DynamicProgramming-handouts-2

DynamicProgramming-handouts-2 - Dynamic Programming...

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Dynamic Programming 3/7/2006 7:16 PM 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)
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Dynamic Programming 3/7/2006 7:16 PM 2 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 AC B dd 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
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Dynamic Programming 3/7/2006 7:16 PM 3 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 3/7/2006 7:16 PM 4 Dynamic Programming 7 Another Greedy Approach Idea #2: repeatedly select the product that uses the fewest operations.
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DynamicProgramming-handouts-2 - Dynamic Programming...

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