Kolmogorov is one of the coolest men I've heard of. Admitting defeat and then anonymously supporting the kid. wild

Props to Kolmogorov, he could have sent the paper in his own name without giving credit to an unknown student and take all the merit. The academic world is sometimes ruthless

I think the fact we don't teach fast fourier transform in elementary school says a lot about society.

For anyone curious at 13:38 N^1.6 is used as an approximation. It's really N ^ log base 2 of 3. If you want to enter it into a calculator use the change of base formula. Log(3) / Log(2)

It's amazing how a simple problem like multiplication can devolve into such complex mathematical discoveries. Who would have thought that multiplying optimally is insanely more difficult than adding.

Between Karatsuba and FFT there is a Toom-Cook algorithm, from 1963-66. As FFT, it treats both numbers as polynomials, evaluate the values naivly in some points (for small numbers! Like 0,+-1, -2,+inf), multiply them and then interpolate it back to polynomial form. "2 way" toom-cook recreates Karatsuba. The original "3 way" and "for way" have the complexity O(N^1.465) and O(N^1.404). The GMP library (a hefty library for big numbers) uses naive, Karatsuba, "3","4","6.5" and "8.5-way" toom-cook, and fft, using each algorithm for numbers of different lengths.

I love how you bring nearly-unreachable knowledge to the community through interesting and easy-to-understand videos. I would never know this bit of theoretical CS otherwise. Keep up the good work!!!

Kolmogorov also advanced the study of fluid flow turbulence so much that they named a constant after him and still refer to his work to this day!

I have just discovered this channel and the animations and the gradients are so beautiful, the content, so mesmerising, that I instantly subscribed. Thank you.

The legend is back

I loved fast inverse square root and finally you've released some more videos! Makes my day. Take however long you want, they're worth it.

14:47 That was honest of Kolmogorov. I have met a few people in my career who would pretend to have done work which was actually done by someone else. They would then take the credit for the other person's work.

This is what I studied in my 200-level, 300-level, and 400-level computer science algorithms class. Good explanation!

Even if the lower bound is Ω(N × log N), there is still mathematical progress to be made (or disproven) in finding an algorithm which is that efficient with smaller and smaller inputs.

Fun fact: When computers are multiplying whole numbers, the compiler will often optimize the code, so it doubles (or halves) the number one or more times (which is a single operation in the computer, known as bit shifting) and then add or subtract a konstant to achieve the result. So a code of x * 9 (which is (x * 8) + 1) would be compiled as an equivalent to (x

Wonderful video :) I am writing an Algorithms exam next week and wanted to take a break from learning but ended up learning about the algorithm more than in my lecture and in a more exciting and relaxing way. Thank you for this masterpiece and wonderful editing!

I really enjoyed this, good to see more coming from this channel. Excitedly looking forward for more!

what an incredible video, you have a real talent for getting at the core of these ideas and showcasing the clear arguments which easily get muddled by technicalities

I'm super excited to watch this later when I'm done with work. Thank you for the amazing content!

I'm so happy that you've made another video, this makes my hyped to learn again. It's great motivation!

Content like this is why I still pay my internet bill. Thoughtfully presented, beautifully explained, and utterly fascinating even to a cynical math-o-phobe like me. Eighteen minutes well spent. I look forward to future content as a new subscriber. Bravo!

Glad I had my notifications on, Welcome back! Thanks for yet another informative video that is surprisingly easy to understand :DD

17:20 note that also loglogN is practically constant like the k^log*(n) since loglog(N) where N is the numbers of atoms in the observable universe is around 8. If N is the number of atoms in the observable universe then loglogN is actually smaller than 4^log*(N).

My goodness the explanation and visuals are amazing! Glad I came across this channel.

Thanks for explaining this. I vaguely remember running into this algorithm, but discarded it because it recursed without really reducing complexity. After watching your explanation, I realized that if I restrict the recursion depth, I might get something usable.

Thank you for educating us with this beautiful video of yours. The way you put them together is just perfect, thank you again.

Really glad you're back! Can't wait to see more of your content.

Glad to have you back! Some of the highest quality content on the platform

I have to say, I can't even remember how many times I have learned Big O notation already, but it's the first time in my life I heard about Linear speedup theorem. Like, it suddenly explained everything. I suddenly understand why the linear magnitude does not matter.

Amazingly beautiful review and info! Thanks for making this! All the best

I really love your videos so far, clear and somewhat concise Really instructive, thanks. I hope you make more

The Quality and The Content is top notch! Thanks for sharing!

Omg he published again!!!! The god returned yeeeesss Your content is so high quality, can't emphasize this enough.

Absolutely fascinating, please keep them coming.

1:01 In Russian it is better to use the word "сложение" instead of "дополнение" to denote addition.

I really like that the best multiplication algorithm uses the Ramanujan-Hardy number.

Fascinating algorithm and historical context. Thanks for sharing this and for explaining it so lucidly. For those who aren't old enough to remember the old days of computing, one of the reasons multiplication was of such interest is that early CPUs did not have a multiply instruction in hardware. They relied on repeated addition, so if you wanted 58 * 37 it was computed as 58 + 58 + 58 ..... (37 times total), or vice-versa. I'm not sure if the first computers even had the hardware smarts to swap the numbers so they added the larger number a smaller number of times. Repeated addition is often even slower than the O(N**2) elementary school algorithm, so computer scientists were eager for anything that could improve upon that. Also for the non-computer folks, Nemean makes the comment that subtraction is essentially the same problem as addition. You know from grade school that subtracting N is the same as adding -N, of course, but it might occur to you that -N is defined as -1 * N, which seems to imply a hidden multiplication step. Fortunately, since computers work in binary, we avoid that by using the "twos complement". In binary, this means flip every bit of the original number, which gives you the "ones complement", then add one. Adding the twos complement of N to another number, say M, is the same as computing M - N. Here's an example using 8-bit integers, a common size for early CPUs, to compute 100 - 35. 100 is 64 + 32 + 4, or 01100100 binary. 35 is 32 + 3, or 00100011 binary. Take the ones complement of 00100011 to get 11011100, then add one for the twos complement of 11011101. Adding 01100100 to 11011101 gives (1)01000001. The parentheses are around the carry bit, which in this situation we ignore (see note below). 01000001 is 64 + 1, or 65 decimal, the answer we expect. Even in very early computers, the operations to invert every bit (ones complement) and to add one (increment) were single hardware instructions, so the twos complement took at most two steps (and some CPUs had a single instruction to combine them). So subtraction, even on an early CPU with no subtract instruction, was not significantly more difficult than addition. The use of twos complement binary arithmetic does imply a need to keep track of that leftmost bit and being aware of whether it is being used as a sign (1 for negative, 0 for positive) or simply as another binary digit. Programmers can define "signed integers" which cut the value's range in half but allow negative numbers, or "unsigned integers" which allow the full range but cannot be less than zero. For instance, a 16-bit unsigned integer can be 0 to 65535, inclusive, while a 16-bit signed integer can instead be -32768 to +32767, inclusive. The CPU hardware, generally, handles the raw bits the same, but the programming language and compiler help the programmer avoid misinterpreting the data. I hope this side-trip into computer history and binary math is useful to readers who aren't computer specialists.

9:30 and 16:00 I think it would've been better if you used actual numbers and showed a practical example of the calculation instead of empty digit boxes/partially filled circle shapes, it would be easier to keep track on and follow what you're talking about. Since the video started with practical examples for the easier algorithms I also was expecting practical examples for the more complicated algorithms. Having to follow where you put which blank box or which abstract circle is filled by how much and trying to find out why you gave the circles these fill values while at the same time also trying to listen to what you are saying is rather irritating.

Really great explanations! And I really appreciate the overview of recent developments

This video is truly amazing, it mixes the beauty of computer science and math

This is FIRE! What a spectacular journey. This is how it should be taught. Can't wait for more videos from you!

That was a joy to watch. Thank you!

Thank you for a fantastically well put together video.

Thanks for making computational mathematics accessible. The last summary is pure gold.

I have seen fast multiplication on Commodore 64 (6502 processor without a built-in multiplier) based on a similar idea. a*b = ( (a+b)/2 )^2 - ( (a-b)/2 )^2. For all possible values of a+b and a-b, the square of a half is precalculated in a table; so for 8-bit numbers, 512 precalculated table entries are needed. This is easily a few times faster than trivial multiplication.

I feel like when you were talking about big O, there were some big aspects you missed. In particular, one of the big reasons big O is important is that it better measures how an algorithm scales to extremely large inputs. While the big O might not be able to tell you an exact runtime, it can tell you how that runtime changes when you change the input. For example, for an O(n) algorithm, doubling the size of the input make it take twice as long as before, while an O(n^2) algorithm will take 4 times as long, and an O(log n) algorithm will only take a constant amount of time more. The ways that different algorithms scale tends to be more important than any constant factors when n is extremely large. For example, the runtime of an O(n) algorithm might be like 10n, while an O(n^2) algorithm might be n^2/10; with small n, the O(n) algorithm is slower due to the high overhead (for n=4, the first algorithm is 40 while the second is 1.6), but as n increases, the difference in powers rapidly overcomes the constant factors (for n=10,000, it’s 100,000 versus 10,000,000, so the first is a hundred times faster). That’s why we talk about big O in algorithms; when the input is big enough that runtime is a concern, that’s what gives you a real idea of the runtime.

The cadence and tone of your voice is very pleasant to listen to. It reminds me of the JCS channel. Thank you very much for teaching me with such detailed and illustrative information.

i've never seen a youtube account with 3 videos that makes such high quality videos. seriously well done man

I love this subject, mainly because it is both quite recent and revolutionary, in a way, as well as rather easily understood by a teenager. Every now and again I talk about it to my students, and it is usually well received.

This whole video is incredibly interesting and explains lots of things very well, but I am laughing so hard at 17:00 . The deadpan delivery of that line “log star of the number of atoms in the universe... is five.”

i kind of understood this. thank you for this video. I often come back to your first video when i need inspiration how to change way of thinking when i search for answer.

Just loved the video man! awesome it was.

I'm not into computer science or even math, yet still here to watch video until finished.

This story about Kolmogorov and Karatsuba should be made into a film so that more people know it

Excellent discussion, thank you. Also what a mensch Kolmogorov is. Good to see, and thanks for telling us about it.

Outstanding explanation! Thank you!

Just a heads up - there's a terminology mistake at 1:00 . "Addition" should be translated as "Сложение" in Russian, while "Дополнение" in English would be "Complement" term from Set Theory.

I believe Karatsuba's algorithm is used in quantum computing as the current fastest/most efficient method of multiplication.

This was super interesting thanks for uploading, I will be watching you form now on

Nice video! Really liked the smooth animation

I looked at this video on a random whim and I'm glad i did! Very well explained video on a topic that can be difficult to follow. As others have mentioned, practical examples may have worked better than the colored blocks used, as this would allow the audience to follow along in an easier fashion. Thanks!

This algorithm is used in Java's implementation of BigDecimals (or BigIntegers ?) for very big numbers.

Even if this video doesn't blow up it is still amazing content, thanks!

Great video Please make more videos about algorithms and the history behind them perfecto

That was fascinating, thank you

Really cool seeing that discoveries in mathematics are still being done to this day :)

Amazing video…..so beautifully explained!

We learned the Karatsuba Method last week in Numerics, so this was an interesting new take on the way we learned it

Thank you ... great explanation ... interesting history ...

In the time when Kolmogorov was at the age of Karatsuba (when they met) there was no Fast Fourier Transform, but on the other hand Parseval theorem was already stated in the 18th century - kids read the books and study them ! :)

Great explanation of a complex subject ... well done!

we learn more in this video ...thnx 4 posting .

i thought this account exists just to post one vid and nothing else, nice to see a new upload!

Since your fast SQRT video I was waiting until your next lauch. This video gave me a thousand goosebumps, incredible! Good job

These series are addictive. Please more!!

This is one of the best explainers on fast multiplication algorithms out there. Much excellent!

Hmmm…reminds me of another algorithm dealing with linear programming (LP). LP is theoretically a N^x steps where x is the number variables. There is a Russian algorithm that has O(N) steps, but the slope of T (time) is so steep it might as well be a quadratic equation. I wrote a paper on this 40 years ago for one of CS Master’s classes after reading about it in a programming journal. The math was so obscure (or maybe the Russian was so obscure) that I had to go back to the original paper to get the algorithm correct. It was a fun project, as I was really interested in linear programming at the time. Seems I fell in love with CAD, though.

That was brilliant. And actually taught me new maths too.

Awesome video! I just found this channel now thanks to the algorithm. I'm subscribing right away!

I'm just discovering this channel now. Nice stuff!

Great video! And as a Russian-speaking person I want to notice that mathematical operation "addition" is called "сложение" not "дополнение". The term's you used meaning is "a minor member of a sentence, usually expressed as a noun". Best Regards!

Excellent video. I was always fascinated by these "dynamic programming" style algorithms. They appear everywhere. Of course I'm just a rando on the internet, but I think such a video on the evolution of matrix multiplication would be equally if not more interesting. Strassen (the same one you mentioned) played a role there too, and a similar (sort of) trick is used in that case. However such a video would probably be extremely long since it's still an open question. Then again, even this video covers an impressive scope.

Hahaha I loved your inverse FFT notation. Well played, well played.

Legend , Pls do more of these type.

I just had an assignment implementing school method addition, subtraction and then karatsuba. Although trivial I probably would have enjoyed it more having watched this video. Also, if anyone knows what software this guy uses for his visualisations it'd be greatly appreciated, I feel like some evolutionary computation concepts could make really good videos.

I smashed subscribe button. No doubt. What a quality algorithm explanation.

I really enjoyed your video, I can only encourage you to publish more content like this.

I loved this video! Inmediatly suscribed!

very interesting and informative video

Excellent video!!

18:00 - For smaller numbers... Lololllol 🤣 I love your delivery. Pure gold.

You delivered! Thanks this is awesome.

I love the super subtle humor here, hahaha. The explanatory visuals are also great even though I feel it still quickly becomes a lot to digest for us lower peasants. YT needs more of this. Thank you.

When some individuals multiply 6 digits numbers fast mentally, do they use any of those algorithms intuitively? Is something known about the phenomenon from that point of view nowadays? I know (from discussions with my friend mathematician), that usually those individuals are lacking logic till the extend that they can't do anything in the area of mathematics. Not always it is the case though. And probably the brightest example of combination of both skills (means lightning mental computations ability and logic) would be the famous mathematician John von Newmann. His mental computational abilities were such that people who had a lucky opportunity to communicate with him had impression that they are dealing with an extraterrestrial (he could add up series mentally and everything alike) and not a human. Thank you very much for this great film.

Fascinating! Keep 'em coming! :D

Thank you for showing me how the computer works using algorithms.

Using boxes for some strange reason instead of actual numbers made this needlessly more confusing to follow.

Omg, Kolmogorov is the real MVP! So impressed that he actually went so far for a student rather than claiming all the credit for himself!

Thank you for posting this.

Awesome video so far. 48 seconds in and started with a verse, had some code in there... Kool stuff!