Minimum Path Sum in Grid in DSA C++ - Build from Scratch

Minimum Path Sum in Grid

📖 Scenario: Imagine you are helping a delivery robot find the easiest path through a city grid. Each block has a cost representing how difficult it is to cross. The robot can only move right or down. Your job is to help the robot find the path with the smallest total cost from the top-left corner to the bottom-right corner.

🎯 Goal: Build a program that calculates the minimum path sum in a grid, moving only right or down, starting from the top-left corner to the bottom-right corner.

📋 What You'll Learn

Create a 2D vector called grid with the exact values given

Create an integer variable called rows and set it to the number of rows in grid

Create an integer variable called cols and set it to the number of columns in grid

Use nested for loops with variables i and j to calculate the minimum path sum

Print the minimum path sum using std::cout

💡 Why This Matters

🌍 Real World

Finding the easiest or cheapest path through a grid is useful in robotics, GPS navigation, and game development.

💼 Career

Understanding grid traversal and dynamic programming is important for software engineers working on pathfinding algorithms and optimization problems.

1

Create the grid

Create a 2D vector called grid with these exact values:
{{1, 3, 1}, {1, 5, 1}, {4, 2, 1}}

DSA C++

#include 
#include 

int main() {
    // Create the grid vector with the given values
    // Your code here

    return 0;
}

Hint

Use std::vector> to create a 2D vector and initialize it with the given values.

2

Set rows and columns variables

Create an integer variable called rows and set it to grid.size(). Then create an integer variable called cols and set it to grid[0].size().

DSA C++

#include 
#include 

int main() {
    std::vector> grid = {{1, 3, 1}, {1, 5, 1}, {4, 2, 1}};
    // Create rows and cols variables
    // Your code here

    return 0;
}

Hint

Use grid.size() to get the number of rows and grid[0].size() to get the number of columns.

3

Calculate minimum path sum

Use nested for loops with variables i and j to update grid[i][j] with the minimum path sum to that cell. Use these rules:
- For the first row (i == 0), add the value from the left cell.
- For the first column (j == 0), add the value from the top cell.
- For other cells, add the minimum of the top or left cell values.
Start loops from i = 0 to rows - 1 and j = 0 to cols - 1.

DSA C++

#include 
#include 

int main() {
    std::vector> grid = {{1, 3, 1}, {1, 5, 1}, {4, 2, 1}};
    int rows = grid.size();
    int cols = grid[0].size();

    // Use nested for loops to update grid with minimum path sums
    // Your code here

    return 0;
}

Hint

Use std::min to find the smaller value between the top and left cells. Update grid[i][j] accordingly.

4

Print the minimum path sum

Print the minimum path sum stored in grid[rows - 1][cols - 1] using std::cout.

DSA C++

#include 
#include 
#include 

int main() {
    std::vector> grid = {{1, 3, 1}, {1, 5, 1}, {4, 2, 1}};
    int rows = grid.size();
    int cols = grid[0].size();

    for (int i = 0; i  0) {
                grid[i][j] += grid[i][j - 1];
            } else if (j == 0 && i > 0) {
                grid[i][j] += grid[i - 1][j];
            } else if (i > 0 && j > 0) {
                grid[i][j] += std::min(grid[i - 1][j], grid[i][j - 1]);
            }
        }
    }

    // Print the minimum path sum stored in grid[rows - 1][cols - 1]
    // Your code here

    return 0;
}

Hint

Use std::cout << grid[rows - 1][cols - 1] << std::endl; to print the result.

Practice

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

easy

A. The largest sum of numbers along any path in the grid.

B. The shortest path length from top-left to bottom-right.

C. The number of paths from top-left to bottom-right.

D. The smallest sum of numbers along a path from top-left to bottom-right, moving only right or down.

Solution

1. Step 1: Understand the problem goal

   The problem asks for the minimum sum of values collected when moving from the top-left corner to the bottom-right corner of a grid, moving only right or down.

2. Step 2: Clarify movement constraints

   Only moves to the right or down are allowed, so the path is restricted and the sum depends on chosen cells.

3. Final Answer:

   The smallest sum of numbers along a path from top-left to bottom-right, moving only right or down. -> Option D

4. Quick Check:

   Minimum path sum = smallest sum path [OK]

Hint: Focus on sum along right/down moves only [OK]

Common Mistakes:

• Thinking diagonal moves are allowed
• Looking for maximum sum instead of minimum
• Counting number of paths instead of sum

2. Which of the following C++ code snippets correctly initializes a 2D vector named grid with 3 rows and 4 columns filled with zeros?

easy

A. std::vector grid(3, std::vector<int>(4, 0));

B. std::vector<int> grid(3, std::vector<int>(4, 0));

C. std::vector grid(4, std::vector<int>(3, 0));

D. std::vector grid(3, 4, 0);

Solution

1. Step 1: Understand 2D vector initialization syntax

   To create a 2D vector with 3 rows and 4 columns filled with zeros, we use vector of vectors: outer vector size 3, inner vector size 4, each element 0.

2. Step 2: Check each option

   std::vector grid(3, std::vector<int>(4, 0)); correctly uses std::vector grid(3, std::vector<int>(4, 0)); which means 3 rows, each row is a vector of 4 zeros.

3. Final Answer:

   std::vector grid(3, std::vector<int>(4, 0)); -> Option A

4. Quick Check:

   Correct 2D vector init = std::vector grid(3, std::vector<int>(4, 0)); [OK]

Hint: Use vector of vectors with inner vector size and default value [OK]

Common Mistakes:

• Using wrong vector type for 2D
• Swapping rows and columns sizes
• Trying to pass three arguments to vector constructor

3. What is the output of the following C++ code that calculates the minimum path sum in a 2x2 grid?

std::vector grid = {{1, 3}, {1, 5}};
int m = grid.size();
int n = grid[0].size();
for (int i = 1; i < m; i++) grid[i][0] += grid[i-1][0];
for (int j = 1; j < n; j++) grid[0][j] += grid[0][j-1];
for (int i = 1; i < m; i++) {
  for (int j = 1; j < n; j++) {
    grid[i][j] += std::min(grid[i-1][j], grid[i][j-1]);
  }
}
std::cout << grid[m-1][n-1] << std::endl;

medium

A. 7

B. 8

C. 9

D. 10

Solution

1. Step 1: Calculate first column cumulative sums

   grid[1][0] = grid[1][0] + grid[0][0] = 1 + 1 = 2

2. Step 2: Calculate first row cumulative sums

   grid[0][1] = grid[0][1] + grid[0][0] = 3 + 1 = 4

3. Step 3: Calculate inner cell minimum path sums

   grid[1][1] = grid[1][1] + min(grid[0][1], grid[1][0]) = 5 + min(4, 2) = 5 + 2 = 7

4. Final Answer:

   7 -> Option A

5. Quick Check:

   Minimum path sum = 7 [OK]

Hint: Update first row and column cumulatively, then inner cells [OK]

Common Mistakes:

• Not updating first row and column before inner cells
• Adding max instead of min
• Mixing indices i and j

4. The following code attempts to compute the minimum path sum in a grid but has a bug. What is the error?

std::vector grid = {{1, 2, 3}, {4, 5, 6}};
int m = grid.size();
int n = grid[0].size();
for (int i = 1; i < m; i++) grid[i][0] += grid[i-1][0];
for (int j = 1; j < n; j++) grid[0][j] += grid[0][j-1];
for (int i = 1; i < m; i++) {
  for (int j = 1; j < n; j++) {
    grid[i][j] += std::max(grid[i-1][j], grid[i][j-1]);
  }
}
std::cout << grid[m-1][n-1] << std::endl;

medium

A. Wrong loop ranges for i and j

B. Not initializing first row and column

C. Using std::max instead of std::min in inner loop

D. Incorrect grid initialization syntax

Solution

1. Step 1: Check the inner loop operation

   The code uses std::max to update grid[i][j], but the problem requires minimum path sum, so std::min should be used.

2. Step 2: Verify other parts

   First row and column are correctly initialized, loops have correct ranges, and grid initialization is valid.

3. Final Answer:

   Using std::max instead of std::min in inner loop -> Option C

4. Quick Check:

   Use min for minimum path sum [OK]

Hint: Use min, not max, when updating path sums [OK]

Common Mistakes:

• Confusing max and min functions
• Skipping first row/column initialization
• Off-by-one errors in loops

5. Given the grid below, what is the minimum path sum from top-left to bottom-right?

grid = {
  {1, 3, 1, 2},
  {1, 5, 1, 1},
  {4, 2, 1, 3}
};

You can only move right or down. Calculate the minimum sum.

hard

A. 8

B. 10

C. 9

D. 11

Solution

1. Step 1: Compute first row and first column cumulative sums

   First row: [1, 4, 5, 7] (1, 1+3=4, 4+1=5, 5+2=7) First column: [1, 2, 6] (1, 1+1=2, 2+4=6)

2. Step 2: Calculate inner cells minimum path sums

   Row 1, col 1: 5 + min(4,2) = 5 + 2 = 7 Row 1, col 2: 1 + min(5,7) = 1 + 5 = 6 Row 1, col 3: 1 + min(7,6) = 1 + 6 = 7 Row 2, col 1: 2 + min(7,6) = 2 + 6 = 8 Row 2, col 2: 1 + min(6,8) = 1 + 6 = 7 Row 2, col 3: 3 + min(7,7) = 3 + 7 = 10

3. Final Answer:

   10 -> Option B

4. Quick Check:

   Minimum path sum = 10 [OK]

Hint: Fill first row/column, then inner cells with min sums [OK]

Common Mistakes:

• Adding max instead of min
• Incorrect cumulative sums for first row/column
• Misindexing grid cells