Author: Dann Corbit
Date: 15:12:29 03/29/04
Go up one level in this thread
On March 29, 2004 at 18:06:22, martin fierz wrote: >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? > >so it's one trillion, then what? >let's try: > >i want to compute your nodes for A/B to depth 2. you say: > >>>>nodes = some_constant * sqrt(mini_max_nodes) >and the constant is one trillion => > >nodes = 10^12*sqrt(35*35) >> minimax_nodes. > >do you see now why your formula is plain wrong? i mean, of course you can define >your constant for every depth, but then it's not a constant :-) > >and the whole point is that this constant *does* scale with depth in a form in >which you can give an analytic expression for it, so why not do so??? > >cheers > martin > > >>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. It was a number from a hat. Look here: http://www.brillianet.com/programming/artificial_intelligence/tutorials/tut1.pdf at figures 3-6 You will see at a glance that every change from random node selection (Hash, history, etc.) all resulted in linear speedups. So the study agrees with my wild guess.
This page took 0 seconds to execute
Last modified: Thu, 15 Apr 21 08:11:13 -0700
Current Computer Chess Club Forums at Talkchess. This site by Sean Mintz.