How many heats?

[dknowles67]

I'm trying to collect some data to show what I believe to be true.
The number of heats required for a "double-elimination" type race will actually be more than some of the PPN, or PN lane rotation methods.
I have the data for the PPN/PN/Stearns methods.
How do you calculate the number of heats for a double-elimination method?
In particular, how many heats would there be for 3,4,5,6,7...25 cars?
( I can probably sit down and figure out the 3,4,5, but by the time you get to 25 cars - that's a lot of work!).
I'm assuming someone else has already figured it out?
Oh yeah, I'm talking about a 3 lane track, where all 3 lanes are used.
The point I'm trying to prove is that using some lane rotation type methods will not take substantially longer than double elimination does.
Thanks.

[Stan Pope]

Look at http://members.aol.com/standcmr/nelimsim.html. This page accepts your parameters, runs through the computations, and displays progress. The right hand column tells the cumulative number of heats.

The basis is no-chart elimination, so charted elimination may differ somewhat, but not badly.

Count also assumes that bye-heats will be run.

[dknowles67]

Thanks, I figured that would be easy.

[dknowles67]

Ouch!
For 25 cars, the PN for 3 lanes is 100 heats!, and CPN is 200!
A Double-Eliminaton would only be 28 heats.
(Triple-Elimination would be 46 heats).
This isn't helping my case much.
It looks like once you get to more than 9 cars, DE is around 1/2 as many heats as PN.
We are trying to hand out 1st-3rd place trophies, so to be fair we really need to do Triple Elimination instead of Double.
This year we had 25 Webelos.
Even at a rapid 40 seconds/heat, PN would take over an hour.
I don't think we are (or can get) organized enough to get to Stan's 34 second/heat rate.

[MERuhl]

You don't need to run a complete perfect-n bracket. In my opinion, a one-round partial perfect bracket beats a double-elimination bracket for fairness and maintaining crowd interest.

If you run just one round of partial-perfect with 25 cars on 3 lanes, you will run 25 heats, with every car racing 3 times, against 6 different cars. Here's why I feel this is better than a double- (or even a triple-) elimination bracket:

1. It's more fair. Each car runs the same number of races, and they all run once in each lane. If one lane is slower than the others, you don't unfairly handicap anyone. Plus, since every car runs the same number of heats, you reduce the overall risk of wear and tear due to handling problems.

2. It keeps everyone's interest up until the very end. Since nothing is final until the last heat is run, crowd interest is much higher than any elimination-type bracket.

I went through this last fall with our Awana Grand Prix. In prior years they ran an ill-conceived elimination-type tournament that I won't even try to describe here. I knew there had to be a better way. Once I found Stan's website, I knew I had the answer.

We ran 90 cars in 3 divisions, on two 4-lane tracks. One division only had 7 cars, so we ran a 7-3 Perfect-N for that. But for the other two divisions (27 and 56 cars), we ran one-round ppn brackets to determine the top 7 cars. We then ran a 7-3 pn on 3 lanes to determine first trough third places.

If you have ties at the top of the qualifying rounds, you can do run-offs to get it down to the top 7 cars. We had to do this. In our larger divisions, we had 9 and 10 cars tied for the first 7 places. Usually it was 3 or 4 cars tied for third or fourth place. We had a run-off with those cars to eliminate them from the bottom up, until we had our 7 to take to the finals.

We completed our racing in less time (about 2 hours) than in past years, and crowd interest was very high (no one left after their car was eliminated, mainly because no one was eliminated until the end of the qualifying heats).

If all you have is 25 cars total, then you could run a 2-round partial perfect, which would mean 50 heats, everyone running in each lane twice, against a total of 12 other cars (never running against the same twice), and most likely come up your top 7 without the need of runoffs.

Then run a perfect 7 bracket to determine first through whatever places your awarding.

You need to sell this on the basis of fairness and fun, and not strictly based on number of heats. However, I believe even 1 round of partial-perfect is more fair that a double-elimination tourney.

Sorry for the disjointed nature of the post - I composed it over an hour or so while at work, with many interruptions.

[Stan Pope]

Ouch!
For 25 cars, the PN for 3 lanes is 100 heats!, and CPN is 200!
A Double-Eliminaton would only be 28 heats.
(Triple-Elimination would be 46 heats).
This isn't helping my case much.
It looks like once you get to more than 9 cars, DE is around 1/2 as many heats as PN.
We are trying to hand out 1st-3rd place trophies, so to be fair we really need to do Triple Elimination instead of Double.
This year we had 25 Webelos.
Even at a rapid 40 seconds/heat, PN would take over an hour.
I don't think we are (or can get) organized enough to get to Stan's 34 second/heat rate.

I wouldn't try to run a full PN or CPN with that many. Use a 1 round PPN (25 heats, three runs per car, 6 opponents on a 3-lane track) or a 2 round PPN (50 heats, 6 runs per car, 12 opponents on a 3-lane track) then take top scorers into a 7 or 13 car PN or CPN final.

PN and CPN seek to get match-ups between every pair of opponents. That is not necessary. Just get enough runs against enough different opponents to give high probability that the top N (where N=number of trophies) cars will be among the finalists.

[MERuhl]

What Stan said...

[Cory]

The point I'm trying to prove is that using some lane rotation type methods will not take substantially longer than double elimination does.

Would like to emphasize one of MERuhl's points: While PPN followed by CPN finals might have more heats than DE (not a lot more, though), it is possible (many would say likely) that you'll run your heats faster with predefined charts.

Knowing exactly which cars are in which heats ahead of time is a BIG advantage. You could well end up running more races in less total time.

[dknowles67]

I need to go back and do some more homework.
Is it correct to assume that for PPN, the number of cars = number of heats?
That's much better, and closely correlates to the DE charts.
I always intended to use (recommend we use) PPN after reading Stan's pages, but somehow I got it mixed up with PN.

I'm also toying with the idea of writing some custom software for running our races, so I really need to understand how to generate the charts.
(Hopefully that's not violating any copywrite laws!)
It is unlikely we would have access to the internet at the race site, and unlikely we will know how many racers we will have before the race starts.
We could print out a bunch of charts beforehand, but I was hoping for something a little more slick.

I'm still trying to wrap my brain around this.
This is almost as hard as my Differential Equations class!
Thanks for all the advice!

[Stan Pope]

I need to go back and do some more homework.
Is it correct to assume that for PPN, the number of cars = number of heats?

True for each "round." 25 cars, 1 round = 25 heats; 2 rounds = 5 = heats. ...

That's much better, and closely correlates to the DE charts.
I always intended to use (recommend we use) PPN after reading Stan's pages, but somehow I got it mixed up with PN.

PN and CPN are special cases of PPN in which extra criteria are satisfied. If I recall correctly, a PN chart can be constructed for any combination of lanes and cars, but many of them require an extraordinary number of rounds.

I'm also toying with the idea of writing some custom software for running our races, so I really need to understand how to generate the charts.
(Hopefully that's not violating any copywrite laws!)
It is unlikely we would have access to the internet at the race site, and unlikely we will know how many racers we will have before the race starts.
We could print out a bunch of charts beforehand, but I was hoping for something a little more slick.

Talk to Cory ... he has the dll for this to incorporate into your software, so you can focus on presentation.

I'm still trying to wrap my brain around this.
This is almost as hard as my Differential Equations class!
Thanks for all the advice!

Yup! More like Theory of Numbers class, where the proofs seemed to come from thin air rather than prior theorems.

[dknowles67]

So is a PPN chart simply the first round of a PN chart for the same number of lanes, same number of cars?
If not, how many rounds does it take to satisfy a PPN chart, or is that a function of the number of lanes, and the number of cars in the race?
After going back and re-reading all the web-pages again, I found a lot of description of how to generate PN, and CPN charts, but nothing on how to generate PPN charts (there is a java-script that does it for you, but I want to know how it works. My mom always used to get mad at me for taking things apart. I think it is a genetic defect).
Thanks again for all the info!

[Stan Pope]

PPN is a definition of a class of racing charts. The definition is given on the left side of the generator page at http://members.aol.com/standcmr/ppngen.html.

"A PPN chart is one which satisfies these conditions:
"1. Each car races the same number of times in each lane (which implies that the number of races is a multiple of the number of cars)
"2. Equality of opposition is optimized, i.e., no head-to-head matchup count exceeds another by more than 1. "

The approach used by the script in that page and in the corresponding dll routine is to apply the same pattern for each heat in a round. Thus, three numbers are needed to describe a one round chart for a given number of cars on a 4-lane track.

If you used random numbers instead of carefully selected numbers as the pattern, then most of the time the charts would be terribly unbalanced, and, in some cases, impossible (such as calling for the same car to appear in two different lanes in the same heat.) Some of the charts might satisfy the PPN criteria, fewer (sometimes none) might satisfy the PN criteria, and even fewer might satisfy the CPN criteria.

Sometimes a one round chart might be augmented by a second round to form a chart that satisfies more stringent criteria. For example, for a 5-car chart on a two lane track, one round is PPN; two rounds, PN; three rounds, PPN and four rounds, CPN.

More?

[dknowles67]

One of the parameters in the PPN chart generator is number of rounds.
So, if I go and generate a 1 round chart, the number of heats will always be equal to the number of cars.
(Our track has 3 lanes)
I did some experiments with your PN chart generator and your PPN chart generator for 3 cars 3 lanes.
The PN chart for this case is 3 heats.
The two charts however are not the same.
I also tried it for 4 cars, with the same results.
So I draw the conclusion that the PPN charts are not simply the first round of a PN chart.
(You didn't directly answer my question).
So I'm still not sure how a PPN chart is generated. (what goes on inside your generator).

[Stan Pope]

One of the parameters in the PPN chart generator is number of rounds.
So, if I go and generate a 1 round chart, the number of heats will always be equal to the number of cars.

Yes!

(Our track has 3 lanes)
I did some experiments with your PN chart generator and your PPN chart generator for 3 cars 3 lanes.
The PN chart for this case is 3 heats.

Yes! BTW, the PN chart generator is the OLD generator... the PPN generator supercedes it. But, I left it up under the "theoretical" heading for curious folks.

The two charts however are not the same.

But they are both PN, aren't they? The PPN Chart Generator encompasses what we thought were the useful cases covered by the PN generator. Not all of the original keys were carried forward... In some cases we decided to replace the keys to make more interesting charts.

I also tried it for 4 cars, with the same results.
So I draw the conclusion that the PPN charts are not simply the first round of a PN chart.
(You didn't directly answer my question).

That is correct. Most of the one-round charts produced by the ppngen satisfy PPN criteria. Some of them, such as the 3 and 4 car charts on a 3 lane track, satisfy PN criteria, as well. And, some of them, such as the 2 car chart on a 2 lane track, satisfy the CPN criteria as well as PN and PPN!

The orientation for the PPN generator is that you know how many cars you want to serve and about how many rounds you want to run. Generate a chart! Note the class of the chart (Misc., PPN, PN, or CPN).

If you like it, set the heat sequence optimizing criteria to your liking and generate the chart again! It will produce the same heats, but sort them according to your criteria.

So I'm still not sure how a PPN chart is generated. (what goes on inside your generator).

The generator took about 40,000 characters of condensed javascript code to describe. (I say "condensed" because some of the javascript interpreters couldn't deal with the program size when descriptive (long) variable names were used.) I've given you in the last two posting an overview of the process.

I don't mind anyone "understanding" the internal operation, but really don't want to make the code "public" and deal with variants of the code floating around. For those who want to implement programs on Windows platform, a dll version (with some additional options) is available. Cory (member of this forum) is managing distribution of the dll. It is "clean", i.e. free of spyware and the like, but it is distributed only in executable form. The license is free, and requires that the dll not be sold.

[dknowles67]

I've contacted Cory privately, and he is helping me with this.
I appreciate your responses, they have been most helpful.
I understand that you want to keep the source confidential, and your reasons for doing so.
One last question, and I won't push it any more.
Would it be accurate to say that the new PPN generator works the way the theory is described on your PN generator web-page, only with a different set of generator keys?
That is to say, a 1 round PPN chart is the first round of a PN chart using a set of keys different from the ones published on your web-page, in order to obtain "more interesting charts"?