Minimum Path Sum in Grid in DSA Go - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the goal of the Minimum Path Sum problem in a grid?

To find the path from the top-left corner to the bottom-right corner of a grid that has the smallest sum of all numbers along the path, moving only down or right.

beginner

Which moves are allowed when finding the minimum path sum in a grid?

You can only move either one step to the right or one step down at each move.

intermediate

How can dynamic programming help solve the Minimum Path Sum problem?

Dynamic programming stores the minimum path sums to each cell, so you don't recalculate paths multiple times, making the solution efficient.

beginner

What is the base case when calculating minimum path sums in a grid?

The top-left cell's minimum path sum is its own value because it is the starting point.

beginner

In Go, how do you typically represent the grid for the Minimum Path Sum problem?

As a two-dimensional slice of integers, where each element represents the cost at that cell.

What is the first step in solving the Minimum Path Sum problem using dynamic programming?

A Ignore the first row and column

B Start from the bottom-right cell

C Move diagonally across the grid

D Initialize the top-left cell with its own value

The top-left cell is the starting point, so its minimum path sum is its own value.

Which of these moves is NOT allowed when finding the minimum path sum in the grid?

A Move right

B Move down

C Move left

D Move down-right diagonally

Only moves to the right or down are allowed, not left or diagonal.

If the grid is [[1,3,1],[1,5,1],[4,2,1]], what is the minimum path sum?

A 7

B 8

C 9

D 10

The path 1→3→1→1→1 sums to 7, which is the minimum.

What data structure is commonly used to store intermediate minimum sums in this problem?

A 2D array or slice

B Queue

C Linked list

D Stack

A 2D array or slice stores minimum sums for each cell efficiently.

Why is dynamic programming preferred over recursion alone for this problem?

A It is easier to write

B It uses less memory

C It avoids recalculating the same paths multiple times

D It uses random moves

Dynamic programming stores results to avoid repeated calculations, improving efficiency.

Explain how to compute the minimum path sum in a grid step-by-step.

Think about how you build the solution from the start to the end.

Describe how you would implement the Minimum Path Sum solution in Go.

Think about how to translate the logic into Go slices and loops.

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