Minimum Path Sum in Grid in DSA Java - 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 possible 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 down or one step right at any point in the grid.

intermediate

Why is dynamic programming useful for solving the Minimum Path Sum problem?

Because it stores the minimum path sums for smaller parts of the grid to avoid recalculating them, making the solution efficient.

intermediate

What does the value at each cell in the dynamic programming table represent in the Minimum Path Sum problem?

It represents the minimum sum of numbers from the start (top-left) to that cell.

intermediate

How do you initialize the first row and first column in the dynamic programming solution for Minimum Path Sum?

The first row is initialized by cumulatively adding the values from left to right, and the first column is initialized by cumulatively adding the values from top to bottom.

In the Minimum Path Sum problem, which directions can you move?

A Any direction

B Up or Left

C Down or Right

D Diagonally

Only moves down or right are allowed to find the minimum path sum.

What does each cell in the DP table store in the Minimum Path Sum problem?

A Maximum sum to reach that cell

B Minimum sum to reach that cell

C Number of paths to that cell

D Value of the cell itself

Each cell stores the minimum sum needed to reach that cell from the start.

How do you compute the minimum path sum for a cell not in the first row or column?

A Add the cell value to the minimum of the top or left cell sums

B Add the cell value to the maximum of the top or left cell sums

C Add the cell value to the sum of top and left cell sums

D Use the cell value only

You add the current cell's value to the smaller of the sums from the cell above or to the left.

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

A O(m * n)

B O(m + n)

C O(m^2 * n^2)

D O(1)

The solution visits each cell once, so time complexity is O(m * n).

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

A 8

B 10

C 9

D 7

The path 1→3→1→1→1 has sum 7, which is the minimum.

Explain how to solve the Minimum Path Sum problem using dynamic programming.

Think about building the solution step-by-step from the start.

Describe how the allowed moves affect the path choices in the Minimum Path Sum problem.

Consider how movement restrictions reduce complexity.

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.

easy

A. The path with the smallest sum of numbers from top-left to bottom-right

B. The path with the largest sum of numbers from top-left to bottom-right

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

D. The maximum number in the grid

Solution

1. Step 1: Understand the problem goal

   The problem asks to find a path from the top-left cell to the bottom-right cell of a grid such that the sum of all numbers on the path is as small as possible.

2. Step 2: Clarify path movement

   Movement is only allowed down or right at each step, so the path is constrained.

3. Final Answer:

   The path with the smallest sum of numbers from top-left to bottom-right -> Option A

4. Quick Check:

   Minimum path sum means smallest total sum [OK]

Hint: Look for smallest sum path moving only right or down [OK]

Common Mistakes:

• Thinking the path can move diagonally or up
• Looking for maximum sum instead of minimum
• Counting number of paths instead of sum

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

easy

A. int grid[][] = new int[3];

B. int[][] grid = new int[3][3];

C. int[] grid = new int[3][3];

D. int[][] grid = new int[3];

Solution

1. Step 1: Understand Java 2D array syntax

   In Java, a 2D array is declared as int[][] name = new int[rows][columns];. This creates a grid with specified rows and columns.

2. Step 2: Check each option

   int[][] grid = new int[3][3]; correctly declares a 3x3 int 2D array. Options A, C, and D have incorrect syntax or dimensions.

3. Final Answer:

   int[][] grid = new int[3][3]; -> Option B

4. Quick Check:

   Correct 2D array syntax = int[][] grid = new int[3][3]; [OK]

Hint: Use int[][] name = new int[rows][cols] for 2D arrays [OK]

Common Mistakes:

• Declaring 2D arrays with wrong brackets
• Initializing only one dimension
• Confusing 1D and 2D array syntax

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

int[][] grid = {{1, 3}, {1, 5}};
int m = grid.length;
int n = grid[0].length;
int[][] dp = new int[m][n];
dp[0][0] = grid[0][0];
for (int i = 1; i < m; i++) dp[i][0] = dp[i-1][0] + grid[i][0];
for (int j = 1; j < n; j++) dp[0][j] = dp[0][j-1] + grid[0][j];
for (int i = 1; i < m; i++) {
  for (int j = 1; j < n; j++) {
    dp[i][j] = Math.min(dp[i-1][j], dp[i][j-1]) + grid[i][j];
  }
}
System.out.println(dp[m-1][n-1]);

medium

A. 7

B. 8

C. 9

D. 10

Solution

1. Step 1: Initialize dp array with first row and column sums

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

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

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

3. Final Answer:

   7 -> Option A

4. Quick Check:

   Minimum path sum = 7 [OK]

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

Common Mistakes:

• Adding max instead of min
• Not initializing first row and column correctly
• Mixing indices in loops

4. Identify the error in the following Java code snippet that tries to compute the minimum path sum in a grid:

int[][] grid = {{1,2,3},{4,5,6}};
int m = grid.length;
int n = grid[0].length;
int[][] dp = new int[m][n];
dp[0][0] = grid[0][0];
for (int i = 1; i < m; i++) {
  dp[i][0] = dp[i-1][0] + grid[i][0];
}
for (int j = 1; j < n; j++) {
  dp[0][j] = dp[0][j-1] + grid[0][j];
}
for (int i = 1; i < m; i++) {
  for (int j = 1; j < n; j++) {
    dp[i][j] = Math.max(dp[i-1][j], dp[i][j-1]) + grid[i][j];
  }
}

medium

A. Grid dimensions are wrong

B. Not initializing dp[0][0]

C. Loop ranges are incorrect

D. Using Math.max instead of Math.min in dp calculation

Solution

1. Step 1: Check dp initialization

   dp[0][0] is correctly initialized to grid[0][0]. Loops for first row and column are correct.

2. Step 2: Identify the error in dp update

   The inner loop uses Math.max instead of Math.min, which finds the maximum path sum, not minimum.

3. Final Answer:

   Using Math.max instead of Math.min in dp calculation -> Option D

4. Quick Check:

   Minimum path sum needs Math.min, not Math.max [OK]

Hint: Use Math.min to find minimum path sum, not Math.max [OK]

Common Mistakes:

• Using max instead of min in dp update
• Forgetting to initialize dp[0][0]
• Incorrect loop boundaries

5. Given a 3x3 grid:

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

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

hard

A. 9

B. 8

C. 7

D. 10

Solution

1. Step 1: Initialize dp first row and column sums

   dp[0][0] = 1
   dp[0][1] = 1 + 2 = 3
   dp[0][2] = 3 + 5 = 8
   dp[1][0] = 1 + 3 = 4
   dp[2][0] = 4 + 4 = 8

2. Step 2: Calculate dp values for inner cells

   dp[1][1] = min(dp[0][1], dp[1][0]) + 2 = min(3,4) + 2 = 5
   dp[1][2] = min(dp[0][2], dp[1][1]) + 1 = min(8,5) + 1 = 6
   dp[2][1] = min(dp[1][1], dp[2][0]) + 1 = min(5,8) + 1 = 6
   dp[2][2] = min(dp[1][2], dp[2][1]) + 1 = min(6,6) + 1 = 7

3. Final Answer:

   7 -> Option C

4. Quick Check:

   Minimum path sum = 7 [OK]

Hint: Fill dp row and column first, then inner cells with min of top and left [OK]

Common Mistakes:

• Not updating dp correctly for inner cells
• Adding max instead of min
• Incorrect initialization of first row or column