Find the Sum of Subsequence Powers

Find the Sum of Subsequence Powers

1. What is this problem about?

The Find the Sum of Subsequence Powers interview question is a complex combinatorial optimization challenge. You are given an array of integers and an integer k. You need to find all subsequences of length k, calculate the "power" of each subsequence (defined as the minimum absolute difference between any two elements in that subsequence), and return the sum of all these powers.

2. Why is this asked in interviews?

Companies like Google and Rubrik ask the Find the Sum of Subsequence Powers coding problem to test a candidate's mastery of Dynamic Programming and Sorting. It is a "Hard" problem because the number of subsequences is exponential. It evaluations your ability to use DP to count subsequences that satisfy a specific minimum-difference property.

3. Algorithmic pattern used

This problem is solved using Sorting and Dynamic Programming.

1. Sort: Start by sorting the array to make the absolute differences easier to calculate.
2. DP State: dp[i][j][diff] represents the number of subsequences of length j ending at index i with a minimum difference of exactly diff.
3. Alternative State: Since diff can be large, it's often easier to define dp[i][j][target_diff] as the count of subsequences with min diff $\geq target\_diff$.
4. Counting: The total sum is $\sum ( ext{target\_diff} imes ext{Count of subsequences with min\_diff } == ext{target\_diff})$.

4. Example explanation

nums = [1, 2, 3, 4], k = 3. Subsequences of length 3:

• [1, 2, 3]: Min diff 1.
• [1, 2, 4]: Min diff 1.
• [1, 3, 4]: Min diff 1.
• [2, 3, 4]: Min diff 1. Sum of powers: $1+1+1+1 = 4$.

5. Common mistakes candidates make

• Brute Force: Trying to generate all $\binom{N}{k}$ subsequences ($O(N^k)$), which will time out for large $N$.
• Inefficient DP: Not recognizing that the minimum difference must be one of the pairwise differences in the original array, which limits the number of diff values to check.
• Modulo: Forgetting to return the result modulo $10^9 + 7$.

6. Interview preparation tip

Master the "Counting Subsequences" DP pattern. If a problem asks for properties of all subsequences, look for a way to build them incrementally. Sorting is almost always the first step for problems involving absolute differences or ranges in Dynamic Programming interview patterns.