| ▲ | AnotherGoodName 4 hours ago |
| The site is taking forever to load but complexity theory doesn't really have the hard walls at things like O(nlogn) that people seem to think. You can always always tradeoff time vs space. It's actually a fundamental part of complexity theory called Blums speedup theorem by the guy who invented complexity theory no less. As an example you give me an nlogn solution and i take a portion of the inner loop, put it in a lookup table (who cares if the lookup table is 2^n in size, that's not what we're measuring here since we're specifically measuring time not space) and suddenly i have something better than nlogn since that lookup table is O(1). That's not to say there's some merit to all of this. It's just that people shouldn't be as astonished by these results as they are. It's complexity theory. There's literally no such thing as a hard wall where something is nlogn in time and can't get lower. It can always get lower. But you need both the time and space complexities together to reason about this. |
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| ▲ | tgv 22 minutes ago | parent | next [-] |
| > You can always always tradeoff time vs space Not always, isn't it? The theorem says that there exists a function that can be sped up, but not that every function can be. A trivial counter example would be the identity function on a TM. It's simply a HALT/ACCEPT. There's no way to reduce that. The space/time trade-off is also not infinitely flexible, since it takes time to fill space. And lookup tables have to be constructed at run-time, or they'll be of finite size, and hence only a constant speed-up. Perhaps you were thinking of that theorem, the linear as wikipedia calls it? https://en.wikipedia.org/wiki/Linear_speedup_theorem |
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| ▲ | cma 4 hours ago | parent | prev | next [-] |
| Wouldn't it take an infinite lookup table to do this satisfying any two integers and still get an order-complexity speedup? The integers can be any size. You can't specify an arbitrarily variable sized lookup table in the normal formulation of these problems without computing the lookup table as part of the computation. |
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| ▲ | sroussey 2 hours ago | parent | next [-] | | You seem to under the impression that we don’t currently use lookup tables for multiplication. But your right in that we don’t have a huge LUT as it’s not worth it. Small one in the ALU though, for partial products to add can be worth the additional wire length. | |
| ▲ | AnotherGoodName 4 hours ago | parent | prev [-] | | Never infinite if you have bounds on integer size. For n bits of input it's always possible to have a 2^n space complexity lookup table that's O(1) speed since 2^n covers all possibilities of n bits. You don't need to go to these extremes regardless though. For small speedups take an nlogn algorithm and just move a very small part of the inner portion out in a similar tradeoff. Eg. "Every nth input is now part of a lookup." If you look at the openAIs <nlogn results there was a series of small improvements of this type (we take this part out, precomputing it, and likewise with this part and that small portion, aha! <nlogn!). I finally loaded the link above and it's 262million lines of such tricks in fact. | | |
| ▲ | vlovich123 3 hours ago | parent | next [-] | | I think the thing non-mathematicians and non-CS people who only know complexity theory vaguely need to remember is that complexity theory has nothing to do with reality. A 2^64bits lookup table only has O(1) lookup if you ignore the impossibility of creating a machine that has 2 exabytes of storage where all those bytes have the same access time. Or you'd have to redefine O(1) to be “at most the worst case”. But strictly speaking from the mathematical definition it’s O(1). Which is why no one in practical engineering takes most complexity results seriously as the entire field maps to practice poorly precisely for these reasons (all our machines use fixed integer math and have limited tape storage) | |
| ▲ | bluecalm 3 hours ago | parent | prev [-] | | The whole point of this problem is that there are no bounds on integer size. The whole point of complexity theory is that there are no bounds on input size. | | |
| ▲ | AnotherGoodName 2 hours ago | parent [-] | | Sure but 2^n space complexity to get O(1) time complexity works for all n even infinite. Thats the end state of Blums speedup theorem in either direction and it doesn’t limit to any bounds. I just stating a limited of bounds above for the sake of giving understanding. Complexity theory doesn’t care that 2^n space complexity doesn’t work when n is infinite anymore than it cares that nlogn also doesn’t work when n is infinite. It’s not the point. | | |
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| ▲ | BrtByte an hour ago | parent | prev [-] |
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