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 <= 201 <= nums[i], k <= 1000In #2597 The Number of Beautiful Subsets, we must count subsets where no two chosen elements differ by exactly k. A common strategy is to treat the problem as a subset generation task with constraints.
One effective approach is backtracking. First sort the array and maintain a hash map or frequency structure that tracks which values are already included in the current subset. While exploring subsets, only add a number if num - k is not currently present. This prunes invalid branches early and ensures we only count valid “beautiful” subsets.
An optimized perspective groups numbers by their remainder relative to k. Within each group, the problem resembles a house-robber style dynamic programming where choosing a number prevents selecting another number exactly k away. Sorting and processing groups independently reduces repeated work.
The backtracking approach explores valid subset combinations, while the grouped DP method improves efficiency for repeated values. Overall complexity is manageable for typical constraints after sorting and pruning invalid states.
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Backtracking with Hash Map | O(2^n) | O(n) |
| Grouped DP / Combinatorics Optimization | O(n log n) | O(n) |
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Use these hints if you're stuck. Try solving on your own first.
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)?
This approach involves using an explicit stack to simulate recursion. We push tasks onto the stack as we go, allowing us to backtrack and manage different states or subproblems efficiently.
Time Complexity: O(n), Space Complexity: O(n)
1function iterativeSolution(arr) {
2 // Your code logic here
3}This JavaScript solution uses an iterative approach with a stack to keep track of tasks.
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Problems involving constrained subset generation, backtracking, and dynamic programming patterns are common in FAANG-style interviews. While this exact question may vary, the underlying techniques such as pruning and DP grouping are frequently tested.
A hash map or frequency array is typically used to track which numbers are already in the current subset during backtracking. This allows quick checks for whether num - k exists. Sorting the array beforehand also helps manage constraints efficiently.
A common optimal strategy combines sorting with backtracking or grouping with dynamic programming. By ensuring that no selected numbers differ by k, the algorithm prunes invalid branches early. Grouping numbers by their modulo relationship with k can further optimize counting.
Sorting ensures numbers are processed in order, making it easier to check constraints like num - k before adding elements. It also helps when grouping numbers or applying DP strategies where relative ordering matters.
This C solution uses recursion with memoization to cache results of subproblems.