<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 DSU:

    def __init__(self, N):
        self._parent = list(range(N))
        self.rank = [1] * N

    def parent(self, u):
        parent_u = self._parent[u]
        if parent_u == u:
            return u
        self._parent[u] = self.parent(parent_u)
        return self._parent[u]

    def is_same_set(self, u, v):
        return self.parent(u) == self.parent(v)

    def unite(self, u, v):
        parent_u = self.parent(u)
        parent_v = self.parent(v)
        if parent_u != parent_v:
            if self.rank[parent_u] < self.rank[parent_v]:
                parent_u, parent_v = parent_v, parent_u
            self._parent[parent_v] = parent_u
            self.rank[parent_u] += self.rank[parent_v]

@cache
def is_horizontally_connected(left, right):
    return left in [1, 4, 6] and right in [1, 3, 5]

@cache
def is_vertically_connected(up, down):
    return up in [2, 3, 4] and down in [2, 5, 6]

class Solution:
    def hasValidPath(self, grid: List[List[int]]) -> bool:
        rows = len(grid)
        columns = len(grid[0])
        dsu = DSU(rows * columns)


        for row_index in range(rows):
            for column_index in range(columns):
                if column_index and is_horizontally_connected(grid[row_index][column_index - 1], grid[row_index][column_index]):
                    dsu.unite(row_index * columns + column_index, row_index * columns + column_index - 1)
                if row_index and is_vertically_connected(grid[row_index - 1][column_index], grid[row_index][column_index]):
                    dsu.unite(row_index * columns + column_index, (row_index - 1) * columns + column_index)
        return dsu.is_same_set(0, rows * columns - 1)
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