Minimum Falling Path Sum
Given an n x n array of integers matrix, return the minimum sum of any falling path through matrix.
A falling path starts at any element in the first row and chooses the element in the next row that is either directly below or diagonally left/right. Specifically, the next element from position (row, col) will be (row + 1, col - 1), (row + 1, col), or (row + 1, col + 1).
Examples
Input: [[2,1,3],[6,5,4],[7,8,9]]
Output: 13
Input: [[-19,57],[-40,-5]]
Output: -59
Hints
Use dynamic programming to build a table where `dp[i][j]` represents the minimum falling path sum ending at `matrix[i][j]`.
For each cell `matrix[i][j]`, compute `dp[i][j]` as `matrix[i][j] + min(dp[i-1][j-1], dp[i-1][j], dp[i-1][j+1])` (handle boundary conditions for `j-1` and `j+1`).
Optimize space by using a 1D array to store the DP values, updating it row by row while only keeping track of the previous row's values.
Related Problems
Minimum Falling Path Sum
Given an `n x n` array of integers `matrix`, return the minimum sum of any falling path through `matrix`.