You are a hero in a game, initially having HP health points and K magic potions.
You have to fight N monsters in the order they are given, knowing the damage each monster deals to you.
damage[i] represents the damage the monster deals to you when you fight them. If the damage is negative, it means your HP increases by that amount.
If you use a magic potion on a monster, it will kill the monster without you fighting or losing health points.
If you reach zero HP, you die. Therefore you can't win agains a monster that has damage[i] >= HP
Find the maximum number of monsters you can kill, considering you are playing optimally.
Example
Input: K = 2, HP = 10,
damage = [-3, 2, 3, -2, 8, 8, 6, 4, 3, 3]
Output: 8
Explanation: You kill the first 8 monsters by using the two potions on the 6th and 7th monsters
The core challenge of this problem is to maximize the number of monsters you can kill while managing your health points (HP) and the limited number of magic potions (K). The significance of this problem lies in its application to resource management and optimization scenarios, which are common in game development and real-world applications.
Potential pitfalls include mismanaging the use of potions or failing to account for the order in which monsters are fought, leading to suboptimal solutions.
To solve this problem, we need to carefully decide when to use the magic potions. A naive approach would be to use potions on the first K monsters, but this is not optimal. Instead, we should use potions on the monsters that deal the most damage, as this will allow us to survive longer and kill more monsters.
We can use a max-heap (priority queue) to keep track of the most damaging monsters we have encountered so far. This allows us to efficiently decide when to use a potion.
The naive solution would involve iterating through the list of monsters and using potions on the first K monsters. This approach is not optimal because it does not consider the damage dealt by each monster.
The optimized solution involves using a max-heap to keep track of the most damaging monsters. We iterate through the list of monsters, and for each monster, we decide whether to fight it or use a potion based on the current state of our health points and the damage dealt by the monster.
1. Initialize a max-heap to keep track of the most damaging monsters.
2. Iterate through the list of monsters.
3. For each monster, check if the damage can be absorbed by the current HP.
4. If the damage can be absorbed, subtract the damage from HP and add the damage to the max-heap.
5. If the damage cannot be absorbed, check if we have any potions left.
6. If we have potions left, use a potion on the most damaging monster encountered so far (from the max-heap) and continue.
7. If we do not have any potions left, break the loop as we cannot kill any more monsters.
// JavaScript implementation of the optimized solution
// Importing the PriorityQueue class from a library like 'js-priority-queue'
const PriorityQueue = require('js-priority-queue');
function maxMonstersKilled(K, HP, damage) {
// Initialize a max-heap (priority queue)
let maxHeap = new PriorityQueue({ comparator: (a, b) => b - a });
let monstersKilled = 0;
for (let i = 0; i < damage.length; i++) {
if (damage[i] < 0) {
// If the damage is negative, it means HP increases
HP += Math.abs(damage[i]);
} else {
// If the damage can be absorbed by the current HP
if (HP > damage[i]) {
HP -= damage[i];
maxHeap.queue(damage[i]);
} else if (K > 0) {
// If we have potions left, use a potion on the most damaging monster
if (maxHeap.length > 0 && maxHeap.peek() > damage[i]) {
HP += maxHeap.dequeue();
HP -= damage[i];
maxHeap.queue(damage[i]);
}
K--;
} else {
// If we cannot absorb the damage and have no potions left, break the loop
break;
}
}
monstersKilled++;
}
return monstersKilled;
}
// Example usage
const K = 2;
const HP = 10;
const damage = [-3, 2, 3, -2, 8, 8, 6, 4, 3, 3];
console.log(maxMonstersKilled(K, HP, damage)); // Output: 8
The time complexity of this solution is O(N log K) because we are using a max-heap to keep track of the most damaging monsters, and each insertion and deletion operation in the heap takes O(log K) time. The space complexity is O(K) due to the storage of the heap.
Potential edge cases include:
These edge cases can be tested by creating specific test cases and verifying the output.
To test the solution comprehensively, we can use a variety of test cases:
We can use testing frameworks like Jest or Mocha for automated testing.
When approaching such problems, it is important to:
In this blog post, we discussed the problem of killing monsters in a game while managing health points and magic potions. We explored a naive solution and an optimized solution using a max-heap. We also analyzed the complexity, discussed edge cases, and provided tips for problem-solving. Understanding and solving such problems is crucial for developing efficient algorithms and improving coding skills.