<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:
        m, n = len(grid), len(grid[0])

        def in_bounds(i, j) -> bool:
            if i < 0 or j < 0 or i >= m or j >= n:
                return False
            return True
        
        seen = set()

        def dfs(i, j) -> bool:
            # invalid states
            if (i, j) in seen or not in_bounds(i, j):
                return False
            seen.add((i, j))

            if i == m - 1 and j == n - 1:
                return True

            res = False
            if grid[i][j] == 1 and in_bounds(i, j - 1) and grid[i][j - 1] in (1, 4, 6):
                res = res or dfs(i, j - 1)
            if grid[i][j] == 1 and in_bounds(i, j + 1) and grid[i][j + 1] in (1, 3, 5):
                res = res or dfs(i, j + 1)

            if grid[i][j] == 2 and in_bounds(i - 1, j) and grid[i - 1][j] in (2, 3, 4):
                res = res or dfs(i - 1, j)
            if grid[i][j] == 2 and in_bounds(i + 1, j) and grid[i + 1][j] in (2, 5, 6):
                res = res or dfs(i + 1, j)

            if grid[i][j] == 3 and in_bounds(i, j - 1) and grid[i][j - 1] in (1, 4, 6):
                res = res or dfs(i, j - 1)
            if grid[i][j] == 3 and in_bounds(i + 1, j) and grid[i + 1][j] in (2, 5, 6):
                res = res or dfs(i + 1, j)

            if grid[i][j] == 4 and in_bounds(i + 1, j) and grid[i + 1][j] in (6, 2, 5):
                res = res or dfs(i + 1, j)
            if grid[i][j] == 4 and in_bounds(i, j + 1) and grid[i][j + 1] in (1, 3, 5):
                res = res or dfs(i, j + 1)

            if grid[i][j] == 5 and in_bounds(i - 1, j) and grid[i - 1][j] in (2, 3, 4):
                res = res or dfs(i - 1, j)
            if grid[i][j] == 5 and in_bounds(i, j - 1) and grid[i][j - 1] in (1, 4, 6):
                res = res or dfs(i, j - 1)

            if grid[i][j] == 6 and in_bounds(i - 1, j) and grid[i - 1][j] in (2, 3, 4):
                res = res or dfs(i - 1, j)
            if grid[i][j] == 6 and in_bounds(i, j + 1) and grid[i][j + 1] in (1, 3, 5):
                res = res or dfs(i, j + 1)

            return res

        return dfs(0, 0)


# problem:
# m x n grid
# grid[i][j] representa a stree
# 1: street connecting left and right cell
# 2: connect upper and lower
# 3: left cell and lower cell
# 4: right cell and lower cell
# 5: left cell and upper cell
# 6: right cell and upper cell
# start at (0, 0)
# valid path in grid is a path that ends in (m - 1, n - 1)
# path can only follow streets
# return TRUE if bottom right can be reached

# solution:
# dfs based
# 1 valid moves: if left cell is (1, 4, 6), if right cell is (1, 3, 5)
# 2 valid moves: if up cell is (2, 3, 4), if bottom cell is (2, 5, 6)
# 3 valid moves: if left cell is (1, 4, 6), if bottom cell is (2, 5, 6)
# 4 valid moves: if bottom cell is (6, 2, 5), if right cell is (1, 3, 5)
# 5 valid moves: if up cell is (2, 3, 4), if left cell is (1, 4, 6)
# 6 valid moves: if up cell is (2, 4, 3), if right cell is (1, 3, 5)
73018