Problem Description
Given an n x n integer matrix grid, return the minimum sum of a falling path with non-zero shifts. A falling path with non-zero shifts is a choice of exactly one element from each row of grid such that no two elements chosen in adjacent rows are in the same column.
Key Insights
- Each element in the falling path must be chosen from a different row.
- No two elements can be from the same column in adjacent rows.
- The problem can be solved using dynamic programming to keep track of the minimum sums while considering the constraints.
Space and Time Complexity
Time Complexity: O(n^2)
Space Complexity: O(n)
Solution
The solution uses a dynamic programming approach. We will maintain a 1D array to keep track of the minimum costs for each row while ensuring that no two elements from adjacent rows are selected from the same column. For each row, we find the two smallest values from the previous row that are not in the same column, and use them to compute the minimum falling path sum for the current row.