Minimum Falling Path Sum II

Minimum Falling Path Sum II

1. What is this problem about?

The Minimum Falling Path Sum II is a challenging variation of the falling path problem. Given an $n \times n$ integer matrix, you need to find the minimum sum of a falling path such that no two elements in adjacent rows are in the same column. In the standard version, you can move to $(i+1, j-1), (i+1, j), (i+1, j+1)$; here, you can move from $(i, j)$ to any $(i+1, k)$ where $k \neq j$. This constraint significantly changes the optimization strategy.

2. Why is this asked in interviews?

This problem is a favorite at top companies like Microsoft, Amazon, and Google because it tests your ability to optimize a Dynamic Programming solution. The naive DP approach would be $O(N^3)$, but by observing the structure of the problem, you can optimize it to $O(N^2)$. It evaluates whether you can identify that you only ever need to know the two smallest values from the previous row to satisfy the "different column" constraint.

3. Algorithmic pattern used

The algorithmic pattern used is Optimized Dynamic Programming. Let dp[i][j] be the minimum path sum ending at row i, column j. To calculate dp[i][j], we need the minimum value from row i-1 that is not in column j. Instead of searching the whole row, we precalculate the two smallest values (let's call them min1 and min2) and their column indices. If j is the same as the index of min1, we use min2; otherwise, we use min1. This "Matrix, Dynamic Programming interview pattern" reduces the transition time from $O(N)$ to $O(1)$.

4. Example explanation

Matrix: [1, 2, 3] [4, 5, 6] [7, 8, 9]

• Row 0: [1, 2, 3]. min1 = 1 (col 0), min2 = 2 (col 1).
• Row 1:
  • dp[1][0] = 4 + min2 = 4 + 2 = 6 (since current col is 0, we use min2)
  • dp[1][1] = 5 + min1 = 5 + 1 = 6
  • dp[1][2] = 6 + min1 = 6 + 1 = 7 Row 1: [6, 6, 7]. min1 = 6 (col 0), min2 = 6 (col 1).
• Row 2:
  • dp[2][0] = 7 + min2 = 7 + 6 = 13
  • dp[2][1] = 8 + min1 = 8 + 6 = 14
  • dp[2][2] = 9 + min1 = 9 + 6 = 15 Min in last row is 13.

5. Common mistakes candidates make

In the Minimum Falling Path Sum II coding problem, the most common mistake is implementing the $O(N^3)$ solution and failing due to time limits. Another error is not correctly handling the case where multiple columns have the same minimum value. Candidates also sometimes fail to initialize their tracking variables for min1 and min2 correctly, leading to incorrect path sums. It is also important to handle matrices of size $1 \times 1$ as a special case.

6. Interview preparation tip

Whenever a DP state depends on "anything but the current index" from the previous state, think about keeping track of the best and second-best results. This is a very common "Dynamic Programming optimization pattern" that frequently appears in hard-level interview questions. It allows you to break a bottleneck that would otherwise require an extra loop.