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Comment by amirhirsch | Hacker News Reader

This is an awesome result.

For those unfamiliar: NC is the class of problems which can be solved in polylogarthmic depth with polynomial number of logic gates. It is unproven if NC != P similar to P != NP.

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Yes, but logic gates with constant fan-in, crucially, otherwise that's called AC.

Yes.

There is a beautiful proof of the disjunction between AC0 and NC showing parity cannot be done in AC0 using harmonic analysis of Boolean functions

This one? https://www.cs.huji.ac.il/~nati/PAPERS/lmn.pdf

https://en.wikipedia.org/wiki/Switching_lemma

That paper is in the wiki refs but Hastad’s original is from 1986

I never studied these specific classes, but my immediate intuition is that an n-input fan-in AND or OR gate can be reduced to a tree of 2-input gates with depth O(log(n)), which preserves polylog complexity, so surely AC = NC.

Wikipedia agrees :)

If you specify the exponent of the log, you get a different answer.

So is it a class of problems that can be parallelized well?

no (in both directions). lots of np/exp problems paralize well and you can be in NC and parallelize really inefficiently (e.g. you can get a 10x speedup, but you need 1000000x the hardware). the better framing is that NC is the class of efficient algorithms that can be sped up near arbitrarily by parallelization

Hmm your last sentence seems to exactly agree that it's a class of algos that parallelize well? What does sped up arbitrarily mean? It's still polynomial speed up right?

It's a difference of degree. People expect something that "parallelizes well" to show near 1-to-1 speedup. Double the hardware, double the speed. This is "you can always speed it up, but the hardware requirements can increase at any polynomial rate".

Ah got it. Reread previous comment and that makes sense.

Yeah it's more of "on a hypothetical infinitely parallel computer, you'll get a big speedup'.

Which is still useful if you can prove a problem is in NC. It's just not quite as strong as people make it out to be.

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