Matrix Chain Multiplication Lecture Notes

Introduction

• Topic: Matrix Chain Multiplication
• Purpose: To clarify missing details from a previous video on the topic
• Contents:
  • What is matrix multiplication?
  • What is matrix chain multiplication?
  • Applying dynamic programming to solve the problem
  • Solving example problems

Matrix Multiplication

• Condition for Multiplication: Number of columns of the first matrix must equal the number of rows of the second matrix
• Dimension of Resultant Matrix: Number of rows of the first matrix by number of columns of the second matrix
• Multiplications Required: Calculated by dimensions (e.g., for 2x3 and 3x2 matrices, 2 * 3 * 2 = 12 multiplications)
• **Key Points: **
  • Multiplication condition: Columns of first matrix = Rows of second matrix
  • Resultant dimension is (rows of first matrix) x (columns of second matrix)
  • Calculate multiplications by multiplying dimensions

Matrix Chain Multiplication

• Goal: To find the most efficient way to multiply a chain of matrices
• Example: Multiplying matrices A1, A2, and A3 in different orders gives different number of multiplications
  • Method 1: (A1 * A2) * A3
  • Method 2: A1 * (A2 * A3)
  • Associative property holds, but efforts (number of multiplications) vary
• Dynamic Programming Approach: Find minimal multiplication cost by considering multiple ways to parenthesize multiplication
• Dynamic Programming Formula:
  • C(i, j) = min {C(i, k) + C(k+1, j) + d(i-1) * d(k) * d(j)} for i <= k < j
• Algorithm Steps:
  • Calculate cost for smaller values first (i.e., smaller subproblems)
  • Use a tabulated approach to store and reuse computed values

Step-by-Step Example

1. Set Dimensions: Define dimensions for matrices involved in the problem
2. Cost Calculation:

• Start with smallest subproblems (individual matrices, cost = 0)
• Move to subproblems with difference of 1, 2, etc.

3. Fill and Use Tables:

• Cost table to store minimum multiplication costs
• K table to store points of optimal matrix splits for minimum cost

4. Formula Application:

• For each C(i, j), consider all possible k values
• Choose k yielding the minimum cost

5. Track Parentheses: Use K table to reconstruct optimal parenthesis structure

Example Problem Solved

• Dimensions: e.g., 3x2, 2x4, 4x2, 2x5
• Stepwise Calculation:
  • Difference 0, cost 0 for each individual matrix
  • Difference 1, difference 2, and so on...
  • Final result derived from optimal subproblem solutions

Complexity Analysis

• Table Cells to Fill: N * (N-1) / 2 for N matrices
• Time per Cell: Each cell potentially requires evaluating N splits
• Overall Complexity: O(N^3)

Conclusion

• Dynamic Programming: Efficient approach to solving matrix chain multiplication problem
• Optimal Order: Derived using cost and K tables
• Multiplication Cost: Minimized using formulated approach