Minimum Path Sum in Grid in DSA C - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Minimum Path Sum Calculation

What is the output of the following C code that calculates the minimum path sum in a 3x3 grid?

DSA C

int grid[3][3] = {
  {1, 3, 1},
  {1, 5, 1},
  {4, 2, 1}
};

int dp[3][3];

// Initialize dp
for (int i = 0; i < 3; i++) {
  for (int j = 0; j < 3; j++) {
    dp[i][j] = 0;
  }
}

// Base case
 dp[0][0] = grid[0][0];

// First row
for (int j = 1; j < 3; j++) {
  dp[0][j] = dp[0][j-1] + grid[0][j];
}

// First column
for (int i = 1; i < 3; i++) {
  dp[i][0] = dp[i-1][0] + grid[i][0];
}

// Fill dp
for (int i = 1; i < 3; i++) {
  for (int j = 1; j < 3; j++) {
    dp[i][j] = (dp[i-1][j] < dp[i][j-1] ? dp[i-1][j] : dp[i][j-1]) + grid[i][j];
  }
}

printf("%d\n", dp[2][2]);

A 7

B 8

C 9

D 6

💡 Hint

Trace the path from top-left to bottom-right choosing the minimum sum at each step.

The minimum path sum is calculated by dynamic programming. The path is 1 -> 3 -> 1 -> 1 -> 1 with sum 7.

❓ Predict Output

intermediate

Minimum Path Sum with Negative Values

What is the output of this C code calculating minimum path sum in a 2x3 grid with negative values?

DSA C

int grid[2][3] = {
  {1, -3, 2},
  {4, 5, -1}
};

int dp[2][3];

// Initialize dp
for (int i = 0; i < 2; i++) {
  for (int j = 0; j < 3; j++) {
    dp[i][j] = 0;
  }
}

// Base case
 dp[0][0] = grid[0][0];

// First row
for (int j = 1; j < 3; j++) {
  dp[0][j] = dp[0][j-1] + grid[0][j];
}

// First column
for (int i = 1; i < 2; i++) {
  dp[i][0] = dp[i-1][0] + grid[i][0];
}

// Fill dp
for (int i = 1; i < 2; i++) {
  for (int j = 1; j < 3; j++) {
    dp[i][j] = (dp[i-1][j] < dp[i][j-1] ? dp[i-1][j] : dp[i][j-1]) + grid[i][j];
  }
}

printf("%d\n", dp[1][2]);

A 3

B 1

C 4

D 2

💡 Hint

Consider negative values carefully when adding.

The dp array is: 1 -2 0 5 3 2 Minimum path sum to bottom-right is 2.

🔧 Debug

advanced

Identify the Error in Minimum Path Sum Code

What error does the following C code produce when calculating minimum path sum in a 2x2 grid?

DSA C

int grid[2][2] = {{1,2},{1,1}};
int dp[2][2];

for (int i = 0; i <= 2; i++) {
  for (int j = 0; j <= 2; j++) {
    dp[i][j] = 0;
  }
}

dp[0][0] = grid[0][0];

for (int i = 1; i < 2; i++) {
  dp[i][0] = dp[i-1][0] + grid[i][0];
}

for (int j = 1; j < 2; j++) {
  dp[0][j] = dp[0][j-1] + grid[0][j];
}

for (int i = 1; i < 2; i++) {
  for (int j = 1; j < 2; j++) {
    dp[i][j] = (dp[i-1][j] < dp[i][j-1] ? dp[i-1][j] : dp[i][j-1]) + grid[i][j];
  }
}

printf("%d\n", dp[1][1]);

A Compilation error due to missing semicolon

B Prints incorrect minimum path sum

C Segmentation fault due to out-of-bounds array access

D No error, prints correct result

💡 Hint

Check the loop conditions for array indexing.

The loops use i <= 2 and j <= 2 which access dp[2][2], out of bounds for 2x2 arrays.

❓ Predict Output

advanced

Minimum Path Sum with Single Row Grid

What is the output of this C code calculating minimum path sum in a 1x4 grid?

DSA C

int grid[1][4] = {{2, 3, 1, 5}};
int dp[1][4];

// Initialize dp
for (int j = 0; j < 4; j++) {
  dp[0][j] = 0;
}

// Base case
 dp[0][0] = grid[0][0];

// First row
for (int j = 1; j < 4; j++) {
  dp[0][j] = dp[0][j-1] + grid[0][j];
}

printf("%d\n", dp[0][3]);

A 11

B 10

C 9

D 8

💡 Hint

Sum all elements in the single row.

The minimum path sum is sum of all elements: 2+3+1+5=11.

🧠 Conceptual

expert

Minimum Path Sum Algorithm Complexity

What is the time complexity of the dynamic programming solution to the Minimum Path Sum problem in an m x n grid?

A O(m + n)

B O(m * n)

C O(m^2 * n^2)

D O(log(m * n))

💡 Hint

Consider how many cells are processed and how many operations per cell.

The algorithm fills an m by n dp table once, so time complexity is O(m*n).

Practice

1. What does the "Minimum Path Sum in Grid" problem ask you to find?
Given a grid of numbers, find the path from top-left to bottom-right with the smallest sum. Which is true?

easy

A. Find the path with the smallest sum moving only right or down.

B. Find the path with the largest sum moving in any direction.

C. Find the path with the smallest sum moving diagonally only.

D. Find the path with the smallest sum moving left or up.

Solution

1. Step 1: Understand the movement constraints

   The problem allows moving only right or down from each cell in the grid.

2. Step 2: Identify the goal

   The goal is to find the path from the top-left corner to the bottom-right corner with the minimum sum of all numbers on the path.

3. Final Answer:

   Find the path with the smallest sum moving only right or down. -> Option A

4. Quick Check:

   Minimum path sum = smallest sum with right/down moves [OK]

Hint: Remember: only right or down moves allowed [OK]

Common Mistakes:

• Thinking diagonal moves are allowed
• Looking for maximum sum instead of minimum
• Moving left or up which is not allowed

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

easy

A. int grid[3][3] = {1};

B. int grid[3][3] = {{1}};

C. int grid[3][3]; memset(grid, 0, sizeof(grid));

D. int grid[3][3] = { {0,0,0}, {0,0,0}, {1,1,1} };

Solution

1. Step 1: Understand array initialization in C

   Initializing a 2D array with zeros can be done by setting all elements to zero using memset after declaration.

2. Step 2: Check each option

   int grid[3][3]; memset(grid, 0, sizeof(grid)); declares the array and then uses memset to set all bytes to zero, correctly initializing all elements to zero.

3. Final Answer:

   int grid[3][3]; memset(grid, 0, sizeof(grid)); -> Option C

4. Quick Check:

   memset sets all elements to zero [OK]

Hint: Use memset for zeroing arrays in C [OK]

Common Mistakes:

• Using {1} only initializes first element
• Confusing single braces with double braces
• Not including string.h for memset

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

int grid[2][2] = {{1, 3}, {1, 5}};
int dp[2][2];
dp[0][0] = grid[0][0];
for(int i=1; i<2; i++) dp[0][i] = dp[0][i-1] + grid[0][i];
for(int i=1; i<2; i++) dp[i][0] = dp[i-1][0] + grid[i][0];
for(int i=1; i<2; i++) {
  for(int j=1; j<2; j++) {
    dp[i][j] = (dp[i-1][j] < dp[i][j-1] ? dp[i-1][j] : dp[i][j-1]) + grid[i][j];
  }
}
printf("%d\n", dp[1][1]);

medium

A. 10

B. 8

C. 9

D. 7

Solution

1. Step 1: Calculate first row and column sums

   dp[0][0] = 1
   dp[0][1] = 1 + 3 = 4
   dp[1][0] = 1 + 1 = 2

2. Step 2: Calculate dp[1][1]

   Minimum of dp[0][1] = 4 and dp[1][0] = 2 is 2
   dp[1][1] = 2 + 5 = 7

3. Final Answer:

   7 -> Option D

4. Quick Check:

   Minimum path sum = 7 [OK]

Hint: Fill dp row and column first, then inner cells [OK]

Common Mistakes:

• Adding max instead of min
• Incorrect dp initialization
• Off-by-one errors in loops

4. The following code attempts to compute the minimum path sum in a 3x3 grid but has a bug. What is the bug?

int grid[3][3] = {{1,2,3},{4,5,6},{7,8,9}};
int dp[3][3];
dp[0][0] = grid[0][0];
for(int i=1; i<3; i++) dp[0][i] = dp[0][i-1] + grid[0][i];
for(int i=1; i<3; i++) dp[i][0] = dp[i-1][0] + grid[i][0];
for(int i=1; i<3; i++) {
  for(int j=1; j<3; j++) {
    dp[i][j] = dp[i-1][j] + dp[i][j-1] + grid[i][j];
  }
}

medium

A. The dp update adds both dp[i-1][j] and dp[i][j-1] instead of taking the minimum.

B. The loops start from 1 instead of 0.

C. The dp array is not initialized to zero.

D. The grid values are not accessed correctly.

Solution

1. Step 1: Analyze dp update formula

   The code adds dp[i-1][j] and dp[i][j-1] together plus grid[i][j], which is incorrect.

2. Step 2: Correct dp update

   It should take the minimum of dp[i-1][j] and dp[i][j-1], then add grid[i][j].

3. Final Answer:

   The dp update adds both dp[i-1][j] and dp[i][j-1] instead of taking the minimum. -> Option A

4. Quick Check:

   Use min(dp[i-1][j], dp[i][j-1]) + grid[i][j] [OK]

Hint: Use min, not sum, of top and left dp cells [OK]

Common Mistakes:

• Adding dp values instead of min
• Wrong loop ranges
• Not initializing dp[0][0]

5. Given a 3x3 grid:

int grid[3][3] = {
  {1, 3, 1},
  {1, 5, 1},
  {4, 2, 1}
};

Write the minimum path sum from top-left to bottom-right moving only right or down. What is the minimum sum?

hard

A. 8

B. 7

C. 9

D. 10

Solution

1. Step 1: Calculate dp first row and column

   dp[0][0] = 1
   dp[0][1] = 1+3=4
   dp[0][2] = 4+1=5
   dp[1][0] = 1+1=2
   dp[2][0] = 2+4=6

2. Step 2: Calculate inner dp cells

   dp[1][1] = min(dp[0][1], dp[1][0]) + 5 = min(4,2)+5=7
   dp[1][2] = min(dp[0][2], dp[1][1]) + 1 = min(5,7)+1=6
   dp[2][1] = min(dp[1][1], dp[2][0]) + 2 = min(7,6)+2=8
   dp[2][2] = min(dp[1][2], dp[2][1]) + 1 = min(6,8)+1=7

3. Final Answer:

   7 -> Option B

4. Quick Check:

   Minimum path sum = 7 [OK]

Hint: Fill dp row-wise and column-wise, then inner cells [OK]

Common Mistakes:

• Choosing wrong min between top and left
• Incorrect dp initialization
• Misreading grid values