<p>Given an <code>m x n</code> integer matrix <code>heightMap</code> representing the height of each unit cell in a 2D elevation map, return <em>the volume of water it can trap after raining</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/08/trap1-3d.jpg" style="width: 361px; height: 321px;" />
<pre>
<strong>Input:</strong> heightMap = [[1,4,3,1,3,2],[3,2,1,3,2,4],[2,3,3,2,3,1]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> After the rain, water is trapped between the blocks.
We have two small ponds 1 and 3 units trapped.
The total volume of water trapped is 4.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/08/trap2-3d.jpg" style="width: 401px; height: 321px;" />
<pre>
<strong>Input:</strong> heightMap = [[3,3,3,3,3],[3,2,2,2,3],[3,2,1,2,3],[3,2,2,2,3],[3,3,3,3,3]]
<strong>Output:</strong> 10
</pre>

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

<ul>
	<li><code>m == heightMap.length</code></li>
	<li><code>n == heightMap[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 200</code></li>
	<li><code>0 &lt;= heightMap[i][j] &lt;= 2 * 10<sup>4</sup></code></li>
</ul>

auto foo = std::ios_base::sync_with_stdio (false);

class Solution {
private:
    template <class  M>
    void print(const M& mmap)
    {
        for (const auto& e : mmap)
            std::cout << '{' << e.first << ":(" << e.second.first << "," << e.second.second << ")} ";
        std::cout << '\n';
    }

public:
    int trapRainWater(vector<vector<int>>& heightMap) {
        int r_len = heightMap.size();
        int c_len = heightMap[0].size();

        vector<vector<bool>> visited(r_len, vector<bool>(c_len, false));
        std::multimap<int, pair<int, int>> mmap; // height -> x,y

        // add the bounding box
        for(int i=0;i<r_len; i++) {
            for (int j=0; j<c_len; j++) {
                if (i==0 || i==r_len-1 || j==0 || j==c_len-1) {
                    mmap.insert({heightMap[i][j], {i,j} });
                    visited[i][j] = true;
                }
            }
        }
        
        int total_water = 0;
        int minHeight = 0;
        while(mmap.size()) {
            const auto &cur_min = mmap.begin();
            int x = cur_min->second.first;
            int y = cur_min->second.second;
            int height = cur_min->first;
            minHeight=max(minHeight,height);
            mmap.erase(cur_min);

            int delta_r[4] = {+1,0,-1,0};
            int delta_c[4] = {0,1,0,-1};
            for(int i=0; i<4; i++){
                int cur_x = x+delta_r[i];
                int cur_y = y+delta_c[i];
                if (cur_x >=0 && cur_x < r_len && cur_y >=0 && cur_y < c_len && !visited[cur_x][cur_y]) {
                    mmap.insert({heightMap[cur_x][cur_y], {cur_x,cur_y}});
                    visited[cur_x][cur_y] = true;

                    if (heightMap[cur_x][cur_y] < minHeight)
                        total_water += minHeight - heightMap[cur_x][cur_y];

                }
            }
        }

        return total_water;
        
    }
};
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