[Verse 1]
Matrices lined up like dominoes in a row
Multiplying pairs, but which order to go?
Parentheses placement determines the cost
Choose wrong and your processing power gets lost
A times B times C, three ways to compute
Left-to-right or right-to-left, which route?
Dimensions matter when you're counting operations
Matrix multiplication needs optimization
[Chorus]
Split and conquer, find the seams
Optimal subproblems fuel our schemes
Memorize the minimum cost you find
Bottom-up builds the algorithmic mind
Split and conquer, find the seams
Optimal subproblems fuel our schemes
[Verse 2]
Dynamic programming saves the day
Build a table, store results along the way
Bottom triangle filled with calculated scores
Each cell holds the minimum that computation stores
Start with chains of length two, then expand
Three matrices, four, five - understand
The recurrence relation breaks it down clean
Minimum of all splits in the range between
[Chorus]
Split and conquer, find the seams
Optimal subproblems fuel our schemes
Memorize the minimum cost you find
Bottom-up builds the algorithmic mind
Split and conquer, find the seams
Optimal subproblems fuel our schemes
[Bridge]
Cubic time complexity, that's the price we pay
N cubed operations, but we save the day
Exponential brute force would crush your machine
Polynomial solution keeps your system lean
Track the split points, reconstruct the tree
Optimal parenthesization, finally free
[Verse 3]
Chain of matrices A through Z in line
Each one's dimensions, rows and columns defined
Position i to j, what's the cheapest way?
Check every split point k between today
Add up left side cost plus right side cost
Plus the multiplication when boundaries are crossed
Fill that table diagonal by diagonal
Mathematical beauty, computational
[Outro]
From chaos to order, matrices aligned
Optimal solutions algorithmically designed
Chain multiplication, problem now solved
Dynamic programming, efficiency evolved