Minimum Health to Beat Game

Minimum Health to Beat Game

1. What is this problem about?

The Minimum Health to Beat Game is a practical optimization problem based on a game scenario. You have an array of damage values representing enemies you must face in order. You also have a "shield" or "armor" that can reduce the damage of exactly one enemy by a certain amount (up to a given armor value). You start with some amount of health and lose health equal to the damage you take. You win if your health stays at least 1 at all times. The goal is to find the minimum starting health required to survive all encounters.

2. Why is this asked in interviews?

Amazon asks the Minimum Health to Beat Game interview question to test basic Greedy reasoning and careful handling of constraints. It evaluates whether you can identify the optimal time to use a limited resource (the armor). In this case, the optimal strategy is intuitive, but the problem requires precision in calculating the reduction and the final health requirement.

3. Algorithmic pattern used

The algorithmic pattern used is a Greedy approach. To minimize the starting health, you must maximize the benefit of your armor. Since the armor can only be used once, you should apply it to the enemy that deals the most damage. The total damage you take will be (Sum of all damages) - (Minimum of max_damage and armor_value). Your starting health must be this total damage plus 1 (to remain alive with at least 1 health). This "Array, Greedy interview pattern" is efficient and runs in $O(N)$ time.

4. Example explanation

Damage: [2, 7, 4, 3], Armor: 4.

1. Total sum = $2 + 7 + 4 + 3 = 16$.
2. Max damage = 7.
3. Benefit from armor: min(7, 4) = 4.
4. Net damage taken = $16 - 4 = 12$.
5. Starting health needed = $12 + 1 = 13$. You use the armor on the enemy that deals 7 damage, reducing that encounter's damage to 3.

5. Common mistakes candidates make

In the Minimum Health to Beat Game coding problem, a frequent error is using the armor on the first enemy that is larger than the armor value, which might not be the largest overall. Another mistake is forgetting the "+1" requirement to survive (starting with health equal to total damage would leave you at 0). Candidates also sometimes use a long loop to simulate the game instead of just calculating the sum and the max, although the complexity remains the same.

6. Interview preparation tip

In problems where you have a "one-time use" power-up, always look for the global maximum or minimum to apply it to. This is the essence of "Greedy interview patterns." Also, be mindful of integer overflow—if the damage values or their sum can exceed $2^{31}-1$, use a larger data type like long long.