Matrix Chain Multiplication in DSA C# - Build from Scratch

Matrix Chain Multiplication

📖 Scenario: You work in a company that optimizes the order of multiplying several matrices to reduce the total number of calculations. Each matrix has specific dimensions, and the order in which you multiply them affects the total work done.

🎯 Goal: You will write a program to find the minimum number of scalar multiplications needed to multiply a chain of matrices in the most efficient order.

📋 What You'll Learn

Create an integer array called dimensions representing the dimensions of matrices in the chain.

Create an integer variable called n to store the number of matrices.

Implement a method called MatrixChainOrder that takes dimensions and n as parameters and returns the minimum multiplication cost.

Print the minimum number of multiplications required.

💡 Why This Matters

🌍 Real World

Matrix chain multiplication optimization is used in computer graphics, scientific computing, and database query optimization to reduce computation time.

💼 Career

Understanding dynamic programming and optimization techniques is valuable for software engineers working on performance-critical applications.

1

Create the dimensions array

Create an integer array called dimensions with these exact values: 40, 20, 30, 10, 30.

DSA C#

// Create the dimensions array with the given values
// Your code here

Hint

Use int[] dimensions = {40, 20, 30, 10, 30}; to create the array.

2

Create the variable for number of matrices

Create an integer variable called n and set it to the length of dimensions minus 1.

DSA C#

int[] dimensions = {40, 20, 30, 10, 30};
// Create the variable n as dimensions.Length - 1
// Your code here

Hint

Use int n = dimensions.Length - 1; to get the number of matrices.

3

Implement the MatrixChainOrder method

Implement a method called MatrixChainOrder that takes int[] dimensions and int n as parameters and returns an integer. Use dynamic programming to calculate the minimum number of multiplications needed to multiply the chain of matrices.

DSA C#

int[] dimensions = {40, 20, 30, 10, 30};
int n = dimensions.Length - 1;

// Implement MatrixChainOrder method below
// Your code here

Hint

Use a 2D array m to store minimum costs. Use nested loops for chain length L, start index i, and split point k. Calculate cost and update minimum.

4

Print the minimum multiplication cost

Call the MatrixChainOrder method with dimensions and n, store the result in minCost, and print it using Console.WriteLine.

DSA C#

int[] dimensions = {40, 20, 30, 10, 30};
int n = dimensions.Length - 1;

int MatrixChainOrder(int[] dimensions, int n)
{
    int[,] m = new int[n, n];

    for (int i = 0; i < n; i++)
        m[i, i] = 0;

    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] + dimensions[i] * dimensions[k + 1] * dimensions[j + 1];
                if (cost < m[i, j])
                    m[i, j] = cost;
            }
        }
    }

    return m[0, n - 1];
}

// Call the method and print the result
// Your code here

Hint

Call MatrixChainOrder(dimensions, n), store in minCost, then print with Console.WriteLine(minCost);.

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