No fastest algorithm

Dragonfall · May 18, 2009

It has been conjectured that there is no fastest algorithm for multiplication, among other things. Can somebody give me an example of something that provably has no fastest algorithm for?

CRGreathouse · May 18, 2009

Here's a partial answer -- the best I could do at the moment. Coppersmith & Winograd show that there is no fastest "Strassen-type bilinear" matrix multiplication.

D. Coppersmith and S. Winograd, "On the Asymptotic Complexity of Matrix Multiplication", SIAM J. Comput. 11:3 (1982), pp. 472-492.

Dragonfall · May 19, 2009

But is there a simple problem, even something completely artificial, that shows this CAN happen?

CRGreathouse · May 19, 2009

Matrix multiplication not simple enough for you? Maybe you need to say what you mean by 'simple'.

Dragonfall · May 19, 2009

Matrix multiplication has not been proven either way, and in fact according to http://arxiv.org/abs/math.GR/0511460, if a certain conjecture is true, then there is a fatest algorithm that's in O(n^2).

Twinbee · May 24, 2009

Surely there has to be a 'fastest' algorithm, or at least an algorithm that's joint fastest. Perhaps that's what it means?

Dragonfall · May 24, 2009

Here are some heuristics on why there might not be a fastest algorithm for something. For any given computable task, there are always infinitely many algorithms to solve it, because you can always add "futile" steps that just waste time, adding to the complexity. Therefore, there always exists an infinite ascending chain of slower and slower complexity classes which contain these algorithms. The question is, is it possible that infinite descending chains also exist?

EDIT: I want something computable, because any uncomputable problem trivially has no fastest solution.

Moo Of Doom · May 24, 2009

Look here, about 2/3 of the way down the page (you can just Ctrl+F for fastest algorithm).

http://www.scottaaronson.com/democritus/lec5.html

Indeed, there may not be a fastest algorithm for certain problems.

Twinbee · May 24, 2009

Oh right I think I see. But since there only a finite number of ways of arranging code, then the fastest algorithm has to exist anyway. Unless you're talking about infinite length of code as well?

CRGreathouse · May 24, 2009

Twinbee said: ↑

Oh right I think I see. But since there only a finite number of ways of arranging code, then the fastest algorithm has to exist anyway. Unless you're talking about infinite length of code as well?

?

The number of finite programs is infinite.

CRGreathouse · May 24, 2009

Dragonfall said: ↑

The question is, is it possible that infinite descending chains also exist?

Of course. The different types of Toom-Cook multiplication are an easy example. Of course there are algorithms that outperform all of them, so they don't satisfy your original question.

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No fastest algorithm

Dragonfall

CRGreathouse

Dragonfall

CRGreathouse

Dragonfall

Twinbee

Dragonfall

Moo Of Doom

Twinbee

CRGreathouse

CRGreathouse

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