Matrix Chain Multiplication in DSA C - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the goal of the Matrix Chain Multiplication problem?

To find the most efficient way to multiply a given sequence of matrices by minimizing the total number of scalar multiplications.

beginner

What does the 'chain' in Matrix Chain Multiplication refer to?

It refers to the sequence of matrices to be multiplied in order, where the order of multiplication affects the total cost.

intermediate

What is the time complexity of the dynamic programming solution for Matrix Chain Multiplication?

O(n^3), where n is the number of matrices in the chain.

intermediate

In the dynamic programming approach, what does the table entry m[i][j] represent?

The minimum number of scalar multiplications needed to multiply matrices from index i to j in the chain.

beginner

Why is the order of matrix multiplication important in Matrix Chain Multiplication?

Because matrix multiplication is associative but not commutative, different parenthesizations can lead to different computation costs.

What does the Matrix Chain Multiplication problem aim to minimize?

A Number of matrices

B Number of scalar multiplications

C Matrix dimensions

D Number of additions

The problem focuses on minimizing the total scalar multiplications needed to multiply the chain.

Which technique is commonly used to solve the Matrix Chain Multiplication problem efficiently?

A Dynamic Programming

B Greedy Algorithm

C Divide and Conquer without memoization

D Brute Force only

Dynamic Programming stores intermediate results to avoid repeated calculations.

If you have matrices A (10x30), B (30x5), and C (5x60), what is the cost of multiplying (A*B)*C?

A 4500 scalar multiplications

B 27000 scalar multiplications

C 15000 scalar multiplications

D 18000 scalar multiplications

First, A*B: 10*30*5 = 1500. Then (10x5)*C: 10*5*60 = 3000. Total: 4500 scalar multiplications.

What does the entry m[2][4] in the DP table represent?

A Total number of matrices

B Dimensions of matrix 2

C Minimum cost to multiply matrices from index 2 to 4

D Maximum cost to multiply matrices from index 2 to 4

m[i][j] stores the minimum multiplication cost for matrices i through j.

Matrix multiplication is associative. What does this mean for Matrix Chain Multiplication?

A Multiplication is commutative

B Order of multiplication cannot be changed

C Matrices must be square

D Order of multiplication can be changed without changing the result

Associative means (A*B)*C = A*(B*C), so we can change parenthesization.

Explain how dynamic programming helps solve the Matrix Chain Multiplication problem.

Think about how you can remember answers to smaller parts to build the full solution.

Describe why the order of multiplying matrices affects the total number of scalar multiplications.

Consider how multiplying a 10x30 matrix with a 30x5 matrix differs from multiplying a 30x5 matrix with a 5x60 matrix.

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 so that the total number of scalar multiplications is minimized.

2. Step 2: Identify the correct goal

   Among the options, only To find the minimum number of scalar multiplications needed to multiply a chain of matrices describes minimizing scalar multiplications, which is the main goal.

3. Final Answer:

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

4. Quick Check:

   Goal = minimize multiplications [OK]

Hint: Remember: minimize multiplication cost, not count matrices [OK]

Common Mistakes:

• Thinking the goal is to multiply matrices directly without optimization
• Confusing counting matrices with minimizing operations
• Assuming maximum matrices multiplication is the goal

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

easy

A. int p[5] = {10, 20, 30, 40};

B. int p[4] = {10, 20, 30, 40};

C. int p[4] = {10, 20, 30, 40, 50};

D. int p[5] = {10, 20, 30, 40, 50};

Solution

1. Step 1: Understand the dimensions array size

   For n matrices, the dimensions array size is n+1. Here, 4 matrices need an array of size 5.

2. Step 2: Check the array values

   The array should list the row count of the first matrix and then the column counts of each matrix in order: {10, 20, 30, 40, 50}.

3. Final Answer:

   int p[5] = {10, 20, 30, 40, 50}; -> Option D

4. Quick Check:

   Array size = matrices + 1, values match dimensions [OK]

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

Common Mistakes:

• Using array size equal to number of matrices instead of number + 1
• Missing the last dimension in the array
• Including extra or fewer elements than needed

3. What is the output of the following C code snippet for Matrix Chain Multiplication cost calculation?

int p[] = {5, 10, 3, 12, 5};
int n = 4;
int m[5][5] = {0};

for (int L = 2; L <= n; L++) {
    for (int i = 1; i <= n - L + 1; i++) {
        int j = i + L - 1;
        m[i][j] = 1000000;
        for (int k = i; k < j; k++) {
            int q = m[i][k] + m[k+1][j] + p[i-1]*p[k]*p[j];
            if (q < m[i][j])
                m[i][j] = q;
        }
    }
}
printf("%d", m[1][n]);

medium

A. 201

B. 405

C. 270

D. 150

Solution

1. Step 1: Understand the matrix dimensions and initialization

   p = {5,10,3,12,5} means matrices: 5x10, 10x3, 3x12, 12x5. The code calculates minimum multiplication cost using dynamic programming.

2. Step 2: Calculate minimum cost m[1][4]

   Calculations:
   - m[1][2] = 5*10*3 = 150
   - m[2][3] = 10*3*12 = 360
   - m[3][4] = 3*12*5 = 180
   Then:
   - m[1][3] = min( m[1][1]+m[2][3]+5*10*12=0+360+600=960, m[1][2]+m[3][3]+5*3*12=150+0+180=330 ) = 330
   - m[2][4] = min( m[2][2]+m[3][4]+10*3*5=0+180+150=330, m[2][3]+m[4][4]+10*12*5=360+0+600=960 ) = 330
   Finally:
   - m[1][4] = min( m[1][1]+m[2][4]+5*10*5=0+330+250=580, m[1][2]+m[3][4]+5*3*5=150+180+75=405, m[1][3]+m[4][4]+5*12*5=330+0+300=630 ) = 405

3. Final Answer:

   405 -> Option B

4. Quick Check:

   Minimum cost = 405 [OK]

Hint: Trace inner loops carefully to find minimum cost [OK]

Common Mistakes:

• Misindexing arrays starting from 0 instead of 1
• Incorrectly calculating multiplication costs
• Ignoring initialization of m[i][j]

4. Identify the error in the following C code snippet for Matrix Chain Multiplication and select the fix:

int p[] = {2, 3, 4};
int n = 2;
int m[3][3] = {0};
for (int L = 2; L <= n; L++) {
    for (int i = 1; i <= n - L + 1; i++) {
        int j = i + L - 1;
        m[i][j] = INT_MAX;
        for (int k = i; k < j; k++) {
            int q = m[i][k] + m[k+1][j] + p[i]*p[k]*p[j];
            if (q < m[i][j])
                m[i][j] = q;
        }
    }
}
printf("%d", m[1][n]);

medium

A. Change p[i]*p[k]*p[j] to p[i-1]*p[k]*p[j]

B. Initialize m array with -1 instead of 0

C. Change loop to start i from 0 instead of 1

D. Use float instead of int for m array

Solution

1. Step 1: Identify the indexing error in cost calculation

   The multiplication cost formula uses p[i]*p[k]*p[j], but correct indexing for dimensions array is p[i-1]*p[k]*p[j] because p has length n+1 and matrices are from i to j.

2. Step 2: Fix the formula to use correct indices

   Changing to p[i-1]*p[k]*p[j] fixes the dimension mismatch and ensures correct cost calculation.

3. Final Answer:

   Change p[i]*p[k]*p[j] to p[i-1]*p[k]*p[j] -> Option A

4. Quick Check:

   Dimension indexing = p[i-1]*p[k]*p[j] [OK]

Hint: Remember dimension array indexing starts at i-1 for cost [OK]

Common Mistakes:

• Using wrong indices in multiplication cost formula
• Not initializing m array properly
• Incorrect loop ranges causing out-of-bound errors

5. Given matrices with dimensions p = {40, 20, 30, 10, 30}, what is the minimum number of scalar multiplications needed to multiply the chain? Use the Matrix Chain Multiplication approach.

hard

A. 18000

B. 30000

C. 26000

D. 24000

Solution

1. Step 1: Understand matrix dimensions and problem size

   p = {40, 20, 30, 10, 30} means matrices: 40x20, 20x30, 30x10, 10x30. We need minimum multiplication cost for 4 matrices.

2. Step 2: Calculate minimum multiplication cost using DP

   Calculate costs:
   - m[1][2] = 40*20*30 = 24000
   - m[2][3] = 20*30*10 = 6000
   - m[3][4] = 30*10*30 = 9000
   Calculate m[1][3]:
   - (m[1][1]+m[2][3]+40*20*10) = 0 + 6000 + 8000 = 14000
   - (m[1][2]+m[3][3]+40*30*10) = 24000 + 0 + 12000 = 36000
   Minimum = 14000
   Calculate m[2][4]:
   - (m[2][2]+m[3][4]+20*30*30) = 0 + 9000 + 18000 = 27000
   - (m[2][3]+m[4][4]+20*10*30) = 6000 + 0 + 6000 = 12000
   Minimum = 12000
   Calculate m[1][4]:
   - (m[1][1]+m[2][4]+40*20*30) = 0 + 12000 + 24000 = 36000
   - (m[1][2]+m[3][4]+40*30*30) = 24000 + 9000 + 36000 = 69000
   - (m[1][3]+m[4][4]+40*10*30) = 14000 + 0 + 12000 = 26000
   Minimum = 26000

3. Final Answer:

   26000 -> Option C

4. Quick Check:

   Minimum scalar multiplications = 26000 [OK]

Hint: Calculate costs for all splits, pick minimum [OK]

Common Mistakes:

• Calculating only one split instead of all
• Mixing up matrix dimensions in multiplication
• Forgetting to add multiplication costs properly