Incorrect Placement visited node set in Graph traversal DFS

Question

I am having trouble understanding why placing visited.append((i,j)) in the else case gives me the right output, while keeping it as shown below gives the incorrect output. I tried to debug with a simpler example [[0,1], [1,1]] But I am not able to figure out why keeping it outside the else gives wrong result. Code:

from typing import List


class Solution:
    def islandPerimeter(self, grid: List[List[int]]) -> int:
        ni, nj = len(grid), len(grid[0])
        visited = set()
        def dfs(i, j):
            nonlocal visited

            if (i,j) in visited:
                return 0
            
            visited.add((i,j))  # Placing this inside else gives me right result

            if i < 0 or j < 0 or i >= ni or j >= nj or grid[i][j] == 0:
                return 1
            
            else:
                return dfs(i-1, j) + dfs(i+1, j) + dfs(i, j-1) + dfs(i, j+1)
                # U D L R

        for i in range(len(grid)):
            for j in range(len(grid[i])):
                if grid[i][j] == 1:
                    return dfs(i, j)
                
print(Solution().islandPerimeter([[0,1], 
                                  [1,1]]))

Answer 1

If I had to guess, it probably has to do with the fact that you're returning 1 instead of 0 depending on whether empty/out-of-bounds positions have been visited.

Taking this example:

 01
 11

You're returning 1 in the following positions:

  *
 *1*
*11*
 **

How many positions is that? Well, that depends on how you count them. If you count each one exactly once (visited is outside the else), there will be 7 positions.

If you count each one each time it's adjacent to a 1 (visited is inside the else), there will be 8:

  A
 G1B
F11C
 ED

A through F will be counted once each (for a total of 6), but G will be counted twice (once when visiting from its right, once when visiting from its bottom).

This is why it's so important to be clear about what you're counting. I imagine you're being asked to determine the perimeter of this shape, which will be 8, not 7:

   _
 _| |
|_ _|