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DynamicProgramming - Dynamic Programming 1:46 AM Matrix...

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Dynamic Programming 2/24/2005 1:46 AM 1 Dynamic Programming 1 Dynamic Programming Dynamic Programming 2 Outline and Reading Matrix Chain-Product (§5.3.1) The General Technique (§5.3.2) 0-1 Knapsack Problem (§5.3.3) Dynamic Programming 3 Matrix Chain-Products Dynamic Programming is a general algorithm design paradigm. Rather than give the general structure, let us first give a motivating example: Matrix Chain-Products Review: Matrix Multiplication. C = A * B A is d × e and B is e × f O ( d e f ) time A C B d d f e f e i j i,j = = 1 0 ] , [ * ] , [ ] , [ e k j k B k i A j i C Dynamic Programming 4 Matrix Chain-Products Matrix Chain-Product: Compute A=A 0 *A 1 *…*A n-1 A i is d i × d i+1 Problem: How to parenthesize? Example B is 3 × 100 C is 100 × 5 D is 5 × 5 (B*C)*D takes 1500 + 75 = 1575 ops B*(C*D) takes 1500 + 2500 = 4000 ops Dynamic Programming 5 Enumeration Approach Matrix Chain-Product Alg.: Try all possible ways to parenthesize A=A 0 *A 1 *…*A n-1 Calculate number of ops for each one Pick the one that is best Running time: The number of parenthesizations is equal to the number of binary trees with n nodes This is exponential ! It is called the Catalan number, and it is almost 4 n . This is a terrible algorithm! Dynamic Programming 6 Greedy Approach Idea #1: repeatedly select the product that uses (up) the most operations. Counter-example : A is 10 × 5 B is 5 × 10 C is 10 × 5 D is 5 × 10 Greedy idea #1 gives (A*B)*(C*D), which takes 500+1000+500 = 2000 ops A*((B*C)*D) takes 500+250+250 = 1000 ops
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Dynamic Programming 2/24/2005 1:46 AM 2 Dynamic Programming 7 Another Greedy Approach Idea #2: repeatedly select the product that uses the fewest operations.
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