将所有可能的点组合分组为m组

Question

I want to group all possible combinations of points taken m at a time into r groups.

Points = [A,B,C,D........] Total n points

Combinations of these points taken m at a time will be a list l

l = list(itertools.combinations(points,m))

How can I further group it into r groups such that i-th element of each group has no similar points.

For example,

Points = [A,B,C,D] m = 2 and r = 2

l = [[A,B],[A,C],[A,D],[B,C],[B,D],[C,D]]

So groups will be

Group 1 = [[A,B],[A,C],[A,D]] and corresponding to it Group 2 = [[C,D],[B,D],[B,C]]

Note : The points in the i-th index of group 1 and group 2 have no similar points.

I want to do it for n number of points taken m at a time and grouping it into r groups.

Please provide me the algorithm.

Also note that when the number of points increases, the possible combinations also increase if we want more than 2 groups.

Answer 1

This is a tough problem! I've implemented a naive brute force solution. This is probably too slow but it's a starting point.

from itertools import combinations, permutations

points = 'ABCDEF'
r = 3
m = len(points) // r
all_combinations = list(combinations(points, m))
group_length = len(all_combinations) // r
assert r * group_length == len(all_combinations)
found = set()

for combinations_permutation in permutations(all_combinations):
    groups = [combinations_permutation[group_length * i: group_length * (i + 1)]
              for i in range(r)]
    transpose = zip(*groups)

    # Avoid very similar-looking solutions
    canonical = frozenset(map(frozenset, transpose))

    if (canonical not in found and
            all(len(col) == len(set(col))
                for col in (sum(column, ()) for column in transpose))):
        found.add(canonical)
        for group in groups:
            print ', '.join(map(''.join, group))
        print '----'

Here's the start of the output:

AB, AC, AD, AE, AF
CD, BE, BF, BD, BC
EF, DF, CE, CF, DE
----
AB, AC, AD, AE, AF
CD, BF, BE, BC, BD
EF, DE, CF, DF, CE
----
AB, AC, AD, AE, AF
CE, BD, BE, BF, BC
DF, EF, CF, CD, DE
----
AB, AC, AD, AE, AF
CE, BF, BC, BD, BE
DF, DE, EF, CF, CD