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1. Matrix Chain Multiplication
Using Dynamic Programming
Name Anum Rehman
Subject Advance algorithm

2. What is the Matrix Chain Multiplication Problem?
● Problem:
Given a sequence of matrices, determine the most efficient way to multiply
them.
● Goal:
Minimize the total number of scalar multiplications.
● Important:
Matrix multiplication is associative but not commutative
→ (A×B)×C ≠ A×(B×C) in terms of cost

3. Problem Input Format
You are given n matrices: A , A , ..., A
₁ ₂ ₙ
Each matrix A has dimensions:
ᵢ p[i-1] × p[i]
Your job is to parenthesize the product to minimize
cost

4. Example:
Matrix dimensions: [10, 20, 30, 40]
Represents matrices:
● A = 10×20
₁
● A = 20×30
₂
● A = 30×40
₃

5. Brute Force Approach
● Try all possible ways of placing parentheses
● Number of ways grows exponentially (~Catalan
number)
● Inefficient for n > 10

6. Why Use Dynamic Programming?
● Overlapping subproblems: same sub-chain multiplications computed multiple
times
● Optimal substructure: optimal solution contains optimal solutions to
subproblems
● DP saves previously computed costs in a table

7. DP Solution - Key Idea
● Let dp[i][j] = minimum number of scalar multiplications needed
to compute the product of matrices A to A
ᵢ ⱼ
● Try all possible k to split between i and j:
dp[i][j] = min(dp[i][k] + dp[k+1][j] + p[i-1]*p[k]*p[j])

8. Example
Input: p = [10, 20, 30, 40] → Matrices: A1(10×20), A2(20×30), A3(30×40)
We fill dp[1][3] by trying:
● Split at k=1:
Cost = dp[1][1] + dp[2][3] + (10×20×40)
● Split at k=2:
Cost = dp[1][2] + dp[3][3] + (10×30×40)
● Take the minimum of the two