What Is Big O Notation?

[@DanGM123]

i love how 3blue1brown has created a new genre

[@vgarzareyna]

I’ve seen a lot of videos similar to 3b1b’s lately and I love them

[@evanward5045]

You commented exactly what I and many others were going to comment

[@S1lentG4mer]

I thought this was 3b1b

[@chrisvinciguerra4128]

Well he did make the Python library that this video used for its graphics so yeah that’s a reason why it’s similar

[@markgross9582]

Yeah. The animation styles seem similar too, so they may be using his library

[@tiff4839]

Soo good. You broke down a commonly misunderstood topic in an intuitive and engaging way. I love it!

[@Reducible]

Thanks Tiffany! I appreciate it!

[@revowolf7413]

I cant understand how youtube isn't recommending these videos. So elegantly explained, and the animations, OMG!

[@sifsif2725]

It just recommended this to me xD

[@astphaire]

I mean his enunciation isn’t great

[@Warpadable]

Well every people watching this had this recommended... You included. Otherwise how did you find it? I agree it is awesome though.

[@robertpoole9258]

A 3Blue1Brown for computer science. Awesome. This might end up being my favourite channel.

[@klaik30]

Wow, I actually think 3B1B has created a revolution in how we should think of structuring educational videos. I'm willing to bet that in the near future his visualization program and video structure are going to be used by MANY more people and hopefully even inspire some teachers to use his methods in class. Amazing work @Reducible! You should think about doing an "Essence of Computer Science" in the future!

[@saulbeck3398]

I came to this video with high hopes as you used Manim, yet, I was blown away with how detailed and thorough you were. I paused the video so many times, yet, not only was told what I was thinking, I was told it was normal, to think and ponder on it. I am actually blown away on how you able to teach something so simple that is a basic need for computer scientises in a way that makes me wonder how tf didn't I think of that!

[@strandedinanisland457]

This is so easy to understand....roughly 2 years ago I tried to gather this information on Computational complexity from a big book and ended up understanding very little.

[@henryroc1969]

Easily some of the best computer science videos on youtube and definitely better than my CS professors :). Thank you for putting in all this work!

[@TheEndermanMob]

I've seen many channels using manim. Beside its creator, this is the most original and really good one. I'm trying to have my own channel where a explain my area with other students, and I'm having trouble programing the animations, so I need to say thanks for this beautiful work of yours.

[@skyslasher6267]

as a junior in cs right now, in my opinion, this is a must watch for all people trying to get a degree in programming

[@nolanfaught6974]

A professor once asked the big O of an algorithm, to which I replied "O(n!^(n!))." He had to admit that I was right, then he rephrased the question as "what is the smallest big-O of the algorithm?" Never forget that a big-O is not necessarily the best bound on an algorithm, just one of many bounds

[@12-343]

If the answer to the question is really O(n!^n!), I think whoever wrote it is having a very rough time

[@Magnogen]

@@12-343 it's probably my code tbh

[@wolfranck7038]

If I'm not mistaken, the "smallest big O" is called big theta !

[@klobiforpresident2254]

@@wolfranck7038 Not quite. There are O and Omega. Big O is *an* upper bound. Often we mean the smallest (known) upper bound. Similarly Ω is *a* lower bound. Often we mean the largest (known) lower bound. When O and Ω are identical then that is called Θ.

[@wolfranck7038]

@@klobiforpresident2254 yeah but if you think about it, the "smallest" big O will be of the same order as the function concerned, thus also being big omega, and it will be in big theta Because smaller than that would be only big omega and bigger thant that would only be big O (It's clearly not a rigorous explanation but can't do much better in a YT comment)

[@garr_inc]

Never was into computer science, but it feels good to finally understand what the notation actually mean after encountering it in my higher-end math classes from time to time. Thank you!

[@caloz.3656]

this is the on video that finally settled my confusion with asymptotic notations. THANK YOU SO. MUCH. THIS IS INSANELY GOOD AND HIGH-QUALITY!!

[@alexplastow9496]

Thanks for the time you put into this video, being able to reduce complex things to something I could explain to a highschool student is thrilling in a nerdy way

[@absolutezero6190]

I noticed a small error in your video. You typed two quotation marks in LaTeX at 9:59. They are both ending quotation marks. The way to fix this is to use this markup: ``this will be surrounded with proper quotation marks when rendered in \LaTeX’’ Notice that I used two single quotes at the end (‘’) not a double quote (“).

[@johnmeyers8542]

Good catch. Although you missed 'defintiion' spelled with the lesser used two 'i's. Neither latex or other rubberised products will help with that one.

[@HienNguyenHMN]

@14:55 "we can actually use this 'symme-tree' to help us"

[@ishikuultra3637]

This is the best explanation I've ever seen.

[@codehorse8843]

Thanks these are lifesavers for my current course.

[@mehtubbhai9709]

Best explanation of Big O I have come across!

[@chrislam1341]

by far it is the best explanation after years of search.

[@1DInciner]

Was expecting several animations about other big O examples, like exponential one. It is very interesting to see them in comparison.

[@hisyamzayd]

Love how to find big o with graph explanations.. thank you 😀😀

[@КузьменкоІгор-к6ы]

At 15:53 table 1->1, 2->2, 3->4, 4->8, 5->16; at 16:15 If n=k, Count = 2^(k-1)= 2^k/2

[@shubhamsingh-xw3tf]

Definitely not forgetting Big O notation for a really long time! Excellent video sir! Thanks

[@suzalwakhley5329]

A very simple and detailed explanation. Thank you very much

[@olanrewajubabalola2322]

BEST VIDIEO ON BIG O NOTATION!!! every other video out there just leaves me more confused. thank you.

[@discreet_boson]

It would be an understatement to say this channel is underrated

[@sharanphadke4954]

just put some thinking pi's and this literally becomes 3blue 1brown video!!! Really great channel

[@playerscience]

Exactly! He is the 3blue 1brown of Computer science. 😎🔥🔥🔥

[@ronaldboulder308]

I remember this topic being explained in algorithm courses very similarly.

[@JonathanMandrake]

As a mathematician, I also have to say that it is easier to find the number of operations made if you try to say what the program does exactly and condensing all the parts that don't really matter to the runtime. For example for this algorithm: define f(n): f(1)=1 f(n)=f(n-1)+f(n-1) Thus, we have 1+2+4+8+...+2^(n-1)=2^n -1 calls, thus O(2^n) calls, as well as around one addition per call, exactly 2^(n-1) -1, being O(2^n). One simple step makes this a linear problem: define f(n): f(1)=1 f(n)=2*f(n-1) Then, we have n-1 multiplications (being O(n)) and n calls of f (also O(n)). But at this point, we can see, that this is a much simpler definition: define f(n): f(n)=2^n Here, we have n-1 multiplications, and 1 call of f, thus we have O(n) multiplications and O(1) calls of the function, making this algorithm much better. And we also have to consider what range of numbers we want to use. It may be great to use a highly complex algorithm for multiplication with an extremely good runtime O(f(n)), but if we only use numbers less than 10^10, it might still be much faster to use the algorithm with the worse O(f(n)) runtime simply because the n is not large enough in this practical application

[@geradoko]

I'm not sure but I think you missed the point. The subject of the video is to explain what O(f(n)) means, not how we can build a better algorithm for a given problem. In this sense, the example is very well-chosen since it is very clear that it's not the best solution to the problem. But there is still one thing in your answer which made me remember of times when processors could only add and multiply (in fact they could only shift bytewise :), and that is that an algorithm of class O(n) which calls an exponential function might not be faster than an algorithm of class O(2^n) which uses only addition. But I don't think these questions are still important for coders nowadays.

[@JonathanMandrake]

@@geradoko I do understand the point quite well agtually, but I was trying to note ways to find out what the actual order of runtime is (i. e. condensing unnecessary code, which also helps with cleaning up and improving the code) and why runtime isn't everything, because often enough the n aren't large enough

[@zapazap]

Which illustrates that (by definition) complexity here describes not the problem but the algorithm used to solve it.

[@JonathanMandrake]

@@zapazap Well, yes and no. If there is an algorithm which has optimal order of operations, then we can say that that problem also has that order. For example, if calculating the n-th term of some specific recursive series is at its best O(n), then we can say that that series has a computational effort of O(n)

[@zapazap]

@@JonathanMandrake Yes, you are correct. The very question of P=NP involves problem classes and not merely particular algorithms. Good call.

[@samuelatienzo4627]

Jeez I learned more here in 15 mins than a few weeks of my statistics and numerical methods class...

[@playerscience]

This is hands down the best explanation of big O notation!!! Instantly subscribed to your channel. 👌👌👌😘😘 BTW your voice is just like 3blue 1brown!

[@dmitriyogureckiy8292]

the best video about O notation, you bring all together, even lim, cool

[@givrally]

Now I want big O of big O notation, to group like growth rates together : polynomials, exponentials, and so on.

[@NovaWarrior77]

ABSOLUTELY AWESOME I-

[@AshutoshKumar-de8wn]

I love your videos for accuracy and clear content. Please upload videos on different algorithms and shortest paths.

[@BeeshoStudying]

you have the best explanation. thanks for the video

[@timhansen5468]

Thank you for saving my discrete math grade

[@the_cheese_cultist]

you probably should've added O(sqrt n) and O(n!) to your "common running times" list since they do appear quite often as well

[@kaleabfetene6258]

Such a great video thank you so much I honestly don’t have enough words to thank you appreciate that

[@Bloody_Mary_]

Beautiful presentation accompanied with fantastic music of enlightenment!

[@ClearerThanMud]

Clearest, most complete Big-O video I've seen. Kudos. It looks like you used Grant's manim package to create this. I have been thinking of learning manim so I can create a video giving some insight into why the radix sort is so fast. Work doesn't leave me much time for such pleasures, though; you do such a good job with this, if you are interested I can privately explain the insight so you could make the video if it appeals to you.

[@lucha6262]

What a great video! Thanks so much!

[@isabellelindblad2835]

This was amaaaazing

[@marpin6162]

who came from 3blue1brown’s post?

[@brooksbryant2478]

Me

[@LeoStaley]

Which one?

[@bazsnell3178]

Me

[@kudzem]

O(2^N) : "Yo, I'm so complex" O(N!) : "What was that, punk?"

[@herlusz]

The second definition is quite understandable

[@hiroyukikuwana3105]

Hope your channel keeps growing!

[@FRANKFIFORM]

Great video!! I’ve always have doubts about this topic.

[@weirongwu4964]

This masterpiece omg thank you so much!

[@ccdavis94303]

O(?) is deep and important in understanding processes in general, not just computing. Social structures scale in scary ways. Some stuff works great at dining room table scale (N~6), is still functional at the scale of a monastery (N ~ 100) but when they scale to small countries ( N ~ 20MM) let alone large countries ( N ~ 300 MM ) or huge countries ( N ~ 1 Billion) it hits the fan. Central planning for example. The Soviets set up an office to make sure everything was fairly priced relative to other stuff. Easy right? But there were N ~ 25 MM items in the economy. So relative fairness is N^2, but everything is fair compared to itself, so N^2 - N. Further comparison is sort of symmetric, so (N^2 -N)/2. Big improvement but when N = 25 MM, O(fair) ~ 312 trillion. Poor aparatchicks. (Tom Sowell cited this case) { if a is fair relative to b and b is fair relative to c, is a fair relative to c? How much would this improve things?}

[@jursamaj]

That argument is obviously fallacious. You don't need to compare the price of every type of watch to every type of car, and every type of bread, etc. The watches only need compared amongst themselves, and to some baseline.

[@tucan1309]

i didnt come from 3b1b but im suprised at quality of these videos

[@danser_theplayer01]

You missed a O(√n) class, which also can be written as O(n^0.5). There is a thing where in the worst case scenario the algorithm will take at most √n "operations" to finish, for any given n, doesn't matter if it's 1 thousand n or 95 billion billion n. (Technically there's also one with √(n)/2 but they're in the same scaling class) Since nobody seems to be mentioning it anywhere I've decided to call it "rooted" running time class. And the use case isn't that uncommon, it's meant to substitute a 2d array to tradeoff lookup time for common insertions/deletions.

[@jayantverma6196]

You are amazing, please upload more videos.

[@danser_theplayer01]

The O notation which drops only small +- values but keeps /* and ^ values is going to tell you the overall time an algorithm will take to run given n input size. E.g. O(n/45) is clearly 45 times faster than O(n). The O notation simplified will tell you the way your algorithm time scales in relation to n input size. E.g. both O(n/45) and O(n) will scale linearly, in the first case every 2 input will need 2/45 (rounded to 1) "operations" to execute, and it's not going to suddenly scale quadratically where for every 2 input you'll need 4 "operations".

[@hotpushupguy4203]

These are so wonderful - thank you !

[@isaacfernandez2243]

Wow, this was incredibly good. Thank you.

[@CheckmatesSpeedruns]

For the problem at the start, there is a better solution: for a in range(n+1): for b in range(n-a): c = n-b-a print(a, b, c) Efficiency: O(n²) It is still faster than Alice's solution, because instead of (n+1)² operations, this one takes n(n+1)/2, or binomial(n+1, 2) operations.

[@berkay453]

it's great and very clear expression thx bro

[@dehilihind2916]

thanks for your efforts , I just discovered your channel and it's INCREDIBLY helpful !

[@_maus]

Thank you so much and I really appreciate the video.

[@anwarulbashirshuaib5673]

How come I noticed this masterpiece so late!? Definitely subscribing!

[@eamonnsiocain6454]

Well presented! Thank you.

[@erv993]

Thanks so much! You have invaluable content!

[@Adomas_B]

I keep reading the text in 3blue1brown's voice

[@joeyhardin5903]

ikr!!

[@snoopy1alpha]

Very well explained! I guess I will quote this video in my next complexity discussion at work :-D

[@joaquingutierrez3072]

I loved this video. Thank you !

[@Starwort]

That last example emphasises the importance of caching - caching would reduce the runtime to O(n) worst-case and O(1) best-case (of course, it could also be reduced to O(n) simply by multiplying the recursive result by 2)

[@manamsetty2664]

The Man.The Myth.The Teacher

[@mychannel-te5ke]

11:45 It's not really true that there should be such a limit. It may not exist. For example g(n) / f(n) may by 1 for even n's and 2 for odd n's. So there's no limit in this case. Even more. It's true that for f(n) = n and g(n) = n^2 it's true that n = O(n^2). But g(n) / f(n) = n has no limit.

[@diegocfq]

Yeah, it's supposed to be lim sup instead of just lim and an absolute value operator should be applied on the numerator of that fraction.

[@DavidPysnik]

How would lim sup help Сергей Обритаев's example? if f(n) = n and g(n) = n^2, the limit as n goes to infinity of g(n)/f(n) would not exist because it goes to infinity, so the lim sup of this expression going to infinity would also not exist. Even with lim sup in that definition shown at 11:45, it doesn't allow n = O(n^2) as the original definition does, so Сергей Обритаев's criticism does not seem fixed.

[@johnmcleodvii]

O notation does not require that the function has a strict limit in the terms that calculus does. If odd numbers for input are one growth in n and even numbers are something other growth in n, the O notation would be for the faster growing one. So in the example where odd numbers had linear growth and even numbers had n² growth, the O notation would be O(n²). Off the top of my head, I can think of no algorithms that have significantly different behaviors for even and odd n.

[@hamzaich7034]

Maaaaan this is amazing ♥️♥️♥️

[@blueskyjavelin2289]

This video helped me alot. Thank you:)

[@rjmbowie]

Amazing video! Love the use of manim.

[@doBobro]

You've got strange timings for O(n^2) and O(n^3). For python code there is a clear n^2 and n^3 relation: For twice the data count_bob times as x8: 165us count_bob(10) 1.02ms count_bob(20) 6.91ms count_bob(40) 51ms count_bob(80) For twice the data count_alice times as x4: 18us count_alice(10) 60.4us count_alice(20) 221us count_alice(40) 848us count_alice(80)

[@indian_otaku2388]

Isn't it machine dependant?

[@doBobro]

@@indian_otaku2388 for some degree yes, but there is a clear error in measurements.

[@AshtonSnapp]

Optimized f(n) from ~16 minutes in: def f(n): if n == 1: return 1 else: return 2 * f(n-1)

[@MuradBeybalaev]

Your passing explanation of machine dependence is incorrect. It's not about a supercomputer being faster. That wouldn't affect the plot shape. It's that different machines can have varying instruction execution costs AND we generally speak of multitasking execution environments that can dynamically tailor performance mid-execution.

[@nomi98]

Glad my teachers taught me well lol.

[@tamptus3479]

What is difficult with this Definition? We do not define O(f) but we define =O(f) the Equal Sign has now no longer the meaning "equal". Example if f = O(g) and h = O(g) then f and h may not equal. if the = has the meaning "equal" O(f) have to be a function, but this is not the case.

[@perrydimes6915]

You're right, it is better to think about it as set inclusion. If you know anything about set theory, this is the "is in" or "is an element of" or simply ∈. So we would say f ∈ O(g), and h ∈ O(g). I believe this is the single biggest issue with understanding, because we're using an equals sign for something that is NOT equality. O(g) is a set. f, g, and h are all functions.

[@tal3541]

There's an inaccuracy in 11:30. If f=O(g) it doesn't necessarily mean that the limit of g(n)/f(n) exists. For example take g(n)=n and f(n) to alternate between n and 2n (for evens and odds). The mathematically correct way to put it is the the liminf of g(n)/f(n) is equal to C which is greater than 0.

[@willnewman9783]

Good point, but I still think this is wrong, at least according to his definition. By what be says, n should be O(n^2), because n0.

[@joaofrancisco8864]

So good!

[@legendgames128]

12:20 I argue that you can cram extra items between constant and logarithmic, by considering the inverse of the Ackermann function. n=3 would equal the cube root of 3. n=2 would be 1 (2/2) and n=1 would be 0 (1-1)

[@jaumm84]

Thanks! I finally undestood it!

[@cross_roadz]

If only I saw this before my discrete mathematics exam

[@Pewpewpew230]

Great work, thanks!

[@wolfVFV]

here is how i would always explain big O notation: lets say you have a program that sorts a 1000 books alphabeticly and that takes 5 seconds. we want to know/extimate how long this program will need to sort 10x as many books. will it need 10x longer? or maybe just twice as long? maybe it needs a 1000 times longer. Big o notation allows use to easily see how much longer a program will need if we increase the input/data. and as a programmer we want to write code in such a way that the big o notation is as low as possible

[@mikesolo7993]

I bring up BigO at work and people's eyes glaze over. I've stopped trying to explain, but damn with this video I think anyone could understand it!

[@ccdavis94303]

This hit home long ago. Looked at a problem - it had a short clean set theoretic algorithm to solve it. Set it up and ran it on a small test dataset. Smoove. Ran on the real data with a counter. After 5 hrs I surrendered. Go O or go home.

[@anasasim3856]

I want to give you a hug bro!

[@nadonadia2521]

Great video i love the topics and the way that have been presented may teacher have not your skills, please more videos on running algorithmes programming ideas, thank you, i have subscribed to tthe channel.

[@shimavalipour5992]

u explained it sooooo goooood, thanks

[@darkelwin02]

Hey good video explaining it. I think including the Factorial class would have been useful. While yes you dont see those outside graphs often, making an algoritmh on your own and recognizing to stay away from that class is practical

[@0_-]

for a in range(n): for b in range(n-a+1): c = n-a-b print(a, b, c)

[@johnmcleodvii]

That is an improvement, but it is still O(n²), but it does have a lower C.

[@0_-]

@@johnmcleodvii yo do you know a O(n) sol.?

[@johnmcleodvii]

No. I don't believe there is an O(n) solution to this problem. I don't believe there is even an O(n*lg(n)) solution.

[@sbsyr5555]

Nicely explained...

[@moonst6020]

I am here to cry over my upcoming tests🙂

[@Pac0110]

I never had a problem understanding the definition of Big O, I feel like it’s not confusing but maybe people aren’t used to think calmly through reading something. Usually definitions we get are “easy” and we don’t need to understand anything in order to use them, in fact, a lot of people don’t even know what they’re doing, they just use them because they work. I think that, in order to improve education overall, we should think a little bit more and try to understand basic logic so we can understand the definitions we get. Pretty nice video though, good explanations

[@General12th]

It would be a good idea to talk to a variety of people and ask them what exactly they find confusing/intimidating about the formal definition. There might be a single reason (perhaps the definition is genuinely hard to grasp) or it might be a variety of reasons. However. It's important to keep in mind that having people think more and understand things better is the _result_ we want. It's not a method of _reaching_ that result. After we figure out where people go wrong with their initial understanding of the definition, we need to teach them specific ideas to shore up their limitations. Just telling them to do better and leaving them to their own devices isn't a useful line of education.

[@Pac0110]

@@General12th I agree with your first paragraph. But I disagree on the second one. I think that making people think by themselves would improve the understanding, thus, making people think would be the result or the objective we want to achieve. Also, I don’t think “just” letting them think more it’s enough, I never said so, I think it would need to be additional to the support they need. But right now, the educational system doesn’t teach you about thinking, it teaches you how to remember stuff or understand some stuff. But if we manage to make people think, they may search for information by themselves and try to figure out a way to get more information or a better understanding about anything. But, of course, we need to provide them of different kind of resources so that they can find them when needed. Finally, I feel like reasoning stuff it’s one of the best things one can do with this kind of definitions. It’s not about just understanding A definition, but the essence of the definition. At least that’s what I think. We might describe the notation with different words, but what’s important is that your definition and mine are essentially the same. Then again, this is just my opinion, and I’ve seen improvement when I taught my students to think rather than just understand the definition.

[@abdelbassetlabbi852]

incridible! just continue bro

[@dwolrdcojp]

I love this channel

[@HamzaQayyum]

Error at 11:37. It should be lim[n->infinity] of f(n)/g(n), not g(n)/f(n).

[@mileslyfe5239]

Is it “O”kay to think about “big O” as the end behavior of the function which represents the number of operations needed for an algorithm to execute?

[@johnmcleodvii]

It's a rough approximation of the number of operations to execute relative to the "size" of the input. What the size of the input is depends on the algorithm. In some cases it is the value of a number, in others it is the count of items. For factorization it is the number to be factored. For sorting, it is the count of items.

[@diegovasquezrevilla]

Great video keep up the good work