The diagonal move is what got me initially.
Clarify the movement directions and coordinate system, then model the problem as a dynamic programming (DP) problem where dp[i][j] represents the number of ways to reach cell (i, j) from the start. Derive the recurrence dp[i][j] = dp[i-1][j] + dp[i][j-1] + dp[i-1][j-1] (with appropriate boundary conditions), implement it with O(mn) time and O(n) space, and test with small cases and edge cases.
Pro tip: Mention that this is a generalization of the classic unique paths problem and that the DP can be optimized to 1D space; also discuss how to handle large numbers (e.g., modulo arithmetic) if needed.
Confirm the matrix dimensions, the starting and ending cells, and the allowed moves (up, right, diagonally up-right). Ensure you understand the coordinate system and whether the matrix is 0-indexed or 1-indexed.
Let dp[i][j] be the number of ways to reach cell (i, j) from the start. Derive the recurrence: dp[i][j] = dp[i-1][j] + dp[i][j-1] + dp[i-1][j-1], with base case dp[start] = 1 and out-of-bounds cells contributing 0.
Write code to compute the DP table, either using a 2D array or optimizing to 1D space. Handle edge cases such as a 1x1 matrix or when start equals end.
Provide test cases: small matrices (e.g., 2x2, 3x3), edge cases (1xN, Nx1), and a larger case to verify performance. Compare with brute force for small inputs if possible.
State the time complexity O(mn) and space complexity O(n) if optimized. Discuss potential improvements or variations (e.g., modulo arithmetic for large results).
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