 30% of elements and not less than 30% elements. So, in worst case, our recusive call to Qselect will have 0.7n elements in subarray. O(n) contains sorting of n/5 arrays of 5 elements (that's essentially n/5 times of O(1)) and the partitioning. So, let's say that these operations' upper bound is D*n. T(n) is bounded by a linear function. T(n) 0 C*n

[@BackToBackSWE]

I would love to dig into this but I can't take the time to (final exams).

[@pashazzubuntu]

@@BackToBackSWE Good luck mate! let me know when (if) you've into this.

[@Strangebeing007]

Dude, you are absolutely precise and amazing. Thanks man.

[@BackToBackSWE]

sure

[@kayamcheung2598]

Thanks for taking the time to teach us! I really want you to know that you definitely teach better than my professor

[@paulstuever8084]

Wow this makes so much sense. I'm in a discrete math course right now and my professor didn't explain any of this. He just threw identities at us an expected it to stick.

[@BackToBackSWE]

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

[@poseidensea7730]

Your Videos are great man. It really helped me with the Exam we took today.

[@BackToBackSWE]

Everyone eats. Eat up.

[@robertculley8447]

Just discovered your channel and I am so happy I did and grateful for you making these! Im a newbie still and these videos clear a lot of things up for me. Thanks!

[@BackToBackSWE]

Nice, thanks, and great

[@alyssarodriguez5891]

I am taking an algorithms class right now and this video helped me understand this topic SO much better. Thank you!!!

[@ThemisTheotokatos]

Thank you. T(n) is the time that a function takes and n is the value passed in the function, like an array. What is the f(n) and why do we use it as a bound? does it have to do anything with the cpu? or the memory ?

[@BackToBackSWE]

f(n) is the base function that describes an asymptotic behavior. Nothing to do with CPU, nothing to do with memory. It is simply a base function that we use a constant to modulate to describe behavior.

[@yfzhai1732]

Wow! Finally, you are back man. I have been confused by a very simple question in leetcode recently, which is a climbing stairs problem. It could not be that hard! But I just can not figure it out by myself! Could you help me out of this? Here is the description. You are climbing a stair case. It takes n steps to reach to the top. Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?

[@BackToBackSWE]

Hahaha...I HAVE DONE THIS. kzbin.info/www/bejne/hHetZKBvlpaigLM

[@yfzhai1732]

@@BackToBackSWE OMG! Terrific! Thank you~

[@BackToBackSWE]

@@yfzhai1732 :)

[@vanreus6150]

Same as everyone else in the comment section I had to stop the video to say this. You are one of the few people on this website that explain things so easy to understand and clear ( and thanks god without an Indian accent :D ). There are so many different people around the globe that watches your videos and appreciates every each of them.

[@BackToBackSWE]

hahaha thanks, means a lot

[@c1inward346]

I understood the concept fairly intuitively - though I need to sort my head a little bit, it read: to better optimize the performance our algorithm we need to have an idea of how it will perform. the big(O) notation essentially upper bounds / cover our function T(n) for some base function if and only if only if T(n) is less than that base function and there exists some value of c that will tightly bound / tuck T(n) below f(n). (I love this idea) The same thing applies to big(omega) where essentially it's just the opposite - it lower bounds the function. Both Big(O) and Big(Omega) gives you a sense of the worse case and best case scenario a particular algorithm takes in terms of time and space. Theta is just a breed of the upper bound and lower bound, though to prove it you need to make sure that it can both upper bound and lower bound our T(n). Is this correct?

[@nagarjunprasad]

Just Wow!!! Really awesome work mate. I recently found your channel and it's become a habit now to search your playlist and find if you have a video on any doubt I have. Keep up the awesome work!

[@BackToBackSWE]

hahaha thanks

[@5qw1rl]

Hey man, I don't usually comment on videos but this is some solid content. I definitely understood it by the first 10-12m but continued watching cause of how straightforward you explain things. Definitely learned more about this than 2 of my 50m lectures.

[@BackToBackSWE]

nice

[@crackthegate6047]

I am preparing for GATE examination, I was struggling with this asymptotic and many more but after watching your explanation I am sure that I can crack any questions. Thank you very much for this wonderful session...

[@BackToBackSWE]

Nice! Good job