The.Algorithm.Design.Manual.Springer-Verlag.1998

Lecture 12 - introduction to dynamic programming
Multiplying an matrix by a matrix (using the common algorithm) takes
multiplications.
We would like to avoid big intermediate matrices, and since matrix multiplication is associative, we can
parenthesise however we want.
Matrix multiplication is not communitive, so we cannot permute the order of the matrices without
changing the result.
Listen To Part 12-12
Example
Consider , where A is , B is , C is , and D is .
There are three possible parenthesizations:
The order makes a big difference in real computation. How do we find the best order?
Let M(i,j) be the minimum number of multiplications necessary to compute .
The key observations are
● The outermost parentheses partition the chain of matricies (i,j) at some k.
● The optimal parenthesization order has optimal ordering on either side of k.
Listen To Part 12-13
A recurrence for this is:
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