5 ms·
Bipartite Matching Is in NC
- kevinten10 4mo ago[flagged]
- amirhirsch 4mo agoThis 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.
- gignico 4mo agoYes, but logic gates with constant fan-in, crucially, otherwise that's called AC.
- amirhirsch 4mo agoYes. 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
- ZeroCool2u 4mo agoThis one? https://www.cs.huji.ac.il/~nati/PAPERS/lmn.pdf https://www.cs.huji.ac.il/~nati/PAPERS/lmn.pdf
- amirhirsch 4mo agohttps://en.wikipedia.org/wiki/Switching_lemma https://en.wikipedia.org/wiki/Switching_lemma That paper is in the wiki refs but Hastad’s original is from 1986
- amluto 4mo agoI 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.
- fleahunter 4mo ago[flagged]
- osti 4mo agoSo is it a class of problems that can be parallelized well?
- adgjlsfhk1 4mo agono (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
- osti 4mo agoHmm 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?
- chowells 4mo agoIt'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".
- osti 4mo agoAh got it. Reread previous comment and that makes sense.
- dragontamer 4mo agoYeah 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.
- vintermann 4mo agoMany years ago, on Boardgamegeek, there was a game trading system called "Math Trades", where participants would list a number of trades they were willing to make, and they ran some complicated calculations to figure out how to let as many as possible trade. CS professor Chris Okasaki, known for his book on purely functional data structures, also played board games and he came across this phenomenon. He realized that it could be modelled as a bipartite matching problem, and wrote a MUCH faster program to manage math trades. https://okasaki.blogspot.com/2008/03/what-heck-is-math-trade.html https://okasaki.blogspot.com/2008/03/what-heck-is-math-trade... I guess it can be made even faster now in theory.
- emil-lp 4mo agoThe kidney exchange problem isn't bipartite matching but a cycle packing problem (or disjoint cycle cover).
- mirashii 4mo agoThe math trades still happen regularly at cons, e.g. Origins had one just last week.
- sigbottle 4mo agoChris okasaki! Was into functional data structures in college, great book and great dude
- throwaway81523 4mo agoI don't think this new result is supposed to be a speedup. It might even be slower than the existing method. Rather, it's a way to get rid of the random number generator that the old algorithm relied on, so it's deterministic unlike the old way. I'm not even sure that it's guaranteed to find the answer, as opposed to finding it with high probability. It's mostly of theoretical interest except for some possible niche applications, I'd say. For a math trade type of problem, you'd just go ahead and use the old algorithm with an RNG. Another famous result of this type was AKS primality testing. Randomized algorithms like Miller-Rabin were known for a long time, very reliable, and quite simple to implement, but AKS was an important theoretical advance because it didn't use random inputs. I think everyone still uses Miller-Rabin in practice.
- deleted 4mo ago[deleted]