Matrix Chain Multiplication in DSA C# - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Matrix Chain Multiplication Cost Calculation

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

DSA C#

using System;

class Program {
    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] = int.MaxValue;
                for (int k = i; k < j; k++) {
                    int cost = m[i, k] + m[k + 1, j] + p[i] * p[k + 1] * p[j + 1];
                    if (cost < m[i, j])
                        m[i, j] = cost;
                }
            }
        }
        return m[0, n - 1];
    }

    static void Main() {
        int[] dims = {40, 20, 30, 10, 30};
        Console.WriteLine(MatrixChainOrder(dims));
    }
}

A 26000

B 30000

C 35000

D 27000

💡 Hint

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

The minimum number of scalar multiplications needed to multiply the chain of matrices with dimensions 40x20, 20x30, 30x10, and 10x30 is 26000.

🧠 Conceptual

intermediate

Understanding the Role of the 'k' Variable in Matrix Chain Multiplication

In the matrix chain multiplication dynamic programming solution, what does the variable 'k' represent in the nested loops?

A The index where the matrix chain is split to minimize multiplication cost

B The total number of matrices in the chain

C The dimension of the current matrix being multiplied

D The maximum multiplication cost found so far

💡 Hint

Think about how the problem breaks down the chain into smaller parts.

The variable 'k' represents the position where the matrix chain is split into two parts to calculate the cost of multiplying those parts separately and then combining them.

🔧 Debug

advanced

Identify the Error in Matrix Chain Multiplication Implementation

What error will the following C# code produce when executed?

DSA C#

using System;

class Program {
    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] = int.MaxValue;
                for (int k = i; k <= j; k++) {
                    int cost = m[i, k] + m[k + 1, j] + p[i] * p[k + 1] * p[j + 1];
                    if (cost < m[i, j])
                        m[i, j] = cost;
                }
            }
        }
        return m[0, n - 1];
    }

    static void Main() {
        int[] dims = {40, 20, 30, 10, 30};
        Console.WriteLine(MatrixChainOrder(dims));
    }
}

A No error, outputs 26000

B StackOverflowException

C IndexOutOfRangeException

D NullReferenceException

💡 Hint

Check the loop bounds for 'k' carefully.

The inner loop uses 'k <= j' which causes 'k + 1' to go out of bounds when k equals j, leading to IndexOutOfRangeException.

🚀 Application

advanced

Minimum Multiplication Cost for a Different Matrix Chain

Given matrices with dimensions 10x30, 30x5, 5x60, and 60x10, what is the minimum number of scalar multiplications needed to multiply the entire chain?

A 4500

B 3500

C 10500

D 5000

💡 Hint

Use the matrix chain multiplication formula and try different parenthesizations.

The minimum cost is 5000 scalar multiplications for the given chain.

🧠 Conceptual

expert

Time Complexity of Matrix Chain Multiplication Algorithm

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

A O(n^2)

B O(n^3)

C O(2^n)

D O(n log n)

💡 Hint

Consider the three nested loops in the algorithm.

The algorithm uses three nested loops each running up to 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 computation.

2. Step 2: Identify the optimization goal

   The goal is to minimize the total scalar multiplications, not just multiply matrices or count them.

3. Final Answer:

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

4. Quick Check:

   Minimum scalar multiplications = To find the minimum number of scalar multiplications needed to multiply a chain of matrices [OK]

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

Common Mistakes:

• Thinking the problem is about counting matrices
• Assuming direct multiplication without optimization
• Confusing maximum with minimum operations

2. Which of the following is the correct way to declare the dimensions array for matrices A1(10x20), A2(20x30), A3(30x40) in C# for Matrix Chain Multiplication?

easy

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

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

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

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

Solution

1. Step 1: Understand the dimensions array format

   For n matrices, dimensions array has length n+1, where dims[i] and dims[i+1] represent matrix i's rows and columns.

2. Step 2: Apply to given matrices

   Given A1(10x20), A2(20x30), A3(30x40), dims = {10, 20, 30, 40} correctly represents all matrices.

3. Final Answer:

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

4. Quick Check:

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

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

Common Mistakes:

• Using array length equal to number of matrices
• Skipping first or last dimension
• Mixing rows and columns order

3. What is the minimum number of scalar multiplications needed to multiply matrices with dimensions {40, 20, 30, 10, 30} using Matrix Chain Multiplication?

medium

A. 18000

B. 26000

C. 24000

D. 30000

Solution

1. Step 1: Understand the matrices and dimensions

   Dimensions array {40, 20, 30, 10, 30} means matrices: A1(40x20), A2(20x30), A3(30x10), A4(10x30).

2. Step 2: Calculate minimum multiplications using DP or manual check

   Optimal parenthesization ((A1 x (A2 x A3)) x A4):
   A2xA3: 20x30x10 = 6000 (result 20x10)
   A1x(20x10): 40x20x10 = 8000 (result 40x10)
   (40x10)xA4(10x30): 40x10x30 = 12000
   Total: 6000 + 8000 + 12000 = 26000.
   Other parenthesizations (e.g., (A1xA2)x(A3xA4) = 69000, ((A1xA2)xA3)xA4 = 48000) are higher.

3. Final Answer:

   26000 -> Option B

4. Quick Check:

   Minimum scalar multiplications = 26000 [OK]

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

Common Mistakes:

• Calculating cost for wrong parenthesization
• Mixing matrix dimensions order
• Ignoring multiplication order impact

4. Identify the error in the following C# code snippet for Matrix Chain Multiplication:

int n = dims.Length;
int[,] dp = new int[n, n];
for (int i = 1; i < n; i++) dp[i, i] = 0;
for (int len = 2; len <= n; len++) {
  for (int i = 1; i <= n - len + 1; i++) {
    int j = i + len - 1;
    dp[i, j] = int.MaxValue;
    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;
    }
  }
}

medium

A. The cost calculation uses wrong indices; it should be 'dims[i] * dims[k+1] * dims[j+1]' instead

B. The initialization 'dp[i, i] = 0;' should start from i=0 instead of i=1

C. The outer loop should run till 'len < n' instead of 'len <= n' to avoid index out of range

D. The dp array should be declared as 'int[,] dp = new int[n-1, n-1];' to match matrix count

Solution

1. Step 1: Check loop bounds for index safety

   n = dims.Length (e.g., 5 for 4 matrices). Outer loop len <= n allows len = n, i=1, j = n, causing dims[j] (dims[n]) index out of range (valid 0 to n-1).

2. Step 2: Other components are correct

   dp initialization i=1 to n-1 covers single matrices 1 to n-1. Cost formula uses correct indices dims[i-1] * dims[k] * dims[j]. dp size n x n sufficient for indices up to n-1.

3. Final Answer:

   The outer loop should run till 'len < n' instead of 'len <= n' to avoid index out of range -> Option C

4. Quick Check:

   len max = dims.Length - 1 to match num matrices [OK]

Hint: Outer loop len < dims.Length to avoid dims[n] access [OK]

Common Mistakes:

• len loop exceeding number of matrices
• Incorrect dp array size
• Misusing dimension indices in cost

5. Given matrices with dimensions {5, 10, 3, 12, 5, 50, 6}, what is the minimum number of scalar multiplications needed to multiply the entire chain using Matrix Chain Multiplication?

hard

A. 2010

B. 2015

C. 2011

D. 2500

Solution

1. Step 1: Understand the matrices and dimensions

   Dimensions array {5, 10, 3, 12, 5, 50, 6} means matrices:
   A1(5x10), A2(10x3), A3(3x12), A4(12x5), A5(5x50), A6(50x6).

2. Step 2: Use dynamic programming to find minimum multiplications

   Calculate minimum cost for all subchains:
   - For example, cost(A1xA2) = 5*10*3 = 150
   - Cost for longer chains computed by checking all splits and picking minimum.
   After full DP computation, minimum cost is 2010 scalar multiplications.

3. Final Answer:

   2010 -> Option A

4. Quick Check:

   Minimum scalar multiplications = 2010 [OK]

Hint: Use DP table to check all splits and pick minimum cost [OK]

Common Mistakes:

• Calculating cost for only one split
• Mixing matrix dimensions order
• Ignoring multiplication order impact