Getting a powerset of a list of lists

Question

I'm given a list of lists s:

s = [["a1", "A"], ["b4", "B"], ["a3", "A"], ["d6", "D"], ["c4", "C"]]

(note that the elements in a list do not necessarily begin with a same letter. I modified the data here for convenience.)

My goal is to sort each list to a category by its second element, and get all possible combinations by picking at most one element in each category.

I first hashed the list of lists to a dictionary:

dic = {i[1]: [] for i in s}
for i in s:
    # set the value of the first item key to the second item
    dic[i[1]].append(i[0])

dic
>>> {'A': ['a1', 'a3'], 'B': ['b4'], 'C': ['c4'], 'D': ['d6']}

The number of all possible combinations, hence the length of a powerset of s, should return 23:

{'a1'},
{'a3'},
{'b4'},
{'c4'}, 
{'d6'}, 
{'a1', 'b4'}, 
{'a1', 'c4'}, 
{'a1', 'd6'}, 
{'a3', 'b4'}, 
{'a3', 'c4'}, 
{'a3', 'd6'}, 
{'b4', 'c4'}, 
{'b4', 'd6'}, 
{'c4', 'd6'}, 
{'a1', 'b4', 'c4'}, 
{'a1', 'b4', 'd6'}, 
{'a1', 'c4', 'd6'}, 
{'a3', 'b4', 'c4'}, 
{'a3', 'b4', 'd6'}, 
{'a3', 'c4', 'd6'}, 
{'b4', 'c4', 'd6'}, 
{'a1', 'b4', 'c4', 'd6'}, 
{'a3', 'b4', 'c4', 'd6'}

I initially was going to put multiple for loops, but since I have no guarantee in how many key I would have in my s (which would also put my time complexity at O(N^x)), I used itertools.chain and itertools.combinations as per this post:

def powerset(s:list):
    return chain.from_iterable(combinations(s, r) for r in range(1, len(s)+1))

The problem is that this only takes elements in a single list to account, hence neglects the constraint: 'not taking more than one element from each list at most'. Flattening a list would disregard the categories, so I've not attempted to do so.

Any insights to tackle this problem would be appreciated.

Answer 1

@don'ttalkjustcode's answer works but unnecessarily incurs the overhead of adding dummy values, and also produces an empty set, which is not required by the question.

A more direct approach would be to use itertools.combinations to pick lists from the dict of lists to pass to itertools.product to produce the desired combinations:

from itertools import product, combinations

print(*(
    set(p)
    for r in range(len(dic))
    for c in combinations(dic.values(), r + 1)
    for p in product(*c)
), sep='\n')

This outputs:

{'a1'}
{'a3'}
{'b4'}
{'c4'}
{'d6'}
{'a1', 'b4'}
{'a3', 'b4'}
{'a1', 'c4'}
{'a3', 'c4'}
{'d6', 'a1'}
{'d6', 'a3'}
{'c4', 'b4'}
{'d6', 'b4'}
{'d6', 'c4'}
{'a1', 'c4', 'b4'}
{'a3', 'c4', 'b4'}
{'d6', 'a1', 'b4'}
{'d6', 'a3', 'b4'}
{'d6', 'a1', 'c4'}
{'d6', 'a3', 'c4'}
{'d6', 'c4', 'b4'}
{'d6', 'a1', 'c4', 'b4'}
{'d6', 'a3', 'c4', 'b4'}