Many 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...

I guess it can be made even faster now in theory.

• emil-lp

  today at 5:53 PM

  The kidney exchange problem isn't bipartite matching but a cycle packing problem (or disjoint cycle cover).

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.

• gignico

  today at 2:22 PM

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

  • amluto

    today at 3:36 PM

    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.

  • amirhirsch

    today at 2:49 PM

    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

    • ZeroCool2u

      today at 4:10 PM

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

      • amirhirsch

        today at 5:15 PM

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

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

• osti

  today at 3:45 PM

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

  • adgjlsfhk1

    today at 4:10 PM

    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

    • osti

      today at 4:21 PM

      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?

      • chowells

        today at 5:08 PM

        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".

        • osti

          today at 5:18 PM

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

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