Integer multiplication is very unexpected, I think most people believed in the n log n lower bound!
replyNote that these are all preprints. None are verified.
replyOther than the by the lean certificate you mean.
replymany of these are not accompanied with leanslop
replyLean has bugs & proofs of ⊥ that have gone undetected previously.
reply> We give a deterministic algorithm that multiplies two n-bit integers in O(n (log n)^(1−κ)) worst-
case time, with κ = 2^(−182).
replyLMAO, I don't think I ever saw such a small number in a CS result.
Yeah its ridiculously small, but any improvement on n log n is wild.
replyLike there is somehow redundancy in a fourier transform that makes it sub Linearithmic?
Which low and behold ->
130. Fourier transforms below n log n.
They also separately give algorithm for Fourier transform over complex number faster than O(n log n)
replyWikipedia just told me there's a galactic algorithm for integer multiplication in O(n log n) based on FFT so I'm guessing those two proofs are related.
replyMultiplication is a lot like convolution, so the connection is natural.
replyIt fascinates me that there's something like this in something as solid and rigid like matrix multiplication. What causes something so rigid to break apart and "leak" at very large scale? Why does the "optimization" appear to be very, very small? Why does galactic algorithm exists? I can't imagine long division suddenly breaking apart after a billion digit, the structure seems very stable? I have heard before that matrix multiplication is apparently optimize-able at very, very large scale.
replyDoes anyone have an intuition to what causes it? What happens at these large scale (or very small)?
One way to think about it is that the classical algorithms are the ones that are fast for small numbers. Galactic algorithms often work for small inputs, it's just that to be faster you need big inputs. A common case of this is a requirement that log(n)<<klog(log(n)). If k=100 then this algorithm will take huge sizes to win
replyI am fully braced for it to be a https://en.wikipedia.org/wiki/Galactic_algorithm
replyVery surprising result though! Multiplication is easier than sorting.
Then 'n' means kind of different things for sorting vs. multiplication though. For example for sorting we assume constant time comparison, which doesn't make sense inputs of O(n) bits
replyIf you sort n k-bit items for a total time of O(nk logn), that scales more poorly in n than multiplying n-word integers. Of course if k is constant you can do radix sort, but I genuinely don't know under what conditions radix sort is more/less galactic than this multiplication algorithm.
replythis alg is way more galactic than radix sort. radix sort often wins in the hundreds of elements. the nlogn multiplication requires numbers with more digits than atoms in the universe (although that could probably be brought down a lot)
replyAh thanks for pointing this out, for some reason I had always equated radix sort and bucket sort (with 2^k buckets) in my head. But I learned today that this isn't true!
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replyIt would be absolutely unbelievable if such an improvement were practical.
replyCan anyone ELI5 to make it make sense?
replyIt seems n would have to be unimaginably large for this to make any difference. What changes about multiplication / FFT at large enough size ?
I guess nobody expected that it did before this result.