<p>Given an <code>m x n</code> grid. Each cell of the grid has a sign pointing to the next cell you should visit if you are currently in this cell. The sign of <code>grid[i][j]</code> can be:</p>

<ul>
	<li><code>1</code> which means go to the cell to the right. (i.e go from <code>grid[i][j]</code> to <code>grid[i][j + 1]</code>)</li>
	<li><code>2</code> which means go to the cell to the left. (i.e go from <code>grid[i][j]</code> to <code>grid[i][j - 1]</code>)</li>
	<li><code>3</code> which means go to the lower cell. (i.e go from <code>grid[i][j]</code> to <code>grid[i + 1][j]</code>)</li>
	<li><code>4</code> which means go to the upper cell. (i.e go from <code>grid[i][j]</code> to <code>grid[i - 1][j]</code>)</li>
</ul>

<p>Notice that there could be some signs on the cells of the grid that point outside the grid.</p>

<p>You will initially start at the upper left cell <code>(0, 0)</code>. A valid path in the grid is a path that starts from the upper left cell <code>(0, 0)</code> and ends at the bottom-right cell <code>(m - 1, n - 1)</code> following the signs on the grid. The valid path does not have to be the shortest.</p>

<p>You can modify the sign on a cell with <code>cost = 1</code>. You can modify the sign on a cell <strong>one time only</strong>.</p>

<p>Return <em>the minimum cost to make the grid have at least one valid path</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/13/grid1.png" style="width: 400px; height: 390px;" />
<pre>
<strong>Input:</strong> grid = [[1,1,1,1],[2,2,2,2],[1,1,1,1],[2,2,2,2]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> You will start at point (0, 0).
The path to (3, 3) is as follows. (0, 0) --&gt; (0, 1) --&gt; (0, 2) --&gt; (0, 3) change the arrow to down with cost = 1 --&gt; (1, 3) --&gt; (1, 2) --&gt; (1, 1) --&gt; (1, 0) change the arrow to down with cost = 1 --&gt; (2, 0) --&gt; (2, 1) --&gt; (2, 2) --&gt; (2, 3) change the arrow to down with cost = 1 --&gt; (3, 3)
The total cost = 3.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/13/grid2.png" style="width: 350px; height: 341px;" />
<pre>
<strong>Input:</strong> grid = [[1,1,3],[3,2,2],[1,1,4]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> You can follow the path from (0, 0) to (2, 2).
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/13/grid3.png" style="width: 200px; height: 192px;" />
<pre>
<strong>Input:</strong> grid = [[1,2],[4,3]]
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 100</code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 4</code></li>
</ul>

class Solution {
public:
    int minCost(vector<vector<int>>& grid) {
        int row = grid.size();
        int col = grid[0].size();
        priority_queue<vector<int>,vector<vector<int>>,greater<vector<int>>>pq;
        vector<vector<int>> dist(row, vector<int>(col, INT_MAX));

        dist[0][0] = 0;
        pq.push({0, 0, 0});
        vector<int> dir = {0, 1, 0, -1, 0};
        map<pair<int,int>,int> dirVal;

        dirVal[{0,1}]  = 1;
        dirVal[{0,-1}] = 2;
        dirVal[{1,0}]  = 3;
        dirVal[{-1,0}] = 4;

        while(!pq.empty()) {
            vector<int>top = pq.top();
            pq.pop();
            int pathDist = top[0];
            int x = top[1];
            int y = top[2];

            if(x == row - 1 && y == col - 1)
                return dist[x][y];

            if(dist[x][y] < pathDist) continue;

            for(int i = 0;i < 4;i++) {
                int new_x = x + dir[i];
                int new_y = y + dir[i + 1];
                if(new_x >= 0 && new_x < row && new_y >= 0 && new_y < col) {
                    int dirCost = 0;
                    if(grid[x][y] != dirVal[{dir[i],dir[i + 1]}])
                        dirCost = 1;
                    int currDist = pathDist + dirCost;

                    if(currDist < dist[new_x][new_y]) {
                        dist[new_x][new_y] = currDist;
                        pq.push({currDist, new_x,new_y});
                    } 
                    
                }
            }
            

        }
    
    return dist[row - 1][col - 1];
    
    }
};
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