Yet another Hacker News

The recently proposed o(n log n) integer multiplication algorithm [pdf]

55 points craigcitro 2 hours ago 28 comments

flebron 1 hour ago | parent

Note in this case the title says "O(n log n)", but the important part is this is "o(n log n)", not just "O(n log n)'. The former is a strict subset of the latter. We already knew how to multiply two n-bit integers in O(n log n) time, via the FFT.

onionisafruit 1 hour ago | parent

For those of us uninformed, what’s the difference between O and o?

theokrueger 1 hour ago | parent

Big O basically drops constants

O(2n+1) = O(2n) = O(n)

o(2n+1) > o(2n) > o(n)

Both define upper boundary, but big O can read like "the function is generally proportional to ___" and little o can read like "the function is never more than _____"

bobmarleybiceps 1 hour ago | parent

that's not the convention I know... (from CLRS algorithms book).

Big O is sort of like <=. little o is sort of like <.

https://dl.acm.org/doi/10.1145/1008328.1008329

npinsker 1 hour ago | parent

yes, GP is slightly off, but the idea of <= and < is accurate.

O(n) is: "there exists a K such that, for sufficiently large n, K*n is always larger".

o(n) is: "for every K, for sufficiently large n, K*n is always larger". (So o(2n+1) and o(2n) and o(n) are the same thing.)

boothby 1 hour ago | parent

If f is O(g) and o(h) then

  Lim f/g -> C a constant

And

  Lim f/h -> 0

appplication 1 hour ago | parent

Lmao! If someone is asking the difference between big and little o notation this is absolutely not the right way to clarify it.

If you were joking then maybe you got me

boothby 1 hour ago | parent

I do hope you can find and reattach your ass; that does sound painful.

adgjlsfhk1 1 hour ago | parent

> Lim f/g -> C a constant.

This isn't quite correct. O(g) doesn't require the limit to exist.

Ygg2 1 hour ago | parent

Both are upper bounds. But small o is strictly smaller than upper bound. Big O is not greater than upper bound.

https://stackoverflow.com/questions/1364444/difference-betwe...

ruined 1 hour ago | parent

f(n) = O(g(n)) means as n increases, f(n)/g(n) must not grow beyond a constant

f(n) = o(g(n)) means as n increases, f(n)/g(n) must shrink

jhanschoo 1 hour ago | parent

Nitpick:

s/must shrink/must eventually shrink, after finitely many n/

Equivalently, must go to zero in the limit.

Davidzheng 33 minutes ago | parent

depends on what you think shrink means. If you mean decreasing, that's false even eventually.

bobmarleybiceps 1 hour ago | parent

Knuth originally defining them https://dl.acm.org/doi/10.1145/1008328.1008329

craigcitro 1 hour ago | parent

ty, fixed. (ahh, copy paste.)

ComplexSystems 1 hour ago | parent

Roughly speaking, O(n log n) means "grows no faster than k(n log n) for some k", whereas o(n log n) means grows strictly more slowly* than k(n log n) for all* choices of k.

jdb1729 1 hour ago | parent

> The new result does not directly imply new complexity bounds for polynomial multiplication over a fixed finite field, say multiplication in F2[x]. The state of the art for this problem is an algorithm that multiplies polynomials of degree n in time O(n log n 4^log*(n)) [HvdH19].

@platypii and I obtained a Lean proof of the same improvement for DFT over prime-order finite fields (yielding polynomial multiplication via the convolution theorem): https://github.com/platypii/integer-mult-bounds-lean/blob/ma...

ajkjk 1 hour ago | parent

I really enjoy the writing style here.

edit: I mean this seriously. This kind of thing

> Some of my colleagues believe that this is a nefarious Company, whose only goal is profit, if necessary at our community’s expense. Perhaps there is some truth to this, but I think it is missing the larger point. The fact is that there now exists an automated system that can perform mathematical research — at least the part that involves “solving hard problems” — far better than any human. Now that such a system has been demonstrated, it is clear that results like these were bound to start appearing soon anyway. I do not blame the Company for getting there first. Who got there first is an irrelevant detail, a historical footnote.

is very pleasantly honest and well-thought-out.

yaoke259 1 hour ago | parent

Awesome write up and intro by a fellow Sydney-sider, kept me hooked! As someone who's not a maths wizard, the technical details flew by my head but the prose allowing me to empathise deeply with those in the field was very enlightening. I never realised how big of a deal the maths dump was and how much it has shaken up the field. I guess we are all just trying to figure out this whole "intelligence on tap" thing together.

m00dy 54 minutes ago | parent

You don’t need to be a math wizard. Just upload the paper to an AI and ask, “Explain this like I’m five.” Then, with each follow-up prompt, increase the age and level of expertise you want it to assume.

trostaft 32 minutes ago | parent

That algorithm won't converge anytime soon. A pre-conditioner (i.e. at least some apprentice wizardy) helps.

Kapura 51 minutes ago | parent

> I have not actually read the paper. In fact, there are three closely related papers [TC26a], [TC26b], [TC26c], and I have not read any of them. I tried, but soon gave up. The math(s) dump does not appear to be strong on exposition. Instead, I asked Fable to explain [TC26a] to me. Fable seems to understand the argument and thinks that it is correct.

this does not feel like a healthy place to be

jrflo 44 minutes ago | parent

Why not? If mathematicians in the field cannot understand the papers, but tools at our disposal can, why not use them? What is "unhealthy" about that? Just seems like it's the reality of the world now.

nico 7 minutes ago | parent

Yup, and it always has been, we don’t even understand how our bodies or minds work, yet we use them everyday

The amount of unimaginable complexity contained in a seemingly very tiny amount of matter to us, is incredible, and it’s all around us in nature

We’ve never had the capacity to fully comprehend neither ourselves nor the environment we live in. But we do our best, and we do survive and sometimes even thrive

6879346626 39 minutes ago | parent

Fable is the "gadget" of language ELI5.

Honestly it is not that much different from the divine inspiration if you have ever experienced it.

I do not know how you view the world or walk through it but language in general is simply subject to interpretation to me, the arrangement of words for the sentence is inspiration in and of itself - though the majority of time it amounts to minimum jumps.

krackers 28 minutes ago | parent

Fable was able to help a human understand the paper though, so the situation is quite a bit better than if we just had a lean proof that was apparently correct but for which all attempts at explanation were "hallucinations". There doesn't seem to be an apparently obstacle to a human fully understanding the proof in several months of focused study.

jdb1729 38 minutes ago | parent

> It would be good to have a formalised proof of the o(n log n) result, for example in Lean. As far as I can tell, one does not yet exist.

The result is fully formalized in Lean by @platypii and myself (on Mathlib's TM2 machine), with ongoing improvements to the exponent (currently at 1-2^-14.38): https://github.com/platypii/integer-mult-bounds-lean

Contributions are welcome!

MiroslavPokorny 27 minutes ago | parent

> Executive summary (longer version) > The bottom line is that I think the claimed result is plausible. I have not actually read the paper.