Matrix Chain Multiplication in DSA Java - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Matrix Chain Multiplication Cost Calculation

What is the output of the following Java code that calculates the minimum multiplication cost for a given matrix chain?

DSA Java

public class MatrixChain {
    public static int matrixChainOrder(int[] p) {
        int n = p.length - 1;
        int[][] m = new int[n][n];
        for (int l = 2; l <= n; l++) {
            for (int i = 0; i <= n - l; i++) {
                int j = i + l - 1;
                m[i][j] = Integer.MAX_VALUE;
                for (int k = i; k < j; k++) {
                    int q = m[i][k] + m[k + 1][j] + p[i] * p[k + 1] * p[j + 1];
                    if (q < m[i][j])
                        m[i][j] = q;
                }
            }
        }
        return m[0][n - 1];
    }

    public static void main(String[] args) {
        int[] dims = {10, 20, 30, 40, 30};
        System.out.println(matrixChainOrder(dims));
    }
}

A 40000

B 36000

C 26000

D 30000

💡 Hint

Trace the cost calculation for each possible split and find the minimum.

The minimum cost to multiply matrices with dimensions 10x20, 20x30, 30x40, and 40x30 is 30000 scalar multiplications.

🧠 Conceptual

intermediate

Understanding Matrix Chain Multiplication Cost Formula

In the matrix chain multiplication problem, what does the term p[i] * p[k+1] * p[j+1] represent in the cost calculation?

A The cost of multiplying two matrices at positions i and j directly

B The dimensions of the matrix resulting from multiplying matrices i through j

C The number of scalar multiplications needed to multiply the resulting matrices from subproblems i to k and k+1 to j

D The cost of splitting the matrix chain at position k

💡 Hint

Think about how matrix multiplication cost depends on matrix dimensions.

The term p[i] * p[k+1] * p[j+1] calculates the scalar multiplications needed to multiply two matrices: one from i to k and the other from k+1 to j, based on their dimensions.

🔧 Debug

advanced

Identify the Error in Matrix Chain Multiplication Implementation

What error will the following Java code produce when run?

DSA Java

public class MatrixChain {
    public static int matrixChainOrder(int[] p) {
        int n = p.length - 1;
        int[][] m = new int[n][n];
        for (int l = 2; l <= n; l++) {
            for (int i = 0; i < n - l + 1; i++) {
                int j = i + l - 1;
                m[i][j] = Integer.MAX_VALUE;
                for (int k = i; k < j; k++) {
                    int q = m[i][k] + m[k + 1][j] + p[i] * p[k + 1] * p[j + 1];
                    if (q < m[i][j])
                        m[i][j] = q;
                }
            }
        }
        return m[0][n - 1];
    }

    public static void main(String[] args) {
        int[] dims = {10, 20, 30, 40, 30};
        System.out.println(matrixChainOrder(dims));
    }
}

A ArrayIndexOutOfBoundsException

B NullPointerException

C No error, outputs 26000

D ArithmeticException

💡 Hint

Check the loop boundaries for k carefully.

The inner loop runs with k <= j, causing m[k+1][j] to access out of bounds when k = j.

🚀 Application

advanced

Minimum Multiplication Cost for Custom Matrix Chain

Given matrices with dimensions 5x10, 10x3, 3x12, 12x5, and 5x50, what is the minimum number of scalar multiplications needed to multiply the entire chain?

A 2010

B 3150

C 2700

D 4050

💡 Hint

Use dynamic programming or carefully consider the best parenthesization.

The optimal order yields 2010 scalar multiplications for the given dimensions.

🧠 Conceptual

expert

Time Complexity of Matrix Chain Multiplication Algorithm

What is the time complexity of the standard dynamic programming solution for the matrix chain multiplication problem with n matrices?

A O(n log n)

B O(n^3)

C O(2^n)

D O(n^2)

💡 Hint

Consider the three nested loops in the algorithm.

The algorithm uses three nested loops over n, resulting in O(n^3) time complexity.

Practice

1. What is the main goal of the Matrix Chain Multiplication problem?

easy

A. To find the minimum number of scalar multiplications needed to multiply a chain of matrices

B. To multiply all matrices in the chain directly without optimization

C. To find the maximum number of matrices that can be multiplied

D. To count the total number of matrices in the chain

Solution

1. Step 1: Understand the problem objective

   The Matrix Chain Multiplication problem aims to find the best way to multiply matrices to minimize cost.

2. Step 2: Identify the cost metric

   The cost is measured by the number of scalar multiplications needed, not just multiplying matrices directly.

3. Final Answer:

   To find the minimum number of scalar multiplications needed to multiply a chain of matrices -> Option A

4. Quick Check:

   Goal = minimum scalar multiplications [OK]

Hint: Focus on minimizing multiplication cost, not counting matrices [OK]

Common Mistakes:

• Thinking the goal is to multiply matrices directly without optimization
• Confusing number of matrices with multiplication cost
• Assuming maximum matrices multiplied is the goal

2. Which of the following is the correct way to declare the dimensions array for 4 matrices with dimensions 10x20, 20x30, 30x40, and 40x50 in Java?

easy

A. int[] dims = {10, 20, 20, 30, 30, 40, 40, 50};

B. int[] dims = {10, 20, 30, 40};

C. int[] dims = {10, 20, 30, 40, 50, 60};

D. int[] dims = {10, 20, 30, 40, 50};

Solution

1. Step 1: Understand the dimensions array format

   For n matrices, the dimensions array has length n+1, storing rows of first matrix and columns of each matrix.

2. Step 2: Apply to given matrices

   Given matrices: 10x20, 20x30, 30x40, 40x50, so dims = {10, 20, 30, 40, 50}.

3. Final Answer:

   int[] dims = {10, 20, 30, 40, 50}; -> Option D

4. Quick Check:

   Length = 5 for 4 matrices [OK]

Hint: Dimensions array length = number of matrices + 1 [OK]

Common Mistakes:

• Using length equal to number of matrices instead of number+1
• Repeating dimensions unnecessarily
• Including extra unrelated dimensions

3. What will be the output of the following Java code snippet for matrix chain multiplication cost calculation?

int[] dims = {5, 10, 3};
int n = dims.length;
int[][] dp = new int[n][n];
for (int l = 2; l < n; l++) {
  for (int i = 1; i < n - l + 1; i++) {
    int j = i + l - 1;
    dp[i][j] = Integer.MAX_VALUE;
    for (int k = i; k < j; k++) {
      int cost = dp[i][k] + dp[k+1][j] + dims[i-1]*dims[k]*dims[j];
      if (cost < dp[i][j]) dp[i][j] = cost;
    }
  }
}
System.out.println(dp[1][n-1]);

medium

A. 0

B. 75

C. 150

D. Syntax Error

Solution

1. Step 1: Understand the input matrices

   Dimensions: 5x10 and 10x3 matrices. Only one way to multiply.

2. Step 2: Calculate multiplication cost

   Cost = 5*10*3 = 150 scalar multiplications.

3. Final Answer:

   150 -> Option C

4. Quick Check:

   5*10*3 = 150 [OK]

Hint: Multiply dims[i-1]*dims[k]*dims[j] for cost [OK]

Common Mistakes:

• Confusing indices in cost calculation
• Assuming zero cost for single multiplication
• Syntax errors from incorrect loops

4. Identify the error in the following Java code snippet for matrix chain multiplication:

int[] dims = {2, 3, 4};
int n = dims.length;
int[][] dp = new int[n][n];
for (int l = 2; l <= n; l++) {
  for (int i = 1; i <= n - l + 1; i++) {
    int j = i + l - 1;
    dp[i][j] = Integer.MAX_VALUE;
    for (int k = i; k < j; k++) {
      int cost = dp[i][k] + dp[k+1][j] + dims[i-1]*dims[k]*dims[j];
      if (cost < dp[i][j]) dp[i][j] = cost;
    }
  }
}
System.out.println(dp[1][n-1]);

medium

A. Loop condition l <= n causes ArrayIndexOutOfBoundsException

B. Incorrect cost calculation formula

C. dp array should be one-dimensional

D. Missing initialization of dp array

Solution

1. Step 1: Analyze loop bounds

   Outer loop runs l from 2 to n inclusive, but dp indices go up to n-1.

2. Step 2: Identify index error

   When l = n, j = i + l - 1 can exceed dp array size causing ArrayIndexOutOfBoundsException.

3. Final Answer:

   Loop condition l <= n causes ArrayIndexOutOfBoundsException -> Option A

4. Quick Check:

   Loop bound off by one causes index error [OK]

Hint: Check loop bounds carefully to avoid index errors [OK]

Common Mistakes:

• Using <= n instead of < n for loop limit
• Ignoring array size limits
• Misunderstanding dp array dimensions

5. Given matrices with dimensions {40, 20, 30, 10, 30}, what is the minimum number of scalar multiplications needed to multiply the entire chain?

hard

A. 30000

B. 26000

C. 18000

D. 35000

Solution

1. Step 1: Understand matrix dimensions

   Matrices: 40x20, 20x30, 30x10, 10x30.

2. Step 2: Calculate minimum cost using dynamic programming

   Possible parenthesizations and costs:
   - ((40x20)x30)x(10x30):
   Cost1 = 40*20*30=24000 + 40*30*10=12000 + 0 = 36000 (too high)
   - (40x(20x30))x(10x30):
   Cost2 = 20*30*10=6000 + 40*20*10=8000 + 40*10*30=12000 = 26000 (minimum)

3. Final Answer:

   26000 -> Option B

4. Quick Check:

   Minimum scalar multiplications = 26000 [OK]

Hint: Try all splits and pick minimum cost [OK]

Common Mistakes:

• Calculating cost for only one parenthesization
• Mixing up matrix dimensions order
• Ignoring intermediate multiplication costs