Problem Description

You're playing a game with n levels (numbered from 0 to n - 1). Each level causes you to lose a certain amount of health, given in an array called damage, where damage[i] represents the health you lose when completing level i.

You have a special armor ability that you can use at most once during the entire game. When you use this armor on any level, it will protect you from up to armor points of damage on that specific level. For example, if a level would deal 10 damage and your armor value is 7, you would only take 3 damage. If your armor value is 15, you would take 0 damage from that level.

The game has the following rules:

• You must complete the levels in order (from level 0 to level n-1)
• Your health must always be greater than 0 (not equal to 0) to continue playing
• You can choose which level to use your armor on (or not use it at all)

Your task is to find the minimum starting health needed to successfully complete all levels of the game.

For example, if damage = [2, 7, 4, 3] and armor = 4, you might choose to use your armor on level 1 (which deals 7 damage), reducing it to 3 damage. So you'd take: 2 + 3 + 4 + 3 = 12 total damage, meaning you need at least 13 health to survive the game.

Intuition

Since we can use the armor ability only once throughout the entire game, we want to maximize its benefit. The key insight is: where should we use the armor to minimize the total damage taken?

Think about it this way - the armor blocks up to armor points of damage on whichever level we choose. To get the maximum benefit, we should use it on the level that deals the most damage. Why? Because that's where we can potentially block the most damage.

Consider this: if we have levels dealing [2, 7, 4, 3] damage and armor = 4:

• Using armor on level 0: blocks 2 damage (the full 2 damage of that level)
• Using armor on level 1: blocks 4 damage (4 out of 7 damage)
• Using armor on level 2: blocks 4 damage (the full 4 damage of that level)
• Using armor on level 3: blocks 3 damage (the full 3 damage of that level)

The maximum damage we can avoid is min(max(damage), armor). This is because:

• We target the level with max(damage)
• But we can only block up to armor amount of damage
• So the actual damage blocked is the minimum of these two values

Once we know the maximum damage we can avoid, the total damage we'll take is: total_damage = sum(damage) - min(max(damage), armor)

Since our health must be greater than 0 after taking all this damage, we need at least: minimum_health = total_damage + 1

This greedy approach of using armor on the highest damage level guarantees we minimize the total damage taken, and therefore minimize the starting health required.

Learn more about Greedy patterns.

Solution Approach

The implementation follows a greedy strategy where we use the armor on the level with maximum damage to minimize the total damage taken.

Here's the step-by-step breakdown:

1. Calculate total damage: First, we compute sum(damage) to get the total damage from all levels if we don't use armor at all.

2. Find maximum damage level: We identify max(damage) to find the level where we should use our armor for maximum benefit.

3. Determine armor effectiveness: We calculate min(max(damage), armor) to find how much damage we can actually block:

   • If max(damage) ≤ armor, we can block all damage from that level
   • If max(damage) > armor, we can only block armor amount of damage

4. Calculate minimum health: The formula becomes:

   minimum_health = sum(damage) - min(max(damage), armor) + 1

   Breaking this down:

   • sum(damage) is the total damage we'd take without armor
   • We subtract min(max(damage), armor) which is the damage avoided by using armor optimally
   • We add 1 because our health must be greater than 0 (not equal to 0) after taking all damage

Example walkthrough: If damage = [2, 7, 4, 3] and armor = 4:

• Total damage without armor: 2 + 7 + 4 + 3 = 16
• Maximum damage level: max(damage) = 7
• Damage blocked by armor: min(7, 4) = 4
• Total damage after using armor: 16 - 4 = 12
• Minimum health needed: 12 + 1 = 13

The solution is implemented in a single line because the greedy approach is straightforward - we don't need to track states or use complex data structures. The time complexity is O(n) for finding the sum and maximum, and space complexity is O(1).

Example Walkthrough

Let's walk through a small example with damage = [3, 5, 2] and armor = 4.

Step 1: Calculate total damage without armor

• Sum of all damage: 3 + 5 + 2 = 10
• This is how much damage we'd take if we don't use armor at all

Step 2: Find the maximum damage level

• Looking at each level: 3, 5, 2
• Maximum damage is 5 (at level 1)
• This is where we should use our armor for maximum benefit

Step 3: Calculate how much damage the armor blocks

• The armor can block up to 4 points of damage
• Level 1 deals 5 damage
• Damage blocked = min(5, 4) = 4
• So level 1 will now deal 5 - 4 = 1 damage instead of 5

Step 4: Calculate total damage after using armor

• Level 0: 3 damage (no armor)
• Level 1: 1 damage (armor used, blocked 4 out of 5)
• Level 2: 2 damage (no armor)
• Total: 3 + 1 + 2 = 6 damage

Step 5: Determine minimum starting health

• We take 6 total damage
• Our health must be > 0 after all levels
• Minimum health = 6 + 1 = 7

Verification: Starting with 7 health:

• After level 0: 7 - 3 = 4 health (> 0 ✓)
• After level 1: 4 - 1 = 3 health (> 0 ✓)
• After level 2: 3 - 2 = 1 health (> 0 ✓)

All levels completed successfully with health > 0!

Solution Implementation

Time and Space Complexity

The time complexity is O(n), where n is the length of the array damage. This is because:

• sum(damage) iterates through all n elements once, taking O(n) time
• max(damage) also iterates through all n elements to find the maximum value, taking O(n) time
• min(max(damage), armor) is a constant time operation O(1)
• The arithmetic operations (subtraction and addition) are O(1)

Since these operations are performed sequentially, the overall time complexity is O(n) + O(n) + O(1) = O(n).

The space complexity is O(1). The algorithm only uses a constant amount of extra space for:

• Storing the sum result
• Storing the maximum value from the damage array
• Storing the minimum between max damage and armor
• Performing the final calculation

No additional data structures that scale with input size are created, so the space complexity remains constant.

Learn more about how to find time and space complexity quickly.

Common Pitfalls

Pitfall 1: Misunderstanding the Health Requirement

The Problem: A common mistake is thinking that having exactly 0 health after all damage is acceptable. Many developers incorrectly calculate the minimum health as just sum(damage) - min(max(damage), armor) without adding 1.

Why it happens: The problem statement says "health must always be greater than 0", but it's easy to misread this as "greater than or equal to 0" or forget this constraint when implementing.

Example of incorrect code:

def minimumHealth(self, damage: List[int], armor: int) -> int:
    return sum(damage) - min(max(damage), armor)  # WRONG: allows health to reach 0

Correct approach: Always add 1 to ensure health remains positive:

def minimumHealth(self, damage: List[int], armor: int) -> int:
    return sum(damage) - min(max(damage), armor) + 1  # Ensures health > 0

Pitfall 2: Not Handling Edge Cases

The Problem: Failing to consider edge cases like empty damage arrays or when armor value is 0.

Why it happens: The greedy solution seems straightforward, but edge cases can cause runtime errors with functions like max() on empty lists.

Example scenarios:

• damage = [] would cause max([]) to throw a ValueError
• armor = 0 means no protection (though the formula handles this correctly)

Solution: Add validation for edge cases:

def minimumHealth(self, damage: List[int], armor: int) -> int:
    if not damage:  # Empty damage array
        return 1  # Need at least 1 health to "survive"
  
    return sum(damage) - min(max(damage), armor) + 1

Pitfall 3: Overthinking the Problem

The Problem: Attempting to use dynamic programming or trying to determine if using armor on a different level (not the maximum) could somehow be better.

Why it happens: The problem seems like it might require exploring different choices, leading to unnecessary complexity.

Incorrect approach example:

def minimumHealth(self, damage: List[int], armor: int) -> int:
    # Unnecessary DP approach trying all possibilities
    min_health = float('inf')
    for i in range(len(damage)):
        # Try using armor on level i
        total = sum(damage) - min(damage[i], armor)
        min_health = min(min_health, total + 1)
    return min_health

Why the greedy approach works: Using armor on the maximum damage level always yields the optimal result because:

• We want to minimize total damage taken
• Armor reduces damage by min(damage[i], armor) for whichever level we choose
• To maximize reduction, we should choose the level where min(damage[i], armor) is largest
• This is always the level with maximum damage (within armor's capacity)