Matrix Chain Multiplication in DSA Typescript - 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 chain of matrices by deciding the order of multiplication that minimizes the total number of scalar multiplications.

beginner

Why do we not multiply matrices in the given order directly in Matrix Chain Multiplication?

Because matrix multiplication is associative, the order of multiplication can change the number of operations needed, so we look for the order that uses the least computations.

intermediate

What does the 'dimensions array' represent in Matrix Chain Multiplication?

An array where each element represents the number of rows or columns of matrices in the chain. For n matrices, the array has n+1 elements, where the i-th matrix has dimensions dimensions[i-1] x dimensions[i].

intermediate

Explain the role of the DP table in Matrix Chain Multiplication.

The DP table stores the minimum number of multiplications needed to multiply matrices from i to j. It helps avoid repeated calculations by saving results of subproblems.

intermediate

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

O(n³), where n is the number of matrices, because it uses three nested loops to compute the minimum multiplication costs for all subchains.

What does the Matrix Chain Multiplication problem aim to minimize?

A Matrix dimensions

B Number of matrices

C Number of additions

D Number of scalar multiplications

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

If you have matrices A (10x30), B (30x5), and C (5x60), which multiplication order is more efficient?

A (A x B) x C

B A x (B x C)

C Both have the same cost

D Cannot multiply these matrices

Multiplying A x B first costs 10*30*5=1500 multiplications, then (A x B) x C costs 10*5*60=3000, total 4500. The other order costs more (27000).

What is the size of the DP table used in Matrix Chain Multiplication for n matrices?

A n x n

B n+1 x n+1

C n x (n+1)

D 2n x 2n

The DP table is a 2D array of size n x n, storing costs for multiplying matrices from i to j.

Which programming technique is primarily used to solve Matrix Chain Multiplication efficiently?

A Greedy algorithm

B Dynamic programming

C Backtracking

D Divide and conquer without memoization

Dynamic programming is used to store and reuse results of subproblems to avoid repeated calculations.

What does the entry m[i][j] in the DP table represent?

A Dimensions of matrix i

B Number of matrices between i and j

C Minimum cost to multiply matrices from i to j

D Maximum cost to multiply matrices from i to j

m[i][j] stores the minimum number of scalar multiplications needed to multiply matrices from i to j.

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

Think about how smaller matrix chains are solved first and used to solve bigger chains.

Explain why the order of multiplying matrices affects the total number of operations.

Consider how multiplying smaller matrices first can reduce work.

Practice

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

easy

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

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

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

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

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 what is minimized

   The cost minimized is the number of scalar multiplications, not the number of matrices or direct multiplication.

3. Final Answer:

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

4. Quick Check:

   Goal = minimum scalar multiplications [OK]

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

Common Mistakes:

• Confusing multiplication count with matrix count
• Thinking the problem multiplies matrices directly
• Assuming maximum matrices multiplication is the goal

2. Which of the following is the correct TypeScript function signature for a recursive Matrix Chain Multiplication function that takes an array of matrix dimensions?

easy

A. function matrixChainOrder(dimensions: string): number

B. function matrixChainOrder(dimensions: number): number[]

C. function matrixChainOrder(dimensions: number[]): number

D. function matrixChainOrder(dimensions: number[][]): void

Solution

1. Step 1: Identify input type for matrix dimensions

   Matrix dimensions are given as an array of numbers representing sizes, so number[] is correct.

2. Step 2: Identify return type for minimum cost

   The function returns the minimum number of scalar multiplications, which is a number.

3. Final Answer:

   function matrixChainOrder(dimensions: number[]): number -> Option C

4. Quick Check:

   Input = number[], Output = number [OK]

Hint: Input is array of numbers, output is a single number [OK]

Common Mistakes:

• Using string instead of number array for dimensions
• Returning an array instead of a number
• Using 2D array or void return incorrectly

3. What is the output of the following TypeScript code snippet?

function matrixChainOrder(p: number[]): number {
  const n = p.length - 1;
  const m: number[][] = Array.from({ length: n }, () => Array(n).fill(0));
  for (let L = 2; L <= n; L++) {
    for (let i = 0; i < n - L + 1; i++) {
      const j = i + L - 1;
      m[i][j] = Number.MAX_SAFE_INTEGER;
      for (let k = i; k < j; k++) {
        const 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];
}

console.log(matrixChainOrder([10, 20, 30, 40]));

medium

A. 18000

B. 24000

C. 30000

D. 60000

Solution

1. Step 1: Understand the input and matrix dimensions

   The input array [10, 20, 30, 40] represents matrices of dimensions 10x20, 20x30, and 30x40.

2. Step 2: Calculate minimum multiplication cost

   Possible parenthesizations:
   - (A1 x (A2 x A3)) = 20*30*40 + 10*20*40 = 24000 + 8000 = 32000
   - ((A1 x A2) x A3) = 10*20*30 + 10*30*40 = 6000 + 12000 = 18000 (minimum)

3. Final Answer:

   18000 -> Option A

4. Quick Check:

   Minimum multiplications = 18000 [OK]

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

Common Mistakes:

• Calculating cost for only one parenthesization
• Mixing matrix dimensions order
• Forgetting to add multiplication costs correctly

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

function matrixChainOrder(p: number[]): number {
  const n = p.length - 1;
  const m: number[][] = Array.from({ length: n }, () => Array(n).fill(0));
  for (let L = 2; L <= n; L++) {
    for (let i = 0; i < n - L + 1; i++) {
      const j = i + L - 1;
      m[i][j] = Infinity;
      for (let k = i; k < j; k++) {
        const q = m[i][k] + m[k + 1][j] + p[i] * p[k] * p[j];
        if (q < m[i][j]) m[i][j] = q;
      }
    }
  }
  return m[0][n - 1];
}

medium

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

B. Initialize m with Infinity instead of 0

C. Change loop for (let k = i; k <= j; k++) to k < j

D. Return m[0][n] instead of m[0][n - 1]

Solution

1. Step 1: Identify the multiplication cost formula error

   The cost formula uses incorrect indices: p[i] * p[k] * p[j] should use p[k + 1] and p[j + 1] to represent matrix dimensions correctly.

2. Step 2: Confirm other parts are correct

   Initialization with 0 is correct for diagonal, loop ranges are correct, and return index is correct.

3. Final Answer:

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

4. Quick Check:

   Correct cost formula indices fix error [OK]

Hint: Check matrix dimension indices carefully in cost formula [OK]

Common Mistakes:

• Using wrong indices for matrix dimensions in cost calculation
• Changing loop bounds unnecessarily
• Returning wrong matrix cell

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? Use the Matrix Chain Multiplication approach.

hard

A. 1510

B. 2010

C. 4050

D. 3500

Solution

1. Step 1: Understand matrix dimensions and chain length

   The array represents matrices: 5x10, 10x3, 3x12, 12x5, 5x50, 50x6 (6 matrices).

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

   By calculating all splits and costs:
   - Minimum cost found is 2010 scalar multiplications.

3. Final Answer:

   2010 -> Option B

4. Quick Check:

   Minimum scalar multiplications = 2010 [OK]

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

Common Mistakes:

• Calculating only one split option
• Mixing up matrix dimensions order
• Forgetting to add multiplication costs for subproblems