将N个元素划分为k个大小的子集

Question

If I have N different numbers, they are to be divided into k sized subsets such that each subset will have at least one element in common. Please help me with the algorithm. the output should be some thing like this if I have elements from 1 to 5 and k = 3 then the output should be:

- S0    =   {0, 1, 2}
- S1    =   {1, 3, 5}
- S2    =   {2, 4, 5}
- S3    =   {0, 3, 4}
- S4    =   {1, 4, 6}
- S5    =   {0, 5, 6}
- S6    =   {2, 3, 6}

Answer 1

1) Notice that if k = 1 and |S| > 1 |S| > 1 then this is impossible (ie k = 1 and S = {1,2} )

2) Otherwise just take the first number, add it always as the first number in each subset, fill in the subsets with remaining numbers. If there's not enough numbers to fill out the last subset just fill it in with random numbers.

As per your example k = 3 and S = {0,1,2,3,4,5,6} we could do:

• S0 = {0,1,2} (first number + next 2)
• S1 = {0,3,4} (first number + next 2 after 1,2 )
• S2 = {0,5,6} (in this case we are lucky since we were able to fill out S2, otherwise should it be only for 0 .. 5 we could add a random number as the last one)

Answer 2

Your example is of 7 points (0,...,6) and 7 "lines" (eg, the "line" {0,1,2}). Note that any two lines "intersect" in a point, and any two points determine a line. (Pick two numbers from 0,...,6 and you'll see that there's exactly one set with both those numbers.)

So the design you presented has more restrictions than you described. If you just want each pair of subsets to have at least one element in common, you could put 0 into each set and get ₆ C ₂ subsets {0,1,2}, {0,1,3}, {0,1,4}, etc., which isn't difficult at all.

The geometric language of points and lines isn't a coincidence. If you want a design where every pair of subsets has exactly one element in common, you probably want a projective plane . The example that you gave is called the Fano plane . Projective planes are not possible for all combinations of N and k. You have to have N = m^2 + m + 1 and k = m + 1 where m is called the order of the plane. Projective planes are easy to construct by a "vector space construction" for m prime using modular arithmetic (eg, m = 2 gives the Fano plane). They are slightly harder to construct for ma power of a prime, m = p^k, using a vector space construction and finite fields. The details can be found in many different references.

For other orders, very little is known, except that through brute force calculations it is known that there are no projective planes of orders 6 and 10.

Is that what you want? If not, there are other "combinatorial block designs" that may have the properties you want, but you have to be precise with your requirements.