Problem Statement
Minimum Path Sum
Given a m x n grid filled with non-negative integers, 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.
Examples
Example 1:
Input: grid = [[1,3,1],[1,5,1],[4,2,1]]
Output: 7
Explanation: Because the path 1 → 3 → 1 → 1 → 1 minimizes the sum.
Example 2:
Input: grid = [[1,2,3],[4,5,6]]
Output: 12
Explanation: The path 1 → 2 → 3 → 6 minimizes the sum.
Constraints
- m == grid.length
- n == grid[i].length
- 1 <= m, n <= 200
- 0 <= grid[i][j] <= 100
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming.
- For each cell, the minimum path sum is the value of the cell plus the minimum of the path sums from the cell above and the cell to the left.
- The base cases are the top-left cell, the first row, and the first column.
- We can use a 2D DP array to store the minimum path sum for each cell.
- We can also optimize the space complexity by using a 1D DP array, as we only need the values from the current row and the row above.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Minimum Path Sum
Given a m x n grid filled with non-negative integers, 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.
Examples
Example 1:
Because the path 1 → 3 → 1 → 1 → 1 minimizes the sum.
Example 2:
The path 1 → 2 → 3 → 6 minimizes the sum.
Problem Breakdown
Constraints:
- m == grid.length
- n == grid[i].length
- 1 <= m, n <= 200
- 0 <= grid[i][j] <= 100
Key Insights:
This problem can be solved using dynamic programming.
For each cell, the minimum path sum is the value of the cell plus the minimum of the path sums from the cell above and the cell to the left.
The base cases are the top-left cell, the first row, and the first column.
We can use a 2D DP array to store the minimum path sum for each cell.
We can also optimize the space complexity by using a 1D DP array, as we only need the values from the current row and the row above.
Real-World Application
This problem has several practical applications:
Route Planning
Finding the most cost-effective route in transportation networks, such as delivery routes or navigation systems.
Cost Optimization
Minimizing costs in various scenarios, such as resource allocation or project planning.
Grid-Based Navigation
Navigating through grid-based environments with varying costs, such as in robotics or video games.