Minimum Falling Path Sum

class Solution {
public:
    int calculateMinPathSum(vector<vector<int>>& grid) {
        vector<int> pathSum(grid.size() + 1, 0);
        for (int i = grid.size() - 1; i >= 0; i--) {
            vector<int> currentPath(grid.size() + 1, 0);
            for (int j = 0; j < grid.size(); j++) {
                if (j == 0) {
                    currentPath[j] = min(pathSum[j], pathSum[j + 1]) + grid[i][j];
                } else if (j == grid.size() - 1) {
                    currentPath[j] = min(pathSum[j], pathSum[j - 1]) + grid[i][j];
                } else {
                    currentPath[j] = min({pathSum[j], pathSum[j - 1], pathSum[j + 1]}) + grid[i][j];
                }
            }
            pathSum = currentPath;
        }
        int minimumPathSum = INT_MAX;
        for (int startIndex = 0; startIndex < grid.size(); startIndex++) {
            minimumPathSum = min(minimumPathSum, pathSum[startIndex]);
        }
        return minimumPathSum;
    }
};

Developing an efficient solution to compute the minimum falling path sum through a matrix arrays, the provided C++ code offers an intuitive yet effective approach using dynamic programming.

• Start by initializing a vector pathSum with length one greater than the matrix size, filled with zeros. This vector will hold the minimum path sum from the last row to the current row during iteration.

• Iterate through the matrix from the bottom row to the top row. For each row, initialize a temporary vector currentPath of the same size as pathSum to store the minimum path sums concluding at each cell of the current row.

• For each element in the current row, calculate the minimum path sum using the results from the previous row (pathSum). If the element is in:

  • The first column, consider the sum coming from directly above or from the upper right.
  • The last column, consider the sum coming from directly above or from the upper left.
  • Any other column, consider sums coming from directly above, upper left, and upper right.

• Update pathSum with currentPath after processing each row, making pathSum ready for the next iteration.

• After completing the iteration over all rows, the final step involves finding the smallest value in pathSum to determine the minimum path sum from top to bottom.

The solution efficiently computes the required result by iterating over the matrix only once and using a dynamic programming array to store interim results, significantly reducing the computational complexity. Ensure to include all necessary headers such as <vector> and <algorithm> to utilize std::min, and define Solution class and calculateMinPathSum method in a manner compatible with your broader program structure.

class Solution {
    public int minimumFallingPathSum(int[][] grid) {
        int n = grid.length;
        int[] previous = new int[n + 1];
        for (int i = n - 1; i >= 0; i--) {
            int[] current = new int[n + 1];
            for (int j = 0; j < n; j++) {
                if (j == 0) {
                    current[j] = Math.min(previous[j], previous[j + 1]) + grid[i][j];
                } else if (j == n - 1) {
                    current[j] = Math.min(previous[j], previous[j - 1]) + grid[i][j];
                } else {
                    current[j] = Math.min(previous[j], Math.min(previous[j + 1], previous[j - 1])) + grid[i][j];
                }
            }
            previous = current;
        }
        int minimumPathSum = Integer.MAX_VALUE;
        for (int col = 0; col < n; col++) {
            minimumPathSum = Math.min(minimumPathSum, previous[col]);
        }
        return minimumPathSum;
    }
}

The provided Java solution efficiently calculates the minimum falling path sum for a given square matrix grid. The approach utilizes dynamic programming to minimize memory usage and complexity. Here’s a breakdown of the implementation:

• A 1D array previous is used to store the minimum path sums of the previous row. This reduces space complexity compared to using a full 2D array.
• The algorithm iterates from the bottom row of the grid upwards. For each cell, it calculates the sum of the cell value and the minimum path sum from the potential previous positions—directly above, above-left, or above-right, managing edge cases for the first and last columns where fewer choices exist.
• After populating the previous array, the algorithm finds the minimum value in previous, which represents the minimum path sum from the top of the matrix to any bottom cell.

The process ensures efficient computation by scanning each column of every row only once and updating the path sum in constant time. This blend of a bottom-up dynamic programming strategy with careful boundary checking ensures both correctness and performance for large matrices.