Enumerate Right-and-Down Paths Through a Binary Matrix

Question

The candidate is asked to find and enumerate all unique paths from the top-left corner to the bottom-right corner of a binary matrix, where movement is restricted to right and down directions.

Codemia · Accepted Answer

To solve the problem of enumerating right-and-down paths through a binary matrix, we can utilize a recursive approach or dynamic programming. The key is to recognize that at each cell in the matrix, we can either move right or down, leading to a combinatorial explosion of possible paths. We will implement a recursive function that explores these paths and collects them as we reach the bottom-right corner.

We will define a recursive function `findPaths(row, col, path)` where `row` and `col` are the current cell coordinates and `path` is a list that stores the current path. The base case will be when we reach the bottom-right corner, at which point we will add the path to our results. The recursive calls will explore moving right and down from the current cell.

Here is a Python code block implementing this logic:

```python
def findPaths(matrix):
    results = []
    def backtrack(row, col, path):
        # Base case: if we reach the bottom-right corner
        if row == len(matrix) - 1 and col == len(matrix[0]) - 1:
            results.append(path[:])  # Append a copy of the current path
            return
        # Move right
        if col + 1 < len(matrix[0]):
            path.append((row, col + 1))
            backtrack(row, col + 1, path)
            path.pop()
        # Move down
        if row + 1 < len(matrix):
            path.append((row + 1, col))
            backtrack(row + 1, col, path)
            path.pop()

backtrack(0, 0, [(0, 0)])
    return results
```

This code initializes the backtracking process from the top-left corner and collects all valid paths.

The time complexity of this approach is O(2^(m+n)), where `m` is the number of rows and `n` is the number of columns in the matrix. This is because, in the worst case, we explore all possible paths. The space complexity is O(m+n) due to the space used by the recursion stack and the storage of paths.

We should consider edge cases such as an empty matrix, a single cell matrix, or matrices where movement is restricted (e.g., all cells are blocked). For these scenarios, we can return an empty list or handle them specifically within our code.

This solution effectively enumerates all unique paths in a binary matrix using a recursive backtracking approach. By clearly defining our base case and recursive steps, we ensure that all paths are explored while maintaining clarity and efficiency in our implementation.

Enumerate Right-and-Down Paths Through a Binary Matrix

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