Subsets

Clarify the problem:

• The problem requires us to generate all possible subsets of a given array.
• A subset is a set of elements that can be obtained by selecting elements from the array without changing their order.
• The order of the elements in the subsets does not matter.
• We need to return all subsets in any order.

Analyze the problem:

• Input: An array of integers.
• Output: A list of lists representing all possible subsets of the array.
• Constraints:
  • The length of the input array is in the range [0, 10].
  • The input array does not contain duplicate elements.

Design an algorithm:

• We can solve this problem using a backtracking algorithm.
• We start with an empty subset and gradually build it by considering each element in the array.
• For each element, we have two choices: either include it in the current subset or exclude it.
• We recursively explore both choices until we have considered all elements.
• At each step, we add the current subset to the result list.
• After exploring all possible choices, we will have generated all subsets of the array.

Explain your approach:

• We will implement a recursive function that performs the backtracking algorithm.
• The function takes parameters including the current subset, the current index in the array, and the result list.
• At each index, we make a choice to either include or exclude the current element.
• If we include the element, we add it to the current subset and recursively call the function for the next index.
• If we exclude the element, we simply move to the next index without modifying the current subset.
• We repeat this process until we have considered all elements in the array.
• Finally, we return the result list containing all generated subsets.

Write clean and readable code:

python

def subsets(nums):
    def backtrack(nums, start, subset, result):
        result.append(subset[:])
        for i in range(start, len(nums)):
            subset.append(nums[i])
            backtrack(nums, i + 1, subset, result)
            subset.pop()

    result = []
    backtrack(nums, 0, [], result)
    return result

Test your code:

• We will test the code with different test cases, including edge cases and corner cases, to ensure its correctness.
• Test Case 1: Empty array

  python

nums = []
print(subsets(nums))
Output: [[]]

python

nums = [1]
print(subsets(nums))
Output: [[], [1]]

python

• nums = [1, 2, 3]
  print(subsets(nums))
  Output: [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]
  

Optimize if necessary:

• The current solution generates all possible subsets, which requires an exponential number of operations.
• Therefore, the current solution is already optimal.

Handle error cases:

• The code handles the case of an empty array by returning a list containing an empty subset.
• There are no other error cases to consider in this problem.

Discuss complexity analysis:

• The time complexity of the solution is O(2^N), where N is the length of the input array.
• This is because there are 2^N possible subsets, and we generate all of them.
• The space complexity is also O(2^N) since we need to store all subsets in the result list.
• The recursive backtracking algorithm visits each element once and generates all possible subsets.
• Overall, the solution is efficient considering the nature of the problem.