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math: all pairs of 1000 things is ~5e5. I want minimum # pools of 10 things that cover all pairs.?
my goal is to cover all pairwise combinations of 1000 entities using much fewer than half a million groups. I figure that my system allows me to pool up to 10 things at once. so how do i configure the minimum number of sets of 10 that cover all pairwise combos.
thx
m
You want to group 1000 items in groups of 10,
such that every item occurs at least once in a group
with every other item ?
So you might do this in rounds.
First round form 100 groups:
1-10, 11-20, 21-30, and so on up to 991-1000.
Each one provides 45 pairings, so you get 4500 from that round.
Then you start regrouping, maybe
1,11,21,31,41,51,61,71,81,91,101 in a new group
2,12,22,32,42,52,62,72,82,92,102
and so on.
Another 4500 pairings.
And for some number of rounds you get 45 new pairings per group,
and 4500 for the whole round.
Since you get 45 pairings from each group of 10,
you would need at least 1000*999 / (2 * 45) groups or 11,100.
After some rounds, though, it gets harder and harder to find all new pairings
and the efficiency goes down.
I think it will turn out to be a multiple of that minimum number, roughly speaking,
but I'm not prepared to guess where it falls.
EDIT:
Further investigation of the 12 : 3 case follows:
Designating the 12 objects as 1-9 and a b c
we can do this arrangment:
123 456 789 abc
147 25a 38b 69c
158 24c 39a 67b
19c 26b 357 48a
168 27a 34c 59b
18a 57c 36b 249
28c 6a3 14b 579
The seven rounds produce this many pairings:
3 3 3 3 = 12
3 3 3 3 = 12
3 3 3 3 = 12
2 2 3 3 = 10
2 2 2 3 = 9
1 2 1 2 = 6
2 1 2 0 = 5
Note that the last two rounds only give us 6 and 5 new pairings.
Some of the last few groups did not offer any new
pairs at all, and if we don't go strictly by rounds,
and just make individual groups, we can do it with
fewer than the 28 trios shown here.
However, we see immediately that we cannot be 100% efficient in all cases:
For a group of 12 items, each item wants to be paired with 11 others. 11 is prime, so no group size below 11 divides it evenly and eventually we will repeat some pairings.
Here is an example for 12 items in groups of 4:
1234 1567 189a 1bc4 < have to repeat a pairing with 1 here
258b 2694 267a
359b 368c 37ab
45ac 478c
796b 29bc
The 14 groups add these numbers of pairings:
6 6 6 5 6 5 4 5 6 4 5 4 2 2
The fact that the last two only add two each
suggest that it might be possible to be more
efficient, and reduce this by one.
Maybe can even change some of the 4's to 5's
and reduce it further, but it gets difficult.
In your case, each item is to be paired with 999 others, and the number of others in a group of 10 is 9, so at least for some of them, membership in 111 groups will work. But then the combining and recombining kicks in, and some redundancy must be introduced.
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