Minimum Path Sum in Grid in DSA C - Execution Trace

Concept Flow - Minimum Path Sum in Grid

Start at top-left cell (0,0)

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At each cell, choose min path sum from top or left

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Add current cell value to min of top or left

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Move right or down until bottom-right cell

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Result is min path sum at bottom-right cell

We start at the top-left cell and move only right or down, accumulating the minimum sum of values along the path until we reach the bottom-right cell.

Execution Sample

DSA C

int minPathSum(int** grid, int rows, int cols) {
  for (int i = 1; i < rows; i++)
    grid[i][0] += grid[i-1][0];
  for (int j = 1; j < cols; j++)
    grid[0][j] += grid[0][j-1];
  for (int i = 1; i < rows; i++) {
    for (int j = 1; j < cols; j++) {
      grid[i][j] += (grid[i-1][j] < grid[i][j-1]) ? grid[i-1][j] : grid[i][j-1];
    }
  }
  return grid[rows-1][cols-1];
}

This code updates the grid in-place to store the minimum path sum to each cell, then returns the bottom-right cell value as the result.

Execution Table

Step	Operation	Cell Updated	Value Added	Grid State	Explanation
1	Initialize first column	(1,0)	grid[0][0] = 1	[[1,3,1],[6,2,1],[1,5,1]]	Add top cell value to current cell in first column
2	Initialize first column	(2,0)	grid[1][0] = 6	[[1,3,1],[6,2,1],[7,5,1]]	Add top cell value to current cell in first column
3	Initialize first row	(0,1)	grid[0][0] = 1	[[1,4,1],[6,2,1],[7,5,1]]	Add left cell value to current cell in first row
4	Initialize first row	(0,2)	grid[0][1] = 4	[[1,4,5],[6,2,1],[7,5,1]]	Add left cell value to current cell in first row
5	Update cell	(1,1)	min(4,6) = 4	[[1,4,5],[6,6,1],[7,5,1]]	Add min of top or left to current cell
6	Update cell	(1,2)	min(5,6) = 5	[[1,4,5],[6,6,6],[7,5,1]]	Add min of top or left to current cell
7	Update cell	(2,1)	min(6,7) = 6	[[1,4,5],[6,6,6],[7,11,1]]	Add min of top or left to current cell
8	Update cell	(2,2)	min(6,11) = 6	[[1,4,5],[6,6,6],[7,11,7]]	Add min of top or left to current cell
9	Return result	(2,2)	7	[[1,4,5],[6,6,6],[7,11,7]]	Minimum path sum is value at bottom-right cell

💡 All cells updated with minimum path sums; bottom-right cell holds final result

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	After Step 6	After Step 7	After Step 8	Final
grid	[[1,3,1],[4,2,1],[1,5,1]]	[[1,3,1],[6,2,1],[1,5,1]]	[[1,3,1],[6,2,1],[7,5,1]]	[[1,4,1],[6,2,1],[7,5,1]]	[[1,4,5],[6,2,1],[7,5,1]]	[[1,4,5],[6,6,1],[7,5,1]]	[[1,4,5],[6,6,6],[7,5,1]]	[[1,4,5],[6,6,6],[7,11,1]]	[[1,4,5],[6,6,6],[7,11,7]]	[[1,4,5],[6,6,6],[7,11,7]]

Key Moments - 3 Insights

Why do we initialize the first row and first column separately before filling the rest?

Because the first row and column can only be reached from one direction (left for row, top for column), so their path sums are cumulative sums along that row or column. This is shown in steps 1-4 in the execution_table.

Why do we add the minimum of the top or left cell to the current cell?

Because the problem allows moving only right or down, so the minimum path to the current cell must come from either the cell above or the cell to the left. This is done in steps 5-8 in the execution_table.

Why is the final answer at the bottom-right cell of the grid?

Because the algorithm accumulates the minimum path sums in-place, so the bottom-right cell contains the minimum sum to reach that cell from the top-left, as shown in step 9.

Visual Quiz - 3 Questions

Look at the execution_table, what is the grid value at cell (1,1) after step 5?

A 6

B 5

C 4

D 7

💡 Hint

Check the 'Grid State' column in row for step 5 in execution_table

At which step does the condition to add min(top, left) to cell (2,2) happen?

A Step 7

B Step 8

C Step 6

D Step 9

💡 Hint

Look for the step updating cell (2,2) in execution_table

If the initial value at grid[0][0] was 2 instead of 1, how would the final minimum path sum change?

A It would increase by 1

B It would stay the same

C It would decrease by 1

D It would double

💡 Hint

Check variable_tracker for how grid[0][0] affects all subsequent sums

Concept Snapshot

Minimum Path Sum in Grid:
- Start at top-left cell
- Initialize first row and column by cumulative sums
- For each other cell, add min(top, left) cell value
- Result is value at bottom-right cell
- Only moves allowed: right or down

Full Transcript

The Minimum Path Sum in Grid problem finds the smallest sum of numbers along a path from the top-left to the bottom-right of a grid, moving only right or down. The approach updates the grid in-place: first, it initializes the first row and column by adding values cumulatively since those cells can only be reached from one direction. Then, for each other cell, it adds the minimum of the top or left neighbor's path sums to the current cell's value. This way, each cell stores the minimum path sum to reach it. Finally, the bottom-right cell contains the minimum path sum for the entire grid. The execution table shows each step updating the grid, and the variable tracker follows the grid's state after each update. Key moments clarify why initialization is separate and why the minimum of top and left is chosen. The visual quiz tests understanding of these steps and the effect of changing initial values.