Dungeon Game

LeetCode 174 | Difficulty: Hard

Hard

Problem Description

The demons had captured the princess and imprisoned her in the bottom-right corner of a dungeon. The dungeon consists of m x n rooms laid out in a 2D grid. Our valiant knight was initially positioned in the top-left room and must fight his way through dungeon to rescue the princess.

The knight has an initial health point represented by a positive integer. If at any point his health point drops to 0 or below, he dies immediately.

Some of the rooms are guarded by demons (represented by negative integers), so the knight loses health upon entering these rooms; other rooms are either empty (represented as 0) or contain magic orbs that increase the knight's health (represented by positive integers).

To reach the princess as quickly as possible, the knight decides to move only rightward or downward in each step.

Return the knight's minimum initial health so that he can rescue the princess.

Note that any room can contain threats or power-ups, even the first room the knight enters and the bottom-right room where the princess is imprisoned.

Example 1:

Input: dungeon = [[-2,-3,3],[-5,-10,1],[10,30,-5]]
Output: 7
Explanation: The initial health of the knight must be at least 7 if he follows the optimal path: RIGHT-> RIGHT -> DOWN -> DOWN.

Example 2:

Input: dungeon = [[0]]
Output: 1

Constraints:

- `m == dungeon.length`

- `n == dungeon[i].length`

- `1 <= m, n <= 200`

- `-1000 <= dungeon[i][j] <= 1000`

Topics: Array, Dynamic Programming, Matrix

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Approach

Dynamic Programming

Break the problem into overlapping subproblems. Define a state (what information do you need?), a recurrence (how does state[i] depend on smaller states?), and a base case. Consider both top-down (memoization) and bottom-up (tabulation) approaches.

When to use

Optimal substructure + overlapping subproblems (counting ways, min/max cost, feasibility).

Matrix

Treat the matrix as a 2D grid. Common techniques: directional arrays (dx, dy) for movement, BFS/DFS for connected regions, in-place marking for visited cells, boundary traversal for spiral patterns.

When to use

Grid traversal, island problems, path finding, rotating/transforming matrices.

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Solutions

Solution 1: C# (Best: 92 ms)

Metric	Value
Runtime	92 ms
Memory	24.2 MB
Date	2019-12-09

Solution

public class Solution {
    public int CalculateMinimumHP(int[][] dungeon) {
        int m = dungeon.Length;
    int n = dungeon[0].Length;
    int[,] minHP = new int[m,n];
    
    for (int i = m-1; i >= 0; i--)
    {
        for (int j = n-1; j >= 0; j--)
        {
            if(i==m-1 && j==n-1)
            {
                minHP[i,j] = Math.Max(1, 1-dungeon[i][j]);
            }
            else if (i == m - 1)
            {
                minHP[i, j] = Math.Max(1, minHP[i,j+1] - dungeon[i][j]);
            }
            else if (j == n - 1)
            {
                minHP[i, j] = Math.Max(1, minHP[i+1,j] - dungeon[i][j]);
            }
            else 
            {
                minHP[i, j] = Math.Max(1, Math.Min(minHP[i,j+1],minHP[i+1,j]) - dungeon[i][j]);
            }
        }
    }
    return minHP[0,0];
    }
}

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Complexity Analysis

Approach	Time	Space
DP (2D)	$O(n × m)$	$O(n × m)$

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Interview Tips

Key Points

• Break the problem into smaller subproblems. Communicate your approach before coding.
• Define the DP state clearly. Ask: "What is the minimum information I need to make a decision at each step?"
• Consider if you can reduce space by only keeping the last row/few values.