Count the Number of K-Free Subsets

Count the Number of K-Free Subsets

What is this problem about?

The Count the Number of K-Free Subsets interview question involves an array of unique integers and an integer $k$. A subset is "k-free" if it does not contain any two elements $x$ and $y$ such that $|x - y| == k$.

Your goal is to find the total number of such subsets. Note that the empty subset is always k-free.

Why is this asked in interviews?

Amazon uses this to test combinatorics and dynamic programming on independent groups. The core insight is that elements $x, x+k, x+2k, \dots$ form a chain of dependencies. Elements in different chains (e.g., $\{1, 3\}$ and $\{2, 4\}$ for $k=2$) do not restrict each other. This problem evaluates whether a candidate can decompose a global constraint into local independent sub-problems.

Algorithmic pattern used

This is a Grouped Dynamic Programming problem.

1. Group elements: Group numbers by their remainder modulo $k$. Within each group, sort the numbers.
2. Identify Chains: Within a group, identify contiguous chains where each element is exactly $k$ greater than the previous one (e.g., $1, 4, 7$ for $k=3$).
3. DP on Chains: For a chain of length $L$, the number of ways to pick elements such that no two are adjacent is the $(L+2)$-th Fibonacci number. This is exactly the "House Robber" problem logic:
   • dp[i] = dp[i-1] + dp[i-2] (where dp[i] is ways for first i elements).
4. Multiply: Since chains are independent, multiply the number of ways found for each chain to get the total result.

Example explanation

nums = [1, 2, 3], k = 1

• Remainder mod 1: all in one group. Sorted: [1, 2, 3].
• This is one chain of length 3.
• DP for length 3:
  • dp[0] = 1 (empty)
  • dp[1] = 2 (empty, {1})
  • dp[2] = dp[1] + dp[0] = 3 (empty, {1}, {2})
  • dp[3] = dp[2] + dp[1] = 5 (empty, {1}, {2}, {3}, {1,3}) Total = 5.

Common mistakes candidates make

• Sorting: Forgetting to sort the elements before identifying chains.
• Independence: Not realizing that chains like $\{1, 2, 3\}$ and $\{10, 11\}$ are independent if $k=1$.
• Base cases: Getting the Fibonacci sequence or House Robber base cases wrong.

Interview preparation tip

When you see a constraint like $|x - y| == K$, always think about remainder groups. Grouping by x % K is a standard way to break dependency chains in number theory and array problems.