Minimum Path Sum - Video Solutions

Problem Overview: You are given an m x n grid where each cell contains a non‑negative cost. Starting from the top-left corner, move only right or down until reaching the bottom-right cell. The goal is to compute the minimum possible sum of values along that path.

Approach 1: Dynamic Programming with Extra Space (Time: O(m*n), Space: O(m*n))

The classic solution uses a 2D DP table where dp[i][j] represents the minimum path sum required to reach cell (i, j). Start by initializing the first row and first column because those cells can only be reached from one direction. For every other cell, compute dp[i][j] = grid[i][j] + min(dp[i-1][j], dp[i][j-1]). This works because every valid path must come from either the cell above or the cell to the left. The approach is straightforward and easy to reason about, which makes it a good starting point when practicing dynamic programming on grid problems.

Approach 2: Dynamic Programming with In-Place Modification (Time: O(m*n), Space: O(1))

The DP state can be stored directly inside the input grid to remove the extra memory. Iterate through the matrix row by row and update each cell to represent the minimum cost to reach that position. For the first row, accumulate values from the left; for the first column, accumulate from above. For all other cells, update using grid[i][j] += min(grid[i-1][j], grid[i][j-1]). After processing the entire grid, the bottom-right cell contains the answer. This version keeps the same time complexity but reduces space to constant because it reuses the matrix. It is a common optimization pattern in matrix DP problems.

Alternative Thought Process: Recursive Exploration

A naive recursive strategy explores both choices (move right or move down) from every cell and returns the minimum path sum. Without memoization this leads to repeated recomputation and exponential time complexity O(2^(m+n)). Adding memoization converts the recursion into the same dynamic programming structure described above. Understanding this transition from brute-force recursion to DP is key when solving many array and grid optimization problems.

Recommended for interviews: Interviewers expect the dynamic programming solution with O(m*n) time. Starting with the DP table version clearly shows the recurrence relation and state definition. Once that is established, optimizing to the in-place version demonstrates strong problem-solving skills and awareness of space optimization.