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

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

<p>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&#39;s health (represented by positive integers).</p>

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

<p>Return <em>the knight&#39;s minimum initial health so that he can rescue the princess</em>.</p>

<p><strong>Note</strong> 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.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/03/13/dungeon-grid-1.jpg" style="width: 253px; height: 253px;" />
<pre>
<strong>Input:</strong> dungeon = [[-2,-3,3],[-5,-10,1],[10,30,-5]]
<strong>Output:</strong> 7
<strong>Explanation:</strong> The initial health of the knight must be at least 7 if he follows the optimal path: RIGHT-&gt; RIGHT -&gt; DOWN -&gt; DOWN.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> dungeon = [[0]]
<strong>Output:</strong> 1
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == dungeon.length</code></li>
	<li><code>n == dungeon[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 200</code></li>
	<li><code>-1000 &lt;= dungeon[i][j] &lt;= 1000</code></li>
</ul>

class Solution {
public:
    int m, n;
    void print(vector<vector<int>>& dp) {
        for(int i = 0; i < m; i++) {
            for(int j = 0; j < n; j++) {
                cout << dp[i][j] << " ";
            }
            cout << endl;
        }
    }
    bool isPossible(int ans, vector<vector<int>>& dungeon) {
        vector<vector<int>> dp(m, vector<int>(n, 0));
        dp[0][0] = ans + dungeon[0][0];
        for(int i = 1; i < m; i++) {
            if(dp[i-1][0] > 0)
                dp[i][0] = dungeon[i][0] + dp[i-1][0];
        }
        for(int i = 1; i < n; i++) {
            if(dp[0][i-1] > 0)
                dp[0][i] = dungeon[0][i] + dp[0][i-1];
        }
        for(int i = 1; i < m; i++) {
            for(int j = 1; j < n; j++) {
                if(dp[i-1][j] > 0 && dp[i][j-1] > 0)
                    dp[i][j] = max(dp[i-1][j], dp[i][j-1]) + dungeon[i][j];
                else if(dp[i-1][j] > 0) 
                    dp[i][j] = dp[i-1][j] + dungeon[i][j];
                else if(dp[i][j-1] > 0)
                    dp[i][j] = dp[i][j-1] + dungeon[i][j];
            }
        }
        return dp[m-1][n-1] > 0;
    }
    int calculateMinimumHP(vector<vector<int>>& dungeon) {
        m = dungeon.size(), n = dungeon[0].size();
        int l = 1, h = m * n * 1000, ans = -1;
        while(l <= h) {
            int mid = (l + h) / 2;
            if(isPossible(mid, dungeon)) {
                ans = mid;
                h = mid - 1;
            } else {
                l = mid + 1;
            }
        }
        return ans;
    }
};
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