Minimum Path Sum III in C++ (Time Complexity: O(m*n))

Given a m x n grid filled with non-negative numbers, find a path from top left to bottom right which minimizes the sum of all numbers along its path.

Note: You can only move either down or right at any point in time.

Example:

Input:
[
  [1,3,1],
  [1,5,1],
  [4,2,1]
]
Output: 7
Explanation: Because the path 1→3→1→1→1 minimizes the sum.

Understanding the Problem

The core challenge of this problem is to find the path from the top-left corner to the bottom-right corner of a grid that results in the minimum sum of the numbers along the path. This problem is significant in various applications such as robotics, game development, and pathfinding algorithms where finding the optimal path is crucial.

Potential pitfalls include misunderstanding the movement constraints (only right or down) and not considering all possible paths.

Approach

To solve this problem, we can use dynamic programming. The idea is to build a 2D array dp where dp[i][j] represents the minimum path sum to reach cell (i, j). We can initialize the first cell with the value of the grid's first cell and then fill the array by considering the minimum path sum to reach each cell from either the top or the left cell.

Naive Solution

A naive solution would involve exploring all possible paths from the top-left to the bottom-right corner, which would be highly inefficient due to the exponential number of paths.

Optimized Solution

The optimized solution uses dynamic programming to build the solution iteratively. This approach ensures that each cell is computed only once, resulting in a time complexity of O(m*n).

Algorithm

1. Initialize a 2D array dp with the same dimensions as the grid.

2. Set dp[0][0] to grid[0][0].

3. Fill the first row and first column of dp since they can only be reached from one direction.

4. For each cell (i, j) in the grid, set dp[i][j] to the minimum of dp[i-1][j] and dp[i][j-1] plus grid[i][j].

5. The value at dp[m-1][n-1] will be the minimum path sum.

Code Implementation

#include <vector>
#include <algorithm>
#include <iostream>

using namespace std;

int minPathSum(vector<vector<int>>& grid) {
    int m = grid.size();
    int n = grid[0].size();
    vector<vector<int>> dp(m, vector<int>(n, 0));

    dp[0][0] = grid[0][0];

    // Initialize the first row
    for (int i = 1; i < n; ++i) {
        dp[0][i] = dp[0][i - 1] + grid[0][i];
    }

    // Initialize the first column
    for (int i = 1; i < m; ++i) {
        dp[i][0] = dp[i - 1][0] + grid[i][0];
    }

    // Fill the rest of the dp array
    for (int i = 1; i < m; ++i) {
        for (int j = 1; j < n; ++j) {
            dp[i][j] = min(dp[i - 1][j], dp[i][j - 1]) + grid[i][j];
        }
    }

    return dp[m - 1][n - 1];
}

int main() {
    vector<vector<int>> grid = {
        {1, 3, 1},
        {1, 5, 1},
        {4, 2, 1}
    };
    cout << "Minimum Path Sum: " << minPathSum(grid) << endl;
    return 0;
}

Complexity Analysis

The time complexity of this approach is O(m*n) because we iterate through each cell of the grid once. The space complexity is also O(m*n) due to the additional 2D array dp used for storing the minimum path sums.

Edge Cases

Potential edge cases include:

Empty grid: The function should handle this gracefully.
Single row or single column grid: The function should correctly compute the path sum for these cases.

For example, for a single row grid [[1, 2, 3]], the output should be 6.

Testing

To test the solution comprehensively, consider the following test cases:

General case with multiple rows and columns.
Single row grid.
Single column grid.
Empty grid.

Using a testing framework like Google Test can help automate and manage these test cases effectively.

Thinking and Problem-Solving Tips

When approaching such problems, it's essential to:

Understand the problem constraints and requirements thoroughly.
Break down the problem into smaller subproblems.
Consider edge cases and how your solution handles them.
Practice similar problems to improve problem-solving skills.

Conclusion

In this blog post, we discussed the problem of finding the minimum path sum in a grid. We explored a dynamic programming approach to solve the problem efficiently and provided a detailed explanation of the algorithm and its implementation in C++. Understanding and solving such problems is crucial for developing strong problem-solving skills and preparing for technical interviews.

Additional Resources

For further reading and practice, consider the following resources: