# lect06 - Lecture 6 Dynamic Programming Matrix-chain...

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Lecture 6 Dynamic Programming

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Matrix-chain Multiplication tion. multiplica scaler of number the minimizes way that a in product the ze parenthesi fully , dimension has matrix , ,..., 2 , 1 for where matrices, of } ,..., , { chain a Given 2 1 1 2 1 n i i i n A A A p p A n i n A A A × = -
Fully Parenthesize )) )( (( ) )) ( (( products. zed parenthesi fully two of product or the matrix single a either is it if zed parenthesi fully is product A 4 3 2 1 4 3 2 1 A A A A A A A A

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Scalar Multiplications . is product matrix this do to tions multiplica scalar of number the Thus, ) ( Then ). ( matrix and ) ( matrix Consider 1 pqr b a AB b B r q a A q p q j r p jk ij jk ij = × = = × = ×
tions. multiplica scalar of numbers different give may products zed parenthesi fully Different # of scalar multiplications 3 2 0 2 1 0 3 2 1 3 2 1 3 1 0 3 2 1 ) ) (( )) ( ( p p p p p p A A A p p p p p p A A A + + e.g.,

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ly. respective , and of products zed parenthesi fully optimal are they Then . and of products zed parenthesi fully two of product the is product zed parenthesi fully optimal Suppose 1 1 1 1 n k k n k k A A A A A A A A + + Step 1. Find recursive structure of optimal solution
. if

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lect06 - Lecture 6 Dynamic Programming Matrix-chain...

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