Grid DP — Minimum Path Sum

Grid DP Pattern

Traverse a 2D grid where each cell has a cost. Each cell's optimal value = its own cost + min of the cell above or left. First row and column only have one path (cumulative sum).

Recurrence

dp[0][0] = grid[0][0]
dp[i][0] = dp[i-1][0] + grid[i][0]
dp[0][j] = dp[0][j-1] + grid[0][j]
dp[i][j] = grid[i][j] + min(dp[i-1][j], dp[i][j-1])

Interactive Visualization

Computing

Source (min of)

Computed

Answer / Path

Step 0 of 0

Cell

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Cost

—

dp Value

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Min Path Sum

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✓ Done

Speed

C++ — Minimum Path Sum

int minPathSum(vector<vector<int>>& grid) {
    int m = grid.size(), n = grid[0].size();
    for (int i = 1; i < m; i++) grid[i][0] += grid[i-1][0];
    for (int j = 1; j < n; j++) grid[0][j] += grid[0][j-1];
    for (int i = 1; i < m; i++)
        for (int j = 1; j < n; j++)
            grid[i][j] += min(grid[i-1][j], grid[i][j-1]);
    return grid[m-1][n-1];
}

Complexity

Time

O(M × N)

Visit each cell once.

Space

O(1)

Modify grid in-place.

📝 Notes

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