Minimum Path Sum in Grid in DSA Go - Execution Trace

Concept Flow - Minimum Path Sum in Grid

Start at top-left cell

↓

Check if at bottom-right cell?

Yes→Return current path sum

No↓

Move Right or Down

↓

Add cell value

↓

Repeat from new cell

↓

Choose path with minimum sum

↓

Done

Start from the top-left cell, move only right or down, add cell values, and choose the path with the smallest sum until reaching the bottom-right cell.

Execution Sample

DSA Go

grid := [][]int{
  {1, 3, 3},
  {1, 5, 1},
  {4, 2, 1},
}
minPathSum(grid)

Calculate the minimum path sum from top-left to bottom-right in a 3x3 grid moving only right or down.

Execution Table

Step	Operation	Current Cell (row,col)	Path Sum at Cell	Decision	Visual State
1	Start	(0,0)	1	Start here	[1, _, _] [_, _, _] [_, _, _]
2	Move Right	(0,1)	4 (1+3)	Choose right or down	[1, 4, _] [_, _, _] [_, _, _]
3	Move Down	(1,1)	9 (4+5)	Choose right or down	[1, 4, _] [_, 9, _] [_, _, _]
4	Move Right	(1,2)	10 (9+1)	Choose right or down	[1, 4, _] [_, 9, 10] [_, _, _]
5	Move Down	(2,2)	11 (10+1)	Reached bottom-right	[1, 4, _] [_, 9, 10] [_, _, 11]
6	Alternative Path	(1,0)	2 (1+1)	Explore down first	[1, 4, _] [2, 9, 10] [_, _, 11]
7	Move Down	(2,0)	6 (2+4)	Choose right or down	[1, 4, _] [2, 9, 10] [6, _, 11]
8	Move Right	(2,1)	8 (6+2)	Choose right or down	[1, 4, _] [2, 9, 10] [6, 8, 11]
9	Move Right	(2,2)	9 (8+1)	Reached bottom-right	[1, 4, _] [2, 9, 10] [6, 8, 9]
10	Compare Paths	-	-	Choose minimum sum path	-
11	Result	-	9	Minimum path sum found	-

💡 Reached bottom-right cell with minimum path sum 9

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	After Step 6	After Step 7	After Step 8	After Step 9	Final
Current Cell	(0,0)	(0,1)	(1,1)	(1,2)	(2,2)	(1,0)	(2,0)	(2,1)	(2,2)	(2,2)
Path Sum	1	4	9	10	11	2	6	8	9	9
Decision	Start	Right or Down	Right or Down	Right or Down	Reached End	Down first	Right or Down	Right or Down	Reached End	Minimum sum chosen

Key Moments - 3 Insights

Why do we consider both moving right and down at each step?

Because the problem allows only moving right or down, we must explore both to find the minimum sum path. See execution_table steps 2 and 6 where both directions are considered.

Why is the path sum updated by adding the current cell's value?

Each cell's value contributes to the total path sum. At each step, we add the current cell's value to the sum from the previous path, as shown in execution_table steps 2, 3, and onwards.

How do we know which path has the minimum sum?

At the end, we compare sums from all possible paths reaching the bottom-right cell and choose the smallest, as shown in execution_table step 10.

Visual Quiz - 3 Questions

Look at the execution_table at step 5, what is the path sum at cell (2,2)?

A 11

B 9

C 10

D 8

💡 Hint

Check the 'Path Sum at Cell' column in execution_table row for step 5.

At which step does the path sum first reach the bottom-right cell with the minimum sum?

A Step 5

B Step 9

C Step 11

D Step 7

💡 Hint

Look for the first time the bottom-right cell (2,2) is reached with the smallest path sum in execution_table.

If the value at cell (1,1) changed from 5 to 1, how would the minimum path sum change?

A It would increase

B It would stay the same

C It would decrease

D It would become zero

💡 Hint

Consider how lowering a cell value on a path affects the total sum, referencing variable_tracker path sums.

Concept Snapshot

Minimum Path Sum in Grid:
- Start at top-left cell (0,0).
- Move only right or down.
- Add current cell value to path sum.
- Choose path with minimum sum at each step.
- Stop at bottom-right cell.
- Result is minimum sum path value.

Full Transcript

The Minimum Path Sum in Grid problem involves starting at the top-left cell of a grid and moving only right or down to reach the bottom-right cell. At each step, the value of the current cell is added to the path sum. We explore both possible moves (right and down) to find all paths and choose the one with the smallest total sum. The execution table traces each step, showing the current cell, path sum, and decisions made. Variable tracking shows how the current cell and path sum change over time. Key moments clarify why both directions are considered, why sums are updated, and how the minimum path is chosen. The visual quiz tests understanding of path sums at specific steps and effects of changing cell values. The concept snapshot summarizes the approach in simple steps.

Practice

1. What does the "Minimum Path Sum in Grid" problem ask you to find?

easy

A. The number of paths from top-left to bottom-right moving only left or up

B. The largest sum of numbers from the top-left to bottom-right moving only diagonally

C. The smallest sum of numbers from the top-left to bottom-right moving only right or down

D. The sum of all numbers in the grid

Solution

1. Step 1: Understand the problem goal

   The problem asks to find the path from the top-left corner to the bottom-right corner of a grid that has the smallest total sum of the numbers on the path.

2. Step 2: Recognize allowed moves

   Only moves to the right or down are allowed, so the path must follow these directions.

3. Final Answer:

   The smallest sum of numbers from the top-left to bottom-right moving only right or down -> Option C

4. Quick Check:

   Minimum path sum = smallest total sum [OK]

Hint: Focus on moves right or down to find smallest sum path [OK]

Common Mistakes:

• Thinking diagonal moves are allowed
• Looking for largest sum instead of smallest
• Counting all numbers instead of path sum

2. Which of the following Go code snippets correctly initializes a 2D grid for the Minimum Path Sum problem?

easy

A. grid := [3][3]int{{1, 3, 1}, {1, 5, 1}, {4, 2, 1}}

B. grid := []int{ {1, 3, 1}, {1, 5, 1}, {4, 2, 1} }

C. grid := [][]int{1, 3, 1, 1, 5, 1, 4, 2, 1}

D. grid := [][]int{{1, 3, 1}, {1, 5, 1}, {4, 2, 1}}

Solution

1. Step 1: Check correct 2D slice initialization

   grid := [][]int{{1, 3, 1}, {1, 5, 1}, {4, 2, 1}} uses [][]int with nested slices, which is the correct way to declare a 2D slice in Go.

2. Step 2: Identify incorrect options

   grid := [3][3]int{{1, 3, 1}, {1, 5, 1}, {4, 2, 1}} uses fixed array type which is valid but less flexible; grid := [][]int{1, 3, 1, 1, 5, 1, 4, 2, 1} is a flat slice, not 2D; grid := []int{ {1, 3, 1}, {1, 5, 1}, {4, 2, 1} } uses []int with nested braces which is invalid syntax.

3. Final Answer:

   grid := [][]int{{1, 3, 1}, {1, 5, 1}, {4, 2, 1}} -> Option D

4. Quick Check:

   2D slice with nested slices = correct grid init [OK]

Hint: Use [][]int with nested braces for 2D slices [OK]

Common Mistakes:

• Using flat slices instead of 2D slices
• Incorrect array type syntax
• Missing nested braces for rows

3. What is the output of the following Go code that calculates the minimum path sum?

func minPathSum(grid [][]int) int {
    m, n := len(grid), len(grid[0])
    for i := 1; i < m; i++ {
        grid[i][0] += grid[i-1][0]
    }
    for j := 1; j < n; j++ {
        grid[0][j] += grid[0][j-1]
    }
    for i := 1; i < m; i++ {
        for j := 1; j < n; j++ {
            if grid[i-1][j] < grid[i][j-1] {
                grid[i][j] += grid[i-1][j]
            } else {
                grid[i][j] += grid[i][j-1]
            }
        }
    }
    return grid[m-1][n-1]
}

func main() {
    grid := [][]int{{1,3,1},{1,5,1},{4,2,1}}
    fmt.Println(minPathSum(grid))
}

medium

A. 7

B. 8

C. 9

D. 10

Solution

1. Step 1: Calculate first column cumulative sums

   grid[1][0] = 1+1=2, grid[2][0] = 2+4=6

2. Step 2: Calculate first row cumulative sums

   grid[0][1] = 1+3=4, grid[0][2] = 4+1=5

3. Step 3: Calculate remaining cells minimum path sums

   grid[1][1] = min(4,2)+5=7, grid[1][2] = min(5,7)+1=6, grid[2][1] = min(7,6)+2=8, grid[2][2] = min(6,8)+1=7

4. Final Answer:

   7 -> Option A

5. Quick Check:

   Minimum path sum = 7 [OK]

Hint: Fill first row and column, then choose min of top or left [OK]

Common Mistakes:

• Not updating first row and column sums
• Adding max instead of min
• Mixing indices in loops

4. Find the error in this Go function for minimum path sum and select the fix:

func minPathSum(grid [][]int) int {
    m, n := len(grid), len(grid[0])
    for i := 1; i <= m; i++ {
        grid[i][0] += grid[i-1][0]
    }
    for j := 1; j <= n; j++ {
        grid[0][j] += grid[0][j-1]
    }
    for i := 1; i < m; i++ {
        for j := 1; j < n; j++ {
            if grid[i-1][j] < grid[i][j-1] {
                grid[i][j] += grid[i-1][j]
            } else {
                grid[i][j] += grid[i][j-1]
            }
        }
    }
    return grid[m-1][n-1]
}

medium

A. Initialize grid with zeros before loops

B. Change loops to use < instead of <= for i and j

C. Add else if condition for equal values

D. Return grid[0][0] instead of grid[m-1][n-1]

Solution

1. Step 1: Identify loop boundary error

   The loops use <= which causes index out of range error because valid indices are 0 to m-1 and 0 to n-1.

2. Step 2: Fix loop conditions

   Changing loops to use < instead of <= prevents out of range access and correctly iterates over grid indices.

3. Final Answer:

   Change loops to use < instead of <= for i and j -> Option B

4. Quick Check:

   Loop indices must be less than length [OK]

Hint: Use < length, not <= length in loops [OK]

Common Mistakes:

• Using <= causing index out of range
• Not checking array bounds
• Returning wrong grid cell

5. Given a grid with some cells having zero cost, how does the presence of zero values affect the minimum path sum calculation? Example grid:

[[1, 0, 3],
 [0, 5, 1],
 [4, 2, 0]]

Choose the correct statement.

hard

A. Zero cost cells can reduce the minimum path sum by providing cheaper paths

B. Zero cost cells are ignored in the calculation

C. Zero cost cells cause the algorithm to fail due to division by zero

D. Zero cost cells increase the minimum path sum

Solution

1. Step 1: Understand zero cost cell effect

   Cells with zero cost add no extra sum, so paths passing through them can have lower total sums.

2. Step 2: Confirm algorithm behavior

   The algorithm sums cell values; zero values reduce total sum, so zero cost cells help find cheaper paths.

3. Final Answer:

   Zero cost cells can reduce the minimum path sum by providing cheaper paths -> Option A

4. Quick Check:

   Zero cost cells lower path sum [OK]

Hint: Zero cells lower path sum by adding no cost [OK]

Common Mistakes:

• Ignoring zero cells in sum
• Assuming zero causes errors
• Thinking zero increases cost