<p>You are given an <code>m x n</code> <code>grid</code>. Each cell of <code>grid</code> represents a street. The street of <code>grid[i][j]</code> can be:</p>

<ul>
	<li><code>1</code> which means a street connecting the left cell and the right cell.</li>
	<li><code>2</code> which means a street connecting the upper cell and the lower cell.</li>
	<li><code>3</code> which means a street connecting the left cell and the lower cell.</li>
	<li><code>4</code> which means a street connecting the right cell and the lower cell.</li>
	<li><code>5</code> which means a street connecting the left cell and the upper cell.</li>
	<li><code>6</code> which means a street connecting the right cell and the upper cell.</li>
</ul>
<img alt="" src="https://assets.leetcode.com/uploads/2020/03/05/main.png" style="width: 450px; height: 708px;" />
<p>You will initially start at the street of 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>. <strong>The path should only follow the streets</strong>.</p>

<p><strong>Notice</strong> that you are <strong>not allowed</strong> to change any street.</p>

<p>Return <code>true</code><em> if there is a valid path in the grid or </em><code>false</code><em> otherwise</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/03/05/e1.png" style="width: 455px; height: 311px;" />
<pre>
<strong>Input:</strong> grid = [[2,4,3],[6,5,2]]
<strong>Output:</strong> true
<strong>Explanation:</strong> As shown you can start at cell (0, 0) and visit all the cells of the grid to reach (m - 1, n - 1).
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/03/05/e2.png" style="width: 455px; height: 293px;" />
<pre>
<strong>Input:</strong> grid = [[1,2,1],[1,2,1]]
<strong>Output:</strong> false
<strong>Explanation:</strong> As shown you the street at cell (0, 0) is not connected with any street of any other cell and you will get stuck at cell (0, 0)
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> grid = [[1,1,2]]
<strong>Output:</strong> false
<strong>Explanation:</strong> You will get stuck at cell (0, 1) and you cannot reach cell (0, 2).
</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;= 300</code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 6</code></li>
</ul>

from typing import List

class Solution:
    def hasValidPath(self, grid: List[List[int]]) -> bool:
        R, C = len(grid), len(grid[0])
        parent = list(range(R * C))
        rank = [0] * (R * C)
        
        def find(x):
            if x == parent[x]:
                return x
            parent[x] = find(parent[x])
            return parent[x]
        
        def union(x, y):
            ux = find(x)
            uy = find(y)
            if ux == uy:
                return
            if rank[ux] > rank[uy]:
                parent[uy] = ux
            elif rank[ux] < rank[uy]:
                parent[ux] = uy
            else:
                parent[uy] = ux
                rank[ux] += 1
        
        def unionkaro(i1, j1, i2, j2):
            if 0 <= i1 < R and 0 <= i2 < R and 0 <= j1 < C and 0 <= j2 < C:
                union(i1 * C + j1, i2 * C + j2)
        
        for i in range(R):
            for j in range(C):
                if grid[i][j] == 1:
                    if j + 1 < C and grid[i][j + 1] in [1, 3, 5]:
                        unionkaro(i, j, i, j + 1)
                    if j - 1 >= 0 and grid[i][j - 1] in [1, 4, 6]:
                        unionkaro(i, j, i, j - 1)
                elif grid[i][j] == 2:
                    if i + 1 < R and grid[i + 1][j] in [2, 5, 6]:
                        unionkaro(i, j, i + 1, j)
                    if i - 1 >= 0 and grid[i - 1][j] in [2, 3, 4]:
                        unionkaro(i, j, i - 1, j)
                elif grid[i][j] == 3:
                    if j - 1 >= 0 and grid[i][j - 1] in [1, 4, 6]:
                        unionkaro(i, j, i, j - 1)
                    if i + 1 < R and grid[i + 1][j] in [2, 5, 6]:
                        unionkaro(i, j, i + 1, j)
                elif grid[i][j] == 4:
                    if j + 1 < C and grid[i][j + 1] in [1, 3, 5]:
                        unionkaro(i, j, i, j + 1)
                    if i + 1 < R and grid[i + 1][j] in [2, 5, 6]:
                        unionkaro(i, j, i + 1, j)
                elif grid[i][j] == 5:
                    if j - 1 >= 0 and grid[i][j - 1] in [1, 4, 6]:
                        unionkaro(i, j, i, j - 1)
                    if i - 1 >= 0 and grid[i - 1][j] in [2, 3, 4]:
                        unionkaro(i, j, i - 1, j)
                elif grid[i][j] == 6:
                    if j + 1 < C and grid[i][j + 1] in [1, 3, 5]:
                        unionkaro(i, j, i, j + 1)
                    if i - 1 >= 0 and grid[i - 1][j] in [2, 3, 4]:
                        unionkaro(i, j, i - 1, j)

        return find(0) == find(R * C - 1)
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