#90

Medium

auto_awesomeBacktracking search with pruning

LeetCode Problem Workspace

Subsets II

Subsets II problem asks to generate unique subsets from an array with possible duplicates using backtracking search with pruning.

array backtracking bit manipulation

Problem Statement

Given an integer array nums, which may contain duplicates, the task is to return all possible subsets (the power set) of the array. The solution should not include duplicate subsets.

The power set is a set of all possible subsets, including the empty set. This problem requires careful handling of duplicates to ensure each subset appears only once in the final result.

Examples

Example 1

Input: nums = [1,2,2]

Output: [[],[1],[1,2],[1,2,2],[2],[2,2]]

Example details omitted.

Example 2

Input: nums = [0]

Output: [[],[0]]

Example details omitted.

Constraints

1 <= nums.length <= 10
-10 <= nums[i] <= 10

Solution Approach

Backtracking with Pruning

This problem can be solved using a backtracking approach with pruning. Starting with an empty subset, we attempt to build subsets by recursively adding elements. At each step, we skip adding duplicate elements that would generate redundant subsets. This ensures all unique subsets are collected while avoiding duplicates.

Sorting the Input Array

Sorting the array helps ensure that duplicates are grouped together. This allows us to easily skip over duplicate elements during backtracking. After sorting, we can compare the current element with the previous one to avoid picking the same element again in the same recursive level.

Efficient Result Generation

To generate the result efficiently, we build subsets progressively. Once a subset is complete, it is added to the result list. During recursion, subsets are built from left to right, ensuring that no subset is generated twice by skipping duplicates at each recursion level.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the number of subsets, which is $O(2^n)$ for an array of size n. The pruning ensures that the algorithm avoids unnecessary exploration of duplicate subsets. The space complexity is also $O(2^n)$ due to the space required for storing the subsets.

What Interviewers Usually Probe

Ability to manage duplicate subsets effectively is key to solving this problem.
Understanding backtracking with pruning demonstrates good problem-solving skills.
Familiarity with subset generation, recursive exploration, and skipping duplicates is crucial.

Common Pitfalls or Variants

Common pitfalls

Forgetting to skip duplicate elements leads to redundant subsets being included in the final result.
Not sorting the input array can make it difficult to recognize and skip duplicates.
Failing to prune the recursive tree efficiently can lead to unnecessary computations.

Follow-up variants

Handling arrays of varying sizes up to 10 elements with more complex duplicates.
Adapting the approach for larger datasets that might not fit in memory or have different constraints.
Modifying the solution to generate subsets of a specific size or meet additional requirements.

FAQ

How do I handle duplicates when solving Subsets II?

Duplicates in Subsets II are managed by sorting the array first, then skipping over duplicate elements during backtracking. This prevents redundant subsets.

What is the best approach for generating subsets without duplicates?

The best approach is to use backtracking with pruning. Sorting the array ensures duplicates are grouped together, and skipping over them during recursion eliminates redundant subsets.

How does backtracking with pruning work in Subsets II?

Backtracking with pruning works by recursively building subsets, and pruning occurs by skipping over duplicate elements to avoid redundant subsets from being added to the result.

Why is sorting the input array important in Subsets II?

Sorting the input array helps group duplicate elements together, making it easier to identify and skip over them during the backtracking process, ensuring unique subsets.

What is the time complexity of the Subsets II problem?

The time complexity of Subsets II is $O(2^n)$ , where n is the length of the input array. The pruning technique reduces redundant calculations, but the overall complexity still depends on the number of subsets.

terminal

Solution

Solution 1: Sorting + DFS

We can first sort the array $\textit{nums}$ to facilitate deduplication.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

class Solution:
    def subsetsWithDup(self, nums: List[int]) -> List[List[int]]:
        def dfs(i: int):
            if i == len(nums):
                ans.append(t[:])
                return
            t.append(nums[i])
            dfs(i + 1)
            x = t.pop()
            while i + 1 < len(nums) and nums[i + 1] == x:
                i += 1
            dfs(i + 1)

        nums.sort()
        ans = []
        t = []
        dfs(0)
        return ans

Solution 2: Sorting + Binary Enumeration

Similar to Solution 1, we first sort the array $\textit{nums}$ to facilitate deduplication.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

class Solution:
    def subsetsWithDup(self, nums: List[int]) -> List[List[int]]:
        def dfs(i: int):
            if i == len(nums):
                ans.append(t[:])
                return
            t.append(nums[i])
            dfs(i + 1)
            x = t.pop()
            while i + 1 < len(nums) and nums[i + 1] == x:
                i += 1
            dfs(i + 1)

        nums.sort()
        ans = []
        t = []
        dfs(0)
        return ans

Continue Practicing

#78 subsets #473 matchsticks to square #491 non decreasing subsequences