Question

Let's say I have a class of 30 students and want generate every possible way in which they can be partitioned into groups of 5 (order is irrelevant).

I know how to find all the combinations of students to form one group individually (http://www.merriampark.com/comb.htm). By using that iterator and some recursion, I can find PERMUTATIONS of the possible group combinations. However, order in which the groups are selected isn't relevant and I'd like to minimize my execution time. So how do I find the unique COMBINATIONS of the possible groups?

The above algorithm uses lexicographical ordering to avoid generating duplicate combinations... is there a way that I can use that idea on groups instead of on objects?

I know Ruby well and Java/Python less well. Thanks in advance for any advice!

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Solution

Well, there's (30C5*25C5*20C5*15C5*10C5*5C5)/6! = 30!/(6!*5!6) = 123,378,675,083,039,376 different partitons of 30 into groups of 5, so generating them all will take some time, no matter what method you use.

In general, though, a good method to selecting such a partition is to use some ordering on the elements, and find the grouping for the highest ungrouped element, and then group the rest.

     find_partition = lambda do |elts|
        if elts.empty?
         [[]]
        else
         highest = elts.pop
         elts.combination(4).map do |others|
            find_partition[elts - others].map { |part| part << [highest,*others] }
         end.inject(:+)
        end
     end
     find_partition[(1..30).to_a]

This way you're only generating each partition once

OTHER TIPS

This is an old question, but anyway, for the record, that's how I would it in Ruby:

class Array
  def groups_of_size(n)
    Enumerator.new do |yielder|
      if self.empty?
        yielder.yield([])
      else
        self.drop(1).combination(n-1).map { |vs| [self.first] + vs }.each do |values|
          (self - values).groups_of_size(n).each do |group|
            yielder.yield([values] + group)
          end   
        end
      end
    end
  end
end

I use an enumerator because the output can grow very quickly, a strict output (an array for example) wouldn't be useful. A usage example:

>> pp [0, 1, 2, 3, 4, 5].groups_of_size(3).to_a
=> 
[[[0, 1, 2], [3, 4, 5]],
 [[0, 1, 3], [2, 4, 5]],
 [[0, 1, 4], [2, 3, 5]],
 [[0, 1, 5], [2, 3, 4]],
 [[0, 2, 3], [1, 4, 5]],
 [[0, 2, 4], [1, 3, 5]],
 [[0, 2, 5], [1, 3, 4]],
 [[0, 3, 4], [1, 2, 5]],
 [[0, 3, 5], [1, 2, 4]],
 [[0, 4, 5], [1, 2, 3]]]

You could do some post-processing on the permutations. Some pseudo-code (implement in the language of your choice...):

// We have a list of lists called 'permutations'
// combinations is an (empty) list of lists
for each permutation in permutations
{
   sortedPermutation = permutation.sort()
   if (! combinations.find(sortedPermutation) )
   {
       combinations.add(sortedPermutation);
   }
}

Probably not the most efficient; I'd add the sort & compare to the code that generates the permutations personally.

One possibility would be to find all combinations to form an individual group, then go through and generate combinations that don't contain members of that individual group. Something like:

List<List<Student>> combinations=Combinations(students);
public void GenerateCombinations(int startingIndex, List<List<Student>> currentGroups, int groupsLeft)
{
    if(groupsLeft==0) ProcessCombination(currentGroups);

    for(int i=startingIndex; i<combinations.Count; i++)
    {
        if combinations[i] does not contain a student in current groups
            GenerateCombinations(i+1, currentGroups + combinations[i], groupsLeft -1);
    }
}

It won't be the most efficient method to go about it, but it should generate all combinations of groups. I suspect better performance could be had if you were to generate temporary lists of combinations, where in all groups that can't occur were removed, but that would be a bit more complex.

As a slight aside, there should be 142,506 combinations of 30 students to form a single group of 5. My <sarcasm> awesome </sarcasm> math skills suggest that there should be about 10^17 = 100 quadrillion combinations of groups of students (30!/((5!^6)*6!); 30! orderings of students, ordering of 6 groups of 5 does not matter, and ordering of those 6 groups doesn't matter). You might be sitting there a while waiting for this to finish.

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