<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:
    bool IsPossibleToReach(int start, vector<vector<int>>& dungeon) {
        // O(m) mem, O(n * m) time
        vector<int> prev(dungeon[0].size());

        prev[0] = start;
        for (int i = 0; i < dungeon.size(); ++i) {
            vector<int> current(dungeon[i].size(), 0);

            if (prev[0] > 0) {
                current[0] = max(0, prev[0] + dungeon[i][0]);
            }

            for (int j = 1; j < current.size(); ++j) {
                int mx_state = max(prev[j], current[j - 1]);
                if (mx_state > 0) {
                    current[j] = max(0, mx_state + dungeon[i][j]);
                }
            }
            prev = current;
        }
        return prev.back() > 0;
    }

    int calculateMinimumHP(vector<vector<int>>& dungeon) {
        int left = 0, right = 1000 * 200 + 1;
        // (left, right]

        // O(log((n + m) maxVal) * n * m), O(m)

        while (right - left > 1) {
            int m = (left + right) / 2;

            if (IsPossibleToReach(m, dungeon)) {
                right = m;
            } else {
                left = m;
            }
        }
        return right;
    }
};
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