Power Set

Power Set P(S) of a set S is the set of all subsets of S. For example S = {a, b, c} then P(s) = {{}, {a}, {b}, {c}, {a,b}, {a, c}, {b, c}, {a, b, c}}. If S has n elements in it then P(s) will have 2ⁿ elements

Examples

Input: s = "ab"
Output: "", "a", "b", "ab"
Explanation: The power set of "ab" includes all possible subsets: empty set, single characters, and full string.

Input: s = "abc"
Output: "", "a", "b", "c", "ab", "bc", "ac", "abc"
Explanation: The power set of "abc" includes all subsets formed by choosing any combination of its characters.

Input: s = "a"
Output: "", "a"
Explanation: The power set of "a" consists of the empty set and the single character itself.

Approach: By Using Binary Representation of Numbers from 0 to 2^n - 1

For a given set[] S, the power set can be found by generating all binary numbers between 0 and 2ⁿ-1, where n is the size of the set. 

Set  = [a,b,c]
power_set_size = pow(2, 3) = 8
Run for binary counter = 000 to 111. We consider a when the last bit is set and do not consider if the last bit is not set. We do the same thing for b and second bit, and for c and first bit.

Value of Counter  Subset
   000                    -> Empty set
   001                    -> a
   010                    -> b
   011                    -> ab
   100                    -> c
   101                    -> ac
   110                    -> bc
   111                    -> abc

Detailed Steps:

• Get the size of power set
  • powet_set_size = pow(2, set_size)
• Loop for counter from 0 to pow_set_size
  • Loop for i = 0 to set_size
    • If ith bit in counter is set. Print ith element from set for this subset
  • Print separator for subsets i.e., newline

a
b
ab
c
ac
bc
abc

Time Complexity: O(n * 2ⁿ)
Auxiliary Space: O(1)

Approach: Power set using Previous Permutation

We can generate power set using previous permutation. In auxiliary array of bool set all elements to 0. That represent an empty set.

• Set first element of auxiliary array to 1 and generate all permutations to produce all subsets with one element.
• Then set the second element to 1 which will produce all subsets with two elements, repeat until all elements are included.

a
b
c
ab
ac
bc
abc

Time Complexity: O(n * 2ⁿ)
Auxiliary Space: O(n)

Related article:
Recursive program to generate power set