Understanding how to generate all possible subsets of a given set is a fundamental problem in computer science with applications in various algorithms and data structures. A subset is a set formed by selecting zero or more elements from another set. For example, if we have the set {1, 2, 3}, its subsets include {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, and `{1, 2, 3}}$. This article explores two primary methods for solving this problem: the iterative approach and the recursive (backtracking) approach in C++.

Iterative Approach

The iterative method builds up the list of subsets incrementally. It starts with an empty set as the only subset. Then, for each number in the input array, it takes all existing subsets and adds the current number to them, generating new subsets. These new subsets are then added to the collection of existing subsets.

Consider the input nums = {1, 2}.

• Initially: `subsets = {{}}`
• Process `num = 1`: Take `{}`, add `1` -> `{{1}}`. New `subsets = {{}, {1}}`
• Process `num = 2`:
  • Take `{}`, add `2` -> `{{2}}`
  • Take `{1}`, add `2` -> `{{1, 2}}`

  New `subsets = {{}, {1}, {2}, {1, 2}}`

Here is the C++ implementation for the iterative solution:

class Solution {
public:
    vector<vector<int>> subsets(vector<int>& nums) {
        int n = nums.size();
        vector<vector<int>> ans;
        ans.push_back({}); // Start with an empty subset
        for (int num : nums) {
            int size = ans.size();
            // Iterate through all currently existing subsets
            for(int i=0; i<size; i++){
                vector<int> currentSubset = ans[i];
                currentSubset.push_back(num); // Add the current number to form a new subset
                ans.push_back(currentSubset); // Add the new subset to our collection
            }
        }
        return ans;
    }
};

The code initializes ans with an empty set. It then iterates through each number num in the input nums. In each iteration, it determines the current size of ans (representing the number of subsets found so far). It then loops from 0 to size-1, taking each existing subset, appending num to it, and adding this new subset back into ans. This effectively doubles the number of subsets at each step, ensuring all combinations are generated.

Recursive Approach (Backtracking)

The recursive approach, often implemented using backtracking, explores all possible paths to build a subset. At each element, we have two choices: either include the element in the current subset or exclude it. By recursively exploring both options for all elements, we guarantee that every unique subset is generated.

The function createSubset takes the input nums, the current index we are considering, the res vector to store all subsets, and the subset currently being built.

• Base Case: If `index` reaches the end of `nums`, it means we have considered all elements, so the `subset` built so far is complete and added to `res`.
• Recursive Step 1 (Include): Add `nums[index]` to `subset` and recursively call `createSubset` for `index + 1`.
• Recursive Step 2 (Exclude): After the \’include\’ branch returns, backtrack by removing `nums[index]` from `subset`. Then, recursively call `createSubset` for `index + 1` *without* `nums[index]`.

Here is the C++ implementation for the recursive solution:

class Solution {
public:
    void createSubset(vector<int>& nums, int index , vector<vector<int>>&res, vector<int>&subset ){
        if(index == nums.size()) {
            res.push_back(subset);
            return;
        }
        // Option 1: Include the current element
        subset.push_back(nums[index]);
        createSubset(nums, index+1, res, subset);

        // Option 2: Exclude the current element (backtrack)
        subset.pop_back();
        createSubset(nums, index+1, res, subset);
    }

    vector<vector<int>> subsets(vector<int>& nums) {
        vector<vector<int>> res;
        vector<int> subset;
        createSubset(nums, 0 , res, subset);
        return res;
    }
};

The subsets function initializes the res (result) and subset (temporary storage for current subset) vectors and calls createSubset starting from index 0. The createSubset function intelligently navigates through all possibilities. The key is the backtracking step (subset.pop_back()) which allows the recursion to explore the \’exclude\’ path after fully exploring the \’include\’ path for a given element, ensuring that all unique combinations are generated without duplicates.

Conclusion

Both iterative and recursive approaches effectively generate all subsets of a given set. The iterative method can be more intuitive for some as it explicitly builds up the solution step by step. The recursive (backtracking) approach, while perhaps requiring a deeper understanding of recursion, often leads to more concise and elegant code for combinatorial problems like this. The choice between them often depends on personal preference, specific problem constraints, or the need for optimization.