In the bottom-up approach, we are starting from the top-left corner of the grid.

We can't immediately fill it with 1, as the cat might be sleeping on it.

This means that:

1. If we find a square where the cat is sleeping, we should assign the value of that square to be 0.

2. If the square doesn't have a cat in it and it is the top-left square (0, 0), then fill the dp array for the square with 1.

3. Otherwise, the square's value will be the number of distinct paths from the top plus the number of distinct paths from the left. If the value is out of bounds of the grid, it is 0.

One final thing we haven't discussed is what data structure we would use for a dp object. Since we are dealing with a grid (nested array), a nested array seems appropriate as a dp object as well.

Let's go through an example grid step-by-step:

Step 1: Since the square (0, 0) doesn't have a sleeping cat in it, we fill the dp array for that square with 1.

Step 2: Next, we'll start iterating through the grid, row by row, and fill in each square by summing the values from the top and left. The first square to consider is (0, 1). The value from the top is 0 as it is out of bounds, and the value from the left is 1, so the value for (0, 1) is 1.

Step 3: In the last square of the first row, we have the same situation as in the previous step. The top is out of bounds, and the value to the left is 1, so the value for the square (0, 2) is 1.

Step 4: In the second row, first column, square (1, 0), the value to the left is out of bounds, thus 0, and the top value is 1, which equals 1.

Step 5: We have a sleeping cat at the next square, and we don't want to disturb it, so we'll leave the value as 0 for that square.

Step 6: Next, the value to the left is 0 as we have a sleeping cat there, and the value to the top is 1, so the sum is 1.

Step 7: In the seventh step, the value to the top is 1, and the value to the left is 0 as it is out of bounds, so the sum is 1.

Step 8: Here we have a sleeping cat at the top, so the value is 0, and 1 to the left, so once again the sum is 1.

Step 9: Finally, we reach the last square where the value from the top is 1 and from the left is 1, so the sum is 2, which is our final result.