Is Count the Number of K-Free Subsets easy or hard?

The problem is rated Medium because the brute force idea is straightforward but the optimal solution requires a key observation about modulo grouping. Recognizing that numbers differing by k form independent chains enables a dynamic programming approach that significantly reduces the search space.

Count the Number of K-Free Subsets Python/Java solution

Most implementations first group numbers by num % k using a dictionary or map. Each group is sorted and processed with a DP recurrence that decides whether to include the current value or skip it when a difference of k exists. The same logic translates cleanly across Python, Java, C++, Go, and TypeScript with O(n log n) time complexity.

How to solve Count the Number of K-Free Subsets in O(n)?

If the input values are already grouped or sorting is avoided with hash maps, the DP processing itself is linear. Group elements by num % k, sort within each group if necessary, then apply a DP transition similar to the House Robber problem where selecting value x prevents selecting x - k. Each element contributes constant-time updates, making the DP stage O(n).

What is the best approach for Count the Number of K-Free Subsets?

The most efficient approach groups numbers by their remainder modulo k and applies dynamic programming inside each group. Values that differ by k must appear in the same remainder group, forming chains that behave like the House Robber problem. After sorting each group, DP counts valid selections while preventing adjacent conflicts. The total number of subsets is the product of valid counts from all groups.

Is Count the Number of K-Free Subsets asked at Google/Amazon/Meta?

Difference-constraint subset counting problems appear in interviews at companies like Google, Amazon, and Meta. Variants often combine arrays, sorting, and dynamic programming to count valid combinations while avoiding specific differences. This problem tests the ability to transform pair constraints into independent DP chains.

What data structure is used in Count the Number of K-Free Subsets?

The solution typically uses hash maps or dictionaries to group numbers by remainder modulo k, arrays or lists to store sorted values, and dynamic programming variables to track valid subset counts. Sorting and frequency maps help structure the data so the DP transitions can be computed efficiently.

What is the time complexity of Count the Number of K-Free Subsets?

The optimal solution runs in O(n log n) time due to sorting the numbers or groups. The dynamic programming step processes each element once, which is O(n). Space complexity is O(n) for storing groups, frequency counts, and DP states.

Description Hints 3 Solution Video 16:19

2638. Count the Number of K-Free Subsets

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You are given an integer array nums, which contains distinct elements and an integer k.

A subset is called a k-Free subset if it contains no two elements with an absolute difference equal to k. Notice that the empty set is a k-Free subset.

Return the number of k-Free subsets of nums.

A subset of an array is a selection of elements (possibly none) of the array.

Example 1:

Input: nums = [5,4,6], k = 1
Output: 5
Explanation: There are 5 valid subsets: {}, {5}, {4}, {6} and {4, 6}.

Example 2:

Input: nums = [2,3,5,8], k = 5
Output: 12
Explanation: There are 12 valid subsets: {}, {2}, {3}, {5}, {8}, {2, 3}, {2, 3, 5}, {2, 5}, {2, 5, 8}, {2, 8}, {3, 5} and {5, 8}.

Example 3:

Input: nums = [10,5,9,11], k = 20
Output: 16
Explanation: All subsets are valid. Since the total count of subsets is 2⁴= 16, so the answer is 16.

Constraints:

1 <= nums.length <= 50
1 <= nums[i] <= 1000
1 <= k <= 1000

Asked at 1 company

Amazon

Topics

Hints open one at a time

Hint 1

Split all numbers into several groups, with each group being an arithmetic sequence with a common difference of k.

Hint 2

How many K-free subsets are there for each group? This can be solved by dp: dp[i] = dp[i-1] + dp[i-2], meaning if we choose ith element, we cannot choose (i-1)th; otherwise we can choose (i-1)th element.

Hint 3

After solving the problem for every group, the final result is just the product of the sub-problems.

16:19

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Case 1Case 2Case 3

Input

[5,4,6]
1

Input

[2,3,5,8]
5

Input

[10,5,9,11]
20

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