DynamicProgramming

DynamicProgramming - Dynamic Programming 2/24/2005 1:46 AM...

This preview shows pages 1–3. Sign up to view the full content.

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 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 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/24/2010 for the course CS cs 5329 taught by Professor Wang during the Spring '10 term at Southwest Texas Junior College.

Page1 / 3

DynamicProgramming - Dynamic Programming 2/24/2005 1:46 AM...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online