Power Set Functions #

Use the position of 1 in bites to calculate all the possibilities of combinations in a list.

like five bits 0 0 0 0 0 to indicate all the items’ positions, and 1 can be every position in the bits. All the combinations should be the Powerset we want to, then use >> to check every combination’s position whether it is 1 or not.

def powerSet(items):
    N = len(items)
    # enumerate the 2**N possible combinations
    for i in range(2**N):
        combo = []
        for j in range(N):
            # test bit jth of integer i
            # >>j. move the bit we want to check to the end
            # %2. remove all the other bits execpt the last one
            # check the one we kept if it is 1 not 0,
            # which means we want to keep the item which on the position
            # example:  0 1 1 0 1
            # we want to check the third "1"
            # first move the second bit to the end(>>j), will be "0 0 0 1 1"
            # then remove all the other bits(%2), we got "0 0 0 0 1"
            # compare it with 1, which is true, 
            # so we take the item with the position, which will be item[2]
            if (i >> j) % 2 == 1:
                combo.append(items[j])
        yield combo

Use itertools.combinations to calculate r length subsequences of elements, then iterate with different length.

the function of combinations is a little complicated, added some notations to help me to understand.

from itertools import chain

def powerset_generator(sets):
    for subset in chain.from_iterable(combinations(sets, r) for r in range(len(sets)+1)):
        yield subset

# the logic of this function is 
#   set a new array with length r
#   loop the last element's index from i to i+n-r(n is the length of pool, r is the length of subsequence). 
#   when hit the maximum which should be n-1, increase the last-1 element's index.
#   loop until the first element's index hit the maximum, 
#   then increase the previous index, and set the last index to previous index + 1, 
#   then back to the loop until all of the indices hit the maximum
# For example: iterable = [1,2,3,4,5], r = 3
#   (1, 2, 3)
#   (1, 2, 4)
#   (1, 2, 5) <-- the last index hit the maximum
#   (1, 3, 4) <-- increase the previous index, and set every one after to previous index + 1,
#   (1, 3, 5) 
#   (1, 4, 5) <-- the (last-1) index hit the maximum
#   (2, 3, 4)
#   (2, 3, 5)
#   (2, 4, 5)
#   (3, 4, 5) <-- the (last-2) index hit the maximum
def combinations(iterable, r):

    pool = tuple(iterable)

    n = len(pool)

    if r > n:
        return
    indices = list(range(r))

    # In the "while" circle, we will start to change the indices by adding 1 consistently.
    # So yield the first permutation before the while start.
    yield tuple(pool[x] for x in indices)

    while True:

        # This 'for' loop is checking whether the index has hit the maximum from the last one to the first one.
        # if it indices[i] >= its maximum, 
        #   set i = i-1, check the previous one
        # if all of the indices has hit the maximum, 
        #   return the function
        for i in reversed(range(r)):

            # let's take an example to explain why using i + n - r
            # pool indices: [0,1,2,3,4]
            # subsequence indices: [0,1,2]
            # so
            #   indices[2] can be one of [2,3,4],
            #   indices[1] can be one of [1,2,3],
            #   indices[0] can be one of [0,1,2],
            # and the range of every index is n-r, like this sample uses: 5-3=2
            # then
            #   indices[2] < 2+2 = i+2 = i+n-r,
            #   indices[1] < 1+2 = i+2 = i+n-r,
            #   indices[0] < 0+2 = i+2 = i+n-r,
            # this can also be: indices[i] != i + n - r
            if indices[i] < i + n - r:
                break
        else:
            # loop finished, return
            return

        # Add one for current indices[i], 
        # (we already yield the first permutation before the loop)
        indices[i] += 1
        # this for loop increases every indices which is after indices[i].
        # cause, current index has increased, and we need to confirm every one behind is initialized again.
        # For example: current we got i = 2, indices[i]+1 will be 3, 
        # so the next loop should start with [1, 3, 4], not [1, 3, 3]
        for j in range(i+1, r):
            indices[j] = indices[j-1] + 1

        yield tuple(pool[x] for x in indices)