Minimum Amount of Damage Dealt to Bob

• Calculate t_i = (health[i] + power - 1) / power for each enemy i.
• Use a recursive helper function to generate all n! permutations of enemy indices (0, 1, ..., n-1).
• For each complete permutation p:
  • Initialize total_damage = 0.
  • Initialize current_dps = sum(damage).
  • For each enemy index j in the permutation p:
    • Add the damage incurred during this step: total_damage += t_j * current_dps.
    • Update the dps for the next step: current_dps -= damage[j].
  • Keep track of the minimum total_damage found across all permutations.
• Return the overall minimum damage.

The core idea of this approach is to explore the entire search space of n! possible orderings of enemies. Since any order is a permutation of the n enemies, we can generate all of them and find the one that yields the minimum damage.

First, we pre-calculate the time t_i required to defeat each enemy i, which is ceil(health[i] / power). This can be computed using integer arithmetic as (health[i] + power - 1) / power.

We then use a recursive function to generate all permutations of the enemy indices [0, 1, ..., n-1]. For each complete permutation, we simulate the battle from start to finish:

1. Initialize total_damage_for_this_order to 0.
2. Initialize current_dps to the sum of all enemies' damages.
3. Iterate through the enemies in the order given by the current permutation.
4. For each enemy, calculate the damage Bob takes while he is focused on defeating this enemy. This damage is t_i * current_dps.
5. Add this to total_damage_for_this_order.
6. After an enemy is defeated, subtract its damage from current_dps for the next stage of the battle.

We maintain a global minimum variable, updating it whenever we find an ordering that results in less total damage. While correct, this method is computationally expensive.

1import java.util.ArrayList;2import java.util.List;3 4class Solution {5    long minDamage = Long.MAX_VALUE;6    int n;7    int[] damage;8    long[] timeToKill;9 10    public long minimumDamage(int power, int[] damage, int[] health) {11        this.n = damage.length;12        this.damage = damage;13        this.timeToKill = new long[n];14        for (int i = 0; i < n; i++) {15            timeToKill[i] = (long)(health[i] + power - 1) / power;16        }17 18        permute(new ArrayList<>(), new boolean[n]);19        return minDamage;20    }21 22    private void permute(List<Integer> currentOrder, boolean[] visited) {23        if (currentOrder.size() == n) {24            calculateDamage(currentOrder);25            return;26        }27 28        for (int i = 0; i < n; i++) {29            if (!visited[i]) {30                visited[i] = true;31                currentOrder.add(i);32                permute(currentOrder, visited);33                currentOrder.remove(currentOrder.size() - 1);34                visited[i] = false; // backtrack35            }36        }37    }38 39    private void calculateDamage(List<Integer> order) {40        long currentTotalDamage = 0;41        long currentDps = 0;42        for (int d : damage) {43            currentDps += d;44        }45 46        for (int enemyIndex : order) {47            currentTotalDamage += timeToKill[enemyIndex] * currentDps;48            currentDps -= damage[enemyIndex];49        }50        minDamage = Math.min(minDamage, currentTotalDamage);51    }52}

• Calculate t_i = (health[i] + power - 1) / power for each enemy i.
• Precompute damageSum[mask] for all 2^n masks, where damageSum[mask] is the sum of damages of enemies in the set represented by mask.
• Initialize a dp array of size 2^n with a large value, and set dp[0] = 0.
• Iterate through masks from 1 to (1 << n) - 1.
• For each mask, iterate through each enemy k from 0 to n-1.
• If enemy k is in the current mask:
  • Let prev_mask = mask ^ (1 << k).
  • The damage incurred to defeat k after prev_mask is t_k * (total_initial_dps - damageSum[prev_mask]).
  • Update dp[mask] with the minimum of its current value and dp[prev_mask] plus the damage to kill k.
• The final answer is dp[(1 << n) - 1].

A more optimized approach than brute force is to use dynamic programming. We can define a state dp[mask] as the minimum total damage Bob receives to defeat the set of enemies represented by the bitmask mask. A mask is an integer where the i-th bit is 1 if enemy i has been defeated, and 0 otherwise.

The transition logic considers how a state mask is reached. It must be reached from a state with one fewer enemy defeated. Let's say enemy k was the last one to be defeated to form the set mask. This means we first defeated all enemies in mask except k (represented by prev_mask = mask ^ (1 << k)), and then we defeated enemy k.

The total damage to reach state mask by defeating k last is:

1. The minimum damage to defeat enemies in prev_mask, which is dp[prev_mask].
2. The damage incurred while defeating enemy k. At this point, all enemies not in prev_mask are alive. The total DPS is the sum of damages of these alive enemies. The time to kill k is t_k. So, this damage is t_k * (sum of damages of all enemies not in prev_mask).

We can precompute the sum of damages for every subset to quickly find the DPS at any stage. The recurrence relation becomes:

1dp[mask] = min_{k in mask} (dp[mask ^ (1 << k)] + t_k * (total_dps - damageSum[mask ^ (1 << k)]))

The base case is dp[0] = 0. The final answer is dp[(1 << n) - 1].

1import java.util.Arrays;2 3class Solution {4    public long minimumDamage(int power, int[] damage, int[] health) {5        int n = damage.length;6        long[] timeToKill = new long[n];7        long totalInitialDps = 0;8        for (int i = 0; i < n; i++) {9            timeToKill[i] = (long)(health[i] + power - 1) / power;10            totalInitialDps += damage[i];11        }12 13        long[] damageSum = new long[1 << n];14        for (int mask = 0; mask < (1 << n); mask++) {15            for (int i = 0; i < n; i++) {16                if ((mask & (1 << i)) != 0) {17                    damageSum[mask] += damage[i];18                }19            }20        }21 22        long[] dp = new long[1 << n];23        Arrays.fill(dp, Long.MAX_VALUE);24        dp[0] = 0;25 26        for (int mask = 1; mask < (1 << n); mask++) {27            for (int i = 0; i < n; i++) {28                if ((mask & (1 << i)) != 0) {29                    int prevMask = mask ^ (1 << i);30                    if (dp[prevMask] != Long.MAX_VALUE) {31                        long damageWhileKillingI = timeToKill[i] * (totalInitialDps - damageSum[prevMask]);32                        long newTotalDamage = dp[prevMask] + damageWhileKillingI;33                        dp[mask] = Math.min(dp[mask], newTotalDamage);34                    }35                }36            }37        }38 39        return dp[(1 << n) - 1];40    }41}

• Create a structure or class, say Enemy, to store each enemy's damage and its calculated timeToKill.
• For each enemy i, calculate timeToKill[i] = (health[i] + power - 1) / power.
• Populate an array of Enemy objects.
• Sort the Enemy array in descending order based on the ratio damage / timeToKill. To avoid floating-point arithmetic and potential precision errors, the comparison d1/t1 > d2/t2 should be implemented as d1 * t2 > d2 * t1. Use long for the cross-multiplication to prevent overflow.
• Initialize total_damage = 0 and current_dps = sum of all damages.
• Iterate through the sorted Enemy array:
  • For the current enemy e:
    • Add damage for this step: total_damage += e.timeToKill * current_dps.
    • Decrease the dps for subsequent steps: current_dps -= e.damage.
• Return total_damage.

This approach is based on a greedy strategy. The key insight comes from an exchange argument. If we consider any two adjacent enemies, A and B, in the defeat sequence, we can analyze how the total damage changes if we swap their order. This analysis reveals that placing the enemy with a higher damage[i] / time_to_kill[i] ratio first always results in a lower or equal total damage. This local optimality can be extended to prove that the global optimal strategy is to sort all enemies in descending order based on this ratio.

The algorithm is as follows:

1. For each enemy i, calculate the time required to defeat it: t_i = (health[i] + power - 1) / power.
2. Create a structure or pair for each enemy to hold its damage[i] and calculated t_i.
3. Sort these enemies in descending order based on the ratio damage[i] / t_i. To avoid floating-point precision issues, the comparison d1/t1 > d2/t2 is rewritten as d1 * t2 > d2 * t1. It's important to use long for this cross-multiplication to prevent potential integer overflow.
4. After sorting, iterate through the sorted list of enemies to calculate the total damage. We maintain a running sum of the DPS of all currently alive enemies (current_dps).
5. For each enemy in the sorted order, we add t_i * current_dps to the total damage, and then subtract damage[i] from current_dps as this enemy is now considered defeated.

This method is efficient and correct for this problem.

1import java.util.Arrays;2 3class Solution {4    class Enemy {5        int damage;6        long timeToKill;7 8        Enemy(int damage, long timeToKill) {9            this.damage = damage;10            this.timeToKill = timeToKill;11        }12    }13 14    public long minimumDamage(int power, int[] damage, int[] health) {15        int n = damage.length;16        Enemy[] enemies = new Enemy[n];17        long totalInitialDps = 0;18 19        for (int i = 0; i < n; i++) {20            long time = (long)(health[i] + power - 1) / power;21            enemies[i] = new Enemy(damage[i], time);22            totalInitialDps += damage[i];23        }24 25        // Sort enemies in descending order of damage/time ratio.26        // To avoid floating point, we compare d1/t1 > d2/t2 as d1*t2 > d2*t1.27        // Use long for multiplication to avoid overflow.28        Arrays.sort(enemies, (e1, e2) -> {29            long val1 = (long)e1.damage * e2.timeToKill;30            long val2 = (long)e2.damage * e1.timeToKill;31            return Long.compare(val2, val1); // Descending order32        });33 34        long totalDamageDealt = 0;35        long currentDps = totalInitialDps;36 37        for (Enemy enemy : enemies) {38            totalDamageDealt += enemy.timeToKill * currentDps;39            currentDps -= enemy.damage;40        }41 42        return totalDamageDealt;43    }44}