<p>Given an array of integers <code>heights</code> representing the histogram&#39;s bar height where the width of each bar is <code>1</code>, return <em>the area of the largest rectangle in the histogram</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/01/04/histogram.jpg" style="width: 522px; height: 242px;" />
<pre>
<strong>Input:</strong> heights = [2,1,5,6,2,3]
<strong>Output:</strong> 10
<strong>Explanation:</strong> The above is a histogram where width of each bar is 1.
The largest rectangle is shown in the red area, which has an area = 10 units.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/01/04/histogram-1.jpg" style="width: 202px; height: 362px;" />
<pre>
<strong>Input:</strong> heights = [2,4]
<strong>Output:</strong> 4
</pre>

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

<ul>
	<li><code>1 &lt;= heights.length &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= heights[i] &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution {
public:
    int largestRectangleArea(vector<int>& heights) {
        stack<int> min_track;
        int res = 0;
        vector<int> reach (heights.size(), 1);
        for (int i=0; i < heights.size(); i++) {
            if (min_track.empty()){
                min_track.push(i);
                //cout << "pushed " << i << " reach "<< reach[i] << endl;
                continue;
            }
            while (!min_track.empty() and heights[min_track.top()] >= heights[i]) {
                int index = min_track.top();
                //cout << "popped " << index << endl;
                min_track.pop();
                reach[index] += i - index -1;
                //cout << "update1 " << index << " reach "<< reach[index] << endl;
                //reach[i] = max(reach[index], reach[i]);
                reach[i] = reach[index]+1;
                //cout << "update2 " << i << " reach "<< reach[i] << endl;
            }
            min_track.push(i);
            //cout << "pushed " << i << " reach "<< reach[i] << endl;
        }
        if (!min_track.empty()){
            int prev = heights.size();
            while (!min_track.empty()) {
                int index = min_track.top();
                reach[index] += max(0, prev - index-1);
                //cout << "final update " << index << " reach "<< reach[index] << endl;
                min_track.pop();
            }
        }
        for (int i=0; i < heights.size(); i++) {
            // cout << "reach " << i << ": " << reach[i] << endl;
        }
        for (int i=0; i < heights.size(); i++) {
            res = max(heights[i] * reach[i], res);
        }
        return res;
    }
};
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