Author: Dann Corbit
Date: 13:58:58 03/29/04
Go up one level in this thread
On March 29, 2004 at 16:35:29, Dann Corbit wrote: >On March 29, 2004 at 16:22:21, martin fierz wrote: > >>On March 29, 2004 at 15:50:10, Dann Corbit wrote: >> >>>On March 29, 2004 at 10:17:18, martin fierz wrote: >>> >>>>aloha! >>>> >>>>i was discussing this somewhere in a thread, but thought i'd like to make this >>>>question more visible in the hope of getting a good answer: >>>> >>>>everybody knows that with plain alpha-beta, a fixed number of moves N per node, >>>>and perfect move ordering a search to depth D needs >>>> >>>>nodes(depth) = sqrt(N)^(D/2) nodes. >>>> >>>>with absolutely imperfect move ordering it needs >>>> >>>>nodes(depth) = N^(D) nodes. >>>> >>>>a typical chess program gets something like 90% move ordering in the sense that >>>>if a cutoff move exists, it will search it as first move in 90% of all cases. >>>>here's my question: >>>> >>>>can anybody give an estimate for nodes(depth) as function of this move ordering >>>>parameter? obviously, this would also depend on when you find the best move in >>>>those cases where you don't find it first. any kind of model is acceptable, e.g. >>>>you always find it on 2nd, always on sqrt(N)th, always last, at a random number, >>>>whatever. i'm just interested in the general behavior of nodes(depth) as a >>>>function of the cutoff-%age. >>>> >>>>i'd be extremely surprised if nobody ever estimated this, so: has any of you >>>>ever seen or calculated such numbers, and if yes, what do they look like? >>>> >>>>and is there any theory how this would apply to a modern chess program with >>>>nullmove and extensions instead of the plain A/B framework above? >>>> >>>>basically this question of course means: do you really gain anything tangible >>>>when improving your MO from say 90% to 92%? >>> >>>I have not done the math, but I am guessing no matter what king of move ordering >>>you have (purely randome or the pv move every time) you will get something like >>>this: >>> >>>nodes = some_constant * sqrt(mini_max_nodes) >>> >>>If you have random move ordering, then the constant will be very large. >>>If you have perfect move ordering, then the constant will be very small. >>> >>>You will never get worst case unless you try very hard to achieve it. >>>It might be possible to degenerate to mini-max (or very close to it) but you >>>will have to choose the worst possible move at every single turn except the >>>leaves. I doubt if anyone can do it. >>>;-) >> >>i disagree with your formula. it is definitely not some_constant. it is a >>constant between 1...sqrt(N) taken to the power of D/2. else there would be no >>point in improving move ordering, or at least, not as much as there is :-) > >What if the constant is one trillion? > >If you choose the node at random, you will still find the right one by random >chance on the first try 1/n times (where n is the number of moves) and on the >second try 1/(n-1) times, etc.. On average, we won't have to try more than half >of the nodes to find it (the best one). This will cause a huge reduction in >the number of nodes. > >As you can see, you would have to put forth a stupendous effort to cause minimax >behavior. Consider also that the 2nd best nodes will prevent lots of searches (all except the best node), as well as the 3rd best (all except the top 2 nodes), etc. So alpha-beta even with very bad move ordering will still cause a huge number of cutoffs.
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