Matrix Chain Multiplication (Dynamic Programming)
Find the optimal way to parenthesize matrix chain multiplication to minimize scalar multiplications
Matrix Chain Multiplication Problem
Determine the optimal order of matrix multiplication to minimize the total number of scalar multiplications required.
Theory & Concepts
Definition:
Given a chain of matrices, find the optimal parenthesization that minimizes the number of scalar multiplications needed to compute the product.
Key Points:
- Matrix multiplication is associative but different parenthesizations have different costs
- Cost of multiplying A(p×q) and B(q×r) is p×q×r scalar multiplications
- Optimal substructure: optimal split for A₁...Aₙ contains optimal splits for subchains
- Overlapping subproblems: same subchains appear in multiple parenthesizations
Applications:
- Computer graphics transformation chains
- Scientific computing optimization
- Database query optimization
- Compiler optimization for expression evaluation
University Guidelines
Exam Weightage
8-10 marks MarksSyllabus Coverage:
- Matrix chain multiplication problem formulation
- Dynamic programming approach
- Recurrence relation development
- Optimal parenthesization construction
- Time and space complexity analysis
Common Questions:
- Find minimum multiplications for given matrix chain
- Construct DP table step by step
- Show optimal parenthesization
- Trace algorithm execution with example
Exam Tips:
Remember recurrence: dp[i][j] = min(dp[i][k] + dp[k+1][j] + p[i]×p[k+1]×p[j+1])
Fill table diagonally by chain length
Show all intermediate calculations
Construct parentheses using backtracking
Algorithm Details
O(n³) where n is number of matrices
Time Complexity
O(n²) for DP table
Space Complexity
Algorithm Steps:
- 1Create DP table dp[1..n][1..n] initialized to 0
- 2For chain length l = 2 to n:
- 3 For i = 1 to n-l+1:
- 4 j = i + l - 1
- 5 For k = i to j-1:
- 6 cost = dp[i][k] + dp[k+1][j] + p[i-1]×p[k]×p[j]
- 7 dp[i][j] = min(dp[i][j], cost)
- 8Construct optimal parentheses using recorded splits
Configure Matrix Chain
For n matrices, enter n+1 dimensions. Each adjacent pair defines a matrix.
Matrix Chain:
M1
→1 × 2
M2
→2 × 3
M3
→3 × 4
M4
→4 × 5
Total matrices: 4
Optimization Results
⛓️
Configure matrix dimensions and click "Find Optimal Order" to see the solution