Minimum Cost to Make at Least One Valid Path in a Grid
<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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1300-1399/1368.Minimum%20Cost%20to%20Make%20at%20Least%20One%20Valid%20Path%20in%20a%20Grid/images/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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1300-1399/1368.Minimum%20Cost%20to%20Make%20at%20Least%20One%20Valid%20Path%20in%20a%20Grid/images/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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1300-1399/1368.Minimum%20Cost%20to%20Make%20at%20Least%20One%20Valid%20Path%20in%20a%20Grid/images/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>
Breadth-First Search; Graph; Array; Matrix; Shortest Path; Heap (Priority Queue)
class Solution {
    public int minCost(int[][] grid) {
        int m = grid.length, n = grid[0].length;
        boolean[][] vis = new boolean[m][n];
        Deque<int[]> q = new ArrayDeque<>();
        q.offer(new int[] {0, 0, 0});
        int[][] dirs = {{0, 0}, {0, 1}, {0, -1}, {1, 0}, {-1, 0}};
        while (!q.isEmpty()) {
            int[] p = q.poll();
            int i = p[0], j = p[1], d = p[2];
            if (i == m - 1 && j == n - 1) {
                return d;
            }
            if (vis[i][j]) {
                continue;
            }
            vis[i][j] = true;
            for (int k = 1; k <= 4; ++k) {
                int x = i + dirs[k][0], y = j + dirs[k][1];
                if (x >= 0 && x < m && y >= 0 && y < n) {
                    if (grid[i][j] == k) {
                        q.offerFirst(new int[] {x, y, d});
                    } else {
                        q.offer(new int[] {x, y, d + 1});
                    }
                }
            }
        }
        return -1;
    }
}