1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. Write clean and readable code:

  6. 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
  7. 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
  8. nums = [] print(subsets(nums)) Output: [[]]
  9. Test Case 2: Array with one element
  10. nums = [1] print(subsets(nums)) Output: [[], [1]]
  11. Test Case 3: Array with multiple elements
    • nums = [1, 2, 3] print(subsets(nums)) Output: [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]
  12. Optimize if necessary:

    • The current solution generates all possible subsets, which requires an exponential number of operations.
    • Therefore, the current solution is already optimal.
  13. 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.
  14. 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.
Next Post Previous Post