Generating Unique Subsets with Duplicates

This challenge involves finding all possible unique subsets of a given set of numbers, where the set might contain duplicate elements. Understanding how to handle duplicates is crucial for efficiently generating all distinct combinations and is a common problem in combinatorial algorithms and data processing.

Problem Description

You are given an array of integers nums that may contain duplicates. Your task is to return a list of all possible unique subsets (the power set) of nums.

Subset Definition: A subset is a collection of elements from the original array, where the order of elements within the subset does not matter.
Uniqueness: The output should not contain any duplicate subsets. For example, if the input array is [1, 2, 2], the subset [1, 2] should only appear once in the output, even though it can be formed in two ways (using the first 2 or the second 2).
Empty Subset: The empty subset (represented as []) is always considered a valid subset.
Order of Subsets: The order in which the subsets appear in the output list does not matter. The order of elements within each subset also does not matter, but for consistency, it's often good practice to keep them sorted.

Examples

Example 1:

Input: nums = [1, 2, 2]
Output: [[], [1], [1, 2], [1, 2, 2], [2], [2, 2]]

Explanation: The unique subsets are:

Empty set: []
Subsets with one element: [1], [2]
Subsets with two elements: [1, 2], [2, 2]
Subsets with three elements: [1, 2, 2]

Example 2:

Input: nums = [0]
Output: [[], [0]]

Explanation: The unique subsets are the empty set and the set containing 0.

Example 3:

Input: nums = [4, 4, 4, 1, 4]
Output: [[], [1], [1, 4], [1, 4, 4], [1, 4, 4, 4], [1, 4, 4, 4, 4], [4], [4, 4], [4, 4, 4], [4, 4, 4, 4]]

Explanation: After sorting, nums becomes [1, 4, 4, 4, 4]. The unique subsets are generated considering the duplicates.

Constraints

1 <= nums.length <= 10
-10 <= nums[i] <= 10
The input array nums can contain duplicate integers.

Notes

Consider sorting the input array nums first. This can significantly simplify the logic for handling duplicates and ensuring subset uniqueness.
Think about recursive approaches (backtracking) or iterative methods. How can you build up the subsets?
When dealing with duplicates, a common strategy is to skip over duplicate elements at the same level of recursion or iteration to avoid generating redundant subsets.
The problem is essentially asking for the power set of a multiset.

Generating Unique Subsets with Duplicates

Problem Description

You are given an array of integers nums that may contain duplicates. Your task is to return a list of all possible unique subsets (the power set) of nums.

Subset Definition: A subset is a collection of elements from the original array, where the order of elements within the subset does not matter.

Uniqueness: The output should not contain any duplicate subsets. For example, if the input array is [1, 2, 2], the subset [1, 2] should only appear once in the output, even though it can be formed in two ways (using the first 2 or the second 2).

Empty Subset: The empty subset (represented as []) is always considered a valid subset.

Order of Subsets: The order in which the subsets appear in the output list does not matter. The order of elements within each subset also does not matter, but for consistency, it's often good practice to keep them sorted.

Examples

Example 1:

Input: nums = [1, 2, 2] Output: [[], [1], [1, 2], [1, 2, 2], [2], [2, 2]]

Explanation: The unique subsets are:

Empty set: []

Subsets with one element: [1], [2]

Subsets with two elements: [1, 2], [2, 2]

Subsets with three elements: [1, 2, 2]

Example 2:

Input: nums = [0] Output: [[], [0]]

Explanation: The unique subsets are the empty set and the set containing 0.

Example 3:

Input: nums = [4, 4, 4, 1, 4] Output: [[], [1], [1, 4], [1, 4, 4], [1, 4, 4, 4], [1, 4, 4, 4, 4], [4], [4, 4], [4, 4, 4], [4, 4, 4, 4]]

Explanation: After sorting, nums becomes [1, 4, 4, 4, 4]. The unique subsets are generated considering the duplicates.

Notes

Consider sorting the input array nums first. This can significantly simplify the logic for handling duplicates and ensuring subset uniqueness.

Think about recursive approaches (backtracking) or iterative methods. How can you build up the subsets?

When dealing with duplicates, a common strategy is to skip over duplicate elements at the same level of recursion or iteration to avoid generating redundant subsets.

The problem is essentially asking for the power set of a multiset.