This Java program generates all possible subsets (the power set) of a given array using backtracking

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This Java program generates all possible subsets (the power set) of a given array using backtracking. Let's break it down step by step.

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Let given array be [1, 2, 3].

Understanding the Code

The program consists of two main functions:

1. subsets(int[] nums):

   • Initializes the result list (list) to store all subsets.
   • Sorts the input array (though sorting is unnecessary in this case).
   • Calls the helper function backtrack() to generate subsets.

2. backtrack(List<List<Integer>> list, List<Integer> tempList, int [] nums, int start):

   • Adds the current subset (tempList) to the result list (list).
   • Iterates through the numbers in nums, adding one element at a time to tempList, and recursively explores further subsets.
   • After recursion, removes the last element to backtrack and explore other possibilities.

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Step-by-Step Execution

Let's assume the input is:

nums = [1, 2, 3]

Step 1: Start Execution

• subsets(nums) is called with nums = [1, 2, 3]
• list is initialized as an empty list: list = []
• backtrack() is called with:

  backtrack(list, tempList = [], nums = [1, 2, 3], start = 0)

Step 2: First Call to backtrack

• tempList = [] is added to list
• list = [[]]
• For Loop Iteration 1 (i = 0):
  • Add 1 to tempList → tempList = [1]
  • Call backtrack(list, tempList = [1], nums, start = 1)

Step 3: Second Call to backtrack

• tempList = [1] is added to list
• list = [[], [1]]
• For Loop Iteration 1 (i = 1):
  • Add 2 to tempList → tempList = [1, 2]
  • Call backtrack(list, tempList = [1, 2], nums, start = 2)

Step 4: Third Call to backtrack

• tempList = [1, 2] is added to list
• list = [[], [1], [1, 2]]
• For Loop Iteration 1 (i = 2):
  • Add 3 to tempList → tempList = [1, 2, 3]
  • Call backtrack(list, tempList = [1, 2, 3], nums, start = 3)

Step 5: Fourth Call to backtrack

• tempList = [1, 2, 3] is added to list
• list = [[], [1], [1, 2], [1, 2, 3]]
• start = 3 (out of bounds), so the function returns
• Backtrack: Remove 3 → tempList = [1, 2]

Step 6: Backtrack to Third Call

• Continue loop in Third Call (i = 2 finished)
• Backtrack: Remove 2 → tempList = [1]
• For Loop Iteration 2 (i = 2):
  • Add 3 to tempList → tempList = [1, 3]
  • Call backtrack(list, tempList = [1, 3], nums, start = 3)

Step 7: Fifth Call to backtrack

• tempList = [1, 3] is added to list
• list = [[], [1], [1, 2], [1, 2, 3], [1, 3]]
• Backtrack: Remove 3 → tempList = [1]
• Backtrack: Remove 1 → tempList = []

Step 8: Backtrack to Second Call

• For Loop Iteration 2 (i = 1):
  • Add 2 to tempList → tempList = [2]
  • Call backtrack(list, tempList = [2], nums, start = 2)

Step 9: Sixth Call to backtrack

• tempList = [2] is added to list
• list = [[], [1], [1, 2], [1, 2, 3], [1, 3], [2]]
• For Loop Iteration 1 (i = 2):
  • Add 3 to tempList → tempList = [2, 3]
  • Call backtrack(list, tempList = [2, 3], nums, start = 3)

Step 10: Seventh Call to backtrack

• tempList = [2, 3] is added to list
• list = [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3]]
• Backtrack: Remove 3 → tempList = [2]
• Backtrack: Remove 2 → tempList = []

Step 11: Backtrack to First Call

• For Loop Iteration 3 (i = 2):
  • Add 3 to tempList → tempList = [3]
  • Call backtrack(list, tempList = [3], nums, start = 3)

Step 12: Eighth Call to backtrack

• tempList = [3] is added to list
• list = [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]
• Backtrack: Remove 3 → tempList = []
• All iterations complete. Return list.

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Final Output

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

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Key Takeaways

1. Backtracking Approach:

   • We explore subsets recursively, adding and removing elements (backtracking).
   • The tempList is used to build subsets, and each state is stored in list.

2. Time Complexity:

   • O(2^n) → Since each element can either be included or not, we generate 2^n subsets.

3. Space Complexity:

   • O(2^n * n) → Each subset requires space, and there are 2^n subsets, each taking up to O(n) space.

4. Recursive Tree Visualization:

   • The recursive function creates a tree where each level corresponds to choosing or skipping an element.

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This step-by-step breakdown explains how subsets are generated recursively using backtracking. 🚀