Problem Statement
Given an integer array nums that may contain duplicates, return all possible subsets (the power set).
- Note: The solution set must not contain duplicate subsets. Return the solution in any order.
Example Inputs
Example 1:
Problem Solution
Approach
To generate all possible subsets of the given array that may contain duplicates, we can use a recursive backtracking approach with the following strategies:
- Sort the Array: Sorting helps in identifying and skipping duplicates.
- Backtracking with Choice: For each element, decide to include it in the current subset or skip it to avoid duplicates.
Steps
- Sort the Array: Begin by sorting the
nums array to bring duplicates together.
- Initialize: Start with an empty subset and begin processing from the first element.
- Decision Point: For each element in the array, decide to either:
- Include the current element in the subset.
- Exclude the current element from the subset.
- Skip Duplicates: If the current element is the same as the previous one and the previous one was not included in the subset, skip the current element to avoid duplicate subsets.
- Recursion: Recursively apply the above decisions to the next element until all elements have been processed.
- Store Subset: Once all decisions have been made for a subset, add it to the list of possible subsets.
- Backtrack: Remove the last element added and explore other possibilities.
Time & Space Complexity
- Time Complexity: O(2^n), where n is the number of elements in
nums. Each element has two choices: include or exclude.
- Space Complexity: O(n), which is the space required for the recursion stack and the space for storing subsets.
Code Snippet
Dry Run
Let's perform a dry run with nums = [1, 2, 2].
- Sort the Array:
nums = [1, 2, 2]
- Initial Call:
backtrack(0, [])
- Level 0:
- Include 1:
current = [1]
- Level 1:
- Include 2:
current = [1, 2]
- Level 2:
- Include 2:
current = [1, 2, 2]
- Level 3: Add
[1, 2, 2] to ans.
- Backtrack:
current = [1, 2]
- Backtrack:
current = [1]
- Exclude 2: (Next 2 is same as previous, skip to avoid duplicate)
- Backtrack:
current = [1]
- Exclude 1:
current = []
- Level 1:
- Include 2:
current = [2]
- Level 2:
- Include 2:
current = [2, 2]
- Level 3: Add
[2, 2] to ans.
- Backtrack:
current = [2]
- Backtrack:
current = [2]
- Exclude 2: (Next 2 is same as previous, skip to avoid duplicate)
- Backtrack:
current = []
- Final Subsets:
[ [], [1], [1, 2], [1, 2, 2], [2], [2, 2] ]
Complexity Analysis
Optimized Time and Space
The recursive backtracking approach efficiently explores all possible subsets while avoiding duplicates by skipping over repeated elements.
- Optimized Time Complexity: O(2^n), where n is the number of elements in
nums.
- Optimized Space Complexity: O(n) for the recursion stack and temporary storage.
Conclusion
Generating all subsets of a given set, especially when duplicates are present, is a classic problem that highlights the importance of handling duplicates to avoid redundant solutions. By sorting the array and employing recursive backtracking with careful decision-making to include or exclude elements, we can efficiently generate all unique subsets. This approach not only solves the problem at hand but also strengthens understanding of recursion, backtracking, and handling edge cases in algorithm design.