Author: Christophe Theron
Date: 15:02:44 03/26/01
I have seen a lot of messages recently from people who want to buy a dual
processor computer or motherboard and are asking what is the best around.
Out of curiousity, I sat down in front a sheet of paper, grabbed a calculator
and started some computations.
I wanted to have a better idea of how much better is a dual processor computer
for chess, over a single processor one.
Here is the result of my homeworks:
HOW MUCH FASTER ?
-----------------
Experts (Bob included) say that with current parallel algorithms, the SMP
efficiency ratio for a dual processor computer running a chess program is 1.7.
That means that a biprocessor computer (a "dual") is going to compute
1.7 times faster than a single processor computer, assuming a SMP chess
program is running on both and that the processors are identical.
From this we can estimate the ELO difference between the two computers.
Assuming the generally accepted fact that doubling the speed of a computer
adds 70 ELO points to its playing strength (and this might actually
be less when we go to higher and higher frequencies), the formula
to compute ELO increase from speed increase is:
ELOdiff = 70 * log( SpeedRatio ) / log(2)
So if SpeedRatio=1.7, we can expect a 53.6 ELO points increase (rounded to
the first decimal) from a chess program running on a dual machine rather than
on a single processor one. Keep in mind: assuming same processors and same
processor frequencies.
****************************************************************************
** We expect a 53.6 ELO points difference between a single processor **
** and a dual processor computer (both running at the same clock freq.) **
****************************************************************************
WHERE CAN I BUY ONE ?
---------------------
Unfortunately, while it is possible to get the highest available frequencies
for single processor computers, it seems that it is not possible (at a non-
astronomic price) to get these frequencies for dual motherboards.
As I understand from discussions on CCC on the topic, while you can easily
get an AMD ThunderBird 1.2GHz single processor machine for a reasonnable price,
it seems to be almost impossible to get a dual MB with this processor. Looks
like you'll get a dual AMD TB 1GHz at best, if you are ready to put your
money on it (but then forget about the latest and sexiest Nokia, the money
for it will go into the dual :).
So now we should re-compute the speed ratio to get real world figures.
On one hand you have a 1.2GHz single processor machine, on the other hand you
have a dual 1GHz one. The dual is much more expensive than the single.
SpeedRatio = 1.7 * 1 / 1.2 = 1.417 (rounded to 3 decimals)
The corresponding ELO increase is:
ELOdiff = 70 * log( SpeedRatio ) / log(2) = 35.2 (rounded to first decimal)
****************************************************************************
** So the expected ELO difference between a single 1.2GHz processor **
** and a dual 1GHz one is about 35 elo points. **
****************************************************************************
35 ELO POINTS BETTER, REALLY ?
------------------------------
There's something else that should be taken into account. Unless your dual
pet has 2 HD controllers and 2 independant hard disks, your processors are
actually going to share a single hard disk when they begin to acess endgame
tablebases.
So in the endgame, you can expect tablebases to be less efficient (twice as
less efficient) on a dual computer. Experts say that tablebases generally
increase the strength of a program by 20 to 40 ELO points. From this it is
possible that you lose 10 ELO points when you are using a dual processor
computer with only one hard disk.
****************************************************************************
** So in the end the ELO difference between the fastest **
** single processor PC and the fastest available dual **
** processor PC might be 25 ELO points, maybe less if **
** doubling the speed does not give a 70 ELO increase **
** on nowaday computers. **
****************************************************************************
NOW LET'S COUNT OUR BEANS
-------------------------
What does it mean, 25 ELO points ? What are we talking about exactly ?
A difference in ELO points in real life turns into a winning percentage.
That's exactly what ELO means, and how it is computed.
For winning percentages above 20% and under 80%, there is an approximated
formula that works pretty well:
ELOdiff = ( WinPercentage - 50 ) * 7
From this you can deduce how to compute WinPercentage if you have the ELOdiff:
WinPercentage = ELOdiff / 7 + 50
If ELOdiff=25, then WinPercentage = 53.57% (we are between 20% and 80%
so our above formula applies).
So we are talking about a difference of 3.5 games each time you play 100.
****************************************************************************
** When you play 100 games with your dual 1GHz against **
** your single 1.2GHz, you can expect the dual to win typically **
** by a 3.5 games margin. **
****************************************************************************
HOW MUCH GAMES SHOULD I PLAY TO DEMONSTRATE THAT MY DUAL IS BETTER ?
--------------------------------------------------------------------
However, you must also take into account randomness. As you know, a chess
match has some randomness (some luck is involved, either in the choice of
the opening, of during the game itself in very unclear positions).
For example, if you want to be 90% sure about the result of your 100 games
match, it is wise to account for a +/-6.5 percent margin of error. If we take
the aforementionned match, which is likely to end in a 53.57% win for the dual
(in a perfect world! :), then it means that you will routinely get results
between 47% (the dual actually loses!) and 60% (the dual wins clearly).
So obviously, running a 100 games match dual against single is not enough
to demonstrate the superiority of the expensive dual.
Let's have a look at statistical tables. What we need, obviously, is to have
a margin of error under 3.57%, so every time we play a match dual against
single, the dual has the most chances to win (more than 50% chances to
win).
Here it is: you need to play at least 400 games. For 400 games, the margin of
error for 90% reliability is 3.34% (for 300 games it is 4.72%, which is still
too high).
****************************************************************************
** If you want the dual to win 90% of the matches against **
** the single processor computer, then you need to play **
** 400 games matches. If you play shorter matches, the **
** single processor computer actually has significant **
** chances to win the match. **
****************************************************************************
Christophe Théron, March 26th, 2001.
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