# 78. Subsets

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## Problem Statement

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(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(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,]
``````

### Code explanation

1. The code initializes an empty vector of vectors called `powerset` to store the subsets. It starts with an initial subset containing an empty vector, representing the empty set: `{{}}`.

2. Within the loop, it creates a new vector of vectors called `newSubsets` to store subsets that include the `i`-th element of `nums`.

3. It iterates through the existing subsets in `powerset`. For each subset, it creates a copy of it (represented by the `subset` variable), adds the `i`-th element from `nums` to the copied subset, and pushes the modified subset into the `newSubsets` vector.

4. After processing all existing subsets, the code adds the contents of `newSubsets` to the `powerset` vector. This effectively combines the current subsets with subsets that include the `i`-th element of `nums`.

5. It repeats steps 2-4 until all elements in `nums` have been considered.

6. Finally, the code returns the `powerset`, which now contains all possible subsets of `nums`.

### Complexity

This code essentially generates subsets by iteratively adding each element of `nums` to the existing subsets and accumulating the results. The time complexity of this code is `O(2^N)`, where `N` is the number of elements in `nums`, as it generates all possible subsets. The space complexity is also `O(2^N)` due to the space required to store the subsets.

• Runtime: `O(2^N)`.

• Extra space: `O(2^N)`.

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