<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>

class UnionFind:
    def __init__(self, n):
        self.parent = [0] * n
        for i in range(n):
            self.parent[i] = i
        self.n = n
        self.rank = [1] * n

    def find(self, x):
        parent = self.parent
        if x != parent[x]:
            parent[x] = self.find(parent[x])
        return parent[x]

    def union(self, p, q):
        root_p = self.find(p)
        root_q = self.find(q)
        if root_p == root_q:
            return 0
        self.parent[root_p] = root_q
        return 1


class Solution:
    def hasValidPath(self, grid: List[List[int]]) -> bool:
        right = {
            1: [1, 4, 6],
            2: [],
            3: [1, 4, 6],
            4: [],
            5: [1, 4, 6],
            6: []
        }

        down = {
            1: [],
            2: [2, 3, 4],
            3: [],
            4: [],
            5: [2, 3, 4],
            6: [2, 3, 4]
        }
        m, n = len(grid), len(grid[0])
        uf = UnionFind(m * n)
        for i in range(m):
            for j in range(n):
                # if i>0, we can union the up
                if i > 0 and grid[i - 1][j] in down[grid[i][j]]:
                    uf.union(n * i + j, n * i + j - n)
                if j > 0 and grid[i][j - 1] in right[grid[i][j]]:
                    uf.union(n * i + j, n * i + j - 1)

        return uf.find(0) == uf.find(m * n - 1)
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