Minimum Operations to Make Numbers Non-positive

Minimum Operations to Make Numbers Non-positive - Problem

You are given a 0-indexed integer array nums and two integers x and y.

In one operation, you must choose an index i such that 0 <= i < nums.length and perform the following:

Decrement nums[i] by x
Decrement values by y at all indices except the i-th one

Return the minimum number of operations to make all the integers in nums less than or equal to zero.

Input & Output

Example 1 — Basic Case

$ Input: nums = [3,4,1], x = 3, y = 1

› Output: 3

💡 Note: Operation 1: Choose i=0, nums becomes [0,3,0]. Operation 2: Choose i=1, nums becomes [-1,0,-1]. Operation 3: Choose i=1, nums becomes [-2,-3,-2]. All numbers ≤ 0 after 3 operations.

Example 2 — Equal x and y

$ Input: nums = [1,2,1], x = 2, y = 2

› Output: 1

💡 Note: Since x = y = 2, each operation decreases all numbers by 2. After 1 operation: nums becomes [-1,0,-1]. All numbers ≤ 0.

Example 3 — Large Difference

$ Input: nums = [10], x = 5, y = 1

› Output: 2

💡 Note: Only one element. Operation 1: nums[0] becomes 5. Operation 2: nums[0] becomes 0. Need 2 operations to make 10 ≤ 0.

Constraints

1 ≤ nums.length ≤ 10⁵
1 ≤ nums[i] ≤ 10⁹
1 ≤ x, y ≤ 10⁹

Visualization

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Understanding the Visualization

Input

Array with positive numbers and operation parameters x, y

Operations

Each operation: decrease selected element by x, others by y

Goal

Make all numbers ≤ 0 with minimum operations

Key Takeaway

🎯 Key Insight: Binary search on the answer space and greedily verify feasibility for each candidate

Asked in

G Google 15 M Meta 12 a Amazon 8 m Microsoft 6

The key insight is to use binary search on the answer space. Instead of trying all operation sequences, we binary search on the number of operations k and check if k operations can make all numbers non-positive. Best approach uses binary search with greedy verification in O(n log(max_val)) time and O(1) space.

Common Approaches

Approach	Time	Space	Notes
✓ Brute Force Simulation	O(n^k)	O(k)	Try all possible operation sequences until all numbers are non-positive
Binary Search on Answer	O(n log(max_val))	O(1)	Binary search on the number of operations, then check if feasible

Brute Force Simulation — Algorithm Steps

Step 1: While any number > 0, try each index
Step 2: Apply operation (decrease selected by x, others by y)
Step 3: Count operations until all <= 0

Visualization

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Step-by-Step Walkthrough

Initial State

Array with positive numbers

Try Each Index

For each operation, try selecting every possible index

Recurse

Continue until all numbers are non-positive

Code -

solution.c — C

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

int backtrack(int* arr, int n, int x, int y, int ops) {
    int allNonPositive = 1;
    for (int i = 0; i < n; i++) {
        if (arr[i] > 0) {
            allNonPositive = 0;
            break;
        }
    }
    if (allNonPositive) return ops;
    
    int minOps = INT_MAX;
    for (int i = 0; i < n; i++) {
        int* newArr = malloc(n * sizeof(int));
        memcpy(newArr, arr, n * sizeof(int));
        newArr[i] -= x;
        for (int j = 0; j < n; j++) {
            if (j != i) {
                newArr[j] -= y;
            }
        }
        int result = backtrack(newArr, n, x, y, ops + 1);
        if (result < minOps) minOps = result;
        free(newArr);
    }
    return minOps;
}

int solution(int* nums, int n, int x, int y) {
    return backtrack(nums, n, x, y, 0);
}

int main() {
    char line[1000];
    fgets(line, sizeof(line), stdin);
    int nums[100];
    int n = 0;
    char* token = strtok(line + 1, ",]");
    while (token) {
        nums[n++] = atoi(token);
        token = strtok(NULL, ",]");
    }
    int x, y;
    scanf("%d", &x);
    scanf("%d", &y);
    printf("%d\n", solution(nums, n, x, y));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n^k)

k operations needed, each with n choices, leading to exponential growth

✓ Linear Growth

Space Complexity

O(k)

Recursion stack depth for k operations

✓ Linear Space

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Brute Force Simulation — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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