Find the Sum of Subsequence Powers

Problem Description

You are given an integer array nums of length n, and a positive integer k. The power of a subsequence is defined as the minimum absolute difference between any two elements in the subsequence. Return the sum of powers of all subsequences of nums which have length equal to k. Since the answer may be large, return it modulo 10^9 + 7.

Key Insights

• The problem requires calculating the minimum absolute difference for all possible subsequences of length k.
• The minimum absolute difference can be effectively calculated by sorting the subsequence and analyzing adjacent elements.
• The number of ways to choose subsequences can be managed using combinatorial logic.

Space and Time Complexity

Time Complexity: O(n^2 * k) - for generating all subsequences of length k and calculating their minimum differences. Space Complexity: O(n) - for storing the sorted subsequence and temporary variables.

Solution

The approach to solve the problem involves:

1. Generating all possible combinations of the elements in nums of length k.
2. For each combination, sort the elements to facilitate the calculation of minimum absolute differences.
3. Calculate the minimum absolute difference by iterating through the sorted combination and finding the absolute differences between adjacent elements.
4. Sum these minimum differences for all valid subsequences.
5. Return the final result modulo 10^9 + 7.

This solution utilizes combinatorial methods to enumerate subsequences and sorting to simplify the calculation of differences.

Code Solutions