Dynamic Programming CSL Wiki动态规划 CSL维基.pptVIP

Dynamic Programming ? 2004 Goodrich, Tamassia ? 2004 Goodrich, Tamassia Dynamic Programming Dynamic Programming * Dynamic Programming Dynamic Programming * Matrix Chain-Products (not in book) 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(def ) time A C B d d f e f e i j i,j Dynamic Programming * Matrix Chain-Products Matrix Chain-Product: Compute A=A0*A1*…*An-1 Ai is di × di+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 * An Enumeration Approach Matrix Chain-Product Alg.: Try all possible ways to parenthesize A=A0*A1*…*An-1 Calculate number of ops for each one Pick the one that is best Running time: The number of paranethesizations is equal to the number of binary trees with n nodes This is exponential! It is called the Catalan number, and it is almost 4n. This is a terrible algorithm! Dynamic Programming * A 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 Dynamic Programming * Another Greedy Approach Idea #2: repeatedly select the product that uses the fewest operations. Counter-example: A is 101 × 11 B is 11 × 9 C is 9 × 100 D is 100 × 99 Greedy idea #2 gives A*((B*C)*D)), which takes 109989+9900+108900=228789 ops (A*B)*(C*D) takes 9999+89991+89100=189090 ops The greedy approach is not giving us the optimal value. Dynamic Programming * A “Recursive” Approach Define subproblems: Find the best parenthesization of Ai*Ai+1*…*Aj. Let Ni,j denote the number of operations done by this subproblem. The optimal solution for the whole problem is N0,n-1. S