Algorithm to generate (not quite) spanning set in Python

Question

This follows on from this question:

Algorithm to generate spanning set

Given this input: [1,2,3,4]

I\'d like to generate this set of sets in python:<

Answer 1

You might also enjoy a recursive solution:

def span(lst):
  yield [lst]
  for i in range(1, len(lst)):
    for x in span(lst[i:]):
      yield [lst[:i]] + x

Explanation

We exploit recursion here to break the problem down. The approach is the following:

For every list, the whole list is a valid spanning: [1,2,3,4] => [[1,2,3,4]].

For every list that is longer than size 1, we can use the first item as a group and then apply the same algorithm on the remaining list to get all the combined results:

[1,2,3] => 
  [[1]] + [[2], [3]]  # => [[1], [2], [3]]
  [[1]] + [[2,3]]     # => [[1], [2,3]]

For every list that is longer than size 2, we can just as well use the first two items as a group and then apply the same algorithm on the remaining list and combine the results:

[1,2,3,4,5] =>
  [[1,2]] + [[3], [4], [5]]  # => [[1,2], [3], [4], [5]]
  [[1,2]] + [[3,4], [5]]     # => [[1,2], [3,4], [5]]
  [[1,2]] + [[3], [4,5]]     # => [[1,2], [3], [4,5]]
  [[1,2]] + [[3,4,5]]        # => [[1,2], [3,4,5]]

We can see that the possible combinations on the right side are indeed all possible groupings of the remainder of the list, [3,4,5].

For every list that is longer than ... etc. Thus, the final algorithm is the following:

1. yield the whole list (it is always a valid spanning, see above)
2. For every possible splitting of the list, yield the left-hand part of the list combined with all possible spannings of the right-hand part of the list.

yield is a special keyword in Python that make the function a generator, which means that it returns a iterable object that can be used to enumerate all results found. You can transform the result into a list using the list constructor function: list(span([1,2,3,4])).