K Items With the Maximum Sum

K Items With the Maximum Sum - Problem

Math Greedy Easy

There is a bag that consists of items, each item has a number 1, 0, or -1 written on it.

You are given four non-negative integers numOnes, numZeros, numNegOnes, and k.

The bag initially contains:

numOnes items with 1s written on them.
numZeros items with 0s written on them.
numNegOnes items with -1s written on them.

We want to pick exactly k items among the available items. Return the maximum possible sum of numbers written on the items.

Input & Output

Example 1 — Basic Greedy Selection

$ Input: numOnes = 3, numZeros = 2, numNegOnes = 0, k = 4

› Output: 3

💡 Note: Take all 3 items with value 1, then 1 item with value 0. Total sum = 3×1 + 1×0 = 3

Example 2 — Need Negative Items

$ Input: numOnes = 3, numZeros = 2, numNegOnes = 1, k = 6

› Output: 2

💡 Note: Take all 3 ones, all 2 zeros, and 1 negative one. Total sum = 3×1 + 2×0 + 1×(-1) = 2

Example 3 — Only Ones Needed

$ Input: numOnes = 5, numZeros = 3, numNegOnes = 2, k = 3

› Output: 3

💡 Note: Only need 3 items, take 3 ones. Total sum = 3×1 = 3

Constraints

0 ≤ numOnes, numZeros, numNegOnes ≤ 50
1 ≤ k ≤ numOnes + numZeros + numNegOnes ≤ 50

Visualization

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Asked in

A Amazon 15 M Microsoft 12 G Google 8

The key insight is to use a greedy strategy - always pick the highest value items first to maximize the sum. The optimal approach takes items in priority order: 1s first, then 0s, finally -1s only if necessary. Time: O(1), Space: O(1)

Common Approaches

✓ Brute Force - Try All Combinations

⏱️ Time: O(C(n,k) × k) Space: O(n + k)

Create an array with all items, then generate every possible combination of k items. Calculate the sum for each combination and return the maximum.

Greedy Strategy

⏱️ Time: O(1) Space: O(1)

Since we want maximum sum, greedily pick items with highest values first. Take as many 1s as possible, then 0s if needed, finally -1s if we still need more items.

Brute Force - Try All Combinations — Algorithm Steps

Create array with numOnes 1s, numZeros 0s, numNegOnes -1s
Generate all combinations of k items
Calculate sum for each combination
Return maximum sum

Visualization

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Step-by-Step Walkthrough

Create Items Array

Build array with all available items

Generate Combinations

Try every possible selection of k items

Find Maximum

Calculate sum for each and return maximum

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <limits.h>

int generateCombinations(int* items, int n, int k, int start, int* current, int currentSize) {
    if (currentSize == k) {
        int sum = 0;
        for (int i = 0; i < k; i++) {
            sum += current[i];
        }
        return sum;
    }
    
    int maxSum = INT_MIN;
    for (int i = start; i < n; i++) {
        current[currentSize] = items[i];
        int result = generateCombinations(items, n, k, i + 1, current, currentSize + 1);
        if (result > maxSum) maxSum = result;
    }
    return maxSum;
}

int solution(int numOnes, int numZeros, int numNegOnes, int k) {
    int n = numOnes + numZeros + numNegOnes;
    int* items = malloc(n * sizeof(int));
    int* current = malloc(k * sizeof(int));
    
    int idx = 0;
    for (int i = 0; i < numOnes; i++) items[idx++] = 1;
    for (int i = 0; i < numZeros; i++) items[idx++] = 0;
    for (int i = 0; i < numNegOnes; i++) items[idx++] = -1;
    
    int result = generateCombinations(items, n, k, 0, current, 0);
    
    free(items);
    free(current);
    return result;
}

int main() {
    int numOnes, numZeros, numNegOnes, k;
    scanf("%d", &numOnes);
    scanf("%d", &numZeros);
    scanf("%d", &numNegOnes);
    scanf("%d", &k);
    
    printf("%d\n", solution(numOnes, numZeros, numNegOnes, k));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(C(n,k) × k)

C(n,k) combinations where n = total items, each taking O(k) to sum

✓ Linear Growth

Space Complexity

O(n + k)

Array to store all items plus space for current combination

⚡ Linearithmic Space

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Input & Output

Constraints

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Related Problems

Common Approaches

Brute Force - Try All Combinations — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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