LeetCode #3428 — MEDIUM

Maximum and Minimum Sums of at Most Size K Subsequences

Move from brute-force thinking to an efficient approach using array strategy.

Solve on LeetCode
The Problem

Problem Statement

You are given an integer array nums and a positive integer k. Return the sum of the maximum and minimum elements of all subsequences of nums with at most k elements.

Since the answer may be very large, return it modulo 109 + 7.

Example 1:

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

Output: 24

Explanation:

The subsequences of nums with at most 2 elements are:

Subsequence Minimum Maximum Sum
[1] 1 1 2
[2] 2 2 4
[3] 3 3 6
[1, 2] 1 2 3
[1, 3] 1 3 4
[2, 3] 2 3 5
Final Total     24

The output would be 24.

Example 2:

Input: nums = [5,0,6], k = 1

Output: 22

Explanation:

For subsequences with exactly 1 element, the minimum and maximum values are the element itself. Therefore, the total is 5 + 5 + 0 + 0 + 6 + 6 = 22.

Example 3:

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

Output: 12

Explanation:

The subsequences [1, 1] and [1] each appear 3 times. For all of them, the minimum and maximum are both 1. Thus, the total is 12.

Constraints:

  • 1 <= nums.length <= 105
  • 0 <= nums[i] <= 109
  • 1 <= k <= min(70, nums.length)
Patterns Used
📊Dynamic Programming→

Roadmap

  1. Brute Force Baseline
  2. Core Insight
  3. Algorithm Walkthrough
  4. Edge Cases
  5. Full Annotated Code
  6. Interactive Study Demo
  7. Complexity Analysis
Step 01

Brute Force Baseline

Problem summary: You are given an integer array nums and a positive integer k. Return the sum of the maximum and minimum elements of all subsequences of nums with at most k elements. Since the answer may be very 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 · Math · Dynamic Programming

Example 1

[1,2,3]
2

Example 2

[5,0,6]
1

Example 3

[1,1,1]
2
Step 02

Core Insight

What unlocks the optimal approach

  • Sort the array.
Interview move: turn each hint into an invariant you can check after every iteration/recursion step.
Step 03

Algorithm Walkthrough

Iteration Checklist

  1. Define state (indices, window, stack, map, DP cell, or recursion frame).
  2. Apply one transition step and update the invariant.
  3. Record answer candidate when condition is met.
  4. 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
Upper-end input sizes
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 #3428: Maximum and Minimum Sums of at Most Size K Subsequences
class Solution {
  public int minMaxSums(int[] nums, int k) {
    // In a sorted array, nums[i] will be
    //   1. The maximum for subsequences formed by nums[0..i].
    //   2. The minimum for subsequences formed by nums[i..n - 1].
    //
    // The number of times nums[i] is the maximum is the same as the number of
    // times nums[n - 1 - i] is the minimum, due to the symmetry in subsequences
    // derived from the sorted order.
    //
    // To calculate the contribution of nums[i], we need to find the number of
    // ways to select at most (k - 1) elements from the range of indices where
    // nums[i] is the smallest or nums[n - 1 - i] is the largest.
    final int n = nums.length;
    final int[][] comb = getComb(n, k - 1);
    long ans = 0;

    Arrays.sort(nums);

    // i: available numbers from the left of nums[i] or
    //    available numbers from the right of nums[n - 1 - i]
    for (int i = 0; i < n; ++i) {
      int count = 0;
      for (int j = 0; j <= k - 1; ++j) // selected numbers
        count = (count + comb[i][j]) % MOD;
      ans += (long) nums[i] * count;
      ans += (long) nums[n - 1 - i] * count;
      ans %= MOD;
    }

    return (int) ans;
  }

  private static final int MOD = 1_000_000_007;

  // C(n, k) = C(n - 1, k) + C(n - 1, k - 1)
  private int[][] getComb(int n, int k) {
    int[][] comb = new int[n + 1][k + 1];
    for (int i = 0; i <= n; ++i)
      comb[i][0] = 1;
    for (int i = 1; i <= n; ++i)
      for (int j = 1; j <= k; ++j)
        comb[i][j] = (comb[i - 1][j] + comb[i - 1][j - 1]) % MOD;
    return comb;
  }
}
Step 06

Interactive Study Demo

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

Press Step or Run All to begin.
Step 07

Complexity Analysis

Time
O(n × m)
Space
O(n × m)

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.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.

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.

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