Minimum Health to Beat Game - Practice Coding Problems

Minimum Health to Beat Game - Problem

Array Greedy Medium

🎮 Minimum Health to Beat Game

You're playing an RPG-style game with n levels numbered from 0 to n-1. Each level deals a specific amount of damage to your character, and you must complete all levels in order to win the game.

Given:

A damage array where damage[i] represents health lost at level i
An armor value that can protect you from damage exactly once

Your goal is to find the minimum starting health needed to survive all levels. Remember:

You can use armor on any one level to reduce damage by up to armor points
Your health must remain greater than 0 at all times
You must complete levels sequentially

Example: If levels deal [3, 5, 2] damage and you have 4 armor, you should use armor on level 1 (damage 5) to minimize starting health needed.

Input & Output

example_1.py — Basic Case

$ Input: damage = [2, 7, 4, 3], armor = 4

› Output: 13

💡 Note: Total damage = 16. Use armor on level 1 (damage 7) to reduce by 4. Minimum health = 16 - 4 + 1 = 13.

example_2.py — Armor Exceeds Max Damage

$ Input: damage = [2, 5, 3, 4], armor = 7

› Output: 10

💡 Note: Total damage = 14. Max damage is 5, so armor benefit = min(7, 5) = 5. Minimum health = 14 - 5 + 1 = 10.

example_3.py — Single Level

$ Input: damage = [10], armor = 5

› Output: 6

💡 Note: Only one level with damage 10. Use armor to reduce by 5. Minimum health = 10 - 5 + 1 = 6.

Constraints

1 ≤ n ≤ 10⁵
1 ≤ damage[i] ≤ 10⁴
0 ≤ armor ≤ 10⁴
Health must remain > 0 at all times

Visualization

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Understanding the Visualization

Survey All Levels

Calculate total damage you'll face: sum all damage values

Identify Biggest Threat

Find the level with maximum damage - this is where armor will be most effective

Calculate Armor Benefit

Armor can reduce damage by at most its value, but can't reduce below 0

Optimize Starting Health

Minimum health = Total damage - Maximum armor benefit + 1

Key Takeaway

🎯 Key Insight: Always use armor on the level with highest damage to maximize benefit and minimize required starting health. This greedy approach is optimal because reducing the largest damage gives the maximum overall benefit.

Asked in

G Google 42 ⊞ Microsoft 38 a Amazon 35 f Meta 28

The optimal solution uses a greedy approach in O(n) time. Calculate the total damage across all levels, then subtract the maximum possible armor benefit (which is the minimum of armor value and highest single-level damage). Add 1 to ensure health stays positive. The key insight is that armor should always be used on the level with highest damage to minimize starting health requirements.

Common Approaches

Approach	Time	Space	Notes
✓ Greedy Approach (Optimal)	O(n)	O(1)	Calculate total damage and subtract maximum armor benefit in one optimal placement
Brute Force (Try Armor on Each Level)	O(n²)	O(1)	Try using armor on each level and calculate minimum health needed for each scenario

Greedy Approach (Optimal) — Algorithm Steps

Calculate total damage across all levels
Find the maximum damage that can be reduced by armor
Subtract this maximum reduction from total damage
Add 1 to ensure health stays above 0

Visualization

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Step-by-Step Walkthrough

Calculate Total Damage

Sum all damage values: 3 + 5 + 2 = 10

Find Maximum Damage

Highest single level damage: max(3, 5, 2) = 5

Calculate Armor Benefit

min(armor=4, maxDamage=5) = 4

Compute Result

10 - 4 + 1 = 7 minimum starting health

Code -

solution.c — C

long long minimumHealth(int* damage, int damageSize, int armor) {
    long long totalDamage = 0;
    int maxDamage = 0;
    
    for (int i = 0; i < damageSize; i++) {
        totalDamage += damage[i];
        if (damage[i] > maxDamage) {
            maxDamage = damage[i];
        }
    }
    
    // Use armor optimally on the highest damage level
    long long armorBenefit = (armor < maxDamage) ? armor : maxDamage;
    
    // Minimum starting health = total damage - armor benefit + 1
    return totalDamage - armorBenefit + 1;
}

Time & Space Complexity

Time Complexity

⏱️

O(n)

Single pass to calculate total damage and find maximum damage level

✓ Linear Growth

Space Complexity

O(1)

Only using constant extra space for calculations

✓ Linear Space

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🎮 Minimum Health to Beat Game

Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Greedy Approach (Optimal) — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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