Find all partitions from a list of subsets

Question

Given a list of specific subsets like

S = [ {1, 2}, {3, 4}, {1}, {2, 3}, {4}, {3} ]

and a "universe" set like

U = {1, 2, 3, 4}

what elegant and simple algorithm can be used to find all the possible partitions of U made of sets from S? With this example, such partitions include

{1, 2} {3, 4}
{1, 2} {3} {4}

etc.

Answer 1

Use recursion.

Split the problem into two smaller problems based on whether the first element is used or not:

• Partition using {1,2} and any of the the remaining sets.
• Partition without using {1,2} but using any of the the remaining sets.

These two options cover all possibilities.

• The first is solved by partitioning {3,4} using only [ {3, 4}, {1}, {2, 3}, {4}, {3} ].
• The second is solved by partitioning {1,2,3,4} using only [ {3, 4}, {1}, {2, 3}, {4}, {3} ].

To see how to solve these smaller problems refer to this similar question.