How do you solve Minimum Path Sum on LeetCode?

Minimum Path Sum (LeetCode #64) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(m*n) time complexity and O(m*n) space complexity. Key insight: Dynamic programming is a powerful technique for optimization problems.

What is the brute force solution for Minimum Path Sum?

The brute force approach for Minimum Path Sum is Brute Force, with O(2^(m+n)) time complexity and O(m+n) space complexity. The brute-force approach explores all possible paths from the top-left to the bottom-right of the grid. It calculates the sum of each path and returns the minimum sum found. This method is simple but inefficient for larger grids. The optimal Optimal Solution approach improves this to O(m*n).

What algorithm or data structure is used to solve Minimum Path Sum?

Minimum Path Sum uses the following concepts: Array, Dynamic Programming, Matrix. The recommended patterns to study are: Dynamic Programming, Matrix Traversal.

What are the interview tips for Minimum Path Sum?

Explain your thought process clearly as you work through the problem. Discuss the trade-offs between different approaches before diving into coding. Practice similar problems to get comfortable with dynamic programming.

What are common mistakes when solving Minimum Path Sum?

Not considering edge cases, such as single-row or single-column grids. Failing to initialize the dp array correctly.

#64

Minimum Path Sum

Medium

ArrayDynamic ProgrammingMatrixDynamic ProgrammingMatrix Traversal

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(2^(m+n))	O(m*n)
Space	O(m+n)	O(m*n)

💡

Intuition

Time O(m*n)Space O(m*n)

The optimal solution uses dynamic programming to build a table where each cell represents the minimum path sum to that cell. This approach avoids redundant calculations and efficiently finds the minimum path sum.

⚙️

Algorithm

5 steps

1Step 1: Create a 2D array 'dp' of the same size as the grid.
2Step 2: Initialize dp[0][0] with grid[0][0].
3Step 3: Fill the first row and first column of dp with cumulative sums.
4Step 4: For each cell in the grid, calculate dp[i][j] as the minimum of the sum from the top or left cell plus the current cell value.
5Step 5: Return the value in dp[m-1][n-1] as the result.

solution.py12 lines

1def minPathSum(grid):
2    m, n = len(grid), len(grid[0])
3    dp = [[0] * n for _ in range(m)]
4    dp[0][0] = grid[0][0]
5    for i in range(1, m):
6        dp[i][0] = dp[i-1][0] + grid[i][0]
7    for j in range(1, n):
8        dp[0][j] = dp[0][j-1] + grid[0][j]
9    for i in range(1, m):
10        for j in range(1, n):
11            dp[i][j] = min(dp[i-1][j], dp[i][j-1]) + grid[i][j]
12    return dp[m-1][n-1]

ℹ

Complexity note: The time complexity is O(m*n) because we iterate through each cell in the grid once. The space complexity is also O(m*n) due to the additional dp array used to store intermediate results.

1Dynamic programming is a powerful technique for optimization problems.
2Understanding how to build solutions incrementally can lead to efficient algorithms.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.