Subsets#

Problem Statement#

[1]Given an integer array nums of unique elements, return all possible subsets (the power set).

The solution set must not contain duplicate subsets. Return the solution in any order.

Example 1#

Input: nums = [1,2,3]
Output: [[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]

Example 2#

Input: nums = [1]
Output: [[],[1]]

Constraints#

  • 1 <= nums.length <= 10.

  • -10 <= nums[i] <= 10.

  • All the numbers of nums are unique.

Solution#

You might need to find the relationship between the result of the array nums with the result of itself without the last element.

Example 3#

Input: nums = [1,2]
Output: [[],[1],[2],[1,2]]

You can see the powerset of Example 3 was obtained from the one in Example 2 with additional subsets [2], [1,2]. These new subsets were constructed from subsets [], [1] of Example 2 appended with the new element 2.

Similarly, the powerset of Example 1 was obtained from the one in Example 3 with the additional subsets [3], [1,3], [2,3], [1,2,3]. These new subsets were constructed from the ones of Example 3 appended with the new element 3.

Code#

#include <vector>
#include <iostream>
using namespace std;
vector<vector<int>> subsets(const vector<int>& nums) {
    vector<vector<int>> powerset = {{}};
    int i = 0;
    while (i < nums.size()) {
        vector<vector<int>> newSubsets;
        for (auto subset : powerset) {
            subset.push_back(nums[i]);  
            newSubsets.push_back(subset);
        }
        powerset.insert(powerset.end(), newSubsets.begin(), newSubsets.end());
        i++;
    }
    return powerset;
}
void print(const vector<vector<int>>& powerset) {
    for (auto& set : powerset ) {
        cout << "[";
        for (auto& element : set) {
            cout << element << ",";
        }
        cout << "]";
    }
    cout << endl;
}
int main() {
    vector<int> nums{1,2,3};
    auto powerset = subsets(nums);
    print(powerset);
    nums = {1};
    powerset = subsets(nums);
    print(powerset);
}

Output:
[][1,][2,][1,2,][3,][1,3,][2,3,][1,2,3,]
[][1,]

Complexity#

  • Runtime: O(2^N), where N is the number of elements in nums, as it generates all possible subsets.

  • Extra space: O(2^N) due to the space required to store the subsets.

Conclusion#

This solution generates subsets by iteratively adding each element of nums to the existing subsets and accumulating the results.

Note that in for (auto subset : powerset) you should not use reference auto& because we do not want to change the subsets that have been created.

Exercise#

  • Subsets II[2].