Subsets - Python Solution

1. Introduction

The "Subsets" problem is a classic exercise in combinatorics and algorithms, often encountered in coding interviews. It involves generating all possible subsets (also known as the power set) of a given set of unique integers. This problem is a great opportunity to explore recursive and iterative approaches to generate combinations.

Problem

Given an integer array nums of unique elements, the task is to return all possible subsets of this array. The solution set must not contain duplicate subsets and can be returned in any order.

2. Solution Steps

1. Recognize that a subset of a set can include any combination of the set's elements, including no elements at all.

2. Decide on a method to generate subsets - either iteratively or using recursion.

3. Implement a function to systematically build up subsets.

4. Ensure that each subset is unique.

5. Collect and return all the subsets.

3. Code Program

```Python
def subsets(nums):
    def backtrack(start, path):
        # Add the current path (subset) to the result
        res.append(path[:])
        for i in range(start, len(nums)):
            # Include the current element and move to the next
            path.append(nums[i])
            backtrack(i + 1, path)
            # Exclude the current element (backtrack)
            path.pop()

    res = []
    backtrack(0, [])
    return res

# Testing the function with examples
print(subsets([1, 2, 3]))
print(subsets([0]))

Output:

[[],[1],[1,2],[1,2,3],[1,3],[2],[2,3],[3]]
[[],[0]]

Explanation:

1. Backtracking Function: The backtrack function is used to generate subsets. It takes the starting index and the current path (subset).

2. Subset Formation: For each element in the array, the function recursively considers two scenarios: including the element in the subset or not.

3. Recursive Calls: The function makes a recursive call after including the current element, then backtracks to exclude it and proceed to the next element.

4. Base Case: Each time the function is called, the current subset is added to the result list.

5. Unique Subsets: Since the function explores every combination, all unique subsets are generated.

6. Result: The function returns a list of all possible subsets.