At most you use them for stuff like P vs NP, because Turing machines are 'only' polynomially slower than register machines, so it doesn't matter.
But when you are talking about linear vs quadratic on something resembling real computers, Turing machines are the wrong model.
There is, of course, the very rich field of Numerical Analysis which deals with how we actually represent these things in real life, but for a starting point assuming this sort of thing is par for the course.
Err... doesn't that completely invalidate the big-O analysis!?
I suspected that these Open AI "proofs" of sub-nlogn times were cheating somehow, by sweeping something under a rug. This seems to be it!
This is a common thing in complexity, and I think even the abstract says it is talking about “length n” and not “magnitude n”
As the argmin suggests Writing computational complexity in Big O isn't great here.
since we generally see the interms of no.of complex additions and no.of Multiplications in the FFT algorithm.
p11p•44m ago
PS. I can have this view while simultaneously thinking that Ben is a badass.
skybrian•36m ago
It seems like rather than coming up with more "galactic" algorithms, a more useful thing to do might be to come up with a replacement for big-O that's not so exploitable, and to convince theoreticians to play that game instead.
frotaur•25m ago
Much of our 'practical' understanding of algorithms also in part came from theoretical advances.
adastra22•10m ago
eru•18m ago
You don't need to care. But every so often breakthroughs in theory can be turned into breakthroughs in practice.
> It seems like rather than coming up with more "galactic" algorithms, a more useful thing to do might be to come up with a replacement for big-O that's not so exploitable, and to convince theoreticians to play that game instead.
There's plenty of replacements.
Btw, we already have the optimal (in big-O terms) algorithm for every NP hard problem. We just don't know whether it runs in polynomial time or takes longer. And, yes, that algorithm called Levin's Universal Search is a galactic algorithm. Simplifying a lot, Levin's Universal Search does some clever interleaving to essentially try all possible algorithms in sort-of parallel.
Lo and behold, even the theorists you complain about don't see that 'loophole' as a solution, declare victory and go home. They still keep researching.
cpa•16m ago
Doing theory is what gives you a chance to push the envelope to places you can’t even imagine when you’re knee deep in real world problems. And proving something is possible (if impractical) gives an incentive to find a better algorithm and may attract others.
For example see the first graph here: http://klotza.blogspot.com/2016/06/the-fall-of-twin-prime-ga...
adastra22•11m ago
asdff•5m ago
ViscountPenguin•8m ago
" In 1973, Jacques Morgenstern proved that you couldn’t get a Fourier transform faster than ½ n log n if you required that complex coefficients used in the computation had bounded magnitude"
I highly doubt this is going to somehow inspire people to make faster algorithms than the mathematical bound...
I have no problem with pure mathematics, my training was in it; but it seems very possible to me that most people funding theoretical computer science (as opposed to pure maths) probably assume the work has a more realistic computational model backing it up.
modeless•6m ago
nullc•4m ago
It's extremely hard to know what mathematics may be useful in the future or not.