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Problem 2597 The Number of Beautiful Subsets

Problem Statement

Link - Problem 2597

Question

You are given an array nums of positive integers and a positive integer k.

A subset of nums is beautiful if it does not contain two integers with an absolute difference equal to k.

Return the number of non-empty beautiful subsets of the array nums.

A subset of nums is an array that can be obtained by deleting some (possibly none) elements from nums. Two subsets are different if and only if the chosen indices to delete are different.

Example 1:

Input: nums = [2,4,6], k = 2
Output: 4
Explanation: The beautiful subsets of the array nums are: [2], [4], [6], [2, 6].
It can be proved that there are only 4 beautiful subsets in the array [2,4,6].

Example 2:

Input: nums = [1], k = 1
Output: 1
Explanation: The beautiful subset of the array nums is [1].
It can be proved that there is only 1 beautiful subset in the array [1].

Constraints:

• 1 <= nums.length <= 20
• 1 <= nums[i], k <= 1000

Solution

class Solution {
public:
    void backtrack(const vector<int>& nums, int k, unordered_map<int, int>& flag, int pos, int& result) {
        if (pos == nums.size()) {
            result++;
            return;
        }

        backtrack(nums, k, flag, pos + 1, result);

        int num = nums[pos];
        if (!flag[num - k]) {
            flag[num]++;
            backtrack(nums, k, flag, pos + 1, result);
            flag[num]--;
        }
    }

    int beautifulSubsets(vector<int>& nums, int k) {
        sort(nums.begin(), nums.end());
        unordered_map<int, int> flag;
        int res = 0;
        backtrack(nums, k, flag, 0, res);
        return res - 1;
    }
};

Complexity Analysis

| Algorithm                            | Time Complexity | Space Complexity |
| ------------------------------------ | --------------- | ---------------- |
| Backtracking-based subset generation | O(n(2^n))       | O(n)             |

Explanation

Intial Thoughts

When faced with this problem, my initial thoughts were to think about how subsets work, imagine each number has two options, include or exclude. But here we need to ensure no two numbers have an absolute difference equal to k. At first glance, this seems to be a hard problem, but then I thought of an analogy - thinking of this problem as placing balls in boxes with certain constraints.

Intuitive Analysis

To solve the problem intuitively, try dividing it into two steps. First, understand the concept of beautiful subsets and how adding an element affects this. Think of it as if each element has its own ‘special’ element which cannot be together in the subset. Once you grasp this idea, finding the correct path - whether to include an element or not - can be thought of as similar to the decision process in a recursive approach, checking if adding a new number affects an already added number. The tricky part comes when we backtrack - try thinking of this as like walking through various ‘dead ends’ and correcting our path accordingly to achieve the final result.

1. Intuition

• The problem asks for the number of non-empty beautiful subsets of the array nums, where a subset is beautiful if it does not contain two integers with an absolute difference equal to k.
• To solve this problem, we can use a backtracking approach to generate all possible subsets of the array nums.
• We need to keep track of the numbers that have been included in the current subset to avoid including numbers with an absolute difference equal to k.
• We can use a flag array to keep track of the numbers that have been included in the current subset.
• The key insight is to use the backtracking approach to generate all possible subsets and then filter out the subsets that contain numbers with an absolute difference equal to k.
• This approach works because it allows us to generate all possible subsets and then filter out the subsets that do not meet the condition.
• The time complexity of this approach is O(2^n), where n is the length of the array nums, because we generate all possible subsets of the array nums.

2. Implementation

• The function backtrack is used to generate all possible subsets of the array nums using a backtracking approach.
• The function backtrack takes five parameters: nums, k, flag, pos, and result.
• The flag array is used to keep track of the numbers that have been included in the current subset.
• The pos parameter is used to keep track of the current position in the array nums.
• The result parameter is used to keep track of the number of beautiful subsets found so far.
• In the backtrack function, we first check if the current position pos is equal to the length of the array nums.
• If it is, we increment the result parameter and return.
• Otherwise, we recursively call the backtrack function with the next position pos + 1.
• We then check if the current number nums[pos] can be included in the current subset by checking if flag[nums[pos] - k] is false.
• If it is, we set flag[nums[pos]] to true and recursively call the backtrack function with the next position pos + 1.
• After the recursive call, we set flag[nums[pos]] to false to backtrack and try other possibilities.

Complexity Analysis

Time Complexity:

• In the beautifulSubsets function, backtrack is called recursively to generate all possible subsets, resulting in 2^n possible subsets for n numbers.
• For each subset generated, it iterates through all numbers, hence 2^n _ n possible operations, thus justifying the O(n _ 2^n) time complexity
• The recursive call backtrack(nums, k, flag, pos + 1, result) has n layers due to pos ranging from 0 to nums.size()

Space Complexity:

• The backtrack function call stack can go up to n layers (depth of recursion), hence n space complexity for call stack
• However, recursive call stack aside, additional space usage by the flag map is of size n for storing the count of each number
• Negligible space used by other variables like res and the parameters of backtrack

Footnote

This question is rated as Medium difficulty.

Hints

Sort the array nums and create another array cnt of size nums[i].

Use backtracking to generate all the beautiful subsets. If cnt[nums[i] - k] is positive, then it is impossible to add nums[i] in the subset, and we just move to the next index. Otherwise, it is also possible to add nums[i] in the subset, in this case, increase cnt[nums[i]], and move to the next index.

Bonus: Can you solve the problem in O(n log n)?

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