<p>There are several squares being dropped onto the X-axis of a 2D plane.</p>

<p>You are given a 2D integer array <code>positions</code> where <code>positions[i] = [left<sub>i</sub>, sideLength<sub>i</sub>]</code> represents the <code>i<sup>th</sup></code> square with a side length of <code>sideLength<sub>i</sub></code> that is dropped with its left edge aligned with X-coordinate <code>left<sub>i</sub></code>.</p>

<p>Each square is dropped one at a time from a height above any landed squares. It then falls downward (negative Y direction) until it either lands <strong>on the top side of another square</strong> or <strong>on the X-axis</strong>. A square brushing the left/right side of another square does not count as landing on it. Once it lands, it freezes in place and cannot be moved.</p>

<p>After each square is dropped, you must record the <strong>height of the current tallest stack of squares</strong>.</p>

<p>Return <em>an integer array </em><code>ans</code><em> where </em><code>ans[i]</code><em> represents the height described above after dropping the </em><code>i<sup>th</sup></code><em> square</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/28/fallingsq1-plane.jpg" style="width: 500px; height: 505px;" />
<pre>
<strong>Input:</strong> positions = [[1,2],[2,3],[6,1]]
<strong>Output:</strong> [2,5,5]
<strong>Explanation:</strong>
After the first drop, the tallest stack is square 1 with a height of 2.
After the second drop, the tallest stack is squares 1 and 2 with a height of 5.
After the third drop, the tallest stack is still squares 1 and 2 with a height of 5.
Thus, we return an answer of [2, 5, 5].
</pre>

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

<pre>
<strong>Input:</strong> positions = [[100,100],[200,100]]
<strong>Output:</strong> [100,100]
<strong>Explanation:</strong>
After the first drop, the tallest stack is square 1 with a height of 100.
After the second drop, the tallest stack is either square 1 or square 2, both with heights of 100.
Thus, we return an answer of [100, 100].
Note that square 2 only brushes the right side of square 1, which does not count as landing on it.
</pre>

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

<ul>
	<li><code>1 &lt;= positions.length &lt;= 1000</code></li>
	<li><code>1 &lt;= left<sub>i</sub> &lt;= 10<sup>8</sup></code></li>
	<li><code>1 &lt;= sideLength<sub>i</sub> &lt;= 10<sup>6</sup></code></li>
</ul>

struct State {
    int startX, endX;
    int height;

    State(int a, int b, int c) {
        startX = a;
        endX = b;
        height = c;
    }
};

class Solution {
public:
    vector<int> fallingSquares(vector<vector<int>>& positions) {
        vector<State> states;

        // First square always hits the x axis.
        states.push_back(State(positions[0][0], positions[0][0] + positions[0][1], positions[0][1]));


        vector<int> ans = {positions[0][1]};
        int maxHeight = positions[0][1];
        for(int i = 1; i < positions.size(); i++) {
            // Sort to maintain our invariant for finding the highest square.
            sort(states.begin(), states.end(), [](State& s1, State& s2) {
                return s1.height < s2.height;
            });

            vector<int>& pos = positions[i];
            int startX = pos[0], endX = pos[0] + pos[1], height = pos[1];
            bool found = false;
            // Go backwards in our state array. First square that has overlapping X is the one this square would land on.
            for(int j = states.size()-1; j >= 0; j--) {
                State& prev = states[j];

                if (overlaps(startX, endX, prev.startX, prev.endX)) {
                    found = true;
                    states.push_back(State(startX, endX, height + prev.height));
                    break;
                }
            }
            if (!found) {
                states.push_back(State(startX, endX, height));
            }
            maxHeight = std::max(maxHeight, states[states.size()-1].height);
            ans.push_back(maxHeight);
        }

        return ans;
    }

    bool overlaps(int min1, int max1, int min2, int max2) {
        return (min2 >= min1 && min2 < max1) || (min1 >= min2 && min1 < max2);
    }
};
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