Find the Sum of the Power of All Subsequences

Find the Sum of the Power of All Subsequences

1. What is this problem about?

The Find the Sum of the Power of All Subsequences interview question asks you to find the total sum of "powers" across all possible subsequences. The "power" of a subsequence is defined as the number of its own subsequences that sum up to a target value k. This is a nested combinatorial problem: you are summing counts of sums within subsequences.

2. Why is this asked in interviews?

Companies like Google and DE Shaw use the Find the Sum of the Power coding problem to test advanced Dynamic Programming and mathematical manipulation. It evaluations if you can simplify a nested summation into a single DP process. It tests your ability to track multiple counts (total elements used vs total elements remaining) simultaneously.

3. Algorithmic pattern used

This problem is solved using a variation of the Subset Sum Dynamic Programming pattern.

• The Core Insight: Instead of checking each subsequence, consider each subset that sums to $k$. If a subset of size $m$ sums to $k$, how many global subsequences contain this subset?
• The Answer: If the original array has size $n$, and we picked $m$ elements to sum to $k$, there are $n-m$ remaining elements. Each of these $n-m$ elements can either be in or out of a global subsequence. So, there are $2^{n-m}$ subsequences that include this specific subset.
• DP State: dp[i][target] is the count of subsets of size i that sum to target.
• Summation: Result = $\sum_{i=1}^n dp[i][k] imes 2^{n-i}$.

4. Example explanation

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

• Subset summing to 3: {1, 2} (size 2). Power contribution: $2^{3-2} = 2^1 = 2$.
• Subset summing to 3: {3} (size 1). Power contribution: $2^{3-1} = 2^2 = 4$. Total power: $2 + 4 = 6$.

5. Common mistakes candidates make

• Double Recursion: Trying to iterate through subsequences and then running a subset sum on each one. This is $O(2^N \cdot 2^N)$.
• Ignoring Size: Forgetting that the number of global subsequences depends on the size of the subset that reaches the target $k$.
• Power of 2: Not pre-calculating powers of 2, leading to redundant calculations.

6. Interview preparation tip

Practice the "Contribution of each element/subset" pattern. Instead of iterating over the large space (subsequences), iterate over the property-defining space (subsets that sum to $k$) and calculate how many times each "valid" part appears in the total.