<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 Solution:
    def hasValidPath(self, grid: List[List[int]]) -> bool:
        # Time complexity: O(m*n), space complexity: O(1).
        m, n = len(grid), len(grid[0])
        if m == 1 and n == 1: return True
        adj = [[], [(0,-1),(0,1)], [(-1,0),(1,0)], \
               [(0,-1),(1,0)], [(0,1),(1,0)], \
               [(0,-1),(-1,0)], [(-1,0),(0,1)]]
        
        def helper(r0, c0, v0, h0):
        # Check if there is a path that leads to cell (m - 1,
        # n - 1) starting from cell (r0, c0) with initial
        # direction (v0, h0).
            nonlocal grid, m, n, adj
            i0, j0 = r0, c0
            i1, j1 = r0 + v0, c0 + h0
            while 0 <= i1 < m and 0 <= j1 < n:
            # Cell (i1, j1) is the cell that cell (i0, j0)
            # leads to.
                v1, h1 = adj[grid[i1][j1]][0]
                v2, h2 = adj[grid[i1][j1]][1]
                if (i0, j0) == (i1 + v1, j1 + h1):
                    i2, j2 = i1 + v2, j1 + h2
                elif (i0, j0) == (i1 + v2, j1 + h2):
                    i2, j2 = i1 + v1, j1 + h1
                else: return 0
                # Cell (i0, j0) is connected to cell (i1, j1).
                if (i1, j1) == (m - 1, n - 1): return 1
                elif (i1, j1) == (r0, c0): return -1
                else:
                    i0, j0 = i1, j1
                    i1, j1 = i2, j2
            return 0
        
        for v0, h0 in adj[grid[0][0]]:
            res = helper(0, 0, v0, h0)
            if res == -1: return False
            elif res == 1: return True
        return False
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