Grid 1 - Implementation

Here is the implementation for Grid 1 (AtCoder DP-$H$):

MOD := 1000000007
Input: H, W, grid[1..H][1..W]

dp := 2D array of size H × W, all set to 0

if grid[1][1] = '.' then
    dp[1][1] := 1

for i from 1 to H
    for j from 1 to W
        if grid[i][j] = '#' then
            dp[i][j] := 0
        else
            if i > 1 then
                dp[i][j] := (dp[i][j] + dp[i - 1][j]) mod MOD
            if j > 1 then
                dp[i][j] := (dp[i][j] + dp[i][j - 1]) mod MOD

return dp[H][W]

You fill the grid row by row, left to right. For each empty cell, add the paths from above and from the left. Wall cells stay at $0$, blocking all paths through them. The mod keeps numbers from overflowing. Trace a small $3 \times 3$ grid with one wall to see how blocked cells cut off entire regions.

Time complexity: $O(H \cdot W)$.

Space complexity: $O(H \cdot W)$, reducible to $O(W)$ by keeping only the current and previous rows.