LeetCode #3098 — HARD

Find the Sum of Subsequence Powers

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

The Problem

Problem Statement

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 10⁹+ 7.

Example 1:

Input: nums = [1,2,3,4], k = 3

Output: 4

Explanation:

There are 4 subsequences in nums which have length 3: [1,2,3], [1,3,4], [1,2,4], and [2,3,4]. The sum of powers is |2 - 3| + |3 - 4| + |2 - 1| + |3 - 4| = 4.

Example 2:

Input: nums = [2,2], k = 2

Output: 0

Explanation:

The only subsequence in nums which has length 2 is [2,2]. The sum of powers is |2 - 2| = 0.

Example 3:

Input: nums = [4,3,-1], k = 2

Output: 10

Explanation:

There are 3 subsequences in nums which have length 2: [4,3], [4,-1], and [3,-1]. The sum of powers is |4 - 3| + |4 - (-1)| + |3 - (-1)| = 10.

Constraints:

2 <= n == nums.length <= 50
-10⁸ <= nums[i] <= 10⁸
2 <= k <= n

Step 01

Brute Force Baseline

Problem summary: 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 109 + 7.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Dynamic Programming

Example 1

[1,2,3,4]
3

Example 2

[2,2]
2

Example 3

[4,3,-1]
2

Core Insight

What unlocks the optimal approach

Sort <code>nums</code>.
There are at most <code>n<sup>2</sup></code> distinct differences.
For a particular difference <code>d</code>, let <code>dp[len][i][j]</code> be the number of subsequences of length <code>len</code> in the subarray <code>nums[0..i]</code> where the last element picked was at index <code>j</code>.
For each index, we can check if it can be picked if <code>nums[i] - nums[j] <= d</code>.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Largest constraint values

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #3098: Find the Sum of Subsequence Powers
class Solution {
    private Map<Long, Integer> f = new HashMap<>();
    private final int mod = (int) 1e9 + 7;
    private int[] nums;

    public int sumOfPowers(int[] nums, int k) {
        Arrays.sort(nums);
        this.nums = nums;
        return dfs(0, nums.length, k, Integer.MAX_VALUE);
    }

    private int dfs(int i, int j, int k, int mi) {
        if (i >= nums.length) {
            return k == 0 ? mi : 0;
        }
        if (nums.length - i < k) {
            return 0;
        }
        long key = (1L * mi) << 18 | (i << 12) | (j << 6) | k;
        if (f.containsKey(key)) {
            return f.get(key);
        }
        int ans = dfs(i + 1, j, k, mi);
        if (j == nums.length) {
            ans += dfs(i + 1, i, k - 1, mi);
        } else {
            ans += dfs(i + 1, i, k - 1, Math.min(mi, nums[i] - nums[j]));
        }
        ans %= mod;
        f.put(key, ans);
        return ans;
    }
}

// Accepted solution for LeetCode #3098: Find the Sum of Subsequence Powers
func sumOfPowers(nums []int, k int) int {
	const mod int = 1e9 + 7
	sort.Ints(nums)
	n := len(nums)
	f := map[int]int{}
	var dfs func(i, j, k, mi int) int
	dfs = func(i, j, k, mi int) int {
		if i >= n {
			if k == 0 {
				return mi
			}
			return 0
		}
		if n-i < k {
			return 0
		}
		key := mi<<18 | (i << 12) | (j << 6) | k
		if v, ok := f[key]; ok {
			return v
		}
		ans := dfs(i+1, j, k, mi)
		if j == n {
			ans += dfs(i+1, i, k-1, mi)
		} else {
			ans += dfs(i+1, i, k-1, min(mi, nums[i]-nums[j]))
		}
		ans %= mod
		f[key] = ans
		return ans
	}
	return dfs(0, n, k, math.MaxInt)
}

# Accepted solution for LeetCode #3098: Find the Sum of Subsequence Powers
class Solution:
    def sumOfPowers(self, nums: List[int], k: int) -> int:
        @cache
        def dfs(i: int, j: int, k: int, mi: int) -> int:
            if i >= n:
                return mi if k == 0 else 0
            if n - i < k:
                return 0
            ans = dfs(i + 1, j, k, mi)
            if j == n:
                ans += dfs(i + 1, i, k - 1, mi)
            else:
                ans += dfs(i + 1, i, k - 1, min(mi, nums[i] - nums[j]))
            ans %= mod
            return ans

        mod = 10**9 + 7
        n = len(nums)
        nums.sort()
        return dfs(0, n, k, inf)

// Accepted solution for LeetCode #3098: Find the Sum of Subsequence Powers
// Rust example auto-generated from java reference.
// Replace the signature and local types with the exact LeetCode harness for this problem.
impl Solution {
    pub fn rust_example() {
        // Port the logic from the reference block below.
    }
}

// Reference (java):
// // Accepted solution for LeetCode #3098: Find the Sum of Subsequence Powers
// class Solution {
//     private Map<Long, Integer> f = new HashMap<>();
//     private final int mod = (int) 1e9 + 7;
//     private int[] nums;
// 
//     public int sumOfPowers(int[] nums, int k) {
//         Arrays.sort(nums);
//         this.nums = nums;
//         return dfs(0, nums.length, k, Integer.MAX_VALUE);
//     }
// 
//     private int dfs(int i, int j, int k, int mi) {
//         if (i >= nums.length) {
//             return k == 0 ? mi : 0;
//         }
//         if (nums.length - i < k) {
//             return 0;
//         }
//         long key = (1L * mi) << 18 | (i << 12) | (j << 6) | k;
//         if (f.containsKey(key)) {
//             return f.get(key);
//         }
//         int ans = dfs(i + 1, j, k, mi);
//         if (j == nums.length) {
//             ans += dfs(i + 1, i, k - 1, mi);
//         } else {
//             ans += dfs(i + 1, i, k - 1, Math.min(mi, nums[i] - nums[j]));
//         }
//         ans %= mod;
//         f.put(key, ans);
//         return ans;
//     }
// }

// Accepted solution for LeetCode #3098: Find the Sum of Subsequence Powers
function sumOfPowers(nums: number[], k: number): number {
    const mod = BigInt(1e9 + 7);
    nums.sort((a, b) => a - b);
    const n = nums.length;
    const f: Map<bigint, bigint> = new Map();
    function dfs(i: number, j: number, k: number, mi: number): bigint {
        if (i >= n) {
            if (k === 0) {
                return BigInt(mi);
            }
            return BigInt(0);
        }
        if (n - i < k) {
            return BigInt(0);
        }
        const key =
            (BigInt(mi) << BigInt(18)) |
            (BigInt(i) << BigInt(12)) |
            (BigInt(j) << BigInt(6)) |
            BigInt(k);
        if (f.has(key)) {
            return f.get(key)!;
        }
        let ans = dfs(i + 1, j, k, mi);
        if (j === n) {
            ans += dfs(i + 1, i, k - 1, mi);
        } else {
            ans += dfs(i + 1, i, k - 1, Math.min(mi, nums[i] - nums[j]));
        }
        ans %= mod;
        f.set(key, ans);
        return ans;
    }

    return Number(dfs(0, n, k, Number.MAX_SAFE_INTEGER));
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n^4 × k)

Space

O(n^4 × k)

Approach Breakdown

RECURSIVE

O(2ⁿ) time

O(n) space

Pure recursion explores every possible choice at each step. With two choices per state (take or skip), the decision tree has 2ⁿ leaves. The recursion stack uses O(n) space. Many subproblems are recomputed exponentially many times.

DYNAMIC PROGRAMMING

O(n × m) time

O(n × m) space

Each cell in the DP table is computed exactly once from previously solved subproblems. The table dimensions determine both time and space. Look for the state variables — each unique combination of state values is one cell. Often a rolling array can reduce space by one dimension.

Shortcut: Count your DP state dimensions → that’s your time. Can you drop one? That’s your space optimization.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

State misses one required dimension

Wrong move: An incomplete state merges distinct subproblems and caches incorrect answers.

Usually fails on: Correctness breaks on cases that differ only in hidden state.

Fix: Define state so each unique subproblem maps to one DP cell.

Find the Sum of Subsequence Powers

Problem Statement

Roadmap

Brute Force Baseline

Baseline thinking

Example 1

Example 2

Example 3

Related Problems

Core Insight

What unlocks the optimal approach

Algorithm Walkthrough

Iteration Checklist

Edge Cases

Full Annotated Code

Interactive Study Demo

Complexity Analysis

Approach Breakdown

Common Mistakes

Off-by-one on range boundaries

State misses one required dimension