Author: Dann Corbit
Date: 12:50:10 03/29/04
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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. ;-)
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